Abstract
Abstract This paper concerns with detailed analysis of a reaction-diffusion host-pathogen model with space-dependent parameters in a bounded domain. By considering the fact the mobility of host individuals playing a crucial role in disease transmission, we formulate the model by a system of degenerate reaction-diffusion equations, where host individuals disperse at distinct rates and the mobility of pathogen is ignored in the environment.We first establish the well-posedness of the model, including the global existence of solution and the existence of the global compact attractor. The basic reproduction number is identified, and also characterized by some equivalent principal spectral conditions, which establishes the threshold dynamical result for pathogen extinction and persistence. When the positive steady state is confirmed, we investigate the asymptotic profiles of positive steady state as host individuals disperse at small and large rates. Our result suggests that small and large diffusion rate of hosts have a great impacts in formulating the spatial distribution of the pathogen.
Highlights
In recent years, the studies of some reaction-diffusion host-pathogen models have received much attentions, as the investigation of these systems allow us to get better understanding the interactions between host and pathogens and the mechanisms of the disease spread
By considering the fact the mobility of host individuals playing a crucial role in disease transmission, we formulate the model by a system of degenerate reaction-diffusion equations, where host individuals disperse at distinct rates and the mobility of pathogen is ignored in the environment
When the positive steady state is confirmed, we investigate the asymptotic profiles of positive steady state as host individuals disperse at small and large rates
Summary
The studies of some reaction-diffusion host-pathogen models have received much attentions, as the investigation of these systems allow us to get better understanding the interactions between host and pathogens and the mechanisms of the disease spread. The main concern in the aspect of biological implication is: limiting the flow of susceptible individuals (dS → 0) can eliminate the disease, provided that the disease is of low risk (i.e., β(·) < γ(·) for x ∈ Ω) This pioneering work start up the investigation that how the spatial heterogeneity and the diffusion affect the disease spread and control. Motivated by meaningful and important aspect of spatial heterogeneity of environment and distinct dispersal rates, Wu and Zou [33] further modified the model (1.4) by replacing the frequency-dependent interaction with mass action mechanisms. They showed that an additional condition on the total population is needed for disease control if dS → 0. Detailed conclusions are drawn and some discussion is presented
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