An infinite-dimensional metapopulation SIS model

In this article, we introduce an infinite-dimensional deterministic metapopulation SIS model which takes into account the heterogeneity of the infections and the social network among a large population. We study the long-time behavior of the dynamic. We identify the basic reproduction number R₀ which determines whether there exists a stable endemic steady state (super-critical case: R₀>1) or if the only equilibrium is disease-free (critical and sub-critical case: R₀≤1). As an application of this general study, we prove that the so-called “leaky” and “all-or-nothing” vaccination mechanism have the same effect on R₀. This framework is also very natural and intuitive to model lockdown policies and study their impact.