Dynamics and Profiles of a Diffusive Cholera Model with Bacterial Hyperinfectivity and Distinct Dispersal Rates

Abstract

This paper provides an analysis on global dynamics of a diffusive cholera model. We formulate the model by a reaction–diffusion system with space-dependent parameters and bacterial hyperinfectivity, where susceptible and infectious humans disperse with distinct dispersal rates and the hyper-infectious and lower-infectious vibrios in contaminated water are assumed to be immobile in the domain. We first establish the well-posedness of the model. To cope with the lack of compactness of solution semiflow, we verify the asymptotic smoothness of semiflow implied with $$\kappa $$ -contraction condition. The basic reproduction number, $$\mathfrak {R}_0$$ , is identified as a threshold, predicting whether or not the disease extinction and persistence will occur. $$\mathfrak {R}_0$$ is also equivalently characterized by some principle spectral conditions of an associated elliptic eigenvalue problem. The asymptotic profiles of the positive steady state are investigated for the cases when the dispersal rate of the susceptible humans is small and large. Our theoretical results indicate that: (1) cholera epidemics will be extinct through restricting the flow of susceptible humans within some extend; (2) the risk of infection will be underestimated if hyperinfectivity is not considered. In a homogeneous case, we also confirm the global attractivity of a unique positive equilibrium by utilizing the technique of Lyapunov function.