Stochastic epidemics in a homogeneous community (1808.05350v3)

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.

Overview of the Paper "Stochastic Epidemic Models with Inference"

The paper, "Stochastic Epidemic Models with Inference," edited by Tom Britton and Etienne Pardoux, is a comprehensive exploration of stochastic epidemic models, particularly within a homogeneously mixing community. It is segmented into several parts, each focusing on different model complexities and applications. The contributions underscore the mathematical underpinnings of epidemic modeling, enabling both estimation and control of infectious diseases dynamics.

Stochastic Epidemic Models in Homogenous Settings

In initial sections, the paper focuses on stochastic SEIR models in closed populations, exploring mathematical derivations for epidemic onset and progression. The primary analytical tool is the basic reproduction number, $R_0$, which determines whether an epidemic will grow. Branching processes approximate the early stage of an outbreak, offering insights into the potential size and spread of epidemics. The use of this approximation highlights probabilities for either minor or major outbreaks, quantitatively linking the risk of widespread infection with $R_0$.

The Sellke Construction and Final Size Distributions

The paper introduces the Sellke construction, a stochastic method to simulate epidemic dynamics, which relies on predefined resistance levels against infection for each individual. Through this method, models gain a probabilistic framework for estimating final epidemic sizes and the distribution of infection resistance. This construction permits the handling of various biological complexities beyond the confines of standard deterministic models.

Advanced Topics: Open Populations and Vaccination

Subsequent sections tackle models within open populations, where birth and death processes integrate with infectious dynamics, thus allowing a paper of endemic infections. This expansion to SEIR models in dynamic populations enables a discussion on long-term epidemic behavior, including stable endemic states and extinction probabilities due to demographic stochasticity. The interventions, notably vaccination, alter $R_0$ and serve to demonstrate the models' application in public health strategies.

Asymptotic Analysis: LLN, CLT, and Diffusion Approximation

The paper covers the Law of Large Numbers (LLN) and Central Limit Theorem (CLT) applied to these stochastic models, providing rigorous proof that stochastic epidemics converge to deterministic ODE solutions as population sizes increase. This statistical treatment extends to diffusion approximations, which introduce continuous uncertainties modeled by stochastic differential equations, offering deeper insights into fluctuations around deterministic limits.

Implications and Future Directions

The paper provides significant insights into epidemic modeling, crucial for planning and policy-making in public health. Through stochastic frameworks, it addresses real-world complexities that deterministic models often overlook, such as demographic variance and localized random events leading to disease extinction. The possibility of applying large deviation theory to predict timescales for epidemic extinction further enriches this discourse.

Conclusion

This collection advances the methodological landscape of stochastic epidemics, offering both theoretical constructs and practical applications, particularly in understanding infectious disease dynamics across varying epidemiological settings. Future research is expected to integrate these stochastic models with machine learning to enhance real-time epidemic forecasting and intervention strategies, underscoring the applied potential of these theoretical advancements.