Threshold dynamics of a time-periodic two-strain SIRS epidemic model with distributed delay

In this paper, a two-strain SIRS epidemic model with distributed delay and spatiotemporal heterogeneity is proposed and investigated. We first introduce the basic reproduction number $R_0^i$ and the invasion number ${\hat{R}}_0^i(i=1,2)$ for each strain $i$. Then the threshold dynamics of the model is established in terms of $R_0^i$ and ${\hat{R}}_0^i$ by using the theory of chain transitive sets and persistence. It is shown that if ${\hat{R}}_0^i>1(i=1,2)$, then the disease in two strains is persist uniformly; if $R_0^i>1\ge R_0^j(i\ne j,i,j=1,2)$, then the disease in $i$-th strain is uniformly persist, but the disease in $j$-th strain will disappear; if $R_0^i<1$ or $R_0^i=1(i=1,2)$ and ${\beta }_i(x,t)>0$, then the disease in two strains will disappear.