Threshold and Stability Results of a New Mathematical Model for Infectious Diseases Having Effective Preventive Vaccine

Year 2021, , 56 - 64, 31.08.2021

Sümeyye Çakan

https://doi.org/10.33187/jmsm.884304

Abstract

This paper evaluates the impact of an effective preventive vaccine on the control of some infectious diseases by using a new deterministic mathematical model. The model is based on the fact that the immunity acquired by a fully effective vaccination is permanent. Threshold $\mathcal{R}_{0}$, defined as the basic reproduction number, is critical indicator in the extinction or spread of any disease in any population, and so it has a very important role for the course of the disease that caused to an epidemic. In epidemic models, it is expected that the disease becomes extinct in the population if $\mathcal{R}_{0}<1.$ In addition, when $\mathcal{R}_{0}<1$ it is expected that the disease-free equilibrium point of the model, and so the model, is stable in the sense of local and global. In this context, the threshold value $\mathcal{R}_{0}$ regarding the model is obtained. The local asymptotic stability of the disease-free equilibrium is examined with analyzing the corresponding characteristic equation. Then, by proved the global attractivity of disease-free equilibrium, it is shown that this equilibria is globally asymptotically stable.

Keywords

Vaccine Effect, Diseasefree Equilibrium Point, Basic Reproduction Number, Local Asymptotic Stability, Global Attractivity, Global Asymptotic Stability

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