A Tight Epidemic Threshold for Competing Stochastic Infection Processes with Mutually Exclusive Immunity

Abstract

Stochastic infection processes are continuous-time Markov chains on graphs that assign each vertex one of multiple states, such as susceptible, infected, or recovered. Depending on the model, vertices change their state based on random transition rates and the states of their neighbors, resulting in a variety of complex dynamics. The body of rigorous literature is rich for processes that consider a single infection. In contrast, the setting with at least two infections, where the same state exists for different types, allows for far more transition combinations, leaving several interesting models entirely unexplored. We address this shortcoming in the literature by defining the IRIR process, in which two SIR processes run on the same graph and each vertex is immune only to its most recent infection. We study the survival time of the IRIR process, that is, the time until no infected vertex remains, with mathematical rigor. Our main result is a tight threshold, known as epidemic threshold, where the survival time rapidly changes from at most quasi-linear in the graph size $n$ to at least super-polynomial in $n$. This result is applicable to perfectly mixed graphs, which are graphs where the density of edges between each non-empty subset of vertices is a given value $p \in (0, 1]$. Our super-polynomial lower bound extends to jumbled graphs, which allow for some more flexibility in the density. In particular, this includes with high probability Erdos-Renyi graphs with an average degree of $k\in \omega(\ln^2(n))$. Our proof for the lower bound is based on a potential that transforms the configurations of the IRIR process to a supermartingale with drift in a large region, implying the lower bound. We detail how to systematically derive such a potential, based on a Lyapunov function of the transition equilibrium of the process.