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

TL;DR

This paper develops a stochastic epidemic model incorporating memory of the last infection and waning immunity, deriving PDEs for trait-age density and analyzing conditions for endemic equilibria and stability.

Contribution

It introduces a novel parametric approach with a piecewise deterministic Markov process to model infection memory and derives PDEs for large population limits.

Findings

Endemic equilibrium conditions are characterized.

An endemicity threshold depending on model parameters is established.

Stability analysis of equilibria is performed, including for vaccination scenarios.

Abstract

We adapt the article of Forien, Pang, Pardoux and Zotsa: Arxiv preprint Arxiv2210.04667(2022), on epidemic models with varying infectivity and waning immunity, to incorporate the memory of the last infection. To this end, we introduce a parametric approach and consider a piecewise deterministic Markov process modeling both the evolution of the parameter, also called the trait, and the age of infection of individuals over time. At each new infection, a new trait is randomly chosen for the infected individual according to a Markov kernel, and their age is reset to zero. In the large population limit, we derive a partial differential equation (PDE) that describes the density of traits and ages. The main goal is to study the conditions under which endemic equilibria exist for the deterministic PDE model and to establish an endemicity threshold that depends on the model parameters. The local stability of these equilibria is also analyzed. The endemicity threshold is computed for several examples, including models that incorporate a vaccination policy, and a local stability result is obtained for a memory-free SIS-type model.