Quantitative analysis of a fractional order of the $ SEI_{c}\, I_{\eta} VR $ epidemic model with vaccination strategy

Abstract

<abstract><p>This work focused on studying the effect of vaccination rate $ \kappa $ on reducing the outbreak of infectious diseases, especially if the infected individuals do not have any symptoms. We employed the fractional order derivative in this study since it has a high degree of accuracy. Recently, a lot of scientists have been interested in fractional-order models. It is considered a modern direction in the mathematical modeling of epidemiology systems. Therefore, a fractional order of the SEIR epidemic model with two types of infected groups and vaccination strategy was formulated and investigated in this paper. The proposed model includes the following classes: susceptible $ \mathrm{S}(t) $, exposed $ \mathrm{E}(t) $, asymptomatic infected $ \mathrm{I_{c}}(t) $, symptomatic infected $ \mathrm{I_{\eta}}(t) $, vaccinated $ \mathrm{V}(t) $, and recovered $ \mathrm{R}(t) $. We began our study by creating the existence, non-negativity, and boundedness of the solutions of the proposed model. Moreover, we established the basic reproduction number $ \mathcal{R}_{0} $, that was used to examine the existence and stability of the equilibrium points for the presented model. By creating appropriate Lyapunov functions, we proved the global stability of the free-disease equilibrium point and endemic equilibrium point. We concluded that the free-disease equilibrium point is globally asymptotically stable (GAS) when $ \mathcal{R}_{0}\, \leq \, 1 $, while the endemic equilibrium point is GAS if $ \mathcal{R}_{0} > 1 $. Therefore, we indicated the increasing vaccination rate $ \kappa $ leads to reducing $ \mathcal{R}_0 $. These findings confirm the important role of vaccination rate $ \kappa $ in fighting the spread of infectious diseases. Moreover, the numerical simulations were introduced to validate theoretical results that are given in this work by applying the predictor-corrector PECE method of Adams-Bashforth-Moulton. Further more, the impact of the vaccination rate $ \kappa $ was explored numerically and we found that, as $ \kappa $ increases, the $ \mathcal{R}_{0} $ is decreased. This means the vaccine can be useful in reducing the spread of infectious diseases.</p></abstract>