Stability Analysis for a Fractional HIV Infection Model with Nonlinear Incidence

Abstract

We introduce the fractional-order derivatives into an HIV infection model with nonlinear incidence and show that the established model in this paper possesses nonnegative solution, as desired in any population dynamics. We also deal with the stability of the infection-free equilibrium, the immune-absence equilibrium, and the immune-presence equilibrium. Numerical simulations are carried out to illustrate the results.

Effects of randomness on viral infection model with application

“…We also observed when we simulated the system (22) it seems much effective in comparison with simulation results carried out with Fractional Order as it gives stability conditions in stronger sense. This statement is validated with the simulation results gained in Huang et al in Zhang et al (2015) showed that the with the parameter values λ = 23.3, β = 0.5, a = 0.02, p = 10, q = 0.79, d = 0.09, b = 0.15, c = 0.0031 even with higher efficacy rates of the Lytic and Non-Lytic Immune responses the basic reproduction numbers for both immune absence equilibrium R 0 = 31.4916 > 1 and immune presence equilibrium R 1 = 30.2250 > 1. The graphs in the paper (Zhang et al, 2015) with the above values evidently shows that the endemic equilibrium stabilizes only after 150 days for smaller order of fractions α = 0.56, α = 0.65.…”

“…This statement is validated with the simulation results gained in Huang et al in Zhang et al (2015) showed that the with the parameter values λ = 23.3, β = 0.5, a = 0.02, p = 10, q = 0.79, d = 0.09, b = 0.15, c = 0.0031 even with higher efficacy rates of the Lytic and Non-Lytic Immune responses the basic reproduction numbers for both immune absence equilibrium R 0 = 31.4916 > 1 and immune presence equilibrium R 1 = 30.2250 > 1. The graphs in the paper (Zhang et al, 2015) with the above values evidently shows that the endemic equilibrium stabilizes only after 150 days for smaller order of fractions α = 0.56, α = 0.65. If we look at the empirical study carried out in our paper, it is evident even a smaller rate of efficacies of Lytic and Non-Lytic components, the stabilization is at a much faster rate as our parameter values are nearly closer to those values in Zhang et al (2015).…”

“…Next, we conclude our works in this paper with some discussions based on the results observed. In the mean time, a spark flashed to draw an comparative study between the results observed by Huang et al in Zhang et al (2015) when we surveyed through the simulation results. We found that even though both methods (Fractional Differential Equations (FDE's) and Stochastic Differential Equations (SDE's)) could discover conditions for stability at equilibrium solutions the method by SDE's seems to emphasis early detection of infection outbreak in the host than FDE's.…”

“…The parameter values are taken from the paper cited in Zhang et al (2015) to draw a comparative analysis of the model under different kinds of simulation (see Figs. 8 and 9).…”

“…Modeling the viral infections in vivo has caught the attention of scientists in recent years with the discoveries of many new members in the family of viral diseases like SAARS (Severe Acute Respiratory Syndrome) despite the presence of senior citizens (AIDS -Acquired Immune Deficiency Syndrome, conjunctivitis etc.). The different approaches adopted by mathematical biologists like Huang et al in Huang, Yokoi, Takeuchi, Kajiwara, andSasaki (2011), Herz et.al in Herz, Bonhoeffer, Anderson, May, and tried their hand with Delay Differential Equations and Fractional Differential Equations in Zhang, Huang, Liu, and Fan (2015) with regard to intracellular delay in viral dynamics and HIV infection Models with Non-linear incidence rates respectively while M. Pitchaimani et al in Pitchaimani and Monica (2015) focussed on HIV-1 infection model with three time delays. Also, Pitchaimani and Rajaji contributed works in Stochastic Nowak -May Model analysis as in Pitchaimani and Rajaji (2016).…”