On the stability analysis of solutions of an integral equation with an application in epidemiology

Volume 7, Issue 1, pp 16--25 http://dx.doi.org/10.22436/mns.07.01.02

Publication Date: June 27, 2021 Submission Date: February 03, 2021 Revision Date: May 25, 2021 Accteptance Date: May 27, 2021

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Authors

Ümit Çakan - Department of Mathematics, İnönü University, Malatya, Turkey.

Abstract

This paper concerns a nonlinear integral equation modeling the spread of epidemics in which immunity does not occur after recovery. The model is mainly based on the return of some of the individuals who have been exposed to the pathogen and who have completed the incubation period, into the susceptible class. We first prove the uniqueness of the global solution of the model with the given initial conditions. After determining the positively invariant region for the model, using LaSalle invariance principle [J. P. LaSalle, IRE Trans. CT, \({\bf 7} (1960)\), 520--527] and the concept of persistence we present some results about the stability analysis of the solutions according to the case of the reproduction number \(\mathcal{R}_{0}\) which is a vital threshold in the spread of diseases.

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Keywords

• Global stability analysis
• Lyapunov function
• LaSalle invariance principle
• mathematical epidemiology
• persistence

MSC

•  34D05
•  34D08
•  34D20
•  92B05
•  92D25
•  92D30

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