Numerical Solution of Fractional Model of HIV-1 Infection in Framework of Different Fractional Derivatives

In this paper, we have extended the model of HIV-1 infection to the fractional mathematical model using Caputo-Fabrizio and Atangana-Baleanu fractional derivative operators. A detailed proof for the existence and the uniqueness of the solution of fractional mathematical model of HIV-1 infection in Atangana-Baleanu sense is presented. Numerical approach is used to find and study the behavior of the solution of the stated model using different derivative operators, and the graphical comparison between the solutions obtained for the Caputo-Fabrizio and the Atangana-Baleanu operator is presented to see which fractional derivative operator is more efficient.


Introduction
HIV stands for the Human Immunodeficiency Virus. This virus attacks a person's immune system. The immune system of a person's body works as a defence mechanism against infections such as bacteria and viruses.
All viruses proliferate by infecting the living cells of the body. HIV targets such immune system cells which are meant to defend the body. These cells are known as CD4 + T cells. CD4 + T cells are basically white blood cells that fight infection. CD4 + T cell count is a measure of immune function in a patient with HIV. It is an imperative determinant for the need for opportunistic infection (OI) prophylaxis. CD4 + T cell count is determined from blood as a part of laboratory monitoring for HIV infection.
When HIV takes over these cells, it transforms a cell into a virus factory. It makes the cell to produce thousands of copies of the virus, and these copies then infect the other CD4 + T cells. The infected cells do not function desirably and die early. The paucity of CD4 + T cells deteriorates the immune system and makes it challenging for the body to stay healthy.
CD4 + T cell count is generally measured after being diagnosed with HIV (at baseline). It is measured in every three to six months during the first two years until these cell counts in a patient's body increase above 300 cells/mm 3 ; else, it is measured in every twelve months. Most HIV patients can expect an average increase of about 50-100 cells/mm 3 . Patients who indulge in therapy with low CD4 + T cell count or at an older age may not experience the same increase in their CD4 + T cell count despite virologic suppression.
There are a lot of factors that affect these cell counts. Medication is one of the options to keep these T cell counts high.
The characteristics of Acquired Immunodeficiency Syndrome (AIDS) pathogenesis are progressive depletion of CD4 + T cell population in close association with progressive impairment of cellular immunity and increased susceptibility to opportunistic infections (OI). HIV was originally hypothesized to be a consequence of sluggish CD4 + T cell destruction. However, massive CD4 + memory T cell destruction now occurs at a primitive level of infection. In most individuals, the initial destruction is countered by CD4 + memory T cell regeneration that protects these T cell numbers and functions above the threshold associated with overt immunodeficiency (see [1,2]).
The mathematical model of HIV-1 infection of CD4 + T cells is given as follows here, UðtÞ represents the population with uninfected CD4 + T cells, IðtÞ represents the population with infected CD4 + T cells, and VðtÞ represents the density of virions in plasma at time "t", respectively. Further, ε is the rate of generation of CD4 + T cells, d is the natural death rate, α is the infected death rate of CD4 + T cells, s is the rate by which viruses make the cells dead, c is the rate of formation of virions viruses due to infected CD4 + T cells, and δ gives the death rate of viruses (see [3]). In the current work, we will be concerned with the fractional mathematical model of HIV-1 infection. Most biological systems have aftereffects or memory, so the modeling of such biological systems using fractional order derivatives have many advantages in which the effects like memory are neglected. It has been concluded that there is fractional order electrical conductance in the cell membrane of many biological organisms, and they are classified in groups of noninteger models. Thus, fractional derivatives, see also ( [4][5][6][7][8][9][10][11][12]) study the behavior of these biological models more efficiently. Fractional calculus have been successfully growing and giving fruitful and countless developments in fields of chemistry, physics, biochemistry, medicine, biology, etc. (see [13][14][15][16]). In this work, we will consider the fractional mathematical model of HIV-1 infection in sense of Caputo-Fabrizio derivative operator [17] and Atangana-Baleanu derivative operator [18]. A detailed proof for the existence and the uniqueness of the solution is presented. The numerical solutions are presented for the fractional mathematical model in sense of the Caputo-Fabrizio and the Atangana-Baleanu derivative operator. Further, the results obtained are compared graphically to see which fractional derivative operator gives the better result.
Definition 1 (see [17]). Let f be an integrable function on ℝ, t > 0, 0 < β < 1, the Caputo-Fabrizio fractional derivative of order β is defined as where CF 0 ζ β t represents the Caputo-Fabrizio fractional derivative of order β and NðβÞ is a normalization function and the following holds Nð0Þ = Nð1Þ = 1: Definition 2 (see [19]). Let f be an integrable function on ℝ, t > 0, 0 < β < 1, the Caputo-Fabrizio time fractional integral of order β is given as where NðβÞ is the normalization function and the following holds Nð0Þ = Nð1Þ = 1.
Definition 3 (see [18]). Let f be an integrable function on ℝ, let 0 < β < 1, then the Atangana-Baleanu fractional derivative is given as where ABC 0 ζ β t is the Atangana-Baleanu fractional derivative of order β in Caputo sense, E β is the Mittag-Leffler function, and NðβÞ is the normalization function such that Nð0Þ = N ð1Þ = 1.
Definition 4 (see [18]). Let f be an integrable function on ℝ, the fractional integral of Atangana-Baleanu fractional derivative of order β is given as Theorem 5 (see [18]). The fractional differential equation possesses a solution which is unique given as

Fractional HIV-1 Model in Caputo-Fabrizio Sense
The fractional mathematical model of HIV-1 infection in Caputo-Fabrizio sense is given as follows.

Derivation of Numerical Scheme in Sense of Caputo-Fabrizio Derivative.
To illustrate the method ( [20,21]), we contemplate the following general equation: where ηðtÞ stands for UðtÞ, IðtÞ, VðtÞ and gðt, ηðtÞÞ is an integrable function that stands for ε − dU − αUV, αUV − sI , and cI − δV, respectively. Using the fundamental theorem, the above equation can be written as At point t = t k+1 , for k = 0, 1, 2 ⋯ , equation (10) becomes At point t = t k , for k = 0, 1, 2 ⋯ , equation (10) becomes From the above two equations, we get Considering g(τ, ηðτÞ) through Lagrange polynomial interpolation, where ηðt i Þ is a function at time t i and ηðt i−1 Þ is a function at time t i−1 : Substituting the value of gðτ, ηðτÞÞ in equation (13), we get Substituting h = t i − t i−1 and on solving, we get Using the aforementioned concept of numerical scheme, the numerical scheme for the fractional model of HIV-1 infection in framework of Caputo-Fabrizio derivative operator is given as where where L 2 t, I ð Þ= αUV − s I, where

Fractional HIV-1 Model in Atangana-Baleanu Sense
The fractional mathematical model of HIV-1 infection in Atangana-Baleanu sense is given as follows.

Existence and Uniqueness of the Solution
Theorem 6. The kernels satisfy the Lipschitz condition and contractions if following hold: where k 1 = d + n 1 , k 2 = s, and k 3 = δ.
Proof. Consider the kernel Let U and U 1 be two functions, then Let n 1 = max t kVðtÞk, then where Hence, this proves Lipschitz's condition for L 1 ðt, UÞ, and if 0 < k 1 < 1, then this proves contraction for L 1 ðt, UÞ. Similarly, we can prove the result for the kernels L 2 ðt, IÞ and L 3 ð t, VÞ.
possesses a unique solution under the conditions that we are able to search t max which satisfies where k 1 = d + n 1 , k 2 = s, and k 3 = δ.
Proof. We prove the result for equation (28).
Then, equation (32) can be written as Using Theorem 5, we get

Journal of Function Spaces
Let J = ð0, TÞ and define an operator Z : CðJ, So equation (34) can be seen as Z½UðtÞ = UðtÞ. Define the supremum norm on J as kUk = Sup t∈J jUðtÞj. Then, CðJ, ℝ 3 Þ and k:k define a Banach Space. Finally, consider Taking modulus on both sides of equation (36) and using triangle inequality, we get Lastly, using the fact that the kernel L 1 ðt, UðtÞÞ satisfies Lipschitz condition, we get Hence, using the Banach Fixed Point theorem, we govern the existence of a unique solution for the fractional model of HIV-1 infection in framework of Atangana-Baleanu derivative operator.

Derivation of Numerical Scheme in Sense of ABC
Derivative. Toufik and Atangana ([21,22]) introduced a numerical scheme for solving fractional derivatives having nonsingular and nonlocal kernel. To illustrate the method, contemplate the following fractional differential equation: Using Theorem 5, the above equation can be written as At point t = t k+1 , for k = 0, 1, 2 ⋯ , equation (41) becomes Considering gðτ, ηðτÞÞ through Lagrange polynomial interpolation, Substituting h = t i − t i−1 and on solving, we get Using the aforementioned concept of numerical scheme, the numerical scheme for the fractional model of HIV-1 infection in framework of Atangana-Baleanu derivative operator is given as where