We explore the equilibrium properties of a series of compartmental, ODE models describing the interaction between different strains of pathogen. The interaction is conceptualized as acting through shared antigens: infection and recovery from one strain leaves the host with a primed immune response against subsequent strains. The models consider the effect of this priming on susceptibility (the ability to be infected) and transmission (the ability to infect) in an SIR model. In these models, the specific past history of infection is encapsulated in different susceptible compartments within the model. In a third, SIS, model, specific past history is not included, but strains have differential abilities to infect previously infected hosts. Equilibrium results include criteria for the coexistence of strains. For the SIR models, the region of coexistence defined by parameters shrinks as the effect of strains on each other (increased antigenic similarity) increases. For the SIS model, coexistence depends critically on the rate at which complete susceptibility is recovered following infection, and coexisting strains must have differential abilities to infect completely and partially susceptible hosts. Interestingly, this model provides analogies to commensalism (the first species gains from the presence of the second; the second neither gains nor loses from the interaction) and symbiosis (the presence of both species benefits the other). Additionally, we show that the maximum number of coexisting strains is two in this model. The effect of vaccination depends on the initial strain structure, the ability of vaccination to mount protection to both strains and the coverage. Vaccination may allow a previously excluded strain to coexist or exist alone, and may allow a previously rarer strain to become more common with the possibility of increasing incidence of disease. We discuss the dynamics of these models, compare model results to observed patterns and consider additional model structures. The importance of these results to specific multi-strain pathogens, in particular rotavirus, is considered
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