An Immuno‐Epidemiological Model: From Individual‐Based Approach to Endemic States

ABSTRACT

We consider immuno‐epidemiological SIS models. The models are based on two factors: the dynamics of the immune status and interactions between susceptible and infected individuals. The starting point is an individual‐based model describing interactions between individuals. Through an appropriate limit passage with the number of individuals to infinity, we obtain a model acting on measures describing the distribution of immune status in both epidemic groups. We show that the densities of these measures satisfy a system of partial differential equations with a jump operator. This system generates a continuous semigroup of nonlinear operators on the space of densities. The main results are the existence and uniqueness of an endemic state and a method for determining stationary distributions. We also present simulations of the distribution of the number of infected people over time and the stationary distributions.