Analysis of a reaction-diffusion cholera epidemic model in a spatially heterogeneous environment

Abstract

In this paper, we derive and analyze a reaction-diffusion cholera model in bounded spatial domain with zero-flux boundary condition and general nonlinear incidence functions. The parameters in the model are space-dependent due to the spatial heterogeneity. By applying the theory of monotone dynamical systems and uniform persistence, we prove that the model admits the global threshold dynamics in terms of the basic reproduction number ℜ0, which is defined by the spectral radius of the next generation operator. When all model parameters are strictly positive constants, we study three types of nonlinear incidence functions to achieve the global stability results on the unique positive cholera-endemic steady state (CESS) whenever it exists. For all these examples, the sharp threshold property based on the basic reproduction number was completely established by using Lyapunov functional techniques under some realistic assumptions. Our numerical results reveal that when ℜ0 > 1, the convergence speed of the solution to the CESS becomes faster as the diffusion coefficient d becomes larger in the spatially homogeneous case. While in the spatially heterogeneous case, cholera can not be controlled by limiting the movement of host individuals, and the spatial heterogeneity does not always enhance the disease persistence.