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\geq R_0^j\; (i\neq 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.