Analysis of a reaction–diffusion mosquito-borne model with chemotaxis: well-posedness and global dynamics

In this paper, we construct and analyze a mosquito-borne disease model with chemotaxis in a homogeneous environment. Different from traditional modes that mosquitoes disperse with local diffusion, we correlate the chemotaxis of mosquitoes with the density of the infected human population and assume that mosquito diffusion is a decreasing function with respect to the density of infected human hosts. Our goal of this paper is to investigate how density-dependent diffusion affects the global dynamics of mosquito-borne diseases. We first prove the global existence and ultimate boundedness of the solution through a priori energy estimates. Then, we confirm that the basic reproduction number is a key threshold parameter that predicts whether the disease will persist or not. Moreover, we establish the global attractivity of the unique endemic steady state under some technical conditions. Our results demonstrate that the magnitude of the derivative of density-dependent diffusion function will remarkably affect the stability of the endemic steady state in the sense that when it is large enough, the endemic steady state will no longer be stable. This study also reveals that reducing the mosquito biting rate or controlling the movement of infected individuals can effectively disrupt the transmission chain, thereby providing a theoretical foundation for mosquito-borne disease prevention and control.