Analysis of some generalized models for the cholera disease with applications to Sobolev spaces

Abstract

This dissertation considers general models for the dynamic transmission of cholera disease in two main settings. The first setting involves a dynamical system defined by a system of autonomous nonlinear ordinary differential equations. Starting from simple models for the direct (human-to-human) and indirect (environment-to-human) transmissions of cholera disease, we gradually progressed to a generalized model characterized by general functions for the incidence rate, and the concentration of the pathogen, as proposed by Wang and Liao. For the generalized model, we study its well-posedness in the biologically feasible region. The qualitative analysis of the model begins with the computation of the basic reproduction number using the next-generation matrix approach. The main results are as follows: the unique disease-free equilibrium is globally asymptotically stable whenever the basic reproduction number, R0, is less than one, and unstable when R0 > 1. In the latter case, it is shown that a unique endemic equilibrium exists, which is locally asymptotically stable, using the Routh-Hurwitz criterion. The proof of the global asymptotic stability of the endemic equilibrium is established using the Poincar´e-Bendixson theorem in a particular case. Moreover in the specific case of a linear incidence function and pathogen concentration, the endemic equilibrium is demonstrated to be globally asymptotically stable through Lyapunov function techniques and the LaSalle Invariance Principle. The second setting is a partial differential equation (PDE) system which is an extension of the generalized model investigated above. The PDE has the specific form of a nonlinear reaction-convection-diffusion system for the spread of cholera in both time and space variables as proposed by Yamazaki and Wang. Using the theory of semigroups of bounded linear operators on the space of continuous functions, the problem is formulated as a Volterra integral equation of the second kind, which defines the mild solution of the model. The existence and uniqueness of a local solution are relatively easily established due to Nagumo’s tangent condition Smith. Building on this local solution and the associated conservation laws, we show using the Sobolev embedding theorem that the reaction-convection-diffusion system possesses a unique global mild solution Adams.