An Epidemic Model in Inhomogeneous Environment

#### D. Bertacchi, J. Kampf, E. Sava-Huss, F. Zucca

2022, v.28, Issue 3, 399-442

ABSTRACT

The current work deals with an epidemic model on the complete graph $K_n$ on $n$ vertices in a non-homogeneous setting, where the vertices may have distinct types. Different types differ in the probability of getting infected, and/or in the capacity of infecting other vertices. This generalizes the model in [4]. We prove in Theorem 5.1 and Theorem 6.3 laws of large numbers and central limit theorems for the the total duration of the process and for the number of infected vertices, respectively, when $n\to\infty$. By coupling the epidemic model with a Poisson process, we also obtain continuous-time counterparts of the above-mentioned limit results. Moreover, we also prove that when all individuals have the same spread capacity, then a population with inhomogeneous susceptibility is less affected by the epidemics than a homogeneous population.

Keywords: multitype Galton-Watson process, coupon collector, branching process, limit theorems, Markov chain, Poisson process, random trees