Duality for some models of epidemic spreading

Abstract

We examine the role of boundaries and the structure of nontrivial duality functions for three non conservative interacting particle systems in one dimension that model epidemic spreading: (i) the diffusive contact process (DCP), (ii) a model that we introduce and call generalized diffusive contact process (GDCP), both in finite volume in contact with boundary reservoirs, i.e., with open boundaries, and (iii) the susceptible-infectious-recovered (SIR) model on $\Z$. We establish duality relations for each system through an analytical approach. It turns out that with open boundaries self-duality breaks down and qualitatively different properties compared to closed boundaries (i.e., finite volume without reservoirs) arise: both the DCP and GDCP are ergodic but no longer absorbing, while the respective dual processes are absorbing but not ergodic. We provide expressions for the stationary correlation functions in terms of the dual absorption probabilities. We perform explicit computations for a small sized DCP, and for arbitrary size in a particular setting of the GDCP. The duality function is factorized for the DCP and GDCP, contrary to the SIR model for which the duality relation is nonlocal and yields an explicit expression of the time evolution of some specific correlation functions, describing the time decay of the sizes of clusters of susceptible individuals.