Zika virus dynamics partial differential equations model with sexual transmission route

Inspired by the system of ordinary differential equations in Agusto et al. (2017) that models Zika virus dynamics by taking into account of both sexual and vector-borne transmissions, we furthermore add diffusive terms in order to capture the movement of human hosts and mosquitoes, considering the unique threat of the sexual transmission route of Zika virus. We conduct complete theoretical analysis. In particular, we show that every initial data that is continuous and non-negative admits a unique continuous and non-negative solution for all positive times. Moreover, we derive the basic reproduction number and when it is beneath one, we prove that the disease-free-equilibrium is globally attractive. Finally, when the basic reproduction number is above one, and any of the exposed males or the exposed females or the exposed mosquitoes is not identically zero, we prove the existence of a positive asymptotic lower bound for every component of the solution which in particular implies the uniform persistence of the disease, as well as the existence of at least one positive steady state.