Quantitative analysis of a fractional order of the $SE{I}_c{I}_{\eta }VR$ epidemic model with vaccination strategy

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 ${\mathrm{I}}_{\mathrm{c}}(t)$, 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 ${\mathcal{R}}_0\le 1$, 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.