Competitive exclusion in an infection-age structured vector-host epidemic model

The competitive exclusion principle means that the strain with the largest reproduction number persists while eliminating all other strains with suboptimal reproduction numbers. In this paper, we extend the competitive exclusion principle to a multi-strain vector-borne epidemic model with age-since-infection. The model includes both incubation age of the exposed hosts and infection age of the infectious hosts, both of which describe the different removal rates in the latent period and the variable infectiousness in the infectious period, respectively. The formulas for the reproduction numbers $\mathcal R^j_0$ of strain $j,j=1,2,···, n$ , are obtained from the biological meanings of the model. The strain $j$ can not invade the system if $\mathcal R^j_0<1$ , and the disease free equilibrium is globally asymptotically stable if $\max_j\{\mathcal R^j_0\}<1$ . If $\mathcal R^{j_0}_0>1$ , then a single-strain equilibrium $\mathcal{E}_{j_0}$ exists, and the single strain equilibrium is locally asymptotically stable when $\mathcal R^{j_0}_0>1$ and $\mathcal R^{j_0}_0>\mathcal R^{j}_0,j≠ j_0$ . Finally, by using a Lyapunov function, sufficient conditions are further established for the global asymptotical stability of the single-strain equilibrium corresponding to strain $j_0$ , which means strain $j_0$ eliminates all other stains as long as $\mathcal R^{j}_0/\mathcal R^{j_0}_0<b_j/b_{j_0}<1,j≠ j_0$ , where $b_j$ denotes the probability of a given susceptible vector being transmitted by an infected host with strain $j$ .