Coexistence of Competing Species with a Directly Transmitted Pathogen

In this paper we consider models of two competing species that are both affected by a pathogen which is transmitted directly. We consider both mass action as well as frequency incidence models of disease spread, and Lotka-Volterra competition. Our aim is to address the interaction between competition and disease dynamics. We do this by presenting previously known results along with new results in a unified manner that stresses the role of the basic reproduction number as well as the relative strengths of intra- versus inter-specific competition for both species. For both types of disease models we compute the basic reproduction numbers. For the mass action case we compute all the equilibria except infected coexistence, and analyze the equilibria for their stability. To analyze infected coexistence, we consider a simplified ecologically relevant model and prove a conjecture made in [18, 9] about the stability of the infected coexistence equilibrium. The model with frequency incidence is shown to have a unique endemic equilibrium whose existence and stability depends on the value of the basic reproduction number