Immune level approach for multiple strain pathogens

We introduce a general framework to study the dynamics of multiple strain pathogens that can impart crossimmunity through a variety of structures. We propose two types of immunity and apply them to assess the dynamics of two competing strains. We illustrate this framework using two "visions": the nexttokin protection (NTKP) approach which assumes that strains confer crossimmunity to next in order (neighboring) strains, while providing no protection against all other strains; and the differential protection (DP) approach assumes that individuals randomly gain partial (may be reinfected) and full cross-immunity following an infection with respective probabilities. We show that the risk of infection with a particular strain is significantly higher in the DP model that the NKTP. Moreover, we demonstrate that weaker cross-immunity structures in these models are more likely to lead to instability (sustained oscillations) in the strain coexistence mode. That is, periodic oscillations are sustained in the two-strain DP model for intermediate to weak levels of cross-immunity, while the NTKP model requires at least three strains to support these unstable dynamics.