Stochastic epidemics in a homogeneous community

Abstract

These notes describe stochastic epidemics in a homogenous community. Our main concern is stochastic compartmental models (i.e. models where each individual belongs to a compartment, which stands for its status regarding the epidemic under study : S for susceptible, E for exposed, I for infectious, R for recovered) for the spread of an infectious disease. In the present notes we restrict ourselves to homogeneously mixed communities. We present our general model and study the early stage of the epidemic in chapter 1. Chapter 2 studies the particular case of Markov models, especially in the asymptotic of a large population, which leads to a law of large numbers and a central limit theorem. Chapter 3 considers the case of a closed population, and describes the final size of the epidemic (i.e. the total number of individuals who ever get infected). Chapter 4 considers models with a constant influx of susceptibles (either by birth, immigration of loss of immunity of recovered individuals), and exploits the CLT and Large Deviations to study how long it takes for the stochastic disturbances to stop an endemic situation which is stable for the deterministic epidemic model. The document ends with an Appendix which presents several mathematical notions which are used in these notes, as well as solutions to many of the exercises which are proposed in the various chapters.

Part I of "Stochastic Epidemic Models with Inference", T. Britton & E. Pardoux eds., Lecture Notes in Mathematics 2255, Springer 2019

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• Stochastic epidemics in a heterogeneous community (Part III of the book Stochastic Epidemic Models and Inference)
• A probabilistic framework for particle-based reaction-diffusion dynamics using classical Fock space representations
• Incentives, lockdown, and testing: from Thucydides's analysis to the COVID-19 pandemic
• Stochastic approach to epidemic spreading
• Moderate deviations and extinction of an epidemic
• Dynamics of systems with varying number of particles: from Liouville equations to general master equations for open systems
• Latent event history models for quasi-reaction systems
• A SIR epidemic model on a refining spatial grid II-Central limit theorem