Local Stability Analysis of epidemic models using a Corollary of Gershgorin’s Circle Theorem

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Abstract The techniques and methods that help to obtain necessary and sufficient conditions to determine the local stability of linearized systems are paramount. In this paper, a corollary of the Gershgorin’s circle theorem was used to establish the local stability of different epidemic models with three or more states including, a Tuberculosis model, an SEIRS model, vector-host model and a-Staged HIV/AIDS Model. It was observed that no matter the state or the dimension of the system or matrix, this corollary can be used to analyse the local stability for both disease-free and endemic equilibria, by establishing that when R0 < 1, the Jacobian matrix evaluated at the disease free equilibrium will have negative eigenvalues or negative real part eigenvalues. Thus, the disease-free equilibrium is stable but when R0 > 1, the Jacobian matrix evaluated at the endemic equilibrium will have negative eigenvalues or negative real part eigenvalues making the endemic equilibrium is stable.

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