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

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.