The epidemic threshold and contact number with implications for optimal vaccination strategies

This dissertation explores the relationship between the epidemic threshold and contact number in deterministic infectious disease models. It also investigates some aspects of optimal vaccination strategies. Chapter 1 provides an introduction and summary of results. Chapter 2 discusses the epidemic threshold and contact number in single-population models in which homogeneous mixing is assumed. Examples generally involve only one class of susceptibles, though sometimes more than one class of infectives. It introduces the use of cumulative incidence in a quarantined model as a means of finding the contact number. A threshold quantity is found by equating the equilibrium infection-free susceptible fraction to the equilibrium infection-present susceptible fraction. If the threshold quantity exceeds one then introduced infection will lead to an epidemic and otherwise not. The threshold quantity also can be found from the determinant of the coefficient matrix (the Jacobian matrix for the equations for the exposed and infectious classes evaluated at the infection-free equilibrium). It is shown that the contact number is not always the same as the threshold quantity. Chapter 3 discusses the epidemic threshold and contact number in heterogeneous-population models. It focuses on an age-stratified model, in which the population is divided into three age groups, and an activity-stratified model with two activity groups. For these multi-population models it is seen that the threshold quantity and the contact number in general are not the same. Chapter 4 discusses vaccination strategies. Expressions for the threshold quantity and contact number in models with vaccination are derived. Optimal vaccination strategies in heterogeneous-population models are considered. In the examples discussed, it is shown that subject to a fixed vaccination effort, maximum effort should be placed in the most active group. Minimizing the maximum eigenvalue of the coefficient matrix subject to a fixed vaccination effort does not always result in minimum cumulative incidence (total number of cases up to a specified time). However, in some instances, minimum cumulative incidence can be achieved by minimizing the contact numbers. Cost-effectiveness is highly sensitive to the time interval over which it is assessed.