Analysis of a partially diffusive vector-borne disease model with human-to-human infection in a spatially heterogeneous environment

In this paper, we deal with the sharp threshold results of a mathematical epidemic model for vector-borne diseases. We construct a degenerate reaction–diffusion equation system with spatially heterogeneous parameters to consider the situation where vectors move around randomly in a spatially heterogeneous environment. We then define the basic reproduction number $$\Re _0$$ ℜ 0 as a threshold parameter to predict whether the malaria will spread or not. More precisely, we show that the disease-free equilibrium is globally asymptotically stable if $$\Re _0 < 1$$ ℜ 0 < 1 , whereas the system is uniformly persistent if $$\Re _0>1$$ ℜ 0 > 1 . We also confirm that in the case of $$\Re _0=1$$ ℜ 0 = 1 , the disease will go extinct. In addition, the global asymptotic stability of the unique constant positive equilibrium is investigated by using a Lyapunov function method in a special case where all parameters are spatially homogeneous.