Reproduction Numbers and the Stability of Equilibria of SI Models for Heterogeneous Populations

Source abstract

A major project in deterministic epidemiological modeling of heterogeneous populations is to find conditions for the local and global stability of the equilibria and to work out the relations among these stability conditions, the thresholds for epidemic take-off and endemicity, and the basic reproduction number(s). Most of the work to date has been on models of homogeneous populations of constant size. Motivated by their analysis of models of the dynamics of human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS), the authors carry out this project for SI models of diseases that have multiple stages, which can lead to death, and which infect heterogeneous populations with intricate mixing patterns and varying sizes. In such models, it is even difficult to find the analytical expression for the stability threshold and the reproduction number. The authors show how the number of disease-transmitting contacts by infectives can be used as a Lyapunov function to carry out a systematic stability analysis in these complex models, to compute the expressions for the thresholds and reproductions numbers, and to construct a reproduction function as a cornerstone of the study of the dynamics of disease spread.