Strain coexistence and competition for pathogens with asymmetric cross-immunity and waning immunity

The coexistence of competing pathogen strains is shaped by cross-immunity, the cross-protection that infection with one strain confers against another. Although cross-immunity is often asymmetric between strains, this asymmetry is often neglected in the literature on multi-strain coexistence. The effect on coexistence-exclusion outcomes of waning immunity\textemdash which is particularly relevant for antigenically evolving pathogens\textemdash is also poorly understood. To understand how these factors affect strain coexistence, here we analyze a status-based two-strain SIRS model with asymmetric cross-immunity and strain-specific rates for transmission, recovery, and waning of immunity. We derive explicit invasion thresholds that also determine the feasibility and local stability of a unique coexistence equilibrium. Thus, these thresholds allow us to characterize the region of stable strain coexistence, as a function of the cross-immunities and rates of waning immunity. We also obtain closed-form expressions for the strain prevalences at the coexistence equilibrium, showing that the total prevalence may vary non-monotonically as the basic reproduction number of one strain increases. Finally, we show that a transient reduction in transmission can move a coexisting strain pair across an invasion boundary, driving the weaker strain extinct. Applying this result to influenza B, our analysis offers a parsimonious explanation for the disappearance of the Yamagata lineage during the COVID-19 pandemic.