Stability Analysis of a Fractional Epidemic Model Involving the Vaccination Effect

This paper, by constructing a fractional epidemic model, analyzes the transmission dynamics of some infectious diseases under the effect of vaccination, which is one of the most effective and common control measures. In the model, considering that antibody formation by vaccination may not cause permanent immunity, it has been taken into account that the protection period provided by the vaccine may be finite, in addition to the fact that this period may change according to individuals. The model differs from other SVIR models given in the literature in its progressive process with a distributed delay in the loss of the protective effect provided by the vaccine. To explain this process, the model was constructed by using a system of distributed delay nonlinear fractional integro-differential equations. Thus, the model aims to present a realistic approach to following the course of the disease. Additionally, an analysis was conducted regarding the minimum vaccination ratio of new members required for the elimination of the disease in the population by using the vaccine free basic reproduction number (R0vf). After providing examples for the selection of the distribution function, the variation of R0 was simulated for a specific selection of parameters in the model. Finally, the sensitivity indices of the parameters affecting R0 were calculated, and this situation is been visually supported.