Epidemiology

"The object of this book is to contribute to the study of learning--one of the many fields of psychology. A possible probabilistic framework or model for analyzing data from a variety of experiments on animal and human learning is presented. This model is described in general terms and is applied to a number of experiments. The stress is on the viewpoint that learning is probabilistic or stochastic--that events change the probabilities of the various responses. The book is divided into two main parts. The first presents the general model, describes many of its formal properties, and considers a number of special cases. The second part applies the model to a number of experiments on learning and treats the statistical problems of estimating parameters and measuring goodness of fit. The mathematics used is given in detail in an elementary manner. Serious mathematical difficulties are avoided by Monte Carlo or stat-rat trials that simulate learning experiments by means of routines based on random numbers. Stat-rat trials facilitate the comparison of actual observations with the model. Several special tables will help the reader apply the model to his own experiments"--Jacket. (PsycINFO Database Record (c) 2014 APA, all rights reserved)

@book{doi10103714496000,
    author = "Bush, Robert R. and Mosteller, Frederick",
    title = "Stochastic models for learning.",
    year = "1955",
    abstract = {"The object of this book is to contribute to the study of learning--one of the many fields of psychology. A possible probabilistic framework or model for analyzing data from a variety of experiments on animal and human learning is presented. This model is described in general terms and is applied to a number of experiments. The stress is on the viewpoint that learning is probabilistic or stochastic--that events change the probabilities of the various responses. The book is divided into two main parts. The first presents the general model, describes many of its formal properties, and considers a number of special cases. The second part applies the model to a number of experiments on learning and treats the statistical problems of estimating parameters and measuring goodness of fit. The mathematics used is given in detail in an elementary manner. Serious mathematical difficulties are avoided by Monte Carlo or stat-rat trials that simulate learning experiments by means of routines based on random numbers. Stat-rat trials facilitate the comparison of actual observations with the model. Several special tables will help the reader apply the model to his own experiments"--Jacket. (PsycINFO Database Record (c) 2014 APA, all rights reserved)},
    url = "https://doi.org/10.1037/14496-000",
    doi = "10.1037/14496-000",
    openalex = "W3149412023"
}

@misc{bartlett1960stochastic1,
    author = "Bartlett, M. S",
    title = "Stochastic Population Models in Ecology and Epidemiology",
    year = "1960",
    howpublished = "London, Methuen, 90 p",
    note = "talkorigins\_source = {true}; raw\_reference = {Bartlett, M. S., 1960, Stochastic Population Models in Ecology and Epidemiology: London, Methuen, 90 p.}"
}

@article{bailey1961stochastic,
    author = "Bailey, Norman T. J. and Bartlett, M. S.",
    title = "Stochastic Population Models in Ecology and Epidemiology.",
    year = "1961",
    journal = "Journal of the American Statistical Association",
    url = "https://doi.org/10.2307/2282020",
    doi = "10.2307/2282020",
    number = "296",
    openalex = "W2106418046",
    pages = "1015",
    volume = "56"
}

@article{doi1023072282020,
    author = "Bailey, Norman T. J. and Bartlett, M. S.",
    title = "Stochastic Population Models in Ecology and Epidemiology.",
    year = "1961",
    journal = "Journal of the American Statistical Association",
    url = "https://doi.org/10.2307/2282020",
    doi = "10.2307/2282020",
    openalex = "W2106418046"
}

@article{doi1023072312029,
    author = "Bush, K. A. and Bartlett, M. S.",
    title = "Stochastic Population Models in Ecology and Epidemiology.",
    year = "1961",
    journal = "American Mathematical Monthly",
    url = "https://doi.org/10.2307/2312029",
    doi = "10.2307/2312029",
    openalex = "W2329811423"
}

@article{doi1023072985209,
    author = "Bulmer, M. G. and Bartlett, M. S.",
    title = "Stochastic Population Models in Ecology and Epidemiology.",
    year = "1961",
    journal = "Journal of the Royal Statistical Society Series C (Applied Statistics)",
    url = "https://doi.org/10.2307/2985209",
    doi = "10.2307/2985209",
    openalex = "W2797077302"
}

@article{barton1962stochastic,
    author = "Barton, D. E. and Bartlett, M. S.",
    title = "Stochastic Population Models in Ecology and Epidemiology.",
    year = "1962",
    journal = "Biometrika",
    url = "https://doi.org/10.2307/2333508",
    doi = "10.2307/2333508",
    number = "1/2",
    openalex = "W4252209562",
    pages = "288",
    volume = "49"
}

@article{doi1023071909913,
    author = "Bharucha-Reid, A. T. and Bartlett, M. S.",
    title = "Stochastic Population Models in Ecology and Epidemiology",
    year = "1962",
    journal = "Econometrica",
    url = "https://doi.org/10.2307/1909913",
    doi = "10.2307/1909913",
    openalex = "W2327374781"
}

@article{doi1023072333508,
    author = "Barton, D. E. and Bartlett, M. S.",
    title = "Stochastic Population Models in Ecology and Epidemiology.",
    year = "1962",
    journal = "Biometrika",
    url = "https://doi.org/10.2307/2333508",
    doi = "10.2307/2333508",
    openalex = "W4252209562"
}

@article{doi1023072527465,
    author = "Bharucha-Reid, A. T. and Bartlett, M. S.",
    title = "Stochastic Population Models in Ecology and Epidemiology.",
    year = "1962",
    journal = "Biometrics",
    url = "https://doi.org/10.2307/2527465",
    doi = "10.2307/2527465",
    openalex = "W2797307081"
}

@article{doi1010382231133a0,
    author = "Hassell, M. P. and Varley, G. C.",
    title = "New Inductive Population Model for Insect Parasites and its Bearing on Biological Control",
    year = "1969",
    journal = "Nature",
    url = "https://doi.org/10.1038/2231133a0",
    doi = "10.1038/2231133a0",
    openalex = "W2021265043"
}

The Markov chain and ordinary differential equation models for chemical reaction systems are compared. It is shown that if the volume of the reaction system is taken into account in an appropriate way in the formulation of the Markov chain model, then the o.d.e. model is the infinite volume limit of the Markov chain model. A central limit theorem is also given for the deviation of the Markov chain model from the o.d.e. model.

@article{doi10106311678692,
    author = "Kurtz, Thomas G.",
    title = "The Relationship between Stochastic and Deterministic Models for Chemical Reactions",
    year = "1972",
    journal = "The Journal of Chemical Physics",
    abstract = "The Markov chain and ordinary differential equation models for chemical reaction systems are compared. It is shown that if the volume of the reaction system is taken into account in an appropriate way in the formulation of the Markov chain model, then the o.d.e. model is the infinite volume limit of the Markov chain model. A central limit theorem is also given for the deviation of the Markov chain model from the o.d.e. model.",
    url = "https://doi.org/10.1063/1.1678692",
    doi = "10.1063/1.1678692",
    openalex = "W1976443948"
}

@book{doi101016c20130107362,
    title = "Stochastic Models in Biology",
    year = "1974",
    booktitle = "Elsevier eBooks",
    url = "https://doi.org/10.1016/c2013-0-10736-2",
    doi = "10.1016/c2013-0-10736-2",
    openalex = "W2084206665"
}

@article{doi101007bf01024182,
    author = "Matheson, I. S. and Walls, D. F. and Gardiner, C. W.",
    title = "Stochastic models of firstorder nonequilibrium phase transitions in chemical reactions",
    year = "1975",
    journal = "Journal of Statistical Physics",
    url = "https://doi.org/10.1007/bf01024182",
    doi = "10.1007/bf01024182",
    openalex = "W2063104705"
}

Summary A wide variety of phenomena of geographical spread can be described in terms of a mechanism of “growth” (e.g. birth, infection) and a “contact distribution” which describes how the locations of the individual(s) involved in a migratory move, or infection at a distance, are spatially related. I shall survey work on such models, beginning with an examination of the relations between stochastic and deterministic models; it emerges that both linear and nonlinear deterministic models have close connections with the less interesting “linear” (exponential growth) stochastic models. More realistic models must be nonlinear as well as stochastic; some results are now available for such models. As in the linear case, these deal mainly with asymptotic behaviour. Simulations reveal that nonlinear stochastic processes have a richer spectrum of non-asymptotic behaviour than linear models, though in some circumstances the simpler models may provide an adequate approximation. Thus theoretical study of the short-term behaviour of such processes may be difficult, but should prove rewarding. The other main outstanding problems are those of inference for such models, especially the estimation of contact distributions.

@article{doi101111j251761611977tb01627x,
    author = "Mollison, Denis",
    title = "Spatial Contact Models for Ecological and Epidemic Spread",
    year = "1977",
    journal = "Journal of the Royal Statistical Society Series B (Statistical Methodology)",
    abstract = "Summary A wide variety of phenomena of geographical spread can be described in terms of a mechanism of “growth” (e.g. birth, infection) and a “contact distribution” which describes how the locations of the individual(s) involved in a migratory move, or infection at a distance, are spatially related. I shall survey work on such models, beginning with an examination of the relations between stochastic and deterministic models; it emerges that both linear and nonlinear deterministic models have close connections with the less interesting “linear” (exponential growth) stochastic models. More realistic models must be nonlinear as well as stochastic; some results are now available for such models. As in the linear case, these deal mainly with asymptotic behaviour. Simulations reveal that nonlinear stochastic processes have a richer spectrum of non-asymptotic behaviour than linear models, though in some circumstances the simpler models may provide an adequate approximation. Thus theoretical study of the short-term behaviour of such processes may be difficult, but should prove rewarding. The other main outstanding problems are those of inference for such models, especially the estimation of contact distributions.",
    url = "https://doi.org/10.1111/j.2517-6161.1977.tb01627.x",
    doi = "10.1111/j.2517-6161.1977.tb01627.x",
    openalex = "W76152643",
    references = "bailey1961stochastic, doi1023072282020, doi1023072312029"
}

Journal Article THEORY OF POPULATION GENETICS AND EVOLUTIONARY ECOLOGY: AN INTRODUCTION Get access Henry M. Wilbur, Henry M. Wilbur Department of Zoology Duke University Durham North Carolina 27706 Search for other works by this author on: Oxford Academic Google Scholar Joseph Travis Joseph Travis Department of Zoology Duke University Durham North Carolina 27706 Search for other works by this author on: Oxford Academic Google Scholar Evolution, Volume 34, Issue 5, 1 September 1980, Pages 1032–1033, https://doi.org/10.1111/j.1558-5646.1980.tb04043.x Published: 01 September 1980 Article history Received: 05 May 1980 Published: 01 September 1980

@article{doi101111j155856461980tb04043x,
    author = "Wilbur, Henry M. and Travis, Joseph",
    title = "THEORY OF POPULATION GENETICS AND EVOLUTIONARY ECOLOGY: AN INTRODUCTION",
    year = "1980",
    journal = "Evolution",
    abstract = "Journal Article THEORY OF POPULATION GENETICS AND EVOLUTIONARY ECOLOGY: AN INTRODUCTION Get access Henry M. Wilbur, Henry M. Wilbur Department of Zoology Duke University Durham North Carolina 27706 Search for other works by this author on: Oxford Academic Google Scholar Joseph Travis Joseph Travis Department of Zoology Duke University Durham North Carolina 27706 Search for other works by this author on: Oxford Academic Google Scholar Evolution, Volume 34, Issue 5, 1 September 1980, Pages 1032–1033, https://doi.org/10.1111/j.1558-5646.1980.tb04043.x Published: 01 September 1980 Article history Received: 05 May 1980 Published: 01 September 1980",
    url = "https://doi.org/10.1111/j.1558-5646.1980.tb04043.x",
    doi = "10.1111/j.1558-5646.1980.tb04043.x",
    openalex = "W1988632653"
}

It is shown that many of the asymptotic properties of the Fisher model for population genetics and population ecology can also be derived for a class of models in which time is discrete and space may or may not be discrete. This allows one to discuss the behavior of models in which the data consist of occasional counts on survey tracts, as well as that of computer models.

@article{doi1011370513028,
    author = "Weinberger, Hans F.",
    title = "Long-Time Behavior of a Class of Biological Models",
    year = "1982",
    journal = "SIAM Journal on Mathematical Analysis",
    abstract = "It is shown that many of the asymptotic properties of the Fisher model for population genetics and population ecology can also be derived for a class of models in which time is discrete and space may or may not be discrete. This allows one to discuss the behavior of models in which the data consist of occasional counts on survey tracts, as well as that of computer models.",
    url = "https://doi.org/10.1137/0513028",
    doi = "10.1137/0513028",
    openalex = "W1969504525",
    references = "doi101007bf00275082"
}

@misc{doi101016s0065308x08605618,
    author = "Anderson, Roy M. and May, Robert M.",
    title = "Helminth Infections of Humans: Mathematical Models, Population Dynamics, and Control",
    year = "1985",
    booktitle = "Advances in Parasitology/Advances in parasitology",
    url = "https://doi.org/10.1016/s0065-308x(08)60561-8",
    doi = "10.1016/s0065-308x(08)60561-8",
    openalex = "W101171454",
    references = "doi1023072985209"
}

stage-frequency distribution, a multiple column matrix was used to express the age-stage-structure of animal populations with metamorphosis. Using this new method, the growth process of insect and mite populations (both female and male were included) can be studied with proper stage grouping. Algorithms of population growth and the calculation of intrinsic rate of increase and stable age-stage-distribution were described in detail. For the simulation work of field population ecology, a multidimensional matrix was used to display the spatial distribution of host plants and the age-stage-structure of pest population. Computer programs were designed to simulate the growth and dispersion of field population at the same time. These methods made a way to approach to the agroecosystem simulation. In the last forty years, since the matrix was used in the study of population growth (Lewis 1942, Leslie 1945, 1948), the age structure of a population was given in a

@article{openalexw2182110386,
    author = "Chı, Hsin and Liu, Hsi",
    title = "TWO NEW METHODS FOR THE STUDY OF INSECT POPULATION ECOLOGY",
    year = "1985",
    abstract = "stage-frequency distribution, a multiple column matrix was used to express the age-stage-structure of animal populations with metamorphosis. Using this new method, the growth process of insect and mite populations (both female and male were included) can be studied with proper stage grouping. Algorithms of population growth and the calculation of intrinsic rate of increase and stable age-stage-distribution were described in detail. For the simulation work of field population ecology, a multidimensional matrix was used to display the spatial distribution of host plants and the age-stage-structure of pest population. Computer programs were designed to simulate the growth and dispersion of field population at the same time. These methods made a way to approach to the agroecosystem simulation. In the last forty years, since the matrix was used in the study of population growth (Lewis 1942, Leslie 1945, 1948), the age structure of a population was given in a",
    openalex = "W2182110386",
    references = "doi101038189732a0, doi101093biomet333183"
}

Stochastic factors affecting the demography of a single population are analyzed to determine the relative risks of extinction from demographic stochasticity, environmental stochasticity, and random catastrophes. Relative risks are assessed by comparing asymptotic scaling relationships describing how the average time to extinction, T, increases with the carrying capacity of a population, K, under each stochastic factor alone. Stochastic factors are added to a simple model of exponential growth up to K. A critical parameter affecting the extinction dynamics is $$\tilde r,$$ the long-run growth rate of a population below K, including stochastic factors. If r̃ is positive, with demographic stochasticity T increases asymptotically as a nearly exponential function of K, and with either environmental stochasticity or random catastrophes T increases asymptotically as a power of K. If r̃ is negative, under any stochastic demographic factor, T increases asymptotically with the logarithm of K. Thus, for sufficiently large populations, the risk of extinction from demographic stochasticity is less important than that from either environmental stochasticity or random catastrophes. The relative risks of extinction from environmental stochasticity and random catastrophes depend on the mean and environmental variance of population growth rate, and the magnitude and frequency of catastrophes. Contrary to previous assertions in the literature, a population of modest size subject to environmental stochasticity or random catastrophes can persist for a long time, if r̃ is substantially positive.

@article{doi101086285580,
    author = "Lande, Russell",
    title = "Risks of Population Extinction from Demographic and Environmental Stochasticity and Random Catastrophes",
    year = "1993",
    journal = "The American Naturalist",
    abstract = "Stochastic factors affecting the demography of a single population are analyzed to determine the relative risks of extinction from demographic stochasticity, environmental stochasticity, and random catastrophes. Relative risks are assessed by comparing asymptotic scaling relationships describing how the average time to extinction, T, increases with the carrying capacity of a population, K, under each stochastic factor alone. Stochastic factors are added to a simple model of exponential growth up to K. A critical parameter affecting the extinction dynamics is $$\tilde r,$$ the long-run growth rate of a population below K, including stochastic factors. If r̃ is positive, with demographic stochasticity T increases asymptotically as a nearly exponential function of K, and with either environmental stochasticity or random catastrophes T increases asymptotically as a power of K. If r̃ is negative, under any stochastic demographic factor, T increases asymptotically with the logarithm of K. Thus, for sufficiently large populations, the risk of extinction from demographic stochasticity is less important than that from either environmental stochasticity or random catastrophes. The relative risks of extinction from environmental stochasticity and random catastrophes depend on the mean and environmental variance of population growth rate, and the magnitude and frequency of catastrophes. Contrary to previous assertions in the literature, a population of modest size subject to environmental stochasticity or random catastrophes can persist for a long time, if r̃ is substantially positive.",
    url = "https://doi.org/10.1086/285580",
    doi = "10.1086/285580",
    openalex = "W2065814595",
    references = "doi101016000632078690025x, doi101126science3420403, doi1023071941887"
}

A recurrent problem in ecology and conservation biology is to estimate the risk of population extinctions. Extinction probabilities are not only imperative for conservation and management, but may also elucidate basic mechanisms of the regulation of natural populations. The usual way of modelling stochastic influence on population dynamics has been to assume that the external noise is uncorrelated. This means that each and every randomly drawn noise value is totally independent on previous ones. This is what is generally called 'white' noise. However, the noise itself can be temporally autocorrelated. That is, the values of the random numbers used in the noise process will depend on previous ones. Here we show that the autocorrelation, or colour, of the external noise assumed to influence population dynamics strongly modifies estimated extinction probabilities. For positively autocorrelated ('red') noise, the risk of extinction clearly decreases the stronger the autocorrelation is. Negatively autocorrelated ('blue') noise is more ambiguously related to extinction probability. Thus, the commonly assumed white noise in population modelling will severely bias population extinction risk estimates. Moreover, the extinction probability estimates are also significantly dependent on model structure which calls for a cautious use of traditional discrete-time models.

@article{doi101098rspb19960256,
    author = "Ripa, Jorgen and Lundberg, Per",
    title = "Noise colour and the risk of population extinctions",
    year = "1996",
    journal = "Proceedings of the Royal Society B Biological Sciences",
    abstract = "A recurrent problem in ecology and conservation biology is to estimate the risk of population extinctions. Extinction probabilities are not only imperative for conservation and management, but may also elucidate basic mechanisms of the regulation of natural populations. The usual way of modelling stochastic influence on population dynamics has been to assume that the external noise is uncorrelated. This means that each and every randomly drawn noise value is totally independent on previous ones. This is what is generally called 'white' noise. However, the noise itself can be temporally autocorrelated. That is, the values of the random numbers used in the noise process will depend on previous ones. Here we show that the autocorrelation, or colour, of the external noise assumed to influence population dynamics strongly modifies estimated extinction probabilities. For positively autocorrelated ('red') noise, the risk of extinction clearly decreases the stronger the autocorrelation is. Negatively autocorrelated ('blue') noise is more ambiguously related to extinction probability. Thus, the commonly assumed white noise in population modelling will severely bias population extinction risk estimates. Moreover, the extinction probability estimates are also significantly dependent on model structure which calls for a cautious use of traditional discrete-time models.",
    url = "https://doi.org/10.1098/rspb.1996.0256",
    doi = "10.1098/rspb.1996.0256",
    openalex = "W2059007137"
}

Fluctuations in rates of gene expression can produce highly erratic time patterns of protein production in individual cells and wide diversity in instantaneous protein concentrations across cell populations. When two independently produced regulatory proteins acting at low cellular concentrations competitively control a switch point in a pathway, stochastic variations in their concentrations can produce probabilistic pathway selection, so that an initially homogeneous cell population partitions into distinct phenotypic subpopulations. Many pathogenic organisms, for example, use this mechanism to randomly switch surface features to evade host responses. This coupling between molecular-level fluctuations and macroscopic phenotype selection is analyzed using the phage lambda lysis-lysogeny decision circuit as a model system. The fraction of infected cells selecting the lysogenic pathway at different phage:cell ratios, predicted using a molecular-level stochastic kinetic model of the genetic regulatory circuit, is consistent with experimental observations. The kinetic model of the decision circuit uses the stochastic formulation of chemical kinetics, stochastic mechanisms of gene expression, and a statistical-thermodynamic model of promoter regulation. Conventional deterministic kinetics cannot be used to predict statistics of regulatory systems that produce probabilistic outcomes. Rather, a stochastic kinetic analysis must be used to predict statistics of regulatory outcomes for such stochastically regulated systems.

@article{doi101093genetics14941633,
    author = "Arkin, Adam P. and Ross, John and McAdams, Harley H.",
    title = "Stochastic Kinetic Analysis of Developmental Pathway Bifurcation in Phage λ-Infected Escherichia coli Cells",
    year = "1998",
    journal = "Genetics",
    abstract = "Fluctuations in rates of gene expression can produce highly erratic time patterns of protein production in individual cells and wide diversity in instantaneous protein concentrations across cell populations. When two independently produced regulatory proteins acting at low cellular concentrations competitively control a switch point in a pathway, stochastic variations in their concentrations can produce probabilistic pathway selection, so that an initially homogeneous cell population partitions into distinct phenotypic subpopulations. Many pathogenic organisms, for example, use this mechanism to randomly switch surface features to evade host responses. This coupling between molecular-level fluctuations and macroscopic phenotype selection is analyzed using the phage lambda lysis-lysogeny decision circuit as a model system. The fraction of infected cells selecting the lysogenic pathway at different phage:cell ratios, predicted using a molecular-level stochastic kinetic model of the genetic regulatory circuit, is consistent with experimental observations. The kinetic model of the decision circuit uses the stochastic formulation of chemical kinetics, stochastic mechanisms of gene expression, and a statistical-thermodynamic model of promoter regulation. Conventional deterministic kinetics cannot be used to predict statistics of regulatory systems that produce probabilistic outcomes. Rather, a stochastic kinetic analysis must be used to predict statistics of regulatory outcomes for such stochastically regulated systems.",
    url = "https://doi.org/10.1093/genetics/149.4.1633",
    doi = "10.1093/genetics/149.4.1633",
    openalex = "W1946177469",
    references = "doi1010160021999176900413"
}

Abstract Written by a world renowned biologist, this volume offers a comprehensive synthesis of current research in this rapidly expanding area of population biology. It covers both the essential theory and a wide range of empirical studies, including the author's groundbreaking work on the Glanville fritillary butterfly. It also includes practical applications to conservation biology. The book describes theoretical models for metapopulation dynamics in highly fragmented landscapes and emphasizes spatially realistic models. It presents the incidence function model and includes several detailed examples of its application. Accessible to advanced undergraduate and graduate students, Metapopulation Ecology will be a valuable resource for researchers in population biology, conservation biology, and landscape ecology.

@book{doi101093oso97801985406630010001,
    author = "llkka Hanski",
    title = "Metapopulation Ecology",
    year = "1999",
    abstract = "Abstract Written by a world renowned biologist, this volume offers a comprehensive synthesis of current research in this rapidly expanding area of population biology. It covers both the essential theory and a wide range of empirical studies, including the author's groundbreaking work on the Glanville fritillary butterfly. It also includes practical applications to conservation biology. The book describes theoretical models for metapopulation dynamics in highly fragmented landscapes and emphasizes spatially realistic models. It presents the incidence function model and includes several detailed examples of its application. Accessible to advanced undergraduate and graduate students, Metapopulation Ecology will be a valuable resource for researchers in population biology, conservation biology, and landscape ecology.",
    url = "https://doi.org/10.1093/oso/9780198540663.001.0001",
    doi = "10.1093/oso/9780198540663.001.0001",
    openalex = "W4388324015"
}

@book{doi1010079781461211587,
    author = "Andersson, Håkan and Britton, Tom",
    title = "Stochastic Epidemic Models and Their Statistical Analysis",
    year = "2000",
    booktitle = "Lecture notes in statistics",
    url = "https://doi.org/10.1007/978-1-4612-1158-7",
    doi = "10.1007/978-1-4612-1158-7",
    openalex = "W1608691223"
}

Preface, Creed and Apology. What it is All About and What Not. The Top Down Approach. A Workbook. Portrait of the Reader as a Young Person. A Brief Outline of the Book. Acknowledgments. And What About Reality? I: THE BARE BONES: BASIC ISSUES EXPLAINED IN THE SIMPLEST CONTEXT. The Epidemic in a Closed Population. Heterogeneity: The Art of Averaging. Dynamics at the Demographic Time Scale. II: STRUCTURED POPULATIONS. The Concept of State. The Basic Reproduction Ratio. And Everything Else... Age Structure. Spatial Spread. Macroparasites. What is Contact? III: THE HARD PART: ELABORATIONS (ALMOST) ALL EXERCISES. Elaborations for Part I. Elaborations for Part II. Appendix A: Stochastic Basic of the Kermack-McKendrick ODE Model. Appendix B: Bibliographic Skeleton. Index.

@book{openalexw2104595316,
    author = "Diekmann, Odo and Heesterbeek, Hans",
    title = "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation",
    year = "2000",
    journal = "Socio-Environmental Systems Modeling",
    abstract = "Preface, Creed and Apology. What it is All About and What Not. The Top Down Approach. A Workbook. Portrait of the Reader as a Young Person. A Brief Outline of the Book. Acknowledgments. And What About Reality? I: THE BARE BONES: BASIC ISSUES EXPLAINED IN THE SIMPLEST CONTEXT. The Epidemic in a Closed Population. Heterogeneity: The Art of Averaging. Dynamics at the Demographic Time Scale. II: STRUCTURED POPULATIONS. The Concept of State. The Basic Reproduction Ratio. And Everything Else... Age Structure. Spatial Spread. Macroparasites. What is Contact? III: THE HARD PART: ELABORATIONS (ALMOST) ALL EXERCISES. Elaborations for Part I. Elaborations for Part II. Appendix A: Stochastic Basic of the Kermack-McKendrick ODE Model. Appendix B: Bibliographic Skeleton. Index.",
    openalex = "W2104595316"
}

The stochastic simulation algorithm (SSA) is an essentially exact procedure for numerically simulating the time evolution of a well-stirred chemically reacting system. Despite recent major improvements in the efficiency of the SSA, its drawback remains the great amount of computer time that is often required to simulate a desired amount of system time. Presented here is the “τ-leap” method, an approximate procedure that in some circumstances can produce significant gains in simulation speed with acceptable losses in accuracy. Some primitive strategies for control parameter selection and error mitigation for the τ-leap method are described, and simulation results for two simple model systems are exhibited. With further refinement, the τ-leap method should provide a viable way of segueing from the exact SSA to the approximate chemical Langevin equation, and thence to the conventional deterministic reaction rate equation, as the system size becomes larger.

@article{doi10106311378322,
    author = "Gillespie, Daniel T.",
    title = "Approximate accelerated stochastic simulation of chemically reacting systems",
    year = "2001",
    journal = "The Journal of Chemical Physics",
    abstract = "The stochastic simulation algorithm (SSA) is an essentially exact procedure for numerically simulating the time evolution of a well-stirred chemically reacting system. Despite recent major improvements in the efficiency of the SSA, its drawback remains the great amount of computer time that is often required to simulate a desired amount of system time. Presented here is the “τ-leap” method, an approximate procedure that in some circumstances can produce significant gains in simulation speed with acceptable losses in accuracy. Some primitive strategies for control parameter selection and error mitigation for the τ-leap method are described, and simulation results for two simple model systems are exhibited. With further refinement, the τ-leap method should provide a viable way of segueing from the exact SSA to the approximate chemical Langevin equation, and thence to the conventional deterministic reaction rate equation, as the system size becomes larger.",
    url = "https://doi.org/10.1063/1.1378322",
    doi = "10.1063/1.1378322",
    openalex = "W2084924279",
    references = "doi1010160021999176900413"
}

The majority of pathogens, including many of medical and veterinary importance, can infect more than one species of host. Population biology has yet to explain why perceived evolutionary advantages of pathogen specialization are, in practice, outweighed by those of generalization. Factors that predispose pathogens to generalism include high levels of genetic diversity and abundant opportunities for cross-species transmission, and the taxonomic distributions of generalists and specialists appear to reflect these factors. Generalism also has consequences for the evolution of virulence and for pathogen epidemiology, making both much less predictable. The evolutionary advantages and disadvantages of generalism are so finely balanced that even closely related pathogens can have very different host range sizes.

@article{doi101126science1059026,
    author = "Woolhouse, Mark and Taylor, Louise and Haydon, Daniel T.",
    title = "Population Biology of Multihost Pathogens",
    year = "2001",
    journal = "Science",
    abstract = "The majority of pathogens, including many of medical and veterinary importance, can infect more than one species of host. Population biology has yet to explain why perceived evolutionary advantages of pathogen specialization are, in practice, outweighed by those of generalization. Factors that predispose pathogens to generalism include high levels of genetic diversity and abundant opportunities for cross-species transmission, and the taxonomic distributions of generalists and specialists appear to reflect these factors. Generalism also has consequences for the evolution of virulence and for pathogen epidemiology, making both much less predictable. The evolutionary advantages and disadvantages of generalism are so finely balanced that even closely related pathogens can have very different host range sizes.",
    url = "https://doi.org/10.1126/science.1059026",
    doi = "10.1126/science.1059026",
    openalex = "W2032563322",
    references = "doi101038189732a0"
}

The geographical distribution of a species is determined by a large number of complex processes operating over spatial scales spanning 10 orders of magnitude. Patterns in population processes have been described at numerous scales. We show that two patterns, measured at different scales, jointly allow us to infer heretofore unknown patterns in the distribution of demographic patterns across the geographical range of a species. The resulting model describes three fundamentally different modes of geographical variation in vital rates of populations. One mode is characterized by a positive nonlinear relationship between the maximum rate of population growth and the intensity of intraspecific competition across a geographical range. That is, populations that grow rapidly are also those where individuals experience the greatest per capita negative effect of the presence of other individuals. The second mode of behaviour is described by a negative nonlinear relationship between maximum growth rate and density dependence. Under this scenario, populations with low capacity to grow rapidly have highest intensities of intraspecific competitive effects. A third mode of behaviour is characterized by a weak positive relationship between growth rate and intraspecific competition, with very little geographical variation in maximum growth rate. A survey of studies relating temporal means and variances in population abundance for a variety of species indicate that the second mode of geographical variation in population dynamics across species ranges is the most common, though a few species appear to be characterized by the third mode.

@article{doi101046j14610248200200308x,
    author = "Maurer, Brian A. and Taper, Mark L.",
    title = "Connecting geographical distributions with population processes",
    year = "2002",
    journal = "Ecology Letters",
    abstract = "The geographical distribution of a species is determined by a large number of complex processes operating over spatial scales spanning 10 orders of magnitude. Patterns in population processes have been described at numerous scales. We show that two patterns, measured at different scales, jointly allow us to infer heretofore unknown patterns in the distribution of demographic patterns across the geographical range of a species. The resulting model describes three fundamentally different modes of geographical variation in vital rates of populations. One mode is characterized by a positive nonlinear relationship between the maximum rate of population growth and the intensity of intraspecific competition across a geographical range. That is, populations that grow rapidly are also those where individuals experience the greatest per capita negative effect of the presence of other individuals. The second mode of behaviour is described by a negative nonlinear relationship between maximum growth rate and density dependence. Under this scenario, populations with low capacity to grow rapidly have highest intensities of intraspecific competitive effects. A third mode of behaviour is characterized by a weak positive relationship between growth rate and intraspecific competition, with very little geographical variation in maximum growth rate. A survey of studies relating temporal means and variances in population abundance for a variety of species indicate that the second mode of geographical variation in population dynamics across species ranges is the most common, though a few species appear to be characterized by the third mode.",
    url = "https://doi.org/10.1046/j.1461-0248.2002.00308.x",
    doi = "10.1046/j.1461-0248.2002.00308.x",
    openalex = "W2122270598",
    references = "doi101016016953479390043o, doi101016c20130107362, doi101038189732a0, doi10103847487, doi101046j14610248200000143x, doi101086284267, doi101111j155856461980tb04043x, doi1015159781400881376, doi1023071909913, doi1023071941678, doi105860choice353840"
}

In order to understand the functioning of organisms on the molecular level, we need to know which genes are expressed, when and where in the organism, and to which extent. The regulation of gene expression is achieved through genetic regulatory systems structured by networks of interactions between DNA, RNA, proteins, and small molecules. As most genetic regulatory networks of interest involve many components connected through interlocking positive and negative feedback loops, an intuitive understanding of their dynamics is hard to obtain. As a consequence, formal methods and computer tools for the modeling and simulation of genetic regulatory networks will be indispensable. This paper reviews formalisms that have been employed in mathematical biology and bioinformatics to describe genetic regulatory systems, in particular directed graphs, Bayesian networks, Boolean networks and their generalizations, ordinary and partial differential equations, qualitative differential equations, stochastic equations, and rule-based formalisms. In addition, the paper discusses how these formalisms have been used in the simulation of the behavior of actual regulatory systems.

@article{doi10108910665270252833208,
    author = "de Jong, Hidde",
    title = "Modeling and Simulation of Genetic Regulatory Systems: A Literature Review",
    year = "2002",
    journal = "Journal of Computational Biology",
    abstract = "In order to understand the functioning of organisms on the molecular level, we need to know which genes are expressed, when and where in the organism, and to which extent. The regulation of gene expression is achieved through genetic regulatory systems structured by networks of interactions between DNA, RNA, proteins, and small molecules. As most genetic regulatory networks of interest involve many components connected through interlocking positive and negative feedback loops, an intuitive understanding of their dynamics is hard to obtain. As a consequence, formal methods and computer tools for the modeling and simulation of genetic regulatory networks will be indispensable. This paper reviews formalisms that have been employed in mathematical biology and bioinformatics to describe genetic regulatory systems, in particular directed graphs, Bayesian networks, Boolean networks and their generalizations, ordinary and partial differential equations, qualitative differential equations, stochastic equations, and rule-based formalisms. In addition, the paper discusses how these formalisms have been used in the simulation of the behavior of actual regulatory systems.",
    url = "https://doi.org/10.1089/10665270252833208",
    doi = "10.1089/10665270252833208",
    openalex = "W2058221907",
    references = "doi1010160022519373902087, doi101016b9780444529657x50004, openalexw1576847343"
}

Epidemic dynamics pose a great challenge to stochastic modelling because chance events are major determinants of the size and the timing of the outbreak. Reintroduction of the disease through contact with infected individuals from other areas is an important latent stochastic variable. In this study we model these stochastic processes to explain extinction and recurrence of epidemics observed in measles. We develop estimating functions for such a model and apply the methodology to temporal case counts of measles in 60 cities in England and Wales. In order to estimate the unobserved spatial contact process we suggest a method based on stochastic simulation and marginal densities. The estimation results show that it is possible to consider a unified model for the UK cities where the parameters depend on the city size. Stochastic realizations from the dynamic model realistically capture the transitions from an endemic cyclic pattern in large populations to irregular epidemic outbreaks in small human host populations.

@article{doi101093biostatistics34493,
    author = "Finkenstädt, Bärbel",
    title = "A stochastic model for extinction and recurrence of epidemics: estimation and inference for measles outbreaks",
    year = "2002",
    journal = "Biostatistics",
    abstract = "Epidemic dynamics pose a great challenge to stochastic modelling because chance events are major determinants of the size and the timing of the outbreak. Reintroduction of the disease through contact with infected individuals from other areas is an important latent stochastic variable. In this study we model these stochastic processes to explain extinction and recurrence of epidemics observed in measles. We develop estimating functions for such a model and apply the methodology to temporal case counts of measles in 60 cities in England and Wales. In order to estimate the unobserved spatial contact process we suggest a method based on stochastic simulation and marginal densities. The estimation results show that it is possible to consider a unified model for the UK cities where the parameters depend on the city size. Stochastic realizations from the dynamic model realistically capture the transitions from an endemic cyclic pattern in large populations to irregular epidemic outbreaks in small human host populations.",
    url = "https://doi.org/10.1093/biostatistics/3.4.493",
    doi = "10.1093/biostatistics/3.4.493",
    openalex = "W2135612840",
    references = "doi10100703872725509, doi101007bf00277162, doi1010382231133a0, doi101098rspa19330106, doi1012019781315140919, doi1018900012961520020720169domees20co2, doi1023071909913, doi1023071913974, doi1023072342553, doi1073260003481911721744, doi107326000348199822732"
}

Before the development of mass-vaccination campaigns, measles exhibited persistent fluctuations (endemic dynamics) in large British cities, and recurrent outbreaks (episodic dynamics) in smaller communities. The critical community size separating the two regimes was ∼300 000–500 000. We develop a model, the TSIR (Time-series Susceptible–Infected–Recovered) model, that can capture both endemic cycles and episodic outbreaks in measles. The model includes the stochasticity inherent in the disease transmission (giving rise to a negative binomial conditional distribution) and random immigration. It is thus a doubly stochastic model for disease dynamics. It further includes seasonality in the transmission rates. All parameters of the model are estimated on the basis of time series data on reported cases and reconstructed susceptible numbers from a set of cities in England and Wales in the prevaccination era (1944–1966). The 60 cities analyzed span a size range from London (3.3 × 106 inhabitants) to Teignmouth (10 500 inhabitants). The dynamics of all cities fit the model well. Transmission rates scale with community size, as expected from dynamics adhering closely to frequency dependent transmission (“true mass action”). These rates are further found to reveal strong seasonal variation, corresponding to high transmission during school terms and lower transmission during the school holidays. The basic reproductive ratio, R0, is found to be invariant across the observed range of host community size, and the mean proportion of susceptible individuals also appears to be constant. Through the epidemic cycle, the susceptible population is kept within a 3% interval. The disease is, thus, efficient in “regulating” the susceptible population—even in small cities that undergo recurrent epidemics with frequent extinction of the disease agent. Recolonization is highly sensitive to the random immigration process. The initial phase of the epidemic is also stochastic (due to demographic stochasticity and random immigration). However, the epidemic is nearly “deterministic” through most of the growth and decline phase.

@article{doi1018900012961520020720169domees20co2,
    author = "Bjørnstad, Ottar N. and Finkenstädt, Bärbel and Grenfell, Bryan T.",
    title = "DYNAMICS OF MEASLES EPIDEMICS: ESTIMATING SCALING OF TRANSMISSION RATES USING A TIME SERIES SIR MODEL",
    year = "2002",
    journal = "Ecological Monographs",
    abstract = "Before the development of mass-vaccination campaigns, measles exhibited persistent fluctuations (endemic dynamics) in large British cities, and recurrent outbreaks (episodic dynamics) in smaller communities. The critical community size separating the two regimes was ∼300 000–500 000. We develop a model, the TSIR (Time-series Susceptible–Infected–Recovered) model, that can capture both endemic cycles and episodic outbreaks in measles. The model includes the stochasticity inherent in the disease transmission (giving rise to a negative binomial conditional distribution) and random immigration. It is thus a doubly stochastic model for disease dynamics. It further includes seasonality in the transmission rates. All parameters of the model are estimated on the basis of time series data on reported cases and reconstructed susceptible numbers from a set of cities in England and Wales in the prevaccination era (1944–1966). The 60 cities analyzed span a size range from London (3.3 × 106 inhabitants) to Teignmouth (10 500 inhabitants). The dynamics of all cities fit the model well. Transmission rates scale with community size, as expected from dynamics adhering closely to frequency dependent transmission (“true mass action”). These rates are further found to reveal strong seasonal variation, corresponding to high transmission during school terms and lower transmission during the school holidays. The basic reproductive ratio, R0, is found to be invariant across the observed range of host community size, and the mean proportion of susceptible individuals also appears to be constant. Through the epidemic cycle, the susceptible population is kept within a 3\% interval. The disease is, thus, efficient in “regulating” the susceptible population—even in small cities that undergo recurrent epidemics with frequent extinction of the disease agent. Recolonization is highly sensitive to the random immigration process. The initial phase of the epidemic is also stochastic (due to demographic stochasticity and random immigration). However, the epidemic is nearly “deterministic” through most of the growth and decline phase.",
    url = "https://doi.org/10.1890/0012-9615(2002)072[0169:domees]2.0.co;2",
    doi = "10.1890/0012-9615(2002)072[0169:domees]2.0.co;2",
    openalex = "W1968344341",
    references = "doi101093biostatistics34493, doi1023072527465"
}

Two key linked questions in population dynamics are the relative importance of noise vs. density-dependent nonlinearities and the limits on temporal predictability of population abundance. We propose that childhood microparasitic infections, notably measles, provide an unusually suitable empirical and theoretical test bed for addressing these issues. We base our analysis on a new mechanistic time series model for measles, the TSIR model, which captures the mechanistic essence of epidemic dynamics. The model, and parameter estimates based on short-term fits to prevaccination measles time series for 60 towns and cities in England and Wales, is introduced in a companion paper. Here, we explore how well the model predicts the long-term dynamics of measles and the balance between noise and determinism, as a function of population size. The TSIR model captures the basic dynamical features of the long-term pattern of measles epidemics in large cities remarkably well (based on time and frequency domain analyses). In particular, the model illustrates the impact of secular increases in birth rates, which cause a transition from biennial to annual dynamics. The model also captures the observed increase in epidemic irregularity with decreasing population size and the onset of local extinction below a critical community size. Decreased host population size is shown to be associated with an increased impact of demographic stochasticity. The interaction between nonlinearity and noise is explored using local Lyapunov exponents (LLE). These testify to the high level of stability of the biennial attractor in large cities. Irregularities are due to the limit cycle evolving with changing human birth rates and not due to complex dynamics. The geometry of the dynamics (sign and magnitude of the LLEs across phase space) is similar in the cities and the smaller urban areas. The qualitative difference in dynamics between small and large host communities is that demographic and extinction–recolonization stochasticities are much more influential in the former. The regional dynamics can therefore only be understood in terms of a core–satellite metapopulation structure for this host–enemy system. We also make a preliminary exploration of the model's ability to predict the dynamic consequences of measles vaccination.

@article{doi1018900012961520020720185domesn20co2,
    author = "Grenfell, Bryan T. and Bjørnstad, Ottar N. and Finkenstädt, Bärbel",
    title = "DYNAMICS OF MEASLES EPIDEMICS: SCALING NOISE, DETERMINISM, AND PREDICTABILITY WITH THE TSIR MODEL",
    year = "2002",
    journal = "Ecological Monographs",
    abstract = "Two key linked questions in population dynamics are the relative importance of noise vs. density-dependent nonlinearities and the limits on temporal predictability of population abundance. We propose that childhood microparasitic infections, notably measles, provide an unusually suitable empirical and theoretical test bed for addressing these issues. We base our analysis on a new mechanistic time series model for measles, the TSIR model, which captures the mechanistic essence of epidemic dynamics. The model, and parameter estimates based on short-term fits to prevaccination measles time series for 60 towns and cities in England and Wales, is introduced in a companion paper. Here, we explore how well the model predicts the long-term dynamics of measles and the balance between noise and determinism, as a function of population size. The TSIR model captures the basic dynamical features of the long-term pattern of measles epidemics in large cities remarkably well (based on time and frequency domain analyses). In particular, the model illustrates the impact of secular increases in birth rates, which cause a transition from biennial to annual dynamics. The model also captures the observed increase in epidemic irregularity with decreasing population size and the onset of local extinction below a critical community size. Decreased host population size is shown to be associated with an increased impact of demographic stochasticity. The interaction between nonlinearity and noise is explored using local Lyapunov exponents (LLE). These testify to the high level of stability of the biennial attractor in large cities. Irregularities are due to the limit cycle evolving with changing human birth rates and not due to complex dynamics. The geometry of the dynamics (sign and magnitude of the LLEs across phase space) is similar in the cities and the smaller urban areas. The qualitative difference in dynamics between small and large host communities is that demographic and extinction–recolonization stochasticities are much more influential in the former. The regional dynamics can therefore only be understood in terms of a core–satellite metapopulation structure for this host–enemy system. We also make a preliminary exploration of the model's ability to predict the dynamic consequences of measles vaccination.",
    url = "https://doi.org/10.1890/0012-9615(2002)072[0185:domesn]2.0.co;2",
    doi = "10.1890/0012-9615(2002)072[0185:domesn]2.0.co;2",
    openalex = "W2141438518",
    references = "doi101093biostatistics34493"
}

MrBayes 3 performs Bayesian phylogenetic analysis combining information from different data partitions or subsets evolving under different stochastic evolutionary models. This allows the user to analyze heterogeneous data sets consisting of different data types-e.g. morphological, nucleotide, and protein-and to explore a wide variety of structured models mixing partition-unique and shared parameters. The program employs MPI to parallelize Metropolis coupling on Macintosh or UNIX clusters.

@article{doi101093bioinformaticsbtg180,
    author = "Ronquist, Fredrik and Huelsenbeck, John P.",
    title = "MrBayes 3: Bayesian phylogenetic inference under mixed models",
    year = "2003",
    journal = "Bioinformatics",
    abstract = "MrBayes 3 performs Bayesian phylogenetic analysis combining information from different data partitions or subsets evolving under different stochastic evolutionary models. This allows the user to analyze heterogeneous data sets consisting of different data types-e.g. morphological, nucleotide, and protein-and to explore a wide variety of structured models mixing partition-unique and shared parameters. The program employs MPI to parallelize Metropolis coupling on Macintosh or UNIX clusters.",
    url = "https://doi.org/10.1093/bioinformatics/btg180",
    doi = "10.1093/bioinformatics/btg180",
    openalex = "W2146058063",
    references = "doi101093bioinformatics124357, doi101093bioinformaticsbtg427, doi101093oxfordjournalsmolbeva026160, doi101126science1065889, openalexw2994240441"
}

@article{doi101016jbulm200311007,
    author = "Kirupaharan, Nadarajah",
    title = "Coexistence of multiple pathogen strains in stochastic epidemic models with density-dependent mortality",
    year = "2004",
    journal = "Bulletin of Mathematical Biology",
    url = "https://doi.org/10.1016/j.bulm.2003.11.007",
    doi = "10.1016/j.bulm.2003.11.007",
    openalex = "W1989733110",
    references = "doi1023072527465"
}

Infectious diseases provide a particularly clear illustration of the spatiotemporal underpinnings of consumer-resource dynamics. The paradigm is provided by extremely contagious, acute, immunizing childhood infections. Partially synchronized, unstable oscillations are punctuated by local extinctions. This, in turn, can result in spatial differentiation in the timing of epidemics and, depending on the nature of spatial contagion, may result in traveling waves. Measles epidemics are one of a few systems documented well enough to reveal all of these properties and how they are affected by spatiotemporal variations in population structure and demography. On the basis of a gravity coupling model and a time series susceptible-infected-recovered (TSIR) model for local dynamics, we propose a metapopulation model for regional measles dynamics. The model can capture all the major spatiotemporal properties in prevaccination epidemics of measles in England and Wales.

@article{doi101086422341,
    author = "Xia, Yingcun and Bjørnstad, Ottar N. and Grenfell, Bryan T.",
    title = "Measles Metapopulation Dynamics: A Gravity Model for Epidemiological Coupling and Dynamics",
    year = "2004",
    journal = "The American Naturalist",
    abstract = "Infectious diseases provide a particularly clear illustration of the spatiotemporal underpinnings of consumer-resource dynamics. The paradigm is provided by extremely contagious, acute, immunizing childhood infections. Partially synchronized, unstable oscillations are punctuated by local extinctions. This, in turn, can result in spatial differentiation in the timing of epidemics and, depending on the nature of spatial contagion, may result in traveling waves. Measles epidemics are one of a few systems documented well enough to reveal all of these properties and how they are affected by spatiotemporal variations in population structure and demography. On the basis of a gravity coupling model and a time series susceptible-infected-recovered (TSIR) model for local dynamics, we propose a metapopulation model for regional measles dynamics. The model can capture all the major spatiotemporal properties in prevaccination epidemics of measles in England and Wales.",
    url = "https://doi.org/10.1086/422341",
    doi = "10.1086/422341",
    openalex = "W2029997459",
    references = "doi101093biostatistics34493"
}

The range of potential mechanisms limiting species' distributions in space is nearly as varied and complex as the diversity of life itself. Yet viewed abstractly, a species' border is a geographic manifestation of a species' demographic responses to a spatially and temporally varying world. Population dynamic models provide insight into the different routes by which range limits can arise owing to gradients in demographic rates. In a metapopulation context, for example, range limits may be caused by gradients in extinction rates, colonization rates or habitat availability. We have consider invasion models in uniform and heterogeneous environments as a framework for understanding non‐equilibrium range limits, and explore conditions under which invasions may cease to spread leaving behind a stationary range limit. We conclude that non‐equilibrial range dynamics need further theoretical and empirical attention.

@article{doi101111j00301299200513147x,
    author = "Holt, Robert D. and Keitt, Timothy H. and Lewis, Mark A. and Maurer, Brian A. and Taper, Mark L.",
    title = "Theoretical models of species’ borders: single species approaches",
    year = "2004",
    journal = "Oikos",
    abstract = "The range of potential mechanisms limiting species' distributions in space is nearly as varied and complex as the diversity of life itself. Yet viewed abstractly, a species' border is a geographic manifestation of a species' demographic responses to a spatially and temporally varying world. Population dynamic models provide insight into the different routes by which range limits can arise owing to gradients in demographic rates. In a metapopulation context, for example, range limits may be caused by gradients in extinction rates, colonization rates or habitat availability. We have consider invasion models in uniform and heterogeneous environments as a framework for understanding non‐equilibrium range limits, and explore conditions under which invasions may cease to spread leaving behind a stationary range limit. We conclude that non‐equilibrial range dynamics need further theoretical and empirical attention.",
    url = "https://doi.org/10.1111/j.0030-1299.2005.13147.x",
    doi = "10.1111/j.0030-1299.2005.13147.x",
    openalex = "W2113987013",
    references = "doi101046j14610248200200308x"
}

The Moran effect for populations separated in space states that the autocorrelations in the population fluctuations equal the autocorrelation in environmental noise, assuming the same linear density regulation in all populations. Here we generalize the Moran effect to include also nonlinear density regulation with spatial heterogeneity in local population dynamics as well as in the effects of environmental covariates by deriving a simple expression for the correlation between the sizes of two populations, using diffusion approximation to the theta-logistic model. In general, spatial variation in parameters describing the dynamics reduces population synchrony. We also show that the contribution of a covariate to spatial synchrony depends strongly on spatial heterogeneity in the covariate or in its effect on local dynamics. These analyses show exactly how spatial environmental covariation can synchronize fluctuations of spatially segregated populations with no interchange of individuals even if the dynamics are nonlinear.

@article{doi101086491690,
    author = "Engen, Steinar and Sæther, Bernt‐Erik",
    title = "Generalizations of the Moran Effect Explaining Spatial Synchrony in Population Fluctuations",
    year = "2005",
    journal = "The American Naturalist",
    abstract = "The Moran effect for populations separated in space states that the autocorrelations in the population fluctuations equal the autocorrelation in environmental noise, assuming the same linear density regulation in all populations. Here we generalize the Moran effect to include also nonlinear density regulation with spatial heterogeneity in local population dynamics as well as in the effects of environmental covariates by deriving a simple expression for the correlation between the sizes of two populations, using diffusion approximation to the theta-logistic model. In general, spatial variation in parameters describing the dynamics reduces population synchrony. We also show that the contribution of a covariate to spatial synchrony depends strongly on spatial heterogeneity in the covariate or in its effect on local dynamics. These analyses show exactly how spatial environmental covariation can synchronize fluctuations of spatially segregated populations with no interchange of individuals even if the dynamics are nonlinear.",
    url = "https://doi.org/10.1086/491690",
    doi = "10.1086/491690",
    openalex = "W1996045361",
    references = "barton1962stochastic, doi1023072333508"
}

Networks and the epidemiology of directly transmitted infectious diseases are fundamentally linked. The foundations of epidemiology and early epidemiological models were based on population wide random-mixing, but in practice each individual has a finite set of contacts to whom they can pass infection; the ensemble of all such contacts forms a 'mixing network'. Knowledge of the structure of the network allows models to compute the epidemic dynamics at the population scale from the individual-level behaviour of infections. Therefore, characteristics of mixing networks-and how these deviate from the random-mixing norm-have become important applied concerns that may enhance the understanding and prediction of epidemic patterns and intervention measures. Here, we review the basis of epidemiological theory (based on random-mixing models) and network theory (based on work from the social sciences and graph theory). We then describe a variety of methods that allow the mixing network, or an approximation to the network, to be ascertained. It is often the case that time and resources limit our ability to accurately find all connections within a network, and hence a generic understanding of the relationship between network structure and disease dynamics is needed. Therefore, we review some of the variety of idealized network types and approximation techniques that have been utilized to elucidate this link. Finally, we look to the future to suggest how the two fields of network theory and epidemiological modelling can deliver an improved understanding of disease dynamics and better public health through effective disease control.

@article{doi101098rsif20050051,
    author = "Keeling, Matt J. and Eames, Ken",
    title = "Networks and epidemic models",
    year = "2005",
    journal = "Journal of The Royal Society Interface",
    abstract = "Networks and the epidemiology of directly transmitted infectious diseases are fundamentally linked. The foundations of epidemiology and early epidemiological models were based on population wide random-mixing, but in practice each individual has a finite set of contacts to whom they can pass infection; the ensemble of all such contacts forms a 'mixing network'. Knowledge of the structure of the network allows models to compute the epidemic dynamics at the population scale from the individual-level behaviour of infections. Therefore, characteristics of mixing networks-and how these deviate from the random-mixing norm-have become important applied concerns that may enhance the understanding and prediction of epidemic patterns and intervention measures. Here, we review the basis of epidemiological theory (based on random-mixing models) and network theory (based on work from the social sciences and graph theory). We then describe a variety of methods that allow the mixing network, or an approximation to the network, to be ascertained. It is often the case that time and resources limit our ability to accurately find all connections within a network, and hence a generic understanding of the relationship between network structure and disease dynamics is needed. Therefore, we review some of the variety of idealized network types and approximation techniques that have been utilized to elucidate this link. Finally, we look to the future to suggest how the two fields of network theory and epidemiological modelling can deliver an improved understanding of disease dynamics and better public health through effective disease control.",
    url = "https://doi.org/10.1098/rsif.2005.0051",
    doi = "10.1098/rsif.2005.0051",
    openalex = "W2073627497",
    references = "doi101098rspa19270118"
}

Agent-based complex systems are dynamic networks of many interacting agents; examples include ecosystems, financial markets, and cities. The search for general principles underlying the internal organization of such systems often uses bottom-up simulation models such as cellular automata and agent-based models. No general framework for designing, testing, and analyzing bottom-up models has yet been established, but recent advances in ecological modeling have come together in a general strategy we call pattern-oriented modeling. This strategy provides a unifying framework for decoding the internal organization of agent-based complex systems and may lead toward unifying algorithmic theories of the relation between adaptive behavior and system complexity.

@article{doi101126science1116681,
    author = "Grimm, Volker and Revilla, Eloy and Berger, Uta and Jeltsch, Florian and Mooij, Wolf M. and Railsback, Steven F. and Thulke, Hans‐Hermann and Weiner, Jacob and Wiegand, Thorsten and DeAngelis, Donald L.",
    title = "Pattern-Oriented Modeling of Agent-Based Complex Systems: Lessons from Ecology",
    year = "2005",
    journal = "Science",
    abstract = "Agent-based complex systems are dynamic networks of many interacting agents; examples include ecosystems, financial markets, and cities. The search for general principles underlying the internal organization of such systems often uses bottom-up simulation models such as cellular automata and agent-based models. No general framework for designing, testing, and analyzing bottom-up models has yet been established, but recent advances in ecological modeling have come together in a general strategy we call pattern-oriented modeling. This strategy provides a unifying framework for decoding the internal organization of agent-based complex systems and may lead toward unifying algorithmic theories of the relation between adaptive behavior and system complexity.",
    url = "https://doi.org/10.1126/science.1116681",
    doi = "10.1126/science.1116681",
    openalex = "W2006012747",
    references = "doi101017cbo9780511626135, openalexw1515814298"
}

Individual-based models are an exciting and widely used new tool for ecology. These computational models allow scientists to explore the mechanisms through which population and ecosystem ecology arises from how individuals interact with each other and their environment. This book provides the first in-depth treatment of individual-based modeling and its use to develop theoretical understanding of how ecological systems work, an approach the authors call "individual-based ecology.? Grimm and Railsback start with a general primer on modeling: how to design models that are as simple as possible while still allowing specific problems to be solved, and how to move efficiently through a cycle of pattern-oriented model design, implementation, and analysis. Next, they address the problems of theory and conceptual framework for individual-based ecology: What is "theory"? That is, how do we develop reusable models of how system dynamics arise from characteristics of individuals? What conceptual framework do we use when the classical differential equation framework no longer applies? An extensive review illustrates the ecological problems that have been addressed with individual-based models. The authors then identify how the mechanics of building and using individual-based models differ from those of traditional science, and provide guidance on formulating, programming, and analyzing models. This book will be helpful to ecologists interested in modeling, and to other scientists interested in agent-based modeling.

@book{doi1015159781400850624,
    author = "Grimm, Volker and Railsback, Steven F.",
    title = "Individual-based Modeling and Ecology",
    year = "2005",
    booktitle = "Princeton University Press eBooks",
    abstract = {Individual-based models are an exciting and widely used new tool for ecology. These computational models allow scientists to explore the mechanisms through which population and ecosystem ecology arises from how individuals interact with each other and their environment. This book provides the first in-depth treatment of individual-based modeling and its use to develop theoretical understanding of how ecological systems work, an approach the authors call "individual-based ecology.? Grimm and Railsback start with a general primer on modeling: how to design models that are as simple as possible while still allowing specific problems to be solved, and how to move efficiently through a cycle of pattern-oriented model design, implementation, and analysis. Next, they address the problems of theory and conceptual framework for individual-based ecology: What is "theory"? That is, how do we develop reusable models of how system dynamics arise from characteristics of individuals? What conceptual framework do we use when the classical differential equation framework no longer applies? An extensive review illustrates the ecological problems that have been addressed with individual-based models. The authors then identify how the mechanics of building and using individual-based models differ from those of traditional science, and provide guidance on formulating, programming, and analyzing models. This book will be helpful to ecologists interested in modeling, and to other scientists interested in agent-based modeling.},
    url = "https://doi.org/10.1515/9781400850624",
    doi = "10.1515/9781400850624",
    openalex = "W1483113211"
}

Models that deal with the individual level of populations have shown the importance of stochasticity in ecology, epidemiology and evolution. An increasingly common approach to studying these models is through stochastic (event-driven) simulation. One striking disadvantage of this approach is the need for a large number of replicates to determine the range of expected behaviour. Here, for a class of stochastic models called Markov processes, we present results that overcome this difficulty and provide valuable insights, but which have been largely ignored by applied researchers. For these models, the so-called Kolmogorov forward equation (also called the ensemble or master equation) allows one to simultaneously consider the probability of each possible state occurring. Irrespective of the complexities and nonlinearities of population dynamics, this equation is linear and has a natural matrix formulation that provides many analytical insights into the behaviour of stochastic populations and allows rapid evaluation of process dynamics. Here, using epidemiological models as a template, these ensemble equations are explored and results are compared with traditional stochastic simulations. In addition, we describe further advantages of the matrix formulation of dynamics, providing simple exact methods for evaluating expected eradication (extinction) times of diseases, for comparing expected total costs of possible control programmes and for estimation of disease parameters.

@article{doi101098rsif20071106,
    author = "Keeling, Matt J. and Ross, Joshua V.",
    title = "On methods for studying stochastic disease dynamics",
    year = "2007",
    journal = "Journal of The Royal Society Interface",
    abstract = "Models that deal with the individual level of populations have shown the importance of stochasticity in ecology, epidemiology and evolution. An increasingly common approach to studying these models is through stochastic (event-driven) simulation. One striking disadvantage of this approach is the need for a large number of replicates to determine the range of expected behaviour. Here, for a class of stochastic models called Markov processes, we present results that overcome this difficulty and provide valuable insights, but which have been largely ignored by applied researchers. For these models, the so-called Kolmogorov forward equation (also called the ensemble or master equation) allows one to simultaneously consider the probability of each possible state occurring. Irrespective of the complexities and nonlinearities of population dynamics, this equation is linear and has a natural matrix formulation that provides many analytical insights into the behaviour of stochastic populations and allows rapid evaluation of process dynamics. Here, using epidemiological models as a template, these ensemble equations are explored and results are compared with traditional stochastic simulations. In addition, we describe further advantages of the matrix formulation of dynamics, providing simple exact methods for evaluating expected eradication (extinction) times of diseases, for comparing expected total costs of possible control programmes and for estimation of disease parameters.",
    url = "https://doi.org/10.1098/rsif.2007.1106",
    doi = "10.1098/rsif.2007.1106",
    openalex = "W2132192754",
    references = "doi1010160021999176900413, doi101016b9780444529657x50004, doi101038189732a0, doi101093besa153237, doi101093oso97801985459960010001, doi101098rspa19270118, doi10113719780898719574, doi10560219781421407944"
}

The well–known logistic model has been extensively investigated in deterministic theory. There are numerous case studies where such type of nonlinearities occur in Ecology, Biology and Environmental Sciences. Due to the presence of environmental fluctuations and a lack of precision of measurements, one has to deal with effects of randomness on such models. As a more realistic modeling, we suggest nonlinear stochastic differential equations (SDEs) dX(t) = [(ρ + λX(t))(K − X(t)) − µX(t)]dt + σX(t) α |K − X(t) | β dW (t) of Itô-type to model the growth of populations or innovations X, driven by a Wiener process W and positive real constants ρ, λ, K, µ, α, β ≥ 0. discuss well–posedness, regularity (boundedness) and uniqueness of their solutions. However, explicit expressions for analytical solution of such random logistic equations are rarely known. Therefore one has to resort to numerical solution of SDEs for studying various aspects like the time–evolution of growth patterns, exit frequencies, mean passage times and impact of fluctuating growth parameters. We present some basic aspects of adequate numerical analysis of these random extensions of these models such as numerical regularity and mean square convergence. The problem of keeping reasonable boundaries for analytic solutions under discretization plays an essential role for practically meaningful models, in particular the preservation of intervals with reflecting or absorbing barriers. A discretization of the continuous state space can be circumvented by appropriate methods. Balanced implicit methods (see Schurz, IJNAM 2 (2), p. 197-220, 2005) are used to construct strongly converging approximations with the desired monotone properties. Numerical studies can bring out salient features of the stochastic logistic models (e.g. We almost sure monotonicity, almost sure uniform boundedness, delayed initial evolution or earlier points of inflection compared to deterministic model).

@article{openalexw65299899,
    author = "Schurz, Henri",
    title = "MODELING, ANALYSIS AND DISCRETIZATION OF STOCHASTIC LOGISTIC EQUATIONS",
    year = "2007",
    abstract = "The well–known logistic model has been extensively investigated in deterministic theory. There are numerous case studies where such type of nonlinearities occur in Ecology, Biology and Environmental Sciences. Due to the presence of environmental fluctuations and a lack of precision of measurements, one has to deal with effects of randomness on such models. As a more realistic modeling, we suggest nonlinear stochastic differential equations (SDEs) dX(t) = [(ρ + λX(t))(K − X(t)) − µX(t)]dt + σX(t) α |K − X(t) | β dW (t) of Itô-type to model the growth of populations or innovations X, driven by a Wiener process W and positive real constants ρ, λ, K, µ, α, β ≥ 0. discuss well–posedness, regularity (boundedness) and uniqueness of their solutions. However, explicit expressions for analytical solution of such random logistic equations are rarely known. Therefore one has to resort to numerical solution of SDEs for studying various aspects like the time–evolution of growth patterns, exit frequencies, mean passage times and impact of fluctuating growth parameters. We present some basic aspects of adequate numerical analysis of these random extensions of these models such as numerical regularity and mean square convergence. The problem of keeping reasonable boundaries for analytic solutions under discretization plays an essential role for practically meaningful models, in particular the preservation of intervals with reflecting or absorbing barriers. A discretization of the continuous state space can be circumvented by appropriate methods. Balanced implicit methods (see Schurz, IJNAM 2 (2), p. 197-220, 2005) are used to construct strongly converging approximations with the desired monotone properties. Numerical studies can bring out salient features of the stochastic logistic models (e.g. We almost sure monotonicity, almost sure uniform boundedness, delayed initial evolution or earlier points of inflection compared to deterministic model).",
    openalex = "W65299899",
    references = "doi1023072527465"
}

The extinction time of an isolated population can be exponentially reduced by a periodic modulation of its environment. We investigate this effect using, as an example, a stochastic branching-annihilation process with a time-dependent branching rate. The population extinction is treated in eikonal approximation, where it is described as an instanton trajectory of a proper reaction Hamiltonian. The modulation of the environment perturbs this trajectory and synchronizes it with the modulation phase. We calculate the corresponding change in the action along the instanton using perturbation techniques supported by numerical calculations. The techniques include a first-order theory with respect to the modulation amplitude, a second-order theory in the spirit of the Kapitsa pendulum effect, and adiabatic theory valid for low modulation frequencies.

@article{doi101103physreve78041123,
    author = "Assaf, Michael and Kamenev, Alex and Meerson, Baruch",
    title = "Population extinction in a time-modulated environment",
    year = "2008",
    journal = "Physical Review E",
    abstract = "The extinction time of an isolated population can be exponentially reduced by a periodic modulation of its environment. We investigate this effect using, as an example, a stochastic branching-annihilation process with a time-dependent branching rate. The population extinction is treated in eikonal approximation, where it is described as an instanton trajectory of a proper reaction Hamiltonian. The modulation of the environment perturbs this trajectory and synchronizes it with the modulation phase. We calculate the corresponding change in the action along the instanton using perturbation techniques supported by numerical calculations. The techniques include a first-order theory with respect to the modulation amplitude, a second-order theory in the spirit of the Kapitsa pendulum effect, and adiabatic theory valid for low modulation frequencies.",
    url = "https://doi.org/10.1103/physreve.78.041123",
    doi = "10.1103/physreve.78.041123",
    openalex = "W2075718102",
    references = "barton1962stochastic, doi101103physreve77061107, doi1023072333508"
}

Demographic noise causes unlimited population growth in a broad class of models which, without noise, would predict a stable finite population. We study this effect on the example of a stochastic birth-death model which includes immigration, binary reproduction, and death. The unlimited population growth proceeds as an exponentially slow decay of a metastable probability distribution (MPD) of the population. We develop a systematic WKB theory, complemented by the van Kampen system size expansion, for the MPD and for the decay time. Important signatures of the MPD are a power-law tail (such that all the distribution moments, except the zeroth one, diverge) and the presence in the solution of two different WKB modes.

@article{doi101103physreve78060103,
    author = "Meerson, Baruch and Sasorov, P. V.",
    title = "Noise-driven unlimited population growth",
    year = "2008",
    journal = "Physical Review E",
    abstract = "Demographic noise causes unlimited population growth in a broad class of models which, without noise, would predict a stable finite population. We study this effect on the example of a stochastic birth-death model which includes immigration, binary reproduction, and death. The unlimited population growth proceeds as an exponentially slow decay of a metastable probability distribution (MPD) of the population. We develop a systematic WKB theory, complemented by the van Kampen system size expansion, for the MPD and for the decay time. Important signatures of the MPD are a power-law tail (such that all the distribution moments, except the zeroth one, diverge) and the presence in the solution of two different WKB modes.",
    url = "https://doi.org/10.1103/physreve.78.060103",
    doi = "10.1103/physreve.78.060103",
    openalex = "W1990828655",
    references = "barton1962stochastic, doi1023072333508"
}

Environmental noise can cause an exponential reduction in the mean time to extinction (MTE) of an isolated population. We study this effect on an example of a stochastic birth-death process with rates modulated by a colored (that is, correlated) Gaussian noise. A path integral formulation yields a transparent way of evaluating the MTE and finding the optimal realization of the environmental noise that determines the most probable path to extinction. The population-size dependence of the MTE changes from exponential in the absence of the environmental noise to a power law for a short-correlated noise and to no dependence for long-correlated noise. We also establish the validity domains of the white-noise limit and adiabatic limit.

@article{doi101103physrevlett101268103,
    author = "Kamenev, Alex and Meerson, Baruch and Shklovskiǐ, B. I.",
    title = "How Colored Environmental Noise Affects Population Extinction",
    year = "2008",
    journal = "Physical Review Letters",
    abstract = "Environmental noise can cause an exponential reduction in the mean time to extinction (MTE) of an isolated population. We study this effect on an example of a stochastic birth-death process with rates modulated by a colored (that is, correlated) Gaussian noise. A path integral formulation yields a transparent way of evaluating the MTE and finding the optimal realization of the environmental noise that determines the most probable path to extinction. The population-size dependence of the MTE changes from exponential in the absence of the environmental noise to a power law for a short-correlated noise and to no dependence for long-correlated noise. We also establish the validity domains of the white-noise limit and adiabatic limit.",
    url = "https://doi.org/10.1103/physrevlett.101.268103",
    doi = "10.1103/physrevlett.101.268103",
    openalex = "W1529840611",
    references = "bailey1961stochastic, doi1010079781461211587, doi1010079783662023778, doi1010079783662053898, doi1010160022519381900448, doi1010160169534796810676, doi101016b9780444529657x50004, doi101086285580, doi101098rspb19960256, doi1023071909913"
}

Birth-death processes often exhibit an oscillatory behavior. We investigate a particular case where the oscillation cycles are marginally stable on the mean-field level. An iconic example of such a system is the Lotka-Volterra model of predator-prey interaction. Fluctuation effects due to discreteness of the populations destroy the mean-field stability and eventually drive the system toward extinction of one or both species. We show that the corresponding extinction time scales as a certain power-law of the population sizes. This behavior should be contrasted with the extinction of models stable in the mean-field approximation. In the latter case the extinction time scales exponentially with size.

@article{doi101103physreve80021129,
    author = "Parker, Matthew W. and Kamenev, Alex",
    title = "Extinction in the Lotka-Volterra model",
    year = "2009",
    journal = "Physical Review E",
    abstract = "Birth-death processes often exhibit an oscillatory behavior. We investigate a particular case where the oscillation cycles are marginally stable on the mean-field level. An iconic example of such a system is the Lotka-Volterra model of predator-prey interaction. Fluctuation effects due to discreteness of the populations destroy the mean-field stability and eventually drive the system toward extinction of one or both species. We show that the corresponding extinction time scales as a certain power-law of the population sizes. This behavior should be contrasted with the extinction of models stable in the mean-field approximation. In the latter case the extinction time scales exponentially with size.",
    url = "https://doi.org/10.1103/physreve.80.021129",
    doi = "10.1103/physreve.80.021129",
    openalex = "W2121335398",
    references = "bailey1961stochastic, doi1010079781461211587, doi101016b0123694019006239, doi101016b9780444529657x50004, doi101021j100540a008, doi101073pnas67410, doi101103physreve77061107, doi1023071909913, doi1023073284920, openalexw2104595316"
}

Species range limits involve many aspects of evolution and ecology, from species distribution and abundance to the evolution of niches. Theory suggests myriad processes by which range limits arise, including competitive exclusion, Allee effects, and gene swamping; however, most models remain empirically untested. Range limits are correlated with a number of abiotic and biotic factors, but further experimentation is needed to understand underlying mechanisms. Range edges are characterized by increased genetic isolation, genetic differentiation, and variability in individual and population performance, but evidence for decreased abundance and fitness is lacking. Evolution of range limits is understudied in natural systems; in particular, the role of gene flow in shaping range limits is unknown. Biological invasions and rapid distribution shifts caused by climate change represent large-scale experiments on the underlying dynamics of range limits. A better fusion of experimentation and theory will advance our understanding of the causes of range limits.

@article{doi101146annurevecolsys110308120317,
    author = "Sexton, Jason P. and McIntyre, Patrick J. and Angert, Amy L. and Rice, Kevin J.",
    title = "Evolution and Ecology of Species Range Limits",
    year = "2009",
    journal = "Annual Review of Ecology Evolution and Systematics",
    abstract = "Species range limits involve many aspects of evolution and ecology, from species distribution and abundance to the evolution of niches. Theory suggests myriad processes by which range limits arise, including competitive exclusion, Allee effects, and gene swamping; however, most models remain empirically untested. Range limits are correlated with a number of abiotic and biotic factors, but further experimentation is needed to understand underlying mechanisms. Range edges are characterized by increased genetic isolation, genetic differentiation, and variability in individual and population performance, but evidence for decreased abundance and fitness is lacking. Evolution of range limits is understudied in natural systems; in particular, the role of gene flow in shaping range limits is unknown. Biological invasions and rapid distribution shifts caused by climate change represent large-scale experiments on the underlying dynamics of range limits. A better fusion of experimentation and theory will advance our understanding of the causes of range limits.",
    url = "https://doi.org/10.1146/annurev.ecolsys.110308.120317",
    doi = "10.1146/annurev.ecolsys.110308.120317",
    openalex = "W2138877869",
    references = "doi1010160169534794902488, doi101016s0169534702025545, doi101046j14610248200200297x, doi101046j15231739199206030324x, doi101093biomet3812196, doi101093oso97801985264070010001, doi101098rspa19270118, doi101111j14610248200500739x, doi101111j14610248200801277x, doi101111j146918091937tb02153x, doi101126science2925517673, doi101146annurevecolsys271597, doi101146annurevecolsys37091305110100, doi101146annurevecolsys39110707173430, doi1015159780691209418, doi1018901051076120000100689bicegc20co2, doi1023072408012, doi102307jctvx5wbbh, doi105962bhltitle59991, openalexw2151235472"
}

@article{doi101016jtree201007009,
    author = "Ovaskainen, Otso and Meerson, Baruch",
    title = "Stochastic models of population extinction",
    year = "2010",
    journal = "Trends in Ecology \& Evolution",
    url = "https://doi.org/10.1016/j.tree.2010.07.009",
    doi = "10.1016/j.tree.2010.07.009",
    openalex = "W2130289377",
    references = "doi101103physreve77061107, doi101103physreve80021129, doi101103physrevlett101268103"
}

Momentum-space representation renders an interesting perspective to theory of large fluctuations in populations undergoing Markovian stochastic gain-loss processes. This representation is obtained when the master equation for the probability distribution of the population size is transformed into an evolution equation for the probability generating function. Spectral decomposition then brings about an eigenvalue problem for a non-Hermitian linear differential operator. The ground-state eigenmode encodes the stationary distribution of the population size. For long-lived metastable populations which exhibit extinction or escape to another metastable state, the quasi-stationary distribution and the mean time to extinction or escape are encoded by the eigenmode and eigenvalue of the lowest excited state. If the average population size in the stationary or quasi-stationary state is large, the corresponding eigenvalue problem can be solved via WKB approximation amended by other asymptotic methods. We illustrate these ideas in several model examples.

@article{doi10108817425468201007p07018,
    author = "Assaf, Michael and Meerson, Baruch and Sasorov, P. V.",
    title = "Large fluctuations in stochastic population dynamics: momentum-space calculations",
    year = "2010",
    journal = "Journal of Statistical Mechanics Theory and Experiment",
    abstract = "Momentum-space representation renders an interesting perspective to theory of large fluctuations in populations undergoing Markovian stochastic gain-loss processes. This representation is obtained when the master equation for the probability distribution of the population size is transformed into an evolution equation for the probability generating function. Spectral decomposition then brings about an eigenvalue problem for a non-Hermitian linear differential operator. The ground-state eigenmode encodes the stationary distribution of the population size. For long-lived metastable populations which exhibit extinction or escape to another metastable state, the quasi-stationary distribution and the mean time to extinction or escape are encoded by the eigenmode and eigenvalue of the lowest excited state. If the average population size in the stationary or quasi-stationary state is large, the corresponding eigenvalue problem can be solved via WKB approximation amended by other asymptotic methods. We illustrate these ideas in several model examples.",
    url = "https://doi.org/10.1088/1742-5468/2010/07/p07018",
    doi = "10.1088/1742-5468/2010/07/p07018",
    openalex = "W2021519665",
    references = "barton1962stochastic, doi1023072333508"
}

We investigate the phenomenon of extinction of a long-lived self-regulating stochastic population, caused by intrinsic (demographic) noise. Extinction typically occurs via one of two scenarios depending on whether the absorbing state n=0 is a repelling (scenario A) or attracting (scenario B) point of the deterministic rate equation. In scenario A the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point n=0. In scenario B there is an intermediate repelling point n=n1 between the attracting point n=0 and another attracting point n=n2 in the vicinity of which the metastable population resides. The crux of the theory is a dissipative variant of WKB (Wentzel-Kramers-Brillouin) approximation which assumes that the typical population size in the metastable state is large. Starting from the master equation, we calculate the quasistationary probability distribution of the population sizes and the (exponentially long) mean time to extinction for each of the two scenarios. When necessary, the WKB approximation is complemented (i) by a recursive solution of the quasistationary master equation at small n and (ii) by the van Kampen system-size expansion, valid near the fixed points of the deterministic rate equation. The theory yields both entropic barriers to extinction and pre-exponential factors, and holds for a general set of multistep processes when detailed balance is broken. The results simplify considerably for single-step processes and near the characteristic bifurcations of scenarios A and B.

@article{doi101103physreve81021116,
    author = "Assaf, Michael and Meerson, Baruch",
    title = "Extinction of metastable stochastic populations",
    year = "2010",
    journal = "Physical Review E",
    abstract = "We investigate the phenomenon of extinction of a long-lived self-regulating stochastic population, caused by intrinsic (demographic) noise. Extinction typically occurs via one of two scenarios depending on whether the absorbing state n=0 is a repelling (scenario A) or attracting (scenario B) point of the deterministic rate equation. In scenario A the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point n=0. In scenario B there is an intermediate repelling point n=n1 between the attracting point n=0 and another attracting point n=n2 in the vicinity of which the metastable population resides. The crux of the theory is a dissipative variant of WKB (Wentzel-Kramers-Brillouin) approximation which assumes that the typical population size in the metastable state is large. Starting from the master equation, we calculate the quasistationary probability distribution of the population sizes and the (exponentially long) mean time to extinction for each of the two scenarios. When necessary, the WKB approximation is complemented (i) by a recursive solution of the quasistationary master equation at small n and (ii) by the van Kampen system-size expansion, valid near the fixed points of the deterministic rate equation. The theory yields both entropic barriers to extinction and pre-exponential factors, and holds for a general set of multistep processes when detailed balance is broken. The results simplify considerably for single-step processes and near the characteristic bifurcations of scenarios A and B.",
    url = "https://doi.org/10.1103/physreve.81.021116",
    doi = "10.1103/physreve.81.021116",
    openalex = "W2082327696",
    references = "doi1023072985209"
}

We study several variants of the stochastic four-state rock-paper-scissors game or, equivalently, cyclic three-species predator-prey models with conserved total particle density, by means of Monte Carlo simulations on one- and two-dimensional lattices. Specifically, we investigate the influence of spatial variability of the reaction rates and site occupancy restrictions on the transient oscillations of the species densities and on spatial correlation functions in the quasistationary coexistence state. For small systems, we also numerically determine the dependence of typical extinction times on the number of lattice sites. In stark contrast with two-species stochastic Lotka-Volterra systems, we find that for our three-species models with cyclic competition quenched disorder in the reaction rates has very little effect on the dynamics and the long-time properties of the coexistence state. Similarly, we observe that site restriction only has a minor influence on the system's dynamical properties. Our results therefore demonstrate that the features of the spatial rock-paper-scissors system are remarkably robust with respect to model variations, and stochastic fluctuations as well as spatial correlations play a comparatively minor role.

@article{doi101103physreve82051909,
    author = "He, Qian and Mobilia, Mauro and Täuber, Uwe C.",
    title = "Spatial rock-paper-scissors models with inhomogeneous reaction rates",
    year = "2010",
    journal = "Physical Review E",
    abstract = "We study several variants of the stochastic four-state rock-paper-scissors game or, equivalently, cyclic three-species predator-prey models with conserved total particle density, by means of Monte Carlo simulations on one- and two-dimensional lattices. Specifically, we investigate the influence of spatial variability of the reaction rates and site occupancy restrictions on the transient oscillations of the species densities and on spatial correlation functions in the quasistationary coexistence state. For small systems, we also numerically determine the dependence of typical extinction times on the number of lattice sites. In stark contrast with two-species stochastic Lotka-Volterra systems, we find that for our three-species models with cyclic competition quenched disorder in the reaction rates has very little effect on the dynamics and the long-time properties of the coexistence state. Similarly, we observe that site restriction only has a minor influence on the system's dynamical properties. Our results therefore demonstrate that the features of the spatial rock-paper-scissors system are remarkably robust with respect to model variations, and stochastic fluctuations as well as spatial correlations play a comparatively minor role.",
    url = "https://doi.org/10.1103/physreve.82.051909",
    doi = "10.1103/physreve.82.051909",
    openalex = "W1994337539",
    references = "doi101103physreve80021129"
}

Computational systems biology is concerned with the development of detailed mechanistic models of biological processes. Such models are often stochastic and analytically intractable, containing uncertain parameters that must be estimated from time course data. In this article, we consider the task of inferring the parameters of a stochastic kinetic model defined as a Markov (jump) process. Inference for the parameters of complex nonlinear multivariate stochastic process models is a challenging problem, but we find here that algorithms based on particle Markov chain Monte Carlo turn out to be a very effective computationally intensive approach to the problem. Approximations to the inferential model based on stochastic differential equations (SDEs) are considered, as well as improvements to the inference scheme that exploit the SDE structure. We apply the methodology to a Lotka-Volterra system and a prokaryotic auto-regulatory network.

@article{doi101098rsfs20110047,
    author = "Golightly, Andrew and Wilkinson, Darren J.",
    title = "Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo",
    year = "2011",
    journal = "Interface Focus",
    abstract = "Computational systems biology is concerned with the development of detailed mechanistic models of biological processes. Such models are often stochastic and analytically intractable, containing uncertain parameters that must be estimated from time course data. In this article, we consider the task of inferring the parameters of a stochastic kinetic model defined as a Markov (jump) process. Inference for the parameters of complex nonlinear multivariate stochastic process models is a challenging problem, but we find here that algorithms based on particle Markov chain Monte Carlo turn out to be a very effective computationally intensive approach to the problem. Approximations to the inferential model based on stochastic differential equations (SDEs) are considered, as well as improvements to the inference scheme that exploit the SDE structure. We apply the methodology to a Lotka-Volterra system and a prokaryotic auto-regulatory network.",
    url = "https://doi.org/10.1098/rsfs.2011.0047",
    doi = "10.1098/rsfs.2011.0047",
    openalex = "W2122028584",
    references = "doi101098rsif20090151"
}

@article{doi101016jchaos201112009,
    author = "Pedjeu, Jean-C. and Ladde, G.S.",
    title = "Stochastic fractional differential equations: Modeling, method and analysis",
    year = "2012",
    journal = "Chaos Solitons \& Fractals",
    url = "https://doi.org/10.1016/j.chaos.2011.12.009",
    doi = "10.1016/j.chaos.2011.12.009",
    openalex = "W1974312448",
    references = "doi1023072527465"
}

@article{doi101016jtree201201014,
    author = "Black, Andrew J. and McKane, Alan J.",
    title = "Stochastic formulation of ecological models and their applications",
    year = "2012",
    journal = "Trends in Ecology \& Evolution",
    url = "https://doi.org/10.1016/j.tree.2012.01.014",
    doi = "10.1016/j.tree.2012.01.014",
    openalex = "W2108678981",
    references = "doi101016b9780444529657x50004, doi10108000401706198310487866, doi101093besa153237, doi101093oso97801985406630010001, doi101098rsif20071106, doi101098rstb19520012, doi101126science1116681, doi1018900012965820020831771tuntob20co2, doi1023072312029, doi1023073071998, doi102307jctvcm4gk0"
}

Isolated populations ultimately go extinct because of the intrinsic noise of elementary processes. In multipopulation systems extinction of a population may occur via more than one route. We investigate this generic situation in a simple predator-prey (or infected-susceptible) model. The predator and prey populations may coexist for a long time, but ultimately both go extinct. In the first extinction route the predators go extinct first, whereas the prey thrive for a long time and then also go extinct. In the second route the prey go extinct first, causing a rapid extinction of the predators. Assuming large subpopulation sizes in the coexistence state, we compare the probabilities of each of the two extinction routes and predict the most likely path of the subpopulations to extinction. We also suggest an effective three-state master equation for the probabilities to observe the coexistence state, the predator-free state, and the empty state.

@article{doi101103physreve85021140,
    author = "Gottesman, Omer and Meerson, Baruch",
    title = "Multiple extinction routes in stochastic population models",
    year = "2012",
    journal = "Physical Review E",
    abstract = "Isolated populations ultimately go extinct because of the intrinsic noise of elementary processes. In multipopulation systems extinction of a population may occur via more than one route. We investigate this generic situation in a simple predator-prey (or infected-susceptible) model. The predator and prey populations may coexist for a long time, but ultimately both go extinct. In the first extinction route the predators go extinct first, whereas the prey thrive for a long time and then also go extinct. In the second route the prey go extinct first, causing a rapid extinction of the predators. Assuming large subpopulation sizes in the coexistence state, we compare the probabilities of each of the two extinction routes and predict the most likely path of the subpopulations to extinction. We also suggest an effective three-state master equation for the probabilities to observe the coexistence state, the predator-free state, and the empty state.",
    url = "https://doi.org/10.1103/physreve.85.021140",
    doi = "10.1103/physreve.85.021140",
    openalex = "W2065388471",
    references = "barton1962stochastic, doi1023072333508"
}

Populations of competing biological species exhibit a fascinating interplay between the nonlinear dynamics of evolutionary selection forces and random fluctuations arising from the stochastic nature of the interactions. The processes leading to extinction of species, whose understanding is a key component in the study of evolution and biodiversity, are influenced by both of these factors. Here, we investigate a class of stochastic population dynamics models based on generalized Lotka-Volterra systems. In the case of neutral stability of the underlying deterministic model, the impact of intrinsic noise on the survival of species is dramatic: It destroys coexistence of interacting species on a time scale proportional to the population size. We introduce a new method based on stochastic averaging which allows one to understand this extinction process quantitatively by reduction to a lower-dimensional effective dynamics. This is performed analytically for two highly symmetrical models and can be generalized numerically to more complex situations. The extinction probability distributions and other quantities of interest we obtain show excellent agreement with simulations.

@article{doi101103physreve85051903,
    author = "Dobrinevski, Alexander and Frey, Erwin",
    title = "Extinction in neutrally stable stochastic Lotka-Volterra models",
    year = "2012",
    journal = "Physical Review E",
    abstract = "Populations of competing biological species exhibit a fascinating interplay between the nonlinear dynamics of evolutionary selection forces and random fluctuations arising from the stochastic nature of the interactions. The processes leading to extinction of species, whose understanding is a key component in the study of evolution and biodiversity, are influenced by both of these factors. Here, we investigate a class of stochastic population dynamics models based on generalized Lotka-Volterra systems. In the case of neutral stability of the underlying deterministic model, the impact of intrinsic noise on the survival of species is dramatic: It destroys coexistence of interacting species on a time scale proportional to the population size. We introduce a new method based on stochastic averaging which allows one to understand this extinction process quantitatively by reduction to a lower-dimensional effective dynamics. This is performed analytically for two highly symmetrical models and can be generalized numerically to more complex situations. The extinction probability distributions and other quantities of interest we obtain show excellent agreement with simulations.",
    url = "https://doi.org/10.1103/physreve.85.051903",
    doi = "10.1103/physreve.85.051903",
    openalex = "W2018380000",
    references = "doi101103physreve80021129"
}

The most basic stochastic epidemic models are those involving global transmission, meaning that infection rates depend only on the type and state of the individuals involved, and not on their location in the population. Simple as they are, there are still several open problems for such models. For example, when will such an epidemic go extinct and with what probability (questions depending on the population being fixed, changing or growing)? How can a model be defined explaining the sometimes observed scenario of frequent mid-sized epidemic outbreaks? How can evolution of the infectious agent transmission rates be modelled and fitted to data in a robust way?

@article{doi101016jepidem201405002,
    author = "Britton, Tom and House, Thomas and Lloyd, Alun L. and Mollison, Denis and Riley, Steven and Trapman, Pieter",
    title = "Five challenges for stochastic epidemic models involving global transmission",
    year = "2014",
    journal = "Epidemics",
    abstract = "The most basic stochastic epidemic models are those involving global transmission, meaning that infection rates depend only on the type and state of the individuals involved, and not on their location in the population. Simple as they are, there are still several open problems for such models. For example, when will such an epidemic go extinct and with what probability (questions depending on the population being fixed, changing or growing)? How can a model be defined explaining the sometimes observed scenario of frequent mid-sized epidemic outbreaks? How can evolution of the infectious agent transmission rates be modelled and fitted to data in a robust way?",
    url = "https://doi.org/10.1016/j.epidem.2014.05.002",
    doi = "10.1016/j.epidem.2014.05.002",
    openalex = "W2028249813",
    references = "doi101103physreve77061107"
}

Deterministic models have a long history of being applied to the study of infectious disease epidemiology. We highlight and discuss nine challenges in this area. The first two concern the endemic equilibrium and its stability. We indicate the need for models that describe multi-strain infections, infections with time-varying infectivity, and those where superinfection is possible. We then consider the need for advances in spatial epidemic models, and draw attention to the lack of models that explore the relationship between communicable and non-communicable diseases. The final two challenges concern the uses and limitations of deterministic models as approximations to stochastic systems.

@article{doi101016jepidem201409006,
    author = "Roberts, M. G. and Andreasen, Viggo and Lloyd, Alun L. and Pellis, Lorenzo",
    title = "Nine challenges for deterministic epidemic models",
    year = "2014",
    journal = "Epidemics",
    abstract = "Deterministic models have a long history of being applied to the study of infectious disease epidemiology. We highlight and discuss nine challenges in this area. The first two concern the endemic equilibrium and its stability. We indicate the need for models that describe multi-strain infections, infections with time-varying infectivity, and those where superinfection is possible. We then consider the need for advances in spatial epidemic models, and draw attention to the lack of models that explore the relationship between communicable and non-communicable diseases. The final two challenges concern the uses and limitations of deterministic models as approximations to stochastic systems.",
    url = "https://doi.org/10.1016/j.epidem.2014.09.006",
    doi = "10.1016/j.epidem.2014.09.006",
    openalex = "W2030400104",
    references = "doi10108800344885772026602"
}

We construct a path-integral representation of solutions to a stochastic hybrid system, consisting of one or more continuous variables evolving according to a piecewise-deterministic dynamics. The differential equations for the continuous variables are coupled to a set of discrete variables that satisfy a continuous-time Markov process, which means that the differential equations are only valid between jumps in the discrete variables. Examples of stochastic hybrid systems arise in biophysical models of stochastic ion channels, motor-driven intracellular transport, gene networks, and stochastic neural networks. We use the path-integral representation to derive a large deviation action principle for a stochastic hybrid system. Minimizing the associated action functional with respect to the set of all trajectories emanating from a metastable state (assuming that such a minimization scheme exists) then determines the most probable paths of escape. Moreover, evaluating the action functional along a most probable path generates the so-called quasipotential used in the calculation of mean first passage times. We illustrate the theory by considering the optimal paths of escape from a metastable state in a bistable neural network.

@article{doi101103physreve89042701,
    author = "Bressloff, Paul C. and Newby, Jay",
    title = "Path integrals and large deviations in stochastic hybrid systems",
    year = "2014",
    journal = "Physical Review E",
    abstract = "We construct a path-integral representation of solutions to a stochastic hybrid system, consisting of one or more continuous variables evolving according to a piecewise-deterministic dynamics. The differential equations for the continuous variables are coupled to a set of discrete variables that satisfy a continuous-time Markov process, which means that the differential equations are only valid between jumps in the discrete variables. Examples of stochastic hybrid systems arise in biophysical models of stochastic ion channels, motor-driven intracellular transport, gene networks, and stochastic neural networks. We use the path-integral representation to derive a large deviation action principle for a stochastic hybrid system. Minimizing the associated action functional with respect to the set of all trajectories emanating from a metastable state (assuming that such a minimization scheme exists) then determines the most probable paths of escape. Moreover, evaluating the action functional along a most probable path generates the so-called quasipotential used in the calculation of mean first passage times. We illustrate the theory by considering the optimal paths of escape from a metastable state in a bistable neural network.",
    url = "https://doi.org/10.1103/physreve.89.042701",
    doi = "10.1103/physreve.89.042701",
    openalex = "W2325640490",
    references = "doi101103physreve77061107"
}

Frequency-dependent selection and demographic fluctuations play important roles in evolutionary and ecological processes. Under frequency-dependent selection, the average fitness of the population may increase or decrease based on interactions between individuals within the population. This should be reflected in fluctuations of the population size even in constant environments. Here, we propose a stochastic model that naturally combines these two evolutionary ingredients by assuming frequency-dependent competition between different types in an individual-based model. In contrast to previous game theoretic models, the carrying capacity of the population, and thus the population size, is determined by pairwise competition of individuals mediated by evolutionary games and demographic stochasticity. In the limit of infinite population size, the averaged stochastic dynamics is captured by deterministic competitive Lotka-Volterra equations. In small populations, demographic stochasticity may instead lead to the extinction of the entire population. Because the population size is driven by fitness in evolutionary games, a population of cooperators is less prone to go extinct than a population of defectors, whereas in the usual systems of fixed size the population would thrive regardless of its average payoff.

@article{doi101073pnas1418745112,
    author = "Huang, Weini and Hauert, Christoph and Traulsen, Arne",
    title = "Stochastic game dynamics under demographic fluctuations",
    year = "2015",
    journal = "Proceedings of the National Academy of Sciences",
    abstract = "Frequency-dependent selection and demographic fluctuations play important roles in evolutionary and ecological processes. Under frequency-dependent selection, the average fitness of the population may increase or decrease based on interactions between individuals within the population. This should be reflected in fluctuations of the population size even in constant environments. Here, we propose a stochastic model that naturally combines these two evolutionary ingredients by assuming frequency-dependent competition between different types in an individual-based model. In contrast to previous game theoretic models, the carrying capacity of the population, and thus the population size, is determined by pairwise competition of individuals mediated by evolutionary games and demographic stochasticity. In the limit of infinite population size, the averaged stochastic dynamics is captured by deterministic competitive Lotka-Volterra equations. In small populations, demographic stochasticity may instead lead to the extinction of the entire population. Because the population size is driven by fitness in evolutionary games, a population of cooperators is less prone to go extinct than a population of defectors, whereas in the usual systems of fixed size the population would thrive regardless of its average payoff.",
    url = "https://doi.org/10.1073/pnas.1418745112",
    doi = "10.1073/pnas.1418745112",
    openalex = "W1821620939",
    references = "doi101103physreve80021129"
}

Advances in scientific computing have allowed the development of complex models that are being routinely applied to problems in disease epidemiology, public health and decision making. The utility of these models depends in part on how well they can reproduce empirical data. However, fitting such models to real world data is greatly hindered both by large numbers of input and output parameters, and by long run times, such that many modelling studies lack a formal calibration methodology. We present a novel method that has the potential to improve the calibration of complex infectious disease models (hereafter called simulators). We present this in the form of a tutorial and a case study where we history match a dynamic, event-driven, individual-based stochastic HIV simulator, using extensive demographic, behavioural and epidemiological data available from Uganda. The tutorial describes history matching and emulation. History matching is an iterative procedure that reduces the simulator's input space by identifying and discarding areas that are unlikely to provide a good match to the empirical data. History matching relies on the computational efficiency of a Bayesian representation of the simulator, known as an emulator. Emulators mimic the simulator's behaviour, but are often several orders of magnitude faster to evaluate. In the case study, we use a 22 input simulator, fitting its 18 outputs simultaneously. After 9 iterations of history matching, a non-implausible region of the simulator input space was identified that was 10(11) times smaller than the original input space. Simulator evaluations made within this region were found to have a 65% probability of fitting all 18 outputs. History matching and emulation are useful additions to the toolbox of infectious disease modellers. Further research is required to explicitly address the stochastic nature of the simulator as well as to account for correlations between outputs.

@article{doi101371journalpcbi1003968,
    author = "Andrianakis, Ioannis and Vernon, Ian and McCreesh, Nicky and McKinley, Trevelyan J. and Oakley, Jeremy E. and Nsubuga, Rebecca N. and Goldstein, Michael and White, Richard G.",
    title = "Bayesian History Matching of Complex Infectious Disease Models Using Emulation: A Tutorial and a Case Study on HIV in Uganda",
    year = "2015",
    journal = "PLoS Computational Biology",
    abstract = "Advances in scientific computing have allowed the development of complex models that are being routinely applied to problems in disease epidemiology, public health and decision making. The utility of these models depends in part on how well they can reproduce empirical data. However, fitting such models to real world data is greatly hindered both by large numbers of input and output parameters, and by long run times, such that many modelling studies lack a formal calibration methodology. We present a novel method that has the potential to improve the calibration of complex infectious disease models (hereafter called simulators). We present this in the form of a tutorial and a case study where we history match a dynamic, event-driven, individual-based stochastic HIV simulator, using extensive demographic, behavioural and epidemiological data available from Uganda. The tutorial describes history matching and emulation. History matching is an iterative procedure that reduces the simulator's input space by identifying and discarding areas that are unlikely to provide a good match to the empirical data. History matching relies on the computational efficiency of a Bayesian representation of the simulator, known as an emulator. Emulators mimic the simulator's behaviour, but are often several orders of magnitude faster to evaluate. In the case study, we use a 22 input simulator, fitting its 18 outputs simultaneously. After 9 iterations of history matching, a non-implausible region of the simulator input space was identified that was 10(11) times smaller than the original input space. Simulator evaluations made within this region were found to have a 65\% probability of fitting all 18 outputs. History matching and emulation are useful additions to the toolbox of infectious disease modellers. Further research is required to explicitly address the stochastic nature of the simulator as well as to account for correlations between outputs.",
    url = "https://doi.org/10.1371/journal.pcbi.1003968",
    doi = "10.1371/journal.pcbi.1003968",
    openalex = "W1967579272",
    references = "doi1010079781475737998, doi101016jecolmodel200604023, doi101098rsif20071106, doi1011111467986800294, doi101111j14679868200900736x, doi101214ss1177012413, doi1015159781400850624, doi1023071268522, doi1023071271432, doi1023072288169, doi107551mitpress32060010001"
}

We consider a very general stochastic model for an SIR epidemic on a network which allows an individual's infectious period, and the time it takes to contact each of its neighbours after becoming infected, to be correlated. We write down the message passing system of equations for this model and prove, for the first time, that it has a unique feasible solution. We also generalise an earlier result by proving that this solution provides a rigorous upper bound for the expected epidemic size (cumulative number of infection events) at any fixed time [Formula: see text]. We specialise these results to a homogeneous special case where the graph (network) is symmetric. The message passing system here reduces to just four equations. We prove that cycles in the network inhibit the spread of infection, and derive important epidemiological results concerning the final epidemic size and threshold behaviour for a major outbreak. For Poisson contact processes, this message passing system is equivalent to a non-Markovian pair approximation model, which we show has well-known pairwise models as special cases. We show further that a sequence of message passing systems, starting with the homogeneous one just described, converges to the deterministic Kermack-McKendrick equations for this stochastic model. For Poisson contact and recovery, we show that this convergence is monotone, from which it follows that the message passing system (and hence also the pairwise model) here provides a better approximation to the expected epidemic size at time [Formula: see text] than the Kermack-McKendrick model.

@article{doi101007s0028501711238,
    author = "Wilkinson, Robert R. and Ball, Frank and Sharkey, Kieran J.",
    title = "The relationships between message passing, pairwise, Kermack–McKendrick and stochastic SIR epidemic models",
    year = "2017",
    journal = "Journal of Mathematical Biology",
    abstract = "We consider a very general stochastic model for an SIR epidemic on a network which allows an individual's infectious period, and the time it takes to contact each of its neighbours after becoming infected, to be correlated. We write down the message passing system of equations for this model and prove, for the first time, that it has a unique feasible solution. We also generalise an earlier result by proving that this solution provides a rigorous upper bound for the expected epidemic size (cumulative number of infection events) at any fixed time [Formula: see text]. We specialise these results to a homogeneous special case where the graph (network) is symmetric. The message passing system here reduces to just four equations. We prove that cycles in the network inhibit the spread of infection, and derive important epidemiological results concerning the final epidemic size and threshold behaviour for a major outbreak. For Poisson contact processes, this message passing system is equivalent to a non-Markovian pair approximation model, which we show has well-known pairwise models as special cases. We show further that a sequence of message passing systems, starting with the homogeneous one just described, converges to the deterministic Kermack-McKendrick equations for this stochastic model. For Poisson contact and recovery, we show that this convergence is monotone, from which it follows that the message passing system (and hence also the pairwise model) here provides a better approximation to the expected epidemic size at time [Formula: see text] than the Kermack-McKendrick model.",
    url = "https://doi.org/10.1007/s00285-017-1123-8",
    doi = "10.1007/s00285-017-1123-8",
    openalex = "W2347400488",
    references = "doi101007s002850100344x"
}

There has been a resurgence of interest in non-equilibrium stochastic processes in recent years, driven in part by the observation that the number of molecules (genes, mRNA, proteins) involved in gene expression are often of order 1–1000. This means that deterministic mass-action kinetics tends to break down, and one needs to take into account the discrete, stochastic nature of biochemical reactions. One of the major consequences of molecular noise is the occurrence of stochastic biological switching at both the genotypic and phenotypic levels. For example, individual gene regulatory networks can switch between graded and binary responses, exhibit translational/transcriptional bursting, and support metastability (noise-induced switching between states that are stable in the deterministic limit). If random switching persists at the phenotypic level then this can confer certain advantages to cell populations growing in a changing environment, as exemplified by bacterial persistence in response to antibiotics. Gene expression at the single-cell level can also be regulated by changes in cell density at the population level, a process known as quorum sensing. In contrast to noise-driven phenotypic switching, the switching mechanism in quorum sensing is stimulus-driven and thus noise tends to have a detrimental effect. A common approach to modeling stochastic gene expression is to assume a large but finite system and to approximate the discrete processes by continuous processes using a system-size expansion. However, there is a growing need to have some familiarity with the theory of stochastic processes that goes beyond the standard topics of chemical master equations, the system-size expansion, Langevin equations and the Fokker–Planck equation. Examples include stochastic hybrid systems (piecewise deterministic Markov processes), large deviations and the Wentzel–Kramers–Brillouin (WKB) method, adiabatic reductions, and queuing/renewal theory. The major aim of this review is to provide a self-contained survey of these mathematical methods, mainly within the context of biological switching processes at both the genotypic and phenotypic levels. However, applications to other examples of biological switching are also discussed, including stochastic ion channels, diffusion in randomly switching environments, bacterial chemotaxis, and stochastic neural networks.

@article{doi10108817518121aa5db4,
    author = "Bressloff, Paul C.",
    title = "Stochastic switching in biology: from genotype to phenotype",
    year = "2017",
    journal = "Journal of Physics A Mathematical and Theoretical",
    abstract = "There has been a resurgence of interest in non-equilibrium stochastic processes in recent years, driven in part by the observation that the number of molecules (genes, mRNA, proteins) involved in gene expression are often of order 1–1000. This means that deterministic mass-action kinetics tends to break down, and one needs to take into account the discrete, stochastic nature of biochemical reactions. One of the major consequences of molecular noise is the occurrence of stochastic biological switching at both the genotypic and phenotypic levels. For example, individual gene regulatory networks can switch between graded and binary responses, exhibit translational/transcriptional bursting, and support metastability (noise-induced switching between states that are stable in the deterministic limit). If random switching persists at the phenotypic level then this can confer certain advantages to cell populations growing in a changing environment, as exemplified by bacterial persistence in response to antibiotics. Gene expression at the single-cell level can also be regulated by changes in cell density at the population level, a process known as quorum sensing. In contrast to noise-driven phenotypic switching, the switching mechanism in quorum sensing is stimulus-driven and thus noise tends to have a detrimental effect. A common approach to modeling stochastic gene expression is to assume a large but finite system and to approximate the discrete processes by continuous processes using a system-size expansion. However, there is a growing need to have some familiarity with the theory of stochastic processes that goes beyond the standard topics of chemical master equations, the system-size expansion, Langevin equations and the Fokker–Planck equation. Examples include stochastic hybrid systems (piecewise deterministic Markov processes), large deviations and the Wentzel–Kramers–Brillouin (WKB) method, adiabatic reductions, and queuing/renewal theory. The major aim of this review is to provide a self-contained survey of these mathematical methods, mainly within the context of biological switching processes at both the genotypic and phenotypic levels. However, applications to other examples of biological switching are also discussed, including stochastic ion channels, diffusion in randomly switching environments, bacterial chemotaxis, and stochastic neural networks.",
    url = "https://doi.org/10.1088/1751-8121/aa5db4",
    doi = "10.1088/1751-8121/aa5db4",
    openalex = "W2585592087",
    references = "doi101103physrevlett101268103"
}

Stochasticity can play an important role in the dynamics of biologically relevant populations. These span a broad range of scales: from intra-cellular populations of molecules to population of cells and then to groups of plants, animals and people. Large deviations in stochastic population dynamics - such as those determining population extinction, fixation or switching between different states - are presently in a focus of attention of statistical physicists. We review recent progress in applying different variants of dissipative WKB approximation (after Wentzel, Kramers and Brillouin) to this class of problems. The WKB approximation allows one to evaluate the mean time and/or probability of population extinction, fixation and switches resulting from either intrinsic (demographic) noise, or a combination of the demographic noise and environmental variations, deterministic or random. We mostly cover well-mixed populations, single and multiple, but also briefly consider populations on heterogeneous networks and spatial populations. The spatial setting also allows one to study large fluctuations of the speed of biological invasions. Finally, we briefly discuss possible directions of future work.

@article{doi10108817518121aa669a,
    author = "Assaf, Michael and Meerson, Baruch",
    title = "WKB theory of large deviations in stochastic populations",
    year = "2017",
    journal = "Journal of Physics A Mathematical and Theoretical",
    abstract = "Stochasticity can play an important role in the dynamics of biologically relevant populations. These span a broad range of scales: from intra-cellular populations of molecules to population of cells and then to groups of plants, animals and people. Large deviations in stochastic population dynamics - such as those determining population extinction, fixation or switching between different states - are presently in a focus of attention of statistical physicists. We review recent progress in applying different variants of dissipative WKB approximation (after Wentzel, Kramers and Brillouin) to this class of problems. The WKB approximation allows one to evaluate the mean time and/or probability of population extinction, fixation and switches resulting from either intrinsic (demographic) noise, or a combination of the demographic noise and environmental variations, deterministic or random. We mostly cover well-mixed populations, single and multiple, but also briefly consider populations on heterogeneous networks and spatial populations. The spatial setting also allows one to study large fluctuations of the speed of biological invasions. Finally, we briefly discuss possible directions of future work.",
    url = "https://doi.org/10.1088/1751-8121/aa669a",
    doi = "10.1088/1751-8121/aa669a",
    openalex = "W2560170501",
    references = "doi101103physreve77061107, doi101103physrevlett101268103"
}

Spatially extended population dynamics models that incorporate intrinsic noise serve as case studies for the role of fluctuations and correlations in biological systems. Including spatial structure and stochastic noise in predator-prey competition invalidates the deterministic Lotka-Volterra picture of neutral population cycles. Stochastic models yield long-lived erratic population oscillations stemming from a resonant amplification mechanism. In spatially extended predator-prey systems, one observes noise-stabilized activity and persistent correlations. Fluctuation-induced renormalizations of the oscillation parameters can be analyzed perturbatively. The critical dynamics and the non-equilibrium relaxation kinetics at the predator extinction threshold are characterized by the directed percolation universality class. Spatial or environmental variability results in more localized patches which enhances both species densities. Affixing variable rates to individual particles and allowing for trait inheritance subject to mutations induces fast evolutionary dynamics for the rate distributions. Stochastic spatial variants of cyclic competition with rock-paper-scissors interactions illustrate connections between population dynamics and evolutionary game theory, and demonstrate how space can help maintain diversity. In two dimensions, three-species cyclic competition models of the May-Leonard type are characterized by the emergence of spiral patterns whose properties are elucidated by a mapping onto a complex Ginzburg-Landau equation. Extensions to general food networks can be classified on the mean-field level, which provides both a fundamental understanding of ensuing cooperativity and emergence of alliances. Novel space-time patterns emerge as a result of the formation of competing alliances, such as coarsening domains that each incorporate rock-paper-scissors competition games.

@article{doi10108817518121aa95c7,
    author = "Dobramysl, Ulrich and Mobilia, Mauro and Pleimling, Michel and Täuber, Uwe C",
    title = "Stochastic population dynamics in spatially extended predator–prey systems",
    year = "2017",
    journal = "Journal of Physics A Mathematical and Theoretical",
    abstract = "Spatially extended population dynamics models that incorporate intrinsic noise serve as case studies for the role of fluctuations and correlations in biological systems. Including spatial structure and stochastic noise in predator-prey competition invalidates the deterministic Lotka-Volterra picture of neutral population cycles. Stochastic models yield long-lived erratic population oscillations stemming from a resonant amplification mechanism. In spatially extended predator-prey systems, one observes noise-stabilized activity and persistent correlations. Fluctuation-induced renormalizations of the oscillation parameters can be analyzed perturbatively. The critical dynamics and the non-equilibrium relaxation kinetics at the predator extinction threshold are characterized by the directed percolation universality class. Spatial or environmental variability results in more localized patches which enhances both species densities. Affixing variable rates to individual particles and allowing for trait inheritance subject to mutations induces fast evolutionary dynamics for the rate distributions. Stochastic spatial variants of cyclic competition with rock-paper-scissors interactions illustrate connections between population dynamics and evolutionary game theory, and demonstrate how space can help maintain diversity. In two dimensions, three-species cyclic competition models of the May-Leonard type are characterized by the emergence of spiral patterns whose properties are elucidated by a mapping onto a complex Ginzburg-Landau equation. Extensions to general food networks can be classified on the mean-field level, which provides both a fundamental understanding of ensuing cooperativity and emergence of alliances. Novel space-time patterns emerge as a result of the formation of competing alliances, such as coarsening domains that each incorporate rock-paper-scissors competition games.",
    url = "https://doi.org/10.1088/1751-8121/aa95c7",
    doi = "10.1088/1751-8121/aa95c7",
    openalex = "W2750198471",
    references = "doi101103physreve80021129"
}

Population cycling is a widespread phenomenon, observed across a multitude of taxa in both laboratory and natural conditions. Historically, the theory associated with population cycles was tightly linked to pairwise consumer-resource interactions and studied via deterministic models, but current empirical and theoretical research reveals a much richer basis for ecological cycles. Stochasticity and seasonality can modulate or create cyclic behaviour in non-intuitive ways, the high-dimensionality in ecological systems can profoundly influence cycling, and so can demographic structure and eco-evolutionary dynamics. An inclusive theory for population cycles, ranging from ecosystem-level to demographic modelling, grounded in observational or experimental data, is therefore necessary to better understand observed cyclical patterns. In turn, by gaining better insight into the drivers of population cycles, we can begin to understand the causes of cycle gain and loss, how biodiversity interacts with population cycling, and how to effectively manage wildly fluctuating populations, all of which are growing domains of ecological research.

@article{doi101111ele12789,
    author = "Barraquand, Frédéric and Louca, Stilianos and Abbott, Karen C. and Cobbold, Christina A. and Cordoleani, Flora and DeAngelis, Donald L. and Elderd, Bret D. and Fox, Jeremy W. and Greenwood, Priscilla E. and Hilker, Frank M. and Murray, Dennis L. and Stieha, Christopher R. and Taylor, Rachel A. and Vitense, Kelsey and Wolkowicz, Gail S. K. and Tyson, Rebecca C.",
    title = "Moving forward in circles: challenges and opportunities in modelling population cycles",
    year = "2017",
    journal = "Ecology Letters",
    abstract = "Population cycling is a widespread phenomenon, observed across a multitude of taxa in both laboratory and natural conditions. Historically, the theory associated with population cycles was tightly linked to pairwise consumer-resource interactions and studied via deterministic models, but current empirical and theoretical research reveals a much richer basis for ecological cycles. Stochasticity and seasonality can modulate or create cyclic behaviour in non-intuitive ways, the high-dimensionality in ecological systems can profoundly influence cycling, and so can demographic structure and eco-evolutionary dynamics. An inclusive theory for population cycles, ranging from ecosystem-level to demographic modelling, grounded in observational or experimental data, is therefore necessary to better understand observed cyclical patterns. In turn, by gaining better insight into the drivers of population cycles, we can begin to understand the causes of cycle gain and loss, how biodiversity interacts with population cycling, and how to effectively manage wildly fluctuating populations, all of which are growing domains of ecological research.",
    url = "https://doi.org/10.1111/ele.12789",
    doi = "10.1111/ele.12789",
    openalex = "W2610632796",
    references = "doi101007s1208000900571, doi101016jtree201201014"
}

The presence of protein-DNA binding reactions often leads to analytically intractable models of stochastic gene expression. Here we present the linear-mapping approximation that maps systems with protein-promoter interactions onto approximately equivalent systems with no binding reactions. This is achieved by the marriage of conditional mean-field approximation and the Magnus expansion, leading to analytic or semi-analytic expressions for the approximate time-dependent and steady-state protein number distributions. Stochastic simulations verify the method's accuracy in capturing the changes in the protein number distributions with time for a wide variety of networks displaying auto- and mutual-regulation of gene expression and independently of the ratios of the timescales governing the dynamics. The method is also used to study the first-passage time distribution of promoter switching, the sensitivity of the size of protein number fluctuations to parameter perturbation and the stochastic bifurcation diagram characterizing the onset of multimodality in protein number distributions.

@article{doi101038s41467018058220,
    author = "Cao, Zhixing and Grima, Ramon",
    title = "Linear mapping approximation of gene regulatory networks with stochastic dynamics",
    year = "2018",
    journal = "Nature Communications",
    abstract = "The presence of protein-DNA binding reactions often leads to analytically intractable models of stochastic gene expression. Here we present the linear-mapping approximation that maps systems with protein-promoter interactions onto approximately equivalent systems with no binding reactions. This is achieved by the marriage of conditional mean-field approximation and the Magnus expansion, leading to analytic or semi-analytic expressions for the approximate time-dependent and steady-state protein number distributions. Stochastic simulations verify the method's accuracy in capturing the changes in the protein number distributions with time for a wide variety of networks displaying auto- and mutual-regulation of gene expression and independently of the ratios of the timescales governing the dynamics. The method is also used to study the first-passage time distribution of promoter switching, the sensitivity of the size of protein number fluctuations to parameter perturbation and the stochastic bifurcation diagram characterizing the onset of multimodality in protein number distributions.",
    url = "https://doi.org/10.1038/s41467-018-05822-0",
    doi = "10.1038/s41467-018-05822-0",
    openalex = "W2887216660",
    references = "doi10108817518121aa54d9"
}

Individual-based models provide modularity and structural flexibility necessary for modeling of infectious diseases at the within-host and population levels, but are challenging to implement. Levels of complexity can exceed the capacity and timescales for students and trainees in most academic institutions. Here we describe the process and advantages of a multi-disease framework approach developed with formal software support. The epidemiological modeling software, EMOD, has undergone a decade of software development. It is structured so that a majority of code is shared across disease modeling including malaria, HIV, tuberculosis, dengue, polio and typhoid. In additional to implementation efficiency, the sharing increases code usage and testing. The freely available codebase also includes hundreds of regression tests, scientific feature tests and component tests to help verify functionality and avoid inadvertent changes to functionality during future development. Here we describe the levels of detail, flexible configurability and modularity enabled by EMOD and the role of software development principles and processes in its development.

@article{doi101093femspdfty059,
    author = "Bershteyn, Anna and Gerardin, Jaline and Bridenbecker, Daniel and Lorton, Christopher and Bloedow, Jonathan and Baker, Robert S. and Chabot‐Couture, Guillaume and Chen, Ye and Fischle, Thomas and Frey, Kurt and Gauld, Jillian and Hu, Hao and Izzo, Amanda S and Klein, Daniel J. and Lukacevic, Dejan and McCarthy, Kevin and Miller, Joel C. and Ouédraogo, André Lin and Perkins, T. Alex and Steinkraus, Jeffrey and ten Bosch, Quirine A. and Ting, Hung-Fu and Titova, Svetlana and Wagner, Bradley G. and Welkhoff, Philip A. and Wenger, Edward A. and Wiswell, Christian N",
    title = "Implementation and applications of EMOD, an individual-based multi-disease modeling platform",
    year = "2018",
    journal = "Pathogens and Disease",
    abstract = "Individual-based models provide modularity and structural flexibility necessary for modeling of infectious diseases at the within-host and population levels, but are challenging to implement. Levels of complexity can exceed the capacity and timescales for students and trainees in most academic institutions. Here we describe the process and advantages of a multi-disease framework approach developed with formal software support. The epidemiological modeling software, EMOD, has undergone a decade of software development. It is structured so that a majority of code is shared across disease modeling including malaria, HIV, tuberculosis, dengue, polio and typhoid. In additional to implementation efficiency, the sharing increases code usage and testing. The freely available codebase also includes hundreds of regression tests, scientific feature tests and component tests to help verify functionality and avoid inadvertent changes to functionality during future development. Here we describe the levels of detail, flexible configurability and modularity enabled by EMOD and the role of software development principles and processes in its development.",
    url = "https://doi.org/10.1093/femspd/fty059",
    doi = "10.1093/femspd/fty059",
    openalex = "W2834682931",
    references = "doi101186s1287901726998"
}

Abstract Aim Data on species occurrences are far more common than data on species abundances. However, a central goal of large‐scale ecology is to understand the spatial distribution of abundance. It has been proposed that species distribution models trained on species occurrence records may capture variation in species abundance. Here, we gauge support for relationships between species abundance and predicted climatic suitability from species distribution models, and relate the slope of this relationship to species traits, evolutionary relationships and sampling completeness. Location USA. Time period 1658–2017. Major taxa studied Mammal and tree species. Methods, Results To explore the generality of abundance–suitability relationships, we trained species distribution models on species occurrence and species abundance data for 246 mammal species and 158 tree species, and related model‐predicted occurrence probabilities to population abundance predictions. Further, we related the resulting abundance–suitability relationship coefficients to species traits, geographic range sizes, evolutionary relationships and the number of occurrence records to investigate a potential trait or sampling basis for abundance–suitability relationship detectability. We found little evidence for consistent abundance–suitability relationships in mammal (=.045) or tree (= −.005) species, finding nearly as many negative and positive relationships. These relationships had little explanatory power, and coefficients were unrelated to species traits, range size or evolutionary relationships. Main conclusions Our findings suggest that species climatic suitability based on occurrence data may not be reflected in species abundances, suggesting a need to investigate nonclimatic sources of species abundance variation.

@article{doi101111geb12820,
    author = "Dallas, Tad and Hastings, Alan",
    title = "Habitat suitability estimated by niche models is largely unrelated to species abundance",
    year = "2018",
    journal = "Global Ecology and Biogeography",
    abstract = "Abstract Aim Data on species occurrences are far more common than data on species abundances. However, a central goal of large‐scale ecology is to understand the spatial distribution of abundance. It has been proposed that species distribution models trained on species occurrence records may capture variation in species abundance. Here, we gauge support for relationships between species abundance and predicted climatic suitability from species distribution models, and relate the slope of this relationship to species traits, evolutionary relationships and sampling completeness. Location USA. Time period 1658–2017. Major taxa studied Mammal and tree species. Methods, Results To explore the generality of abundance–suitability relationships, we trained species distribution models on species occurrence and species abundance data for 246 mammal species and 158 tree species, and related model‐predicted occurrence probabilities to population abundance predictions. Further, we related the resulting abundance–suitability relationship coefficients to species traits, geographic range sizes, evolutionary relationships and the number of occurrence records to investigate a potential trait or sampling basis for abundance–suitability relationship detectability. We found little evidence for consistent abundance–suitability relationships in mammal (=.045) or tree (= −.005) species, finding nearly as many negative and positive relationships. These relationships had little explanatory power, and coefficients were unrelated to species traits, range size or evolutionary relationships. Main conclusions Our findings suggest that species climatic suitability based on occurrence data may not be reflected in species abundances, suggesting a need to investigate nonclimatic sources of species abundance variation.",
    url = "https://doi.org/10.1111/geb.12820",
    doi = "10.1111/geb.12820",
    openalex = "W2892452100",
    references = "doi101046j14610248200200308x"
}

Approximate Bayesian Computation (ABC) and other simulation-based inference methods are becoming increasingly used for inference in complex systems, due to their relative ease-of-implementation. We briefly review some of the more popular variants of ABC and their application in epidemiology, before using a real-world model of HIV transmission to illustrate some of challenges when applying ABC methods to high-dimensional, computationally intensive models. We then discuss an alternative approach—history matching—that aims to address some of these issues, and conclude with a comparison between these different methodologies.

@article{doi10121417sts618,
    author = "McKinley, Trevelyan J. and Vernon, Ian and Andrianakis, Ioannis and McCreesh, Nicky and Oakley, Jeremy E. and Nsubuga, Rebecca N. and Goldstein, Michael and White, Richard G.",
    title = "Approximate Bayesian Computation and Simulation-Based Inference for Complex Stochastic Epidemic Models",
    year = "2018",
    journal = "Statistical Science",
    abstract = "Approximate Bayesian Computation (ABC) and other simulation-based inference methods are becoming increasingly used for inference in complex systems, due to their relative ease-of-implementation. We briefly review some of the more popular variants of ABC and their application in epidemiology, before using a real-world model of HIV transmission to illustrate some of challenges when applying ABC methods to high-dimensional, computationally intensive models. We then discuss an alternative approach—history matching—that aims to address some of these issues, and conclude with a comparison between these different methodologies.",
    url = "https://doi.org/10.1214/17-sts618",
    doi = "10.1214/17-sts618",
    openalex = "W2626930952",
    references = "doi101371journalpcbi1003968"
}

Spatially extended population dynamics models that incorporate demographic noise serve as case studies for the crucial role of fluctuations and correlations in biological systems. Numerical and analytic tools from non-equilibrium statistical physics capture the stochastic kinetics of these complex interacting many-particle systems beyond rate equation approximations. Including spatial structure and stochastic noise in models for predator–prey competition invalidates the neutral Lotka–Volterra population cycles. Stochastic models yield long-lived erratic oscillations stemming from a resonant amplification mechanism. Spatially extended predator–prey systems display noise-stabilized activity fronts that generate persistent correlations. Fluctuation-induced renormalizations of the oscillation parameters can be analyzed perturbatively via a Doi–Peliti field theory mapping of the master equation; related tools allow detailed characterization of extinction pathways. The critical steady-state and non-equilibrium relaxation dynamics at the predator extinction threshold are governed by the directed percolation universality class. Spatial predation rate variability results in more localized clusters, enhancing both competing species' population densities. Affixing variable interaction rates to individual particles and allowing for trait inheritance subject to mutations induces fast evolutionary dynamics for the rate distributions. Stochastic spatial variants of three-species competition with 'rock-paper-scissors' interactions metaphorically describe cyclic dominance. These models illustrate intimate connections between population dynamics and evolutionary game theory, underscore the role of fluctuations to drive populations toward extinction, and demonstrate how space can support species diversity. Two-dimensional cyclic three-species May–Leonard models are characterized by the emergence of spiraling patterns whose properties are elucidated by a mapping onto a complex Ginzburg–Landau equation. Multiple-species extensions to general 'food networks' can be classified on the mean-field level, providing both fundamental understanding of ensuing cooperativity and profound insight into the rich spatio-temporal features and coarsening kinetics in the corresponding spatially extended systems. Novel space-time patterns emerge as a result of the formation of competing alliances; e.g. coarsening domains that each incorporate rock-paper-scissors competition games.

@article{doi1017863cam24047,
    author = "Dobramysl, Ulrich and Mobilia, Mauro and Pleimling, Michel and Täuber, Uwe C.",
    title = "Stochastic population dynamics in spatially extended predator–prey systems",
    year = "2018",
    journal = "White Rose Research Online (University of Leeds, The University of Sheffield, University of York)",
    abstract = "Spatially extended population dynamics models that incorporate demographic noise serve as case studies for the crucial role of fluctuations and correlations in biological systems. Numerical and analytic tools from non-equilibrium statistical physics capture the stochastic kinetics of these complex interacting many-particle systems beyond rate equation approximations. Including spatial structure and stochastic noise in models for predator–prey competition invalidates the neutral Lotka–Volterra population cycles. Stochastic models yield long-lived erratic oscillations stemming from a resonant amplification mechanism. Spatially extended predator–prey systems display noise-stabilized activity fronts that generate persistent correlations. Fluctuation-induced renormalizations of the oscillation parameters can be analyzed perturbatively via a Doi–Peliti field theory mapping of the master equation; related tools allow detailed characterization of extinction pathways. The critical steady-state and non-equilibrium relaxation dynamics at the predator extinction threshold are governed by the directed percolation universality class. Spatial predation rate variability results in more localized clusters, enhancing both competing species' population densities. Affixing variable interaction rates to individual particles and allowing for trait inheritance subject to mutations induces fast evolutionary dynamics for the rate distributions. Stochastic spatial variants of three-species competition with 'rock-paper-scissors' interactions metaphorically describe cyclic dominance. These models illustrate intimate connections between population dynamics and evolutionary game theory, underscore the role of fluctuations to drive populations toward extinction, and demonstrate how space can support species diversity. Two-dimensional cyclic three-species May–Leonard models are characterized by the emergence of spiraling patterns whose properties are elucidated by a mapping onto a complex Ginzburg–Landau equation. Multiple-species extensions to general 'food networks' can be classified on the mean-field level, providing both fundamental understanding of ensuing cooperativity and profound insight into the rich spatio-temporal features and coarsening kinetics in the corresponding spatially extended systems. Novel space-time patterns emerge as a result of the formation of competing alliances; e.g. coarsening domains that each incorporate rock-paper-scissors competition games.",
    url = "https://doi.org/10.17863/cam.24047",
    doi = "10.17863/cam.24047",
    openalex = "W3099175199",
    references = "doi101103physreve80021129"
}

Stochasticity is a core component of ecology, as it underlies key processes that structure and create variability in nature. Despite its fundamental importance in ecological systems, the concept is often treated as synonymous with unpredictability in community ecology, and studies tend to focus on single forms of stochasticity rather than taking a more holistic view. This has led to multiple narratives for how stochasticity mediates community dynamics. Here, we present a framework that describes how different forms of stochasticity (notably demographic and environmental stochasticity) combine to provide underlying and predictable structure in diverse communities. This framework builds on the deep ecological understanding of stochastic processes acting at individual and population levels and in modules of a few interacting species. We support our framework with a mathematical model that we use to synthesize key literature, demonstrating that stochasticity is more than simple uncertainty. Rather, stochasticity has profound and predictable effects on community dynamics that are critical for understanding how diversity is maintained. We propose next steps that ecologists might use to explore the role of stochasticity for structuring communities in theoretical and empirical systems, and thereby enhance our understanding of community dynamics.

@article{doi101002ecy2922,
    author = "Shoemaker, Lauren G. and Sullivan, Lauren L. and Donohue, Ian and Cabral, Juliano Sarmento and Williams, Ryan J. and Mayfield, Margaret M. and Chase, Jonathan M. and Chu, Chengjin and Harpole, W. Stanley and Huth, Andreas and HilleRisLambers, Janneke and James, Aubrie R. M. and Kraft, Nathan J. B. and May, Felix and Muthukrishnan, Ranjan and Satterlee, S. Andrew and Taubert, Franziska and Wang, Xugao and Wiegand, Thorsten and Yang, Qiang and Abbott, Karen C.",
    title = "Integrating the underlying structure of stochasticity into community ecology",
    year = "2019",
    journal = "Ecology",
    abstract = "Stochasticity is a core component of ecology, as it underlies key processes that structure and create variability in nature. Despite its fundamental importance in ecological systems, the concept is often treated as synonymous with unpredictability in community ecology, and studies tend to focus on single forms of stochasticity rather than taking a more holistic view. This has led to multiple narratives for how stochasticity mediates community dynamics. Here, we present a framework that describes how different forms of stochasticity (notably demographic and environmental stochasticity) combine to provide underlying and predictable structure in diverse communities. This framework builds on the deep ecological understanding of stochastic processes acting at individual and population levels and in modules of a few interacting species. We support our framework with a mathematical model that we use to synthesize key literature, demonstrating that stochasticity is more than simple uncertainty. Rather, stochasticity has profound and predictable effects on community dynamics that are critical for understanding how diversity is maintained. We propose next steps that ecologists might use to explore the role of stochasticity for structuring communities in theoretical and empirical systems, and thereby enhance our understanding of community dynamics.",
    url = "https://doi.org/10.1002/ecy.2922",
    doi = "10.1002/ecy.2922",
    openalex = "W2981737754",
    references = "doi101002ecm1263, doi101002ecm1302, doi101016jtree201201014"
}

@book{doi1010079781493998289,
    author = "Brauer, Fred and Castillo‐Chávez, Carlos and Feng, Zhilan",
    title = "Mathematical Models in Epidemiology",
    year = "2019",
    booktitle = "Texts in applied mathematics",
    url = "https://doi.org/10.1007/978-1-4939-9828-9",
    doi = "10.1007/978-1-4939-9828-9",
    openalex = "W2980308966",
    references = "bailey1961stochastic, doi1023072282020"
}

@book{doi1010079783030309008,
    author = "Britton, Tom and Pardoux, Etienne",
    title = "Stochastic Epidemic Models with Inference",
    year = "2019",
    booktitle = "Lecture notes in mathematics",
    url = "https://doi.org/10.1007/978-3-030-30900-8",
    doi = "10.1007/978-3-030-30900-8",
    openalex = "W4288083333",
    references = "doi1023072985209"
}

Many cellular processes are governed by stochastic reaction events. These events do not necessarily occur in single steps of individual molecules, and, conversely, each birth or death of a macromolecule (e.g., protein) could involve several small reaction steps, creating a memory between individual events and thus leading to nonmarkovian reaction kinetics. Characterizing this kinetics is challenging. Here, we develop a systematic approach for a general reaction network with arbitrary intrinsic waiting-time distributions, which includes the stationary generalized chemical-master equation (sgCME), the stationary generalized Fokker-Planck equation, and the generalized linear-noise approximation. The first formulation converts a nonmarkovian issue into a markovian one by introducing effective transition rates (that explicitly decode the effect of molecular memory) for the reactions in an equivalent reaction network with the same substrates but without molecular memory. Nonmarkovian features of the reaction kinetics can be revealed by solving the sgCME. The latter 2 formulations can be used in the fast evaluation of fluctuations. These formulations can have broad applications, and, in particular, they may help us discover new biological knowledge underlying memory effects. When they are applied to generalized stochastic models of gene-expression regulation, we find that molecular memory is in effect equivalent to a feedback and can induce bimodality, fine-tune the expression noise, and induce switch.

@article{doi101073pnas1913926116,
    author = "Zhang, Jiajun and Zhou, Tianshou",
    title = "Markovian approaches to modeling intracellular reaction processes with molecular memory",
    year = "2019",
    journal = "Proceedings of the National Academy of Sciences",
    abstract = "Many cellular processes are governed by stochastic reaction events. These events do not necessarily occur in single steps of individual molecules, and, conversely, each birth or death of a macromolecule (e.g., protein) could involve several small reaction steps, creating a memory between individual events and thus leading to nonmarkovian reaction kinetics. Characterizing this kinetics is challenging. Here, we develop a systematic approach for a general reaction network with arbitrary intrinsic waiting-time distributions, which includes the stationary generalized chemical-master equation (sgCME), the stationary generalized Fokker-Planck equation, and the generalized linear-noise approximation. The first formulation converts a nonmarkovian issue into a markovian one by introducing effective transition rates (that explicitly decode the effect of molecular memory) for the reactions in an equivalent reaction network with the same substrates but without molecular memory. Nonmarkovian features of the reaction kinetics can be revealed by solving the sgCME. The latter 2 formulations can be used in the fast evaluation of fluctuations. These formulations can have broad applications, and, in particular, they may help us discover new biological knowledge underlying memory effects. When they are applied to generalized stochastic models of gene-expression regulation, we find that molecular memory is in effect equivalent to a feedback and can induce bimodality, fine-tune the expression noise, and induce switch.",
    url = "https://doi.org/10.1073/pnas.1913926116",
    doi = "10.1073/pnas.1913926116",
    openalex = "W2985088907",
    references = "doi101038ncomms8366, doi10108817518121aa54d9"
}

This work is devoted to study the existence and uniqueness of global positive solution for a stochastic epidemic model with media coverage driven by Lévy noise. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Numerical simulations are presented to confirm the theoretical results.

@article{doi1010801744250820191595622,
    author = "Fatini, Mohamed El and Laaribi, Aziz and Pettersson, Roger and Taki, Regragui",
    title = "Lévy noise perturbation for an epidemic model with impact of media coverage",
    year = "2019",
    journal = "Stochastics",
    abstract = "This work is devoted to study the existence and uniqueness of global positive solution for a stochastic epidemic model with media coverage driven by Lévy noise. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Numerical simulations are presented to confirm the theoretical results.",
    url = "https://doi.org/10.1080/17442508.2019.1595622",
    doi = "10.1080/17442508.2019.1595622",
    openalex = "W2966140820",
    references = "doi1023072527465"
}

implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community.

@article{doi101098rsif20180943,
    author = "Warne, David J. and Baker, Ruth E. and Simpson, Matthew J.",
    title = "Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art",
    year = "2019",
    journal = "Journal of The Royal Society Interface",
    abstract = "implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community.",
    url = "https://doi.org/10.1098/rsif.2018.0943",
    doi = "10.1098/rsif.2018.0943",
    openalex = "W2905450522",
    references = "doi10108817518121aa54d9"
}

BACKGROUND: Mathematical transmission models are increasingly used to guide public health interventions for infectious diseases, particularly in the context of emerging pathogens; however, the contribution of modeling to the growing issue of antimicrobial resistance (AMR) remains unclear. Here, we systematically evaluate publications on population-level transmission models of AMR over a recent period (2006-2016) to gauge the state of research and identify gaps warranting further work. METHODS: We performed a systematic literature search of relevant databases to identify transmission studies of AMR in viral, bacterial, and parasitic disease systems. We analyzed the temporal, geographic, and subject matter trends, described the predominant medical and behavioral interventions studied, and identified central findings relating to key pathogens. RESULTS: We identified 273 modeling studies; the majority of which (> 70%) focused on 5 infectious diseases (human immunodeficiency virus (HIV), influenza virus, Plasmodium falciparum (malaria), Mycobacterium tuberculosis (TB), and methicillin-resistant Staphylococcus aureus (MRSA)). AMR studies of influenza and nosocomial pathogens were mainly set in industrialized nations, while HIV, TB, and malaria studies were heavily skewed towards developing countries. The majority of articles focused on AMR exclusively in humans (89%), either in community (58%) or healthcare (27%) settings. Model systems were largely compartmental (76%) and deterministic (66%). Only 43% of models were calibrated against epidemiological data, and few were validated against out-of-sample datasets (14%). The interventions considered were primarily the impact of different drug regimens, hygiene and infection control measures, screening, and diagnostics, while few studies addressed de novo resistance, vaccination strategies, economic, or behavioral changes to reduce antibiotic use in humans and animals. CONCLUSIONS: The AMR modeling literature concentrates on disease systems where resistance has been long-established, while few studies pro-actively address recent rise in resistance in new pathogens or explore upstream strategies to reduce overall antibiotic consumption. Notable gaps include research on emerging resistance in Enterobacteriaceae and Neisseria gonorrhoeae; AMR transmission at the animal-human interface, particularly in agricultural and veterinary settings; transmission between hospitals and the community; the role of environmental factors in AMR transmission; and the potential of vaccines to combat AMR.

@article{doi101186s1291601913149,
    author = "Niewiadomska, Anna Maria and Jayabalasingham, Bamini and Seidman, Jessica C. and Willem, Lander and Grenfell, Bryan T. and Spiro, David and Viboud, Cécile",
    title = "Population-level mathematical modeling of antimicrobial resistance: a systematic review",
    year = "2019",
    journal = "BMC Medicine",
    abstract = "BACKGROUND: Mathematical transmission models are increasingly used to guide public health interventions for infectious diseases, particularly in the context of emerging pathogens; however, the contribution of modeling to the growing issue of antimicrobial resistance (AMR) remains unclear. Here, we systematically evaluate publications on population-level transmission models of AMR over a recent period (2006-2016) to gauge the state of research and identify gaps warranting further work. METHODS: We performed a systematic literature search of relevant databases to identify transmission studies of AMR in viral, bacterial, and parasitic disease systems. We analyzed the temporal, geographic, and subject matter trends, described the predominant medical and behavioral interventions studied, and identified central findings relating to key pathogens. RESULTS: We identified 273 modeling studies; the majority of which (> 70\%) focused on 5 infectious diseases (human immunodeficiency virus (HIV), influenza virus, Plasmodium falciparum (malaria), Mycobacterium tuberculosis (TB), and methicillin-resistant Staphylococcus aureus (MRSA)). AMR studies of influenza and nosocomial pathogens were mainly set in industrialized nations, while HIV, TB, and malaria studies were heavily skewed towards developing countries. The majority of articles focused on AMR exclusively in humans (89\%), either in community (58\%) or healthcare (27\%) settings. Model systems were largely compartmental (76\%) and deterministic (66\%). Only 43\% of models were calibrated against epidemiological data, and few were validated against out-of-sample datasets (14\%). The interventions considered were primarily the impact of different drug regimens, hygiene and infection control measures, screening, and diagnostics, while few studies addressed de novo resistance, vaccination strategies, economic, or behavioral changes to reduce antibiotic use in humans and animals. CONCLUSIONS: The AMR modeling literature concentrates on disease systems where resistance has been long-established, while few studies pro-actively address recent rise in resistance in new pathogens or explore upstream strategies to reduce overall antibiotic consumption. Notable gaps include research on emerging resistance in Enterobacteriaceae and Neisseria gonorrhoeae; AMR transmission at the animal-human interface, particularly in agricultural and veterinary settings; transmission between hospitals and the community; the role of environmental factors in AMR transmission; and the potential of vaccines to combat AMR.",
    url = "https://doi.org/10.1186/s12916-019-1314-9",
    doi = "10.1186/s12916-019-1314-9",
    openalex = "W2941479653",
    references = "doi101186s1287901726998"
}

The stochasticity of gene expression presents significant challenges to the modeling of genetic networks. A two-state model describing promoter switching, transcription, and messenger RNA (mRNA) decay is the standard model of stochastic mRNA dynamics in eukaryotic cells. Here, we extend this model to include mRNA maturation, cell division, gene replication, dosage compensation, and growth-dependent transcription. We derive expressions for the time-dependent distributions of nascent mRNA and mature mRNA numbers, provided two assumptions hold: 1) nascent mRNA dynamics are much faster than those of mature mRNA; and 2) gene-inactivation events occur far more frequently than gene-activation events. We confirm that thousands of eukaryotic genes satisfy these assumptions by using data from yeast, mouse, and human cells. We use the expressions to perform a sensitivity analysis of the coefficient of variation of mRNA fluctuations averaged over the cell cycle, for a large number of genes in mouse embryonic stem cells, identifying degradation and gene-activation rates as the most sensitive parameters. Furthermore, it is shown that, despite the model's complexity, the time-dependent distributions predicted by our model are generally well approximated by the negative binomial distribution. Finally, we extend our model to include translation, protein decay, and auto-regulatory feedback, and derive expressions for the approximate time-dependent protein-number distributions, assuming slow protein decay. Our expressions enable us to study how complex biological processes contribute to the fluctuations of gene products in eukaryotic cells, as well as allowing a detailed quantitative comparison with experimental data via maximum-likelihood methods.

@article{doi101073pnas1910888117,
    author = "Cao, Zhixing and Grima, Ramon",
    title = "Analytical distributions for detailed models of stochastic gene expression in eukaryotic cells",
    year = "2020",
    journal = "Proceedings of the National Academy of Sciences",
    abstract = "The stochasticity of gene expression presents significant challenges to the modeling of genetic networks. A two-state model describing promoter switching, transcription, and messenger RNA (mRNA) decay is the standard model of stochastic mRNA dynamics in eukaryotic cells. Here, we extend this model to include mRNA maturation, cell division, gene replication, dosage compensation, and growth-dependent transcription. We derive expressions for the time-dependent distributions of nascent mRNA and mature mRNA numbers, provided two assumptions hold: 1) nascent mRNA dynamics are much faster than those of mature mRNA; and 2) gene-inactivation events occur far more frequently than gene-activation events. We confirm that thousands of eukaryotic genes satisfy these assumptions by using data from yeast, mouse, and human cells. We use the expressions to perform a sensitivity analysis of the coefficient of variation of mRNA fluctuations averaged over the cell cycle, for a large number of genes in mouse embryonic stem cells, identifying degradation and gene-activation rates as the most sensitive parameters. Furthermore, it is shown that, despite the model's complexity, the time-dependent distributions predicted by our model are generally well approximated by the negative binomial distribution. Finally, we extend our model to include translation, protein decay, and auto-regulatory feedback, and derive expressions for the approximate time-dependent protein-number distributions, assuming slow protein decay. Our expressions enable us to study how complex biological processes contribute to the fluctuations of gene products in eukaryotic cells, as well as allowing a detailed quantitative comparison with experimental data via maximum-likelihood methods.",
    url = "https://doi.org/10.1073/pnas.1910888117",
    doi = "10.1073/pnas.1910888117",
    openalex = "W3007264533",
    references = "doi10108817518121aa54d9"
}

, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

@article{doi101098rsif20200652,
    author = "Browning, Alexander P. and Warne, David J. and Burrage, Kevin and Baker, Ruth E. and Simpson, Matthew J.",
    title = "Identifiability analysis for stochastic differential equation models in systems biology",
    year = "2020",
    journal = "Journal of The Royal Society Interface",
    abstract = ", we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.",
    url = "https://doi.org/10.1098/rsif.2020.0652",
    doi = "10.1098/rsif.2020.0652",
    openalex = "W3112752205",
    references = "doi10108817518121aa54d9"
}

The bulk of stochastic gene expression models in the literature do not have an explicit description of the age of a cell within a generation and hence they cannot capture events such as cell division and DNA replication. Instead, many models incorporate the cell cycle implicitly by assuming that dilution due to cell division can be described by an effective decay reaction with first-order kinetics. If it is further assumed that protein production occurs in bursts, then the stationary protein distribution is a negative binomial. Here we seek to understand how accurate these implicit models are when compared with more detailed models of stochastic gene expression. We derive the exact stationary solution of the chemical master equation describing bursty protein dynamics, binomial partitioning at mitosis, age-dependent transcription dynamics including replication, and random interdivision times sampled from Erlang or more general distributions; the solution is different for single lineage and population snapshot settings. We show that protein distributions are well approximated by the solution of implicit models (a negative binomial) when the mean number of mRNAs produced per cycle is low and the cell cycle length variability is large. When these conditions are not met, the distributions are either almost bimodal or else display very flat regions near the mode and cannot be described by implicit models. We also show that for genes with low transcription rates, the size of protein noise has a strong dependence on the replication time, it is almost independent of cell cycle variability for lineage measurements, and increases with cell cycle variability for population snapshot measurements. In contrast for large transcription rates, the size of protein noise is independent of replication time and increases with cell cycle variability for both lineage and population measurements.

@article{doi101103physreve101032403,
    author = "Beentjes, Casper H. L. and Perez‐Carrasco, Rubén and Grima, Ramon",
    title = "Exact solution of stochastic gene expression models with bursting, cell cycle and replication dynamics",
    year = "2020",
    journal = "Physical review. E",
    abstract = "The bulk of stochastic gene expression models in the literature do not have an explicit description of the age of a cell within a generation and hence they cannot capture events such as cell division and DNA replication. Instead, many models incorporate the cell cycle implicitly by assuming that dilution due to cell division can be described by an effective decay reaction with first-order kinetics. If it is further assumed that protein production occurs in bursts, then the stationary protein distribution is a negative binomial. Here we seek to understand how accurate these implicit models are when compared with more detailed models of stochastic gene expression. We derive the exact stationary solution of the chemical master equation describing bursty protein dynamics, binomial partitioning at mitosis, age-dependent transcription dynamics including replication, and random interdivision times sampled from Erlang or more general distributions; the solution is different for single lineage and population snapshot settings. We show that protein distributions are well approximated by the solution of implicit models (a negative binomial) when the mean number of mRNAs produced per cycle is low and the cell cycle length variability is large. When these conditions are not met, the distributions are either almost bimodal or else display very flat regions near the mode and cannot be described by implicit models. We also show that for genes with low transcription rates, the size of protein noise has a strong dependence on the replication time, it is almost independent of cell cycle variability for lineage measurements, and increases with cell cycle variability for population snapshot measurements. In contrast for large transcription rates, the size of protein noise is independent of replication time and increases with cell cycle variability for both lineage and population measurements.",
    url = "https://doi.org/10.1103/physreve.101.032403",
    doi = "10.1103/physreve.101.032403",
    openalex = "W2994626354",
    references = "doi10108817518121aa54d9"
}

We present results of a study of a simple, stochastic, agent-based model of influenza A infection, simulating its dynamics over the course of one flu season. Building on an early work of Bartlett, we define a model with a limited number of parameters and rates that have clear epidemiological interpretation and can be constrained by data. We demonstrate the occurrence of recurrent behavior in the infected number [more than one peak in a season], which is observed in data, in our simulations for populations consisting of cohorts with strong intra- and weak inter-cohort transmissibility. We examine the dependence of the results on epidemiological and population characteristics by investigating their dependence on a range of parameter values. Finally, we study infection with two strains of influenza, inspired by observations, and show a counter-intuitive result for the effect of inoculation against the strain that leads to the first wave of infection.

@article{doi101371journalpone0231521,
    author = "Whitman, John and Jayaprakash, C.",
    title = "Stochastic modeling of influenza spread dynamics with recurrences",
    year = "2020",
    journal = "PLoS ONE",
    abstract = "We present results of a study of a simple, stochastic, agent-based model of influenza A infection, simulating its dynamics over the course of one flu season. Building on an early work of Bartlett, we define a model with a limited number of parameters and rates that have clear epidemiological interpretation and can be constrained by data. We demonstrate the occurrence of recurrent behavior in the infected number [more than one peak in a season], which is observed in data, in our simulations for populations consisting of cohorts with strong intra- and weak inter-cohort transmissibility. We examine the dependence of the results on epidemiological and population characteristics by investigating their dependence on a range of parameter values. Finally, we study infection with two strains of influenza, inspired by observations, and show a counter-intuitive result for the effect of inoculation against the strain that leads to the first wave of infection.",
    url = "https://doi.org/10.1371/journal.pone.0231521",
    doi = "10.1371/journal.pone.0231521",
    openalex = "W3016939109",
    references = "barton1962stochastic, doi1023072333508"
}

Non-Markovian models of stochastic biochemical kinetics often incorporate explicit time delays to effectively model large numbers of intermediate biochemical processes. Analysis and simulation of these models, as well as the inference of their parameters from data, are fraught with difficulties because the dynamics depends on the system's history. Here we use an artificial neural network to approximate the time-dependent distributions of non-Markovian models by the solutions of much simpler time-inhomogeneous Markovian models; the approximation does not increase the dimensionality of the model and simultaneously leads to inference of the kinetic parameters. The training of the neural network uses a relatively small set of noisy measurements generated by experimental data or stochastic simulations of the non-Markovian model. We show using a variety of models, where the delays stem from transcriptional processes and feedback control, that the Markovian models learnt by the neural network accurately reflect the stochastic dynamics across parameter space.

@article{doi101038s41467021229191,
    author = "Jiang, Qingchao and Fu, Xiaoming and Yan, Shifu and Li, Runlai and Du, Wenli and Cao, Zhixing and Qian, Feng and Grima, Ramon",
    title = "Neural network aided approximation and parameter inference of non-Markovian models of gene expression",
    year = "2021",
    journal = "Nature Communications",
    abstract = "Non-Markovian models of stochastic biochemical kinetics often incorporate explicit time delays to effectively model large numbers of intermediate biochemical processes. Analysis and simulation of these models, as well as the inference of their parameters from data, are fraught with difficulties because the dynamics depends on the system's history. Here we use an artificial neural network to approximate the time-dependent distributions of non-Markovian models by the solutions of much simpler time-inhomogeneous Markovian models; the approximation does not increase the dimensionality of the model and simultaneously leads to inference of the kinetic parameters. The training of the neural network uses a relatively small set of noisy measurements generated by experimental data or stochastic simulations of the non-Markovian model. We show using a variety of models, where the delays stem from transcriptional processes and feedback control, that the Markovian models learnt by the neural network accurately reflect the stochastic dynamics across parameter space.",
    url = "https://doi.org/10.1038/s41467-021-22919-1",
    doi = "10.1038/s41467-021-22919-1",
    openalex = "W3160417254",
    references = "doi10108817518121aa54d9"
}

While averages and typical fluctuations often play a major role in understanding the behavior of a nonequilibrium system, this nonetheless is not always true. Rare events and large fluctuations are also pivotal when a thorough analysis of the system is being done. In this context, the statistics of extreme fluctuations in contrast to the average plays an important role, as has been discussed in fields ranging from statistical and mathematical physics to climate, finance, and ecology. Herein, we study extreme value statistics (EVS) of stochastic resetting systems, which have recently gained significant interest due to its ubiquitous and enriching applications in physics, chemistry, queuing theory, search processes, and computer science. We present a detailed analysis for the finite and large time statistics of extremals (maximum and arg-maximum, i.e., the time when the maximum is reached) of the spatial displacement in such system. In particular, we derive an exact renewal formula that relates the joint distribution of maximum and arg-maximum of the reset process to the statistical measures of the underlying process. Benchmarking our results for the maximum of a reset trajectory that pertain to the Gumbel class for large sample size, we show that the arg-maximum density attains a uniform distribution independent of the underlying process at a large observation time. This emerges as a manifestation of the renewal property of the resetting mechanism. The results are augmented with a wide spectrum of Markov and non-Markov stochastic processes under resetting, namely, simple diffusion, diffusion with drift, Ornstein-Uhlenbeck process, and random acceleration process in one dimension. Rigorous results are presented for the first two setups, while the latter two are supported with heuristic and numerical analysis.

@article{doi101103physreve103052119,
    author = "Singh, Prashant and Pal, A.",
    title = "Extremal statistics for stochastic resetting systems",
    year = "2021",
    journal = "Physical review. E",
    abstract = "While averages and typical fluctuations often play a major role in understanding the behavior of a nonequilibrium system, this nonetheless is not always true. Rare events and large fluctuations are also pivotal when a thorough analysis of the system is being done. In this context, the statistics of extreme fluctuations in contrast to the average plays an important role, as has been discussed in fields ranging from statistical and mathematical physics to climate, finance, and ecology. Herein, we study extreme value statistics (EVS) of stochastic resetting systems, which have recently gained significant interest due to its ubiquitous and enriching applications in physics, chemistry, queuing theory, search processes, and computer science. We present a detailed analysis for the finite and large time statistics of extremals (maximum and arg-maximum, i.e., the time when the maximum is reached) of the spatial displacement in such system. In particular, we derive an exact renewal formula that relates the joint distribution of maximum and arg-maximum of the reset process to the statistical measures of the underlying process. Benchmarking our results for the maximum of a reset trajectory that pertain to the Gumbel class for large sample size, we show that the arg-maximum density attains a uniform distribution independent of the underlying process at a large observation time. This emerges as a manifestation of the renewal property of the resetting mechanism. The results are augmented with a wide spectrum of Markov and non-Markov stochastic processes under resetting, namely, simple diffusion, diffusion with drift, Ornstein-Uhlenbeck process, and random acceleration process in one dimension. Rigorous results are presented for the first two setups, while the latter two are supported with heuristic and numerical analysis.",
    url = "https://doi.org/10.1103/physreve.103.052119",
    doi = "10.1103/physreve.103.052119",
    openalex = "W3129568006",
    references = "doi101073pnas1213237110"
}

The COVID-19 pandemic has created an urgent need for models that can project epidemic trends, explore intervention scenarios, and estimate resource needs. Here we describe the methodology of Covasim (COVID-19 Agent-based Simulator), an open-source model developed to help address these questions. Covasim includes country-specific demographic information on age structure and population size; realistic transmission networks in different social layers, including households, schools, workplaces, long-term care facilities, and communities; age-specific disease outcomes; and intrahost viral dynamics, including viral-load-based transmissibility. Covasim also supports an extensive set of interventions, including non-pharmaceutical interventions, such as physical distancing and protective equipment; pharmaceutical interventions, including vaccination; and testing interventions, such as symptomatic and asymptomatic testing, isolation, contact tracing, and quarantine. These interventions can incorporate the effects of delays, loss-to-follow-up, micro-targeting, and other factors. Implemented in pure Python, Covasim has been designed with equal emphasis on performance, ease of use, and flexibility: realistic and highly customized scenarios can be run on a standard laptop in under a minute. In collaboration with local health agencies and policymakers, Covasim has already been applied to examine epidemic dynamics and inform policy decisions in more than a dozen countries in Africa, Asia-Pacific, Europe, and North America.

@article{doi101371journalpcbi1009149,
    author = "Kerr, Cliff C. and Stuart, Robyn M. and Mistry, Dina and Abeysuriya, Romesh and Rosenfeld, Katherine and Hart, Gregory R. and Núñez, Rafael C. and Cohen, Jamie A. and Selvaraj, Prashanth and Hagedorn, Brittany and George, Lauren and Jastrzębski, Michał and Izzo, Amanda S and Fowler, Greer and Palmer, Anna and Delport, Dominic and Scott, Nick and Kelly, Sherrie L. and Bennette, Caroline S. and Wagner, Bradley G. and Chang, Stewart T. and Oron, Assaf P. and Wenger, Edward A. and Panovska‐Griffiths, Jasmina and Famulare, Michael and Klein, Daniel J.",
    title = "Covasim: An agent-based model of COVID-19 dynamics and interventions",
    year = "2021",
    journal = "PLoS Computational Biology",
    abstract = "The COVID-19 pandemic has created an urgent need for models that can project epidemic trends, explore intervention scenarios, and estimate resource needs. Here we describe the methodology of Covasim (COVID-19 Agent-based Simulator), an open-source model developed to help address these questions. Covasim includes country-specific demographic information on age structure and population size; realistic transmission networks in different social layers, including households, schools, workplaces, long-term care facilities, and communities; age-specific disease outcomes; and intrahost viral dynamics, including viral-load-based transmissibility. Covasim also supports an extensive set of interventions, including non-pharmaceutical interventions, such as physical distancing and protective equipment; pharmaceutical interventions, including vaccination; and testing interventions, such as symptomatic and asymptomatic testing, isolation, contact tracing, and quarantine. These interventions can incorporate the effects of delays, loss-to-follow-up, micro-targeting, and other factors. Implemented in pure Python, Covasim has been designed with equal emphasis on performance, ease of use, and flexibility: realistic and highly customized scenarios can be run on a standard laptop in under a minute. In collaboration with local health agencies and policymakers, Covasim has already been applied to examine epidemic dynamics and inform policy decisions in more than a dozen countries in Africa, Asia-Pacific, Europe, and North America.",
    url = "https://doi.org/10.1371/journal.pcbi.1009149",
    doi = "10.1371/journal.pcbi.1009149",
    openalex = "W3186633320",
    references = "doi101371journalpcbi1003968"
}

Stochastic kinetic models (SKMs) are increasingly used to account for the inherent stochasticity exhibited by interacting populations of species in areas such as epidemiology, population ecology and systems biology. Species numbers are modelled using a continuous-time stochastic process, and, depending on the application area of interest, this will typically take the form of a Markov jump process or an It\^o diffusion process. Widespread use of these models is typically precluded by their computational complexity. In particular, performing exact fully Bayesian inference in either modelling framework is challenging due to the intractability of the observed data likelihood, necessitating the use of computationally intensive techniques such as particle Markov chain Monte Carlo (particle MCMC). It is proposed to increase the computational and statistical efficiency of this approach by leveraging the tractability of an inexpensive surrogate derived directly from either the jump or diffusion process. The surrogate is used in three ways: in the design of a gradient-based parameter proposal, to construct an appropriate bridge and in the first stage of a delayed-acceptance step. The resulting approach, which exactly targets the posterior of interest, offers substantial gains in efficiency over a standard particle MCMC implementation.

@article{doi101016jcsda2023107760,
    author = "Lowe, Tom and Golightly, A. and Sherlock, C.",
    title = "Accelerating inference for stochastic kinetic models",
    year = "2022",
    journal = "Comput. Stat. Data Anal.",
    abstract = "Stochastic kinetic models (SKMs) are increasingly used to account for the inherent stochasticity exhibited by interacting populations of species in areas such as epidemiology, population ecology and systems biology. Species numbers are modelled using a continuous-time stochastic process, and, depending on the application area of interest, this will typically take the form of a Markov jump process or an It\^o diffusion process. Widespread use of these models is typically precluded by their computational complexity. In particular, performing exact fully Bayesian inference in either modelling framework is challenging due to the intractability of the observed data likelihood, necessitating the use of computationally intensive techniques such as particle Markov chain Monte Carlo (particle MCMC). It is proposed to increase the computational and statistical efficiency of this approach by leveraging the tractability of an inexpensive surrogate derived directly from either the jump or diffusion process. The surrogate is used in three ways: in the design of a gradient-based parameter proposal, to construct an appropriate bridge and in the first stage of a delayed-acceptance step. The resulting approach, which exactly targets the posterior of interest, offers substantial gains in efficiency over a standard particle MCMC implementation.",
    url = "https://doi.org/10.1016/j.csda.2023.107760",
    doi = "10.1016/j.csda.2023.107760",
    is_oa = "true",
    pages = "107760",
    semanticscholar_citation_count = "6",
    semanticscholar_id = "7a075a59c35e53cbeb1ce1ce455f95ee8706ecf8",
    volume = "185"
}

Population growth is a fundamental process in ecology, evolution, and epidemiology. The population size dynamics during growth are often described by deterministic equations derived from kinetic models. Here, we simulate several population growth models and compare the size averaged over many stochastic realizations with the deterministic predictions. We show that these deterministic equations are generically bad predictors of the average stochastic population dynamics. Specifically, deterministic predictions overestimate the simulated population sizes, especially those of populations starting with a small number of individuals. Describing population growth as a stochastic birth process, we prove that the discrepancy between deterministic predictions and simulated data is due to unclosed-moment dynamics. In other words, the deterministic approach does not take into account the variability of birth times, which is particularly important at small population sizes. We evaluate different moment-closure approximations and show that they do not satisfactorily reduce the error between analytical predictions and simulated data. We present two novel solutions to the stochastic growth dynamics, one of which applies to any population growth model. We show that our solution exactly quantifies the dynamics of a community composed of different strains and correctly predicts the fixation probability of a strain in a serial dilution experiment. Our work sets the foundations for a more faithful modeling of community dynamics. It provides tools for a more accurate analysis of experimental results, including the inference of important growth parameters.

@article{doi10110120221115516663,
    author = "Marrec, L. and Bank, Claudia and Bertrand, T.",
    title = "Solving the stochastic dynamics of population growth",
    year = "2022",
    journal = "Ecology and Evolution",
    abstract = "Population growth is a fundamental process in ecology, evolution, and epidemiology. The population size dynamics during growth are often described by deterministic equations derived from kinetic models. Here, we simulate several population growth models and compare the size averaged over many stochastic realizations with the deterministic predictions. We show that these deterministic equations are generically bad predictors of the average stochastic population dynamics. Specifically, deterministic predictions overestimate the simulated population sizes, especially those of populations starting with a small number of individuals. Describing population growth as a stochastic birth process, we prove that the discrepancy between deterministic predictions and simulated data is due to unclosed-moment dynamics. In other words, the deterministic approach does not take into account the variability of birth times, which is particularly important at small population sizes. We evaluate different moment-closure approximations and show that they do not satisfactorily reduce the error between analytical predictions and simulated data. We present two novel solutions to the stochastic growth dynamics, one of which applies to any population growth model. We show that our solution exactly quantifies the dynamics of a community composed of different strains and correctly predicts the fixation probability of a strain in a serial dilution experiment. Our work sets the foundations for a more faithful modeling of community dynamics. It provides tools for a more accurate analysis of experimental results, including the inference of important growth parameters.",
    url = "https://boris.unibe.ch/185230/1/Ecology\_and\_Evolution\_-\_2023\_-\_Marrec.pdf",
    doi = "10.1101/2022.11.15.516663",
    is_oa = "true",
    semanticscholar_citation_count = "18",
    semanticscholar_id = "51c258d10e0db08377217bde22cb35be83e107ec"
}

Population dynamics, the study of how and why populations change over time, is a fundamental aspect of ecology, epidemiology, and resource management. Mathematical modeling provides a powerful framework to simulate and predict population behavior under varying biological and environmental conditions. This paper explores classical and modern mathematical models including exponential, logistic, Lotka–Volterra, age-structured, and stochastic models. By leveraging these frameworks, researchers can analyze complex biological systems, estimate growth rates, model interspecies interactions, and predict the outcomes of interventions. The paper also highlights the importance of integrating real-world data and outlines the future directions of research in population modeling.

@article{doi1071465aljb3279,
    author = "Morales, Dr. Ethan",
    title = "Mathematical Models for Understanding Population Dynamics",
    year = "2022",
    journal = "Austra \& Lian Journal of Basic Sciences",
    abstract = "Population dynamics, the study of how and why populations change over time, is a fundamental aspect of ecology, epidemiology, and resource management. Mathematical modeling provides a powerful framework to simulate and predict population behavior under varying biological and environmental conditions. This paper explores classical and modern mathematical models including exponential, logistic, Lotka–Volterra, age-structured, and stochastic models. By leveraging these frameworks, researchers can analyze complex biological systems, estimate growth rates, model interspecies interactions, and predict the outcomes of interventions. The paper also highlights the importance of integrating real-world data and outlines the future directions of research in population modeling.",
    url = "https://www.semanticscholar.org/paper/fc738a0cc2155a795877c05499e1390f0e9b180b",
    doi = "10.71465/aljb3279",
    is_oa = "true",
    number = "1",
    pages = "1-10",
    semanticscholar_id = "fc738a0cc2155a795877c05499e1390f0e9b180b",
    volume = "3"
}

The R Package IBMPopSim aims to simulate the random evolution of heterogeneous populations using stochastic Individual-Based Models (IBMs). The package enables users to simulate population evolution, in which individuals are characterized by their age and some characteristics, and the population is modified by different types of events, including births/arrivals, death/exit events, or changes of characteristics. The frequency at which an event can occur to an individual can depend on their age and characteristics, but also on the characteristics of other individuals (interactions). Such models have a wide range of applications in fields including actuarial science, biology, ecology or epidemiology. IBMPopSim overcomes the limitations of time-consuming IBMs simulations by implementing new efficient algorithms based on thinning methods, which are compiled using the Rcpp package while providing a user-friendly interface.

@article{doi1048550arxiv230306183,
    author = "Giorgi, Daphné and Kaakai, Sarah and Lemaire, Vincent",
    title = "Efficient simulation of individual-based population models: the R Package IBMPopSim",
    year = "2023",
    journal = "ArXiv",
    publisher = "arXiv",
    abstract = "The R Package IBMPopSim aims to simulate the random evolution of heterogeneous populations using stochastic Individual-Based Models (IBMs). The package enables users to simulate population evolution, in which individuals are characterized by their age and some characteristics, and the population is modified by different types of events, including births/arrivals, death/exit events, or changes of characteristics. The frequency at which an event can occur to an individual can depend on their age and characteristics, but also on the characteristics of other individuals (interactions). Such models have a wide range of applications in fields including actuarial science, biology, ecology or epidemiology. IBMPopSim overcomes the limitations of time-consuming IBMs simulations by implementing new efficient algorithms based on thinning methods, which are compiled using the Rcpp package while providing a user-friendly interface.",
    url = "http://arxiv.org/pdf/2303.06183",
    doi = "10.48550/arXiv.2303.06183",
    is_oa = "true",
    semanticscholar_citation_count = "4",
    semanticscholar_id = "22ed00e440fec302a677c3ea215215504c5179a4"
}

Random walks and related spatial stochastic models have been used in a range of application areas, including animal and plant ecology, infectious disease epidemiology, developmental biology, wound healing and oncology. Classical random walk models assume that all individuals in a population behave independently, ignoring local physical and biological interactions. This assumption simplifies the mathematical description of the population considerably, enabling continuum-limit descriptions to be derived and used in model analysis and fitting. However, interactions between individuals can have a crucial impact on population-level behaviour. In recent decades, research has increasingly been directed towards models that include interactions, including physical crowding effects and local biological processes such as adhesion, competition, dispersal, predation and adaptive directional bias. In this article, we review the progress that has been made with models of interacting individuals. We aim to provide an overview that is accessible to researchers in application areas, as well as to specialist modellers. We focus particularly on derivation of asymptotically exact or approximate continuum-limit descriptions and simplified deterministic models of mean-field behaviour and resulting spatial patterns. We provide worked examples and illustrative results of selected models. We conclude with a discussion of current areas of focus and future challenges.

@article{doi101098rsif20240422,
    author = "Plank, Michael J. and Simpson, Matthew J. and Baker, Ruth E.",
    title = "Random walk models in the life sciences: including births, deaths and local interactions",
    year = "2024",
    journal = "Journal of the Royal Society Interface",
    abstract = "Random walks and related spatial stochastic models have been used in a range of application areas, including animal and plant ecology, infectious disease epidemiology, developmental biology, wound healing and oncology. Classical random walk models assume that all individuals in a population behave independently, ignoring local physical and biological interactions. This assumption simplifies the mathematical description of the population considerably, enabling continuum-limit descriptions to be derived and used in model analysis and fitting. However, interactions between individuals can have a crucial impact on population-level behaviour. In recent decades, research has increasingly been directed towards models that include interactions, including physical crowding effects and local biological processes such as adhesion, competition, dispersal, predation and adaptive directional bias. In this article, we review the progress that has been made with models of interacting individuals. We aim to provide an overview that is accessible to researchers in application areas, as well as to specialist modellers. We focus particularly on derivation of asymptotically exact or approximate continuum-limit descriptions and simplified deterministic models of mean-field behaviour and resulting spatial patterns. We provide worked examples and illustrative results of selected models. We conclude with a discussion of current areas of focus and future challenges.",
    url = "https://www.semanticscholar.org/paper/7bec7d556c7f75a4a4ac6a336f15b5763990b572",
    doi = "10.1098/rsif.2024.0422",
    is_oa = "true",
    number = "222",
    semanticscholar_citation_count = "14",
    semanticscholar_id = "7bec7d556c7f75a4a4ac6a336f15b5763990b572",
    volume = "22"
}

Population dynamics in fields such as molecular biology, epidemiology, and ecology exhibit highly stochastic and non-linear behavior. In gene regulatory systems in particular, oscillations and multistability are especially common. Despite this, none of the currently available stochastic models for population dynamics are both accurate and computationally efficient for long-term predictions. A prominent model in this field, the Linear Noise Approximation (LNA), is computationally efficient for tasks such as simulation, sensitivity analysis, and parameter estimation; however, it is only accurate for linear systems and short-time predictions. Other models may achieve greater accuracy across a broader range of systems, but they sacrifice computational efficiency and analytical tractability. This paper demonstrates that, with specific modifications, the LNA can accurately capture non-linear dynamics in population processes. We introduce a new framework based on centre manifold theory, a classical concept from non-linear dynamical systems. This approach enables the identification of simple, system-specific modifications to the LNA, tailored to classes of qualitatively similar non-linear dynamical systems. With these modifications, the LNA can achieve accurate long-term simulations without compromising computational efficiency. We apply our methodology to classes of oscillatory and bi-stable systems, and present multiple examples from molecular population dynamics that demonstrate accurate long-term simulations alongside significant improvements in computational efficiency.

@misc{s26cb0b23953e382fcc11f728fc36e3f2116c31c99,
    author = "Truman-Williams, Frederick and Minas, G.",
    title = "Simulating stochastic population dynamics: The Linear Noise Approximation can capture non-linear phenomena",
    year = "2025",
    booktitle = "arXiv (Cornell University)",
    abstract = "Population dynamics in fields such as molecular biology, epidemiology, and ecology exhibit highly stochastic and non-linear behavior. In gene regulatory systems in particular, oscillations and multistability are especially common. Despite this, none of the currently available stochastic models for population dynamics are both accurate and computationally efficient for long-term predictions. A prominent model in this field, the Linear Noise Approximation (LNA), is computationally efficient for tasks such as simulation, sensitivity analysis, and parameter estimation; however, it is only accurate for linear systems and short-time predictions. Other models may achieve greater accuracy across a broader range of systems, but they sacrifice computational efficiency and analytical tractability. This paper demonstrates that, with specific modifications, the LNA can accurately capture non-linear dynamics in population processes. We introduce a new framework based on centre manifold theory, a classical concept from non-linear dynamical systems. This approach enables the identification of simple, system-specific modifications to the LNA, tailored to classes of qualitatively similar non-linear dynamical systems. With these modifications, the LNA can achieve accurate long-term simulations without compromising computational efficiency. We apply our methodology to classes of oscillatory and bi-stable systems, and present multiple examples from molecular population dynamics that demonstrate accurate long-term simulations alongside significant improvements in computational efficiency.",
    url = "https://www.semanticscholar.org/paper/6cb0b23953e382fcc11f728fc36e3f2116c31c99",
    doi = "10.48550/arxiv.2504.15166",
    is_oa = "true",
    openalex = "W4417058156",
    semanticscholar_id = "6cb0b23953e382fcc11f728fc36e3f2116c31c99"
}