A note on lienard-chipart criteria and its application to epidemic models

Abstract

An important part of the study of epidemic models is the local stability analysis of the equilibrium points. The linear algebra method which is commonly employed is the well-known Routh-Hurwitz criteria. The criteria give necessary and sufficient c onditions f or a ll of t he r oots of the characteristic polynomial to be negative or have negative real parts. To date, there are no epidemic models in the literature which employ Lienard-Chipart criteria. This note recommends an alternative linear algebra method namely Lienard-Chipart criteria, to significantly s implify t he l ocal s tability analysis of epidemic models. Although Routh-Hurwitz criteria is a correct method for local stability analysis, Lienard-Chipart criteria have advantages over Routh-Hurwitz criteria. Using Lienard-Chipart criteria, only about half of the Hurwitz determinants inequalities are required, with the remaining conditions of each set concern with only the sign of the alternate coefficients of the characteristic polynomial. The Lienard-Chipart criteria are especially useful for polynomials with symbolic coefficients, as the determinants are usually significantly more complicated than original coefficients as degree of the polynomial increases. Lienard- Chipart criteria and Routh-Hurwitz criteria have similar performance for systems of dimension five or less. Theoretically, for systems of dimension higher than five, verifying Lienard-Chipart criteria should be much easier than verifying Routh-Hurwitz criteria and the advantage of Lienard-Chipart criteria may become clear. Examples of local stability analysis using Lienard-Chipart criteria for two recently proposed models are demonstrated to show the advantages of simplified Lienard-Chipart criteria over Routh- Hurwitz criteria.