Epidemics in two competing species

An SIS epidemic model in two competing species with the mass action incidence is formulated and analysed. Thresholds for the existence of boundary equilibria are identified and conditions for their local asymptotic stability or instability are found. By persistence theory, conditions for the persistence of either hosts or pathogens are proved. Using Hopf bifurcation theory and numerical simulations, some aspects of the complicated dynamic behaviours of the model are shown: the system may have zero up to three internal equilibria, may have a stable limit cycle, may have three stable attractors. Through the results on persistence and stability of the boundary equilibria, some important interactions between infection and competition are revealed: (1) a species that would become extinct without the infection, may persist in presence of the infection; (2) a species that would coexist with its competitor without the infection, is driven to extinction by the infection; (3) an infection that would die out in either species without the interinfection of disease, may persist in both species in presence of this factor.