Modeling the dynamics of certain emerging epidemics

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In this study, two general nonlinear mathematical models was iscussed. The first work develops a fractional mathematical model with general incidence rate and time delay in application to COVID-19 in Algeria owing to the disease caused by new coronavirus pandemic that emerged in China in December 2019. Analytically, the well-posedness of this model is established and discussed. Using the theory of fractional order derivative, theequilibrium stability was analyzed. In order to support analytic results, numerical simulations were carried out to identify the factors that significantly affect the disease’s ability to spread Matlab software was used for the numerical simulations. In the second work, to explore the behavior of the solution where the incidence function is more general, a fractional Susceptible, Exposed, Infected and Recovered SEIRepidemicmodel has been presented, where the derivative is the sense of Caputo. After proving the basic proprieties of the solution, we use the next generation matrix approach to get the value of the fundamentalreproduction number noted R . We 0 will demonstrate that if R is smaller than one, then there exists a unique disease- 0 free equilibrium that is both locally asymptotically stable by using the theory tools of fractional calculus, but whenR > 1 heendemicequilibrium is locallyasymptotically 0 stable. Furthermore, using a suitable of Lyapunov function, we will prove the global stability of the healthy equilibrium and establish sufficient requirements for both equilibrium point. Finally, we provide some numerical simulations to demonstrate our main findings.

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