Stochastic and deterministic SIS patch model

一个自然的随机SIS模型、矩分析及不同扰动技术的比较

A Natural Stochastic SIS Model, Analysis of Moments and Comparison of Different Perturbation Techniques

本文提出用几乎必然非负的一维扩散过程替代传播率β来自然构造随机SIS模型，分析感染人群动态的灭绝与持续性，并提供基于修正项幂次展开的矩近似解析方法，进而比较两种不同扰动对模型动态的影响。

Individual and patch behaviour in structured metapopulation models

Density dependent Markov population processes with countably many types can often be well approximated over finite time intervals by the solution of the differential equations that describe their average drift, provided that the total population size is large. They also exhibit diffusive stochastic fluctuations on a smaller scale about this deterministic path. Here, it is shown that the individuals in such processes experience an almost deterministic environment. Small groups of individuals behave almost independently of one another, evolving as Markov jump processes, whose transition rates ar…

Large population asymptotics for a multitype stochastic SIS epidemic model in randomly switched environment

We consider an epidemic SIS model described by a multitype birth-and-death process in a randomly switched environment. That is, the infection and cure rates of the process depend on the state of a finite Markov jump process (the environment), whose transitions also depend on the number of infectives. The total size of the population is constant and equal to some K $\in$ N * , and the number of infectives vanishes almost surely in finite time. We prove that, as K $\rightarrow$ $\infty$, the process composed of the proportions of infectives of each type X^K and the state of the environment $Ξ$^K…

Multi-patch epidemic models with general exposed and infectious periods

We study multi-patch epidemic models where individuals may migrate from one patch to another in either of the susceptible, exposed/latent, infectious and recovered states. We assume that infections occur both locally with a rate that depends on the patch as well as "from distance" from all the other patches. The exposed and infectious periods have general distributions, and are not affected by the possible migrations of the individuals. The migration processes in either of the three states are assumed to be Markovian, and independent of the exposed and infectious periods. We establish a functi…

A stochastic model for speciation by mating preferences

Mechanisms leading to speciation are a major focus in evolutionary biology. In this paper, we present and study a stochastic model of population where individuals, with type a or A, are equivalent from ecological, demographical and spatial points of view, and differ only by their mating preference: two individuals with the same genotype have a higher probability to mate and produce a viable offspring. The population is subdivided in several patches and individuals may migrate between them. We show that mating preferences by themselves, even if they are very small, are enough to entail reproduc…

一个具有最近感染记忆和免疫力衰退的随机流行病模型

A stochastic epidemic model with memory of the last infection and waning immunity

本文改编了关于具有可变感染力和免疫力衰退的流行病模型，引入参数化分段确定性马尔可夫过程以纳入对最近感染的记忆，推导出描述特征与感染年龄密度的PDE，研究地方病平衡的存在条件、阈值及局部稳定性。

Functional Limit Theorems for Non-Markovian Epidemic Models

We study non-Markovian stochastic epidemic models (SIS, SIR, SIRS, and SEIR), in which the infectious (and latent/exposing, immune) periods have a general distribution. We provide a representation of the evolution dynamics using the time epochs of infection (and latency/exposure, immunity). Taking the limit as the size of the population tends to infinity, we prove both a functional law of large number (FLLN) and a functional central limit theorem (FCLT) for the processes of interest in these models. In the FLLN, the limits are a unique solution to a system of deterministic Volterra integral eq…

Functional law of large numbers and PDEs for epidemic models with infection-age dependent infectivity

We study epidemic models where the infectivity of each individual is a random function of the infection age (the elapsed time of infection). To describe the epidemic evolution dynamics, we use a stochastic process that tracks the number of individuals at each time that have been infected for less than or equal to a certain amount of time, together with the aggregate infectivity process. We establish the functional law of large numbers (FLLN) for the stochastic processes that describe the epidemic dynamics. The limits are described by a set of deterministic integral equations, which has a furth…