Analysis of a degenerated reaction–diffusion cholera model with spatial heterogeneity and stabilized total humans

Abstract

In this paper, we perform a complete analysis on a degenerated reaction–diffusion cholera model with stabilizing total humans and non-mobility of cholera bacteria in a spatially heterogeneous bounded domain. The existence of a global attractor is established through introducing the Kuratowski measure of non-compactness. The basic reproduction number and its equivalent characterizations have been used to analyze the threshold-type results, where the persistence and extinction of cholera can also be characterized by the dispersal rate of infected humans. In the homogeneous case, the global attractivity of the unique positive equilibrium is achieved by the Lyapunov function. Moreover, we compare the basic reproduction numbers for the models without and with considering the mobility of cholera bacteria. Our results suggest that: the basic reproduction numbers attain different values as the dispersal rates of infected humans and cholera approach to infinity, while they attain the same value as the dispersal rates of infected humans and cholera approach to zero. Numerical simulations support our analytical results and discuss the impact of the dispersal rate of infected humans on the transmission of cholera.