Dynamics of a Fractional Order HIV Infection Model with Specific Functional Response and Cure Rate

Received Jun 11, 2017; Accepted Jul 31, 2017

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Fractional order differential equations (FDEs) are a generalization of ordinary differential equations (ODEs) and they have many applications in various fields such as mechanics, image processing, viscoelasticity, bioengineering, finance, psychology, and control theory [1–7]. In addition, it has been deduced that the membranes of cells of biological organisms have fractional order electrical conductance [8].

Modeling by FDEs has more advantages to describe the dynamics of phenomena with memory which exists in most biological systems, because fractional order derivatives depend not only on local conditions but also on the past. More precisely, calculating the time-fractional derivative of a function f(t) at some time t=t1 requires all the previous history, that is, all f(t) from t=0 to t=t1. In addition, the region of stability of FDEs is larger than that of ODEs. Moreover, some previous study compared between the results of the fractional order model, the results of the integer model, and the measured real data obtained from 10 patients during primary HIV infection [9]. This study proved that the results of the fractional order model give predictions to the plasma virus load of the patients better than those of the integer model.

From the above biological and mathematical reasons, we propose a fractional order model to describe the dynamics of HIV infection that is given by (1)DαTt=λ-dT-βTV1+α1T+α2V+α3TV+ρI,DαIt=βTV1+α1T+α