Mathematical analysis and numerical simulation of an age-structured model of cholera with vaccination and demographic movements

In this paper, we formulate a deterministic, nonlinear model of cholera with age structure which integrates the direct transmission and the indirect transmission of the disease. The vaccination and the demographic movements are also taken into account in this model. The propounded model is an initial/boundary-value problem constituted of four partial differential equations of first order describing the transmission dynamics of human hosts and of two ordinary differential equations representing the bacterial dynamics in the environment. We conduct a rigorous mathematical analysis of this model and we prove that it admits a unique positive bounded solution. The existence of a unique equilibrium which is infection-free in the absence of the transmission disease and endemic in the presence of the transmission disease is also established. We determine a threshold parameter ℜ0 such that this equilibrium is locally asymptotically stable when ℜ0<1 and unstable when ℜ0>1. Also, a parameter ℜ0⋆ is determined such that when ℜ0>1 and ℜ0⋆<1, the number of the infected individuals of the equilibrium becomes less than 1. At the end, we use Wendland's Compactly Supported Radial Basis Functions (CSRBFs) method to find the numerical solution of the formulated model. This numerical solution is used to conduct the numerical simulation allowing us thus to check our theoretical results.