On a bi-virus epidemic model with partial and waning immunity

We propose a deterministic compartmental model to study the impact of partial and waning immunity on the spread of two competitive epidemic diseases, hereafter termed viruses. Building on a standard bi-virus SIS model, we introduce additional compartments to account for individuals who recovered from each virus, and tunable parameters to capture the level of virus-specific and cross protection acquired after recovery from a specific virus, and the rate at which such immunity could wane. We formalise the model as a system of nonlinear ordinary differential equations, which is amenable to analytical treatment, and we focus our analysis on two specialisations of the model. First, in the absence of waning immunity, we establish a global convergence result showing that, above the epidemic threshold, only the “fittest” virus becomes endemic. Second, in the absence of cross-immunity, we demonstrate instead that long-lasting co-existence of the two viruses may emerge, depending on the model parameters.