Predicting N-strain coexistence from co-colonization interactions: epidemiology meets ecology and the replicator equation

Abstract

Multi-type spreading processes are ubiquitous in ecology, epidemiology and social systems, but remain hard to model mathematically and to understand on a fundamental level. Here, we describe and study a multi-type susceptible-infected-susceptible (SIS) model that allows for up to two co-infections of a host. Fitness differences between N infectious agents are mediated through altered susceptibilities to secondary infections that depend on colonizer- co-colonizer interactions. By assuming small differences between such pairwise traits (and other infection parameters equal), we derive a model reduction framework using separation of timescales. This quasi-neutrality in strain space yields a fast timescale where all types behave as neutral, and a slow timescale where non-neutral dynamics take place. On the slow timescale, N equations govern strain frequencies and accurately approximate the dynamics of the full system with O(N2) variables. We show that this model reduction coincides with a special case of the replicator equation, which, in our system, emerges in terms of the pairwise invasion fitnesses among strains. This framework allows to build the multi-type community dynamics bottom-up from only pairwise outcomes between constituent members. We find that mean fitness of the multi-strain system, changing with individual frequencies, acts equally upon each type, and is a key indicator of system resistance to invasion. Besides efficient computation and complexity reduction, these results open new perspectives into high-dimensional community ecology, detection of species interactions, and evolution of biodiversity, with applications to other multi-type biological contests. By uncovering the link between an epidemiological system and the replicator equation, we also show our co-infection model relates to Fishers fundamental theorem and to conservative Lotka-Volterra systems.