# CAIMS Prize Award 463

Department of Mathematical Sciences, University of Alberta

Supervisor: J. Muldowney

Thesis title: Global Stability in Epidemiological Models

Abstract:

Theoretical concepts related to the stability of continuous dynamical systems are investigated. These concepts are then applied to various epidemiological models. A strategy is presented for showing stability of time-dependent linear systems. The strategy describes how to develop non-absolute norms which are designed to take advantage of both the signs and the magnitudes of the entries in the derivative matrix. Techniques, involving compound matrices, for the analysis of the asymptotic behaviour of a dynamical system are studied. A theorem which can be used to show that a boundary equilibrium is globally stable, is proven. A new technique for analyzing behaviour on an invariant manifold is presented. This involves characterizing all dynamical systems whose restriction to the manifold is the same as the restriction of the system which is under consideration. A test is given that can be used on a homogeneous differential equation in three variables to obtain information about the limiting behaviour of solutions. The method is applied to a general example which includes several epidemiological models.

Several systems are studied which model the interaction of an infectious disease and a gene that confers some protection from the disease. A model of differential infectivity is analyzed. A threshold parameter is calculated and the impact that it has on the dynamics is determined. Global stability is demonstrated for a subset of the parameter space. A model of staged progression and amelioration is presented. When the total population exceeds a certain threshold, there is a globally stable endemic equilibrium. When the total population is below the threshold, the disease-free equilibrium is globally stable. A second model of staged progression and amelioration is also studied. A threshold parameter is calculated and its implications for stability are demonstrated. Global stability is shown for a subset of the parameter space. An advance is made on the global stability problem for the MSEIR model. It is shown that, when present, the unique endemic equilibrium is globally stable if the proportion of the population in the exposed class at the endemic equilibrium is greater than the proportion in the passively immune class. Recommendations are given for future work related to various topics covered in the thesis. The recommendations are both mathematical and biological in nature.