On a two-strain epidemic model involving delay equations.

We propose an epidemiological model for the interaction of either two viruses or viral strains with cross-immunity, where the individuals infected by the first virus cannot be infected by the second one, and without cross-immunity, where a secondary infection can occur. The model incorporates distributed recovery and death rates and consists of integro-differential equations governing the dynamics of susceptible, infectious, recovered, and dead compartments. Assuming that the recovery and death rates are uniformly distributed in time throughout the duration of the diseases, we can simplify the model to a conventional ordinary differential equation (ODE) model. Another limiting case arises if the recovery and death rates are approximated by the delta-function, thereby resulting in a new point-wise delay model that incorporates two time delays corresponding to the durations of the diseases. We establish the positiveness of solutions for the distributed delay models and determine the basic reproduction number and an estimate for the final size of the epidemic for the delay model. According to the results of the numerical simulations, both strains can coexist in the population if the disease transmission rates for them are close to each other. If the difference between them is sufficiently large, then one of the strains dominates and eliminates the other one.