Dynamics and Asymptotic Profiles in a Host‐Pathogen Epidemic Model With Advection and Degenerated Heterogeneous Diffusion

ABSTRACT In this article, we investigate the dynamical behavior and asymptotic profiles for a host‐pathogen epidemic model, where the different advection rates and degenerated heterogeneous diffusions are adopted and the total population is variable. First, the scalar equation of susceptible with diffusion and advection rate is investigated. The existence, global stability and prior estimations of positive steady state are established, and then the asymptotic properties of positive steady state are discussed as and approach to zero or infinity, respectively. Next, the well‐posedness of solutions for the model, including the global existence, nonnegativity and ultimate boundedness of solutions, and the existence of global attractor are established. Following, the display expression of the basic reproduction number is calculated by means of the variational method. The local reproduction number , the special forms , , of and the relationships between these reproduction numbers are presented. Then, the global dynamics of solutions for the model in terms with are established by using the comparison principle, properties of principle eigenvalue and the persistence theory of dynamical systems. That is, when the disease‐free steady state is globally asymptotic stable, otherwise when the disease is uniformly persistent. Furthermore, it is proved that is monotonically decreasing with respect to advection rate of infected individuals. The asymptotic profiles of in relation to the heterogeneous diffusion rates , and advection rates , approaching zero or infinity are discussed in detail by means of the display expression of and the corresponding principal eigenvalue and weight eigenvalue problems, including the eighteen limit cases of , , and , and involving single limits and double limits. Finally, some open questions are proposed for the model, left us to further explore. Compared with the existing results for the constant diffusion rates and common advection rate, our model is more general and more complicated, the results established in this paper are richer and more meaningful.