Local Asymptotic Stability and Sensitivity Analysis of a New Mathematical Epidemic Model Without Immunity

Year 2022, Volume: 10 Issue: 1, 50 - 62, 01.03.2022

Sümeyye Çakan

https://doi.org/10.36753/mathenot.935016

Abstract

With this study it is aimed to introduce and analyze a new SIS epidemic model including vaccination
effect. Vaccination considered in the model provides a temporary protection effect and is administered
to both susceptible and new members of the population. The study provides a different aspect to the
SIS models used to express, mathematically, some infectious diseases which are not eradicated by the
immune system. The model given this study is designed by considering varying processes from person
to person in the disease transmission, the recovery from disease (recovery without immunity) and in the
loss of protective effect provided by the vaccine. The processes that change according to individuals are
explained by distributed delays used in the relevant differential equations that provide the transition
between compartments. The differences in the model are especially evident in these parts. In analyzing
the model, firstly, the disease-free and endemic equilibrium points related to the model are determined.
Then, the basic reproduction number R₀ is calculated with the next generation matrix method. Next, the
dynamics about locally asymptotically stable of the model at the disease-free and endemic equilibriums are examined according to the basic reproduction number R₀. Attempts intended to reduce the spread of the disease are, of course, in the direction supporting the lowering the value R0. In this context, the reducing and enhancing effects of the parameters used in the model on the value R₀ have been interpreted mathematically and suggestions were made to implement control measures in this direction. Also, in order to evaluate the support provided by the vaccine during the spread of the disease, the model has been examined as vaccinated and unvaccinated, and by some mathematical process, it has been seen that the vaccination has a crucial effect on disease control by decreasing the basic reproduction number. In other respects, by explored that the effect of parameters related to vaccination on the change of R₀, a result about the minimum vaccination ratio of new members required for the elimination of the disease in the population within the scope of the target of R₀<1 has been obtained.

Keywords

Local Asymptotic Stability, Sensitivity Analysis, SIS model, Vaccine Effect, Disease-free equilibrium point, Endemic equilibrium point, Basic Reproduction Number

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