Lyapunov Functions In Epidemiological Modeling

Abstract

In this mini thesis, we study the application of Lyapunov functions in epidemiological

modeling. The aim is to give an extensive discussion of Lyapunov functions, and use

some specific classes of epidemiological models to demonstrate the construction of

Lyapunov functions. The study begins with a review of Lyapunov functions in general,

and their usage in global stability analysis. Lyapunov’s “direct method” is used

to analyse the stability of the disease-free equilibrium. Moreover, a matrix-theoretic

method is critically examined for its capability and overall functionality in the construction

and development of an appropriate Lyapunov function for the stability analysis of the

nonlinear dynamical systems. This method additionally demonstrates the construction

of the basic reproduction number for the SEIR model, and it is shown that the disease-free

equilibrium is locally asymptotically stable ifR0 1. Furthermore,

a Lyapunov function is constructed for the Vector-Host model to study the global

stability of the disease-free equilibrium. The results indicate that the disease-free

equilibrium is globally asymptotically stable whenR0 1 (i.e. every solution trajectory

of the Vector-Host model converges to the largest compact invariant setM=f(Sho; Ih;Svo; Iv)g)

and unstable when R0 > 1.