No coexistence for free: neutral null models for multistrain pathogens

Abstract

In most pathogens, multiple strains are maintained within host populations. Quantifying the mechanisms underlying strain coexistence would aid public health planning and improve understanding of disease dynamics. We argue that mathematical models of strain coexistence, when applied to indistinguishable strains, should meet criteria for both ecological neutrality and population genetic neutrality. We show that closed clonal transmission models which can be written in an "ancestor-tracing" form that meets the former criterion will also satisfy the latter. Neutral models can be a parsimonious starting point for studying mechanisms of strain coexistence; implications for past and future studies are discussed.

Fig. 1

(A) structure of the model in Eq. (1). Hosts may become infected by one or two strains (which may be the same or different), but not more; second infections occur at a reduced rate compared to primary infections. Among hosts who have two strains already, it is possible (if c>0) for one strain to “knock out” another. See text for more details (B) General approach for a model with multiple infections. The figure shows compartments with different multiplicities of infection, starting from an uninfected shaded state. Infection from this state leads to an infected state with strain i. Superinfection leads to a dually infected state, a triply infected state, and so on, leading potentially but not necessarily to increased infectiousness. The ‘knock-out’ process is not shown explicitly. If the ‘ladder’ of superinfection is truncated to two co-infections, this model is equivalent to the model of A.

Fig. 2

Evolution of the model in Eq. (1) under various starting conditions. Left ordinate shows the state variables; right ordinate shows f₁ (in black). Colors: I₁: red; I₂: green; I₁₁: blue; I₁₂: tan; I₂₂: magenta. Parameters: c=0.14; k=0.4; q=0.5; β=2; u=1.

Fig. 3

Alternative model structures. (A) Model equations 11; (B) Model equations 12; (C) Model equations 14.

Fig. 4

Models with immunity. (A) A model with strain-specific immunity, Eq. (15). This model is not neutral, and it promotes stable coexistence of the two strains. (B) A neutral model with immunity, Eq. (16). This model has immune states that depend only on the number of times a host has been infected, but not on which strains have infected the host.