Multi-type infection processes are ubiquitous in ecology, epidemiology and social systems, but remain hard to analyze and to understand on a fundamental level. Here, we study a multi-strain susceptible-infected-susceptible model with coinfection. A host already colonized by one strain can become more or less vulnerable to co-colonization by a second strain, as a result of facilitating or competitive interactions between the two. Fitness differences between N strains are mediated through $$N^2$$ N 2 altered susceptibilities to secondary infection that depend on colonizer-cocolonizer identities ($$K_{ij}$$ K ij ). By assuming strain similarity in such pairwise traits, we derive a model reduction for the endemic system using separation of timescales. This ‘quasi-neutrality’ in trait space sets a fast timescale where all strains interact neutrally, and a slow timescale where selective dynamics unfold. We find that these slow dynamics are governed by the replicator equation for N strains. Our framework allows to build the community dynamics bottom-up from only pairwise invasion fitnesses between members. We highlight that mean fitness of the multi-strain network, changes with their individual dynamics, acts equally upon each type, and is a key indicator of system resistance to invasion. By uncovering the link between N-strain epidemiological coexistence and the replicator equation, we show that the ecology of co-colonization relates to Fisher’s fundamental theorem and to Lotka-Volterra systems. Besides efficient computation and complexity reduction for any system size, these results open new perspectives into high-dimensional community ecology, detection of species interactions, and evolution of biodiversity.

Understanding the dynamics of multi-type microbial ecosystems remains a challenge, despite advancing molecular technologies for diversity resolution within and between hosts. Analytical progress becomes difficult when modelling realistic levels of community richness, relying on computationally-intensive simulations and detailed parametrisation. Simplification of dynamics in polymorphic pathogen systems is possible using aggregation methods and the slow-fast dynamics approach. Here, we develop one new such framework, tailored to the epidemiology of an endemic multi-strain pathogen. We apply Goldstone's idea of slow dynamics resulting from spontaneously broken symmetries to study direct interactions in co-colonization, ranging from competition to facilitation between strains. The slow-fast dynamics approach interpolates between a neutral and nonneutral model for multi-strain coexistence, and quantifiesThe original version of this article was revised: Mistakes were introduced during the publishing process. Please refer to the Erratum article for the complete list of changes. Electronic supplementary materialThe online version of this article

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 susceptibleinfected-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-cocolonizer 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(N 2 ) 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 Fisher's fundamental theorem and to conservative Lotka-Volterra systems.Many factors have been shown to be important for ecological biodiversity. These include inter-vs. intra-species 2 interactions (Tilman, 1987), population size (Taylor et al., 2004), number and functional links with resources (Armstrong and McGehee, 1976), and movement in space (Nowak and May, 1992). Diversity of an ecosystem is closely 4 related to its stability, resilience to perturbations and productivity (Hooper et al., 2005). Co-colonization processes appear in many diverse ecological communities, from plant and marine ecosystems to infectious diseases: two species 6 encountering and coexisting together in a unit of space or resource. Interactions between resident and co-colonizer entities upon encounter may exhibit asymmetries, randomness, and special structures, and may range from facilitation 8 to competition. Although special cases for low dimensionality seem to be tractable analytically, the entangled network that arises between N interacting entities in co-colonization, and its consequences for multi-type dynamics over short 42 play. These studies have also considered very low system size (N = 2), and either only cooperative or only compe...