The impact of dual time delay and Caputo fractional derivative on the long-run behavior of a viral system with the non-cytolytic immune hypothesis

Abstract

Abstract 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  ◦ , immune-free steady state  1 ⋆ , and immunity-activated steady point with the humoral feedback  2 ⋆ . By determining two critical values  ◦ and  1 , 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.