An age-space structured cholera model linking within- and between-host dynamics with Neumann boundary condition

In this work, we are concerned with the asymptotical behaviors of a multi-scale cholera model with Neumann boundary condition, which takes account for the heterogeneous decay process of immunity induced by vaccination and its transmission routes consisting of human-to-human or environment-to-human. Based on a nested approach, the linkage law of within- and between-systems is established by the assumption that the between-host transmission parameters are the functions of the within-host variables. By the technique of a generalized fixed theory, we firstly prove the existence, uniqueness, and positivity of the between-host system. Next, we obtain explicit expressions of the basic reproduction numbers of the within- and between-host systems, respectively. Moreover, we employ the technique of the Lyapunov functionals to establish the globally asymptotic stability of the feasible equilibrium. Finally, we perform some numerical experiments to illustrate the theoretical results. Interestingly, numerical evidence shows that the prevalence is a non-monotonous function of pathogen replication rate r p , which means that the use of antimicrobial drugs at certain doses for blocking the growth of the pathogen in an individual may lead to an increase in the prevalence of cholera.