Quantitative analysis of a fractional order of the SEIc IηVR SEI_{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 $\mathrm{I_{c}}(t)$, 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

Global Behavior of a Computer Virus Propagation Model on Multilayer Networks

This paper presents a new linear computer viruses propagation model on multilayer networks to explore the mechanism of computer virus propagation. Theoretical analysis demonstrates that the maximum eigenvalue of the sum of all the subnetworks is a vital factor in determining the viral prevalence. And then, a new sufficient condition for the global stability of virus-free equilibrium has been obtained. The persistence of computer virus propagation system has also been proved. Eventually, some numerical simulation results verify the main conclusions of the theoretical analysis