Global dynamics analysis of a Zika transmission model with environment transmission route and spatial heterogeneity

Zika virus, a recurring mosquito-borne flavivirus, became a global public health agency in 2016. It is mainly transmitted through mosquito bites. Recently, experimental result demonstrated that $Aedes$ mosquitoes can acquire and transmit Zika virus by breeding in contaminated aquatic environments. The environmental transmission route is unprecedented discovery for the Zika virus. Therefore, it is necessary to introduce environment transmission route into Zika model. Furthermore, we consider diffusive terms in order to capture the movement of humans and mosquitoes. In this paper, we propose a novel reaction-diffusion Zika model with environment transmission route in a spatial heterogeneous environment, which is different from all Zika models mentioned earlier. We introduce the basic offspring number $R_0^m$ and basic reproduction number $R_0$ for this spatial model. By using comparison arguments and the theory of uniform persistence, we prove that disease free equilibrium with the absence of mosquitoes is globally attractive when $R_0^m<1$, disease free equilibrium with the presence of mosquitoes is globally attractive when $R_0^m>1$ and $R_0<1$, the model is uniformly persistent when $R_0^m>1$ and $R_0>1$. Finally, numerical simulations conform these analytical results.