Highlights We propose to study an optimal control approach with delay in state and control variables. Numerical simulation of different strategies. The cost of effectivness.

This article deals with a Caputo fractional-order viral model that incorporates the non-cytolytic immune hypothesis and the mechanism of viral replication inhibition. Firstly, we establish the existence, uniqueness, non-negativity, and boundedness of the solutions of the proposed viral model. Then, we point out that our model has the following three equilibrium points: equilibrium point without virus, equilibrium state without immune system, and equilibrium point activated by immunity with humoral feedback. By presenting two critical quantities, the asymptotic stability of all said steady points is examined. Finally, we examine the finesse of our results by highlighting the impact of fractional derivatives on the stability of the corresponding steady points.

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.

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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.