In this paper, we study the stability of a fractional order SIRS epidemic model with nonlinear incidence rate and time delay, where the fractional derivative is defined in the Caputo sense. The delay is introduced into the model in order to modeled the incubation period. Using the stability analysis of delayed fractional order systems, we prove that the disease-free equilibrium is locally asymptotically stable when the basic reproduction number R 0 < 1. Also, we show that if R 0 > 1, the endemic equilibrium is locally asymptotically stable.
Proceeding from the fact that fractional systems can better characterize the virological properties than the ordinary formulation, in the present study, we treat a Caputo fractional order viral formulation under some interesting assumptions. Our model incorporates the time delay hypothesis as well as the non-cytolytic immune mechanism and inhibition of viral replication. Analytically, we show that our enhanced delayed viral model exhibits the following three equilibria: virus-clear steady point D, immune-free steady state D?1, and immunity-activated steady point with the humoral feedback D?2. By determining two critical values S and S1, the asymptotic stability of all said steady points is examined and the dynamical bifurcation associated with time delay is also explored. This theoretical arsenal provides an excellent insight into the long-run behavior of the infection. Numerically, we check the reliability of our results by highlighting the influence of fractional derivatives and time lags on the stability of steady points. We mention that our work enrich and generalize the work of Dhar et al. [11] by considering a general hypothetical setting.
In this paper, we consider an
SEIS
epidemic model with infectious force in latent and infected period, which incorporates by nonlinear incidence rates. The local stability of the equilibria is discussed. By means of
Lyapunov
functionals and
LaSalle’s
invariance principle, we proved the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium. An application is given and numerical simulation results based on real data of COVID-19 in Morocco are performed to justify theoretical findings.
This article studies the controllability and observability of nonlinear positive discrete systems. These properties play a fundamental role in system analysis before controller and observer design is engaged. We solve these problems by a technique based on the fixed point theory.
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