A cholera model linking between-host, within-host and environmental disease dynamics

Cholera is an acute intestinal illness caused by infection with the Vibrio cholerae bacteria. The dynamics of the disease transmission are governed by human-human, environment-human, and within-human sub-dynamics. In this paper, a model is presented to incorporate all three of these dynamical components. The model is divided into three subgroups where the dynamics are analyzed according to their respective time scales. Specifically, the within host system incorporates the interaction of virus and immune cell interaction with the vibrios. For each subgroup, we discuss the existence and uniqueness of a DFE (Disease Free Equilibrium) by calculation of the Basic Reproduction number R_0, when applicable, and the existence and uniqueness of a positive EE (Endemic Equilibrium) solution. We also discuss the conditions needed to achieve local and global stability in each system. Finally, we combine the three smaller models together and discuss the existence and uniqueness of a DFE and EE for the full system