A FRACTIONAL APPROACH TO SOLVE A MATHEMATICAL MODEL OF HIV INFECTION OF CD4+T CELLS

A mathematical model that calculates susceptible CD4+T cells, infected CD4+T cells and virus particles has been examined here using the fractional differential transform method (FDTM) with stability analysis. A stability of the fractional nonlinear model with Hurwitz state matrix is examined using the Lyapunov direct method. A nonlinear mathematical model of differential equations has been put forward and analyzed by applying FDTM. An infinite series solution of the system of differential equation is computed by defining fixed components with different time intervals. Furthermore, the solution calculated through FDTM (integer order) is correlated with the solution calculated using DTM and LADM. The solution is analyzed numerically and graphically by using the software Python.


INTRODUCTION
Human immunodeficiency virus (HIV) infection is a disease which impacts on the immune system of human being caused by the HIV virus(avert.org). HIV corrupts the white blood cells in the immune system when it enters into the body. HIV can not reproduce by its own. The life cycle of HIV goes through the different steps taken by the virus in which it multiplies and make the copies of itself. First, on arriving into the human body the virus gets attached itself to the Thelper cells (also called CD4 + T cells) and gets blended with it. Virus starts regulated the original cells and takes the control of its DNA. The virus starts created copies of itself and discharges more HIV into the blood. This process of multiplying and discharging of virus into the blood never gets stopped and termed as the HIV life cycle.A person can get HIV infection by coming into contact with the bodily fluid like blood, semen,breast milk of someone living with the virus.
HIV can also be transmitted during unprotected sex, through sharing surgical equipment,from mother to baby during pregnancy,at the time of birth and breastfeeding, it also enters into hu- A fractional differentiation in Caputo sense is more applicable to real world problems. Semi analytical numerical technique that is developed to study fractional power series in a same way Differential Transform Method does for Taylor series. A mathematical model of HIV infection of CD4 + T cells given in [1] is reviewed here for finding the solution of it using Fractional Differential Transform Method.The system of differential equation is given as follows; With the initial conditions: Where the terms denote s(t) −→ Concentration of susceptible CD4 + T cells.
i(t) −→ Concentration of CD4 + T cells infected by HIV virus.
v(t) −→ Free HIV virus particles presented in the blood.

PRELIMINARIES
The fractional differential transform method is used in this paper to obtain the solution of mathematical model of HIV Infection of CD4 + T cells. This method has been developed in [3] as follows. The fractional differentiation in Riemann-Liouville sense is defined as where m − 1 ≤ q < m, m is an positive integer and x > x 0 . The expansion of analytic and continuous function f (x) in terms of fractional power series is given as: where λ is the order of the fraction and F(k) is the fractional differential transform of f (x). The relation between the Riemann-Liouville operator and Caputo operator is given as follows; * is used for the Caputo operator to distinguish between Riemann-Liouville operator and Ca- We obtain fractional derivative in the Caputo sense as follows; Since the initial conditions are implemented to the integer order differential equation, the transformed initial conditions can be obtained using the below relation; Where k = 0, 1, 2, − − −, qλ − 1 and q is the order of the FDE. λ is to be chosen such that qλ is a positive integer. The following theorems can be proved using equations 3 and 4.For more detail refer [3].

Lyapunov Stability Analysis of Fractional Nonlinear System:
The stability analysis of the proposed model is studied using Lypunov function in Caputo sense [13].The main purpose is to discuss asymptotic stability of the model. Before approaching to the methodology we will consider few statements of the theorems which will be required for the analysis. Consider the where µ ∈ (0, 1),x ∈ R n represents the state vector of the system, A ∈ R n×n is a constant matrix and a : R + × R n −→ R n is a nonlinear function.
Let that the state matrix A is Hurwitz and the condition a(t, x(t)) < ρ x(t) holds. If there exists a positive definite matrix P such that the following inequality holds ρ < λ min (Q) 2λ max (P) where A T P + PA = −Q, then the trivial solution of the fractional linear system is asymptotically stable.

Theorem 3.3.[13]
If a function a(t, x(t)) is an one -sided Lipschitz and quadratically inner bounded with constants ρ, σ and ω. Assume that there exists two matrices P and Q which verifies: Then the origin of the system is asymptotically stable.

Theorem 3.4.[13]
If the function a(t, x(t)) of the system D α t 0 x(t) = f (t, x(t)) = Ax(t) + Bu + a(t, x(t)) with x(t 0 ) = x 0 is an one -sided Lipschitz and quadratically inner bounded with constants ρ, σ and ω. Assume that there exists a positive symmetric matrix P, a constant matrix K ∈ R p×n and positive constant ε such that; (A + BK) T P + P(A + BK) = −εI [(σ + 1) + ρ(ω + 2)]λ max (P) < λ min (P) + ε, Where B ∈ R n×p is a constant matrix and u ∈ R p is the control input to be defined. Then the control law u(x) = Kx render the above system is fractional asymptotically stable.

Writing the proposed model in the form
Hence we can say the proposed model is fractional asymptotically stable with the defined parameters.

Methodology of the Proposed Model:
Rewriting equation (1) taking fractional order µ.

CONCLUSIONS
A stability of the fractional nonlinear model with Hurwitz state matrix is examined using the Lyapunov direct method. This analysis plays a vital role to describe the behavior of physical systems when modeled by means of FDEs. Lyapunov direct method provides a very effective way to analyze stability of nonlinear systems.In this paper, with stabilty a fractional approach is also used to solve the system of ordinary differential equations. A nonlinear mathematical

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.