Abstract:

This report presents how the matrix tree theorem in algebraic graph theory is used to construct Lyapunov functions in models of infectious disease transmission, such as SEIR, SIS, multi-stage, multi-group, multi-strain models, etc. The basic reproduction number is defined as the expected number of secondary infections produced by a typical infective individual in a complete susceptible population. These Lyapunov functions are used to prove the global stability of a disease free equilibrium when the basic reproduction number is less than 1. However, when the basic reproduction number is greater than 1, the Lyapunov functions help to prove the global stability and uniqueness of an endemic equilibrium.