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

Authors : Sümeyye ÇAKAN

Pages : 56-64

Doi:10.33187/jmsm.884304

View : 22 | Download : 15

Publication Date : 2021-08-31

Article Type : Research Paper

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