Lyapunovfunctions in epidemiological modeling

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, but unstable 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.

Grantor dc:publisher

University of Namibia

Year dc:date.issued

2018

Author dc:creator

• Lazarus, Elisen N.

• Lyapunov function
• Next generation matrix

Language dc:language.iso

en

Handle dc:identifier.uri

http://hdl.handle.net/11070/2251

OAI identifier oai:identifier

oai:repository.unam.edu.na:11070/2251