It has been shown in the past that for the most basic multi-strain ordinary differential equation (ODE) model of SIR-type a competitive exclusion principle holds. The competitive exclusion principle means that the strain with the largest reproduction number persists but eliminates all other strains with suboptimal reproduction numbers. In this paper, we extend the competitive exclusion principle to a multi-strain age-since-infection structured model of SIR/SI-type. We also include environmental transmission for each of the pathogens. The model describes well transmission of avian influenza or cholera. Using a Lyapunov functional, we are able to establish global stability of the disease-free equilibrium if all reproduction numbers are smaller or equal to one. If Rj, the reproduction number of strain j is larger than one, then a single-strain equilibrium, corresponding to strain j exists. This single strain equilibrium is locally stable whenever Rj>1 and Rj is the unique maximal reproduction number. If R1>1 is the maximal reproduction number, using a Lyapunov functional, we establish that the corresponding single-strain equilibrium E1 is globally stable. That is, strain one eliminates all other strains, independently of their reproduction numbers as long as they are smaller than R1. © 2013 Elsevier Ltd.
Martcheva, M., & Li, X. Z. (2013). Competitive exclusion in an infection-age structured model with environmental transmission. Journal of Mathematical Analysis and Applications, 408(1), 225–246. https://doi.org/10.1016/j.jmaa.2013.05.064