System Analysis of HIV Infection Model with CD4+T under Non-Singular Kernel Derivative

  • 1 Balıkesir University, Balıkesir, Turkey
Fırat Evirgen, Sümeyra Uçar and Necati Özdemir

Abstract

Infectious diseases have caused the death of many people throughout the world for centuries. For this purpose, many researchers have investigated these diseases for establishing new treatment and protective measures. The most important of these is HIV disease. In this study, an HIV infection model of CD4+T cells is handled comprehensively with the newly defined Atangana-Baleanu (AB) fractional derivative. The existence and uniqueness of the solutions for fractionalized HIV disease model with the new derivative by considering the Arzela-Ascoli theorem.

1 Introduction

Over the past 50 years, the Human Immunodeficiency Virus (HIV) has become a lethal disease that affects the world in a global sense. HIV is a virus that generates infection by targeting immune system cells and causes fatal consequences if infection progresses. According to the researches conducted by the World Health Organization (WHO) worldwide, approximately 35 million people were affected by this disease, 940000 people died from HIV-related causes at the end of 2017. These statistics are inevitably increasing in spite of all the precaution taken over the years.

For this reason, many scientists worldwide are doing various researches in order to prevent such deadly diseases. Mathematical modeling is the most important part of these researches. With the help of mathematical models, researchers obtain very crucial information about the spread of diseases and measures to be taken [1, 2, 3].

On the other hand, fractional calculus has become a very important tool in mathematical modeling as in many other fields in recent years [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20].

Furthermore, there are many studies in the literature, which are modeled by the fractional concept related to diseases and bad habits [21, 22, 23, 24, 25, 26, 27, 28].

In this paper, by the above motivation, we will handle an HIV infection model of CD4+T cells, considered in [1], for investigating the system components under the effect of non-singular kernel derivative which is defined in [12].

For this purpose, the rest of the paper is divided into 4 Sections. In Section 2, some basic necessary definitions and theorem for Atangana-Baleanu (AB) derivative, which is also named as non-singular Mittag-Leffler kernel derivative, are given. Section 3 is devoted to the model construction. In Section 4, the existence and uniqueness of the solutions for the HIV infection model in the frame of the AB fractional derivative are given. Finally, in Section 5, our results are briefly summarized.

2 Basic definitions and preliminaries

Definition 1

Let f ∈ H1 (a,b), a < b be a function and υ ∈ [0,1]. The Atangana-Baleanu derivative in Caputo type of order υ of f is given by [12]:

aABCDtυ[f(t)]=B(υ)1υatf'(x)Eυ[υ(tx)υ1υ]dx.

Definition 2

Let g ∈ H1 (a,b), a < b be a function and υ ∈ [0,1]. The Atangana-Baleanu derivative in Riemann-Liouville type of order υ of f is defined by [12]:

aABRDtυ[f(t)]=B(υ)1υddtatg(x)Eυ[υ(tx)υ1υ]dx.

Definition 3

The fractional integral related to the fractional derivative is given by [12]:

aABItυ[f(t)]=1υB(υ)f(t)+υB(υ)Γ(υ)atf(λ)(tλ)υ1dλ.

3 Model description

In this section, the fractional order HIV infection model of CD4+T cells by the AB fractional derivative is constructed according to the reference in [1].

This system can be modeled by the three fractional order nonlinear differential equations as follow:

0ABCDtυ(T(t))=qυ¯T(t)+rT(t)(1T(t)+I(t)Tmax)kV(t)T(t),0ABCDtυ(I(t))=kV(t)T(t)βI(t),0ABCDtυ(V(t))=μβI(t)γV(t),
where, T (t) is the number of healty CD4+T cells at time t, I (t) is the concentration of infected CD4+T cells by the HIV viruses at time t, V (t) is the number of HIV viruses at time t and υ¯, r, Tmax, k, β, µ and γ are the estimated data from real world applications.

4 Existence and uniqueness of the solutions

In this section, the existence and uniqueness of the solutions to the model (4) is described and proved under the effect of AB fractional derivative.

For this purpose, first of all, we apply AB fractional integral in Eq. (3) on both sides of the system and we get

T(t)T(0)=1υB(υ){qυ¯T(t)+rT(t)(1T(t)+I(t)Tmax)kV(t)T(t)}+υB(υ)Γ(υ)0t(ty)υ1{qυ¯T(y)+rT(y)(1T(y)+I(y)Tmax)kV(y)T(y)}dy,I(t)I(0)=1υB(υ){kV(t)T(t)βI(t)}+υB(υ)Γ(υ)0t(ty)υ1{kV(y)T(y)βI(y)}dy,V(t)V(0)=1υB(υ){μβI(t)γV(t)}+υB(υ)Γ(υ)0t(ty)υ1{μβI(y)γV(y)}dy.

We can choose our kernels s(t,T (t)), s(t,I (t)), s(t,V (t)) for simplifying the demonstration of the equation (5) as follows,

s(t,T(t))=qυ¯T(t)+rT(t)(1T(t)+I(t)Tmax)kV(t)T(t),s(t,I(t))=kV(t)T(t)βI(t),s(t,V(t))=μβI(t)γV(t).

At first, we need to be able to identify an operator. We will then show that this operator is compact. Now, we consider the operator K : H → H and then we get

KT(t)=1υB(υ)s(t,T(t))+υB(υ)Γ(υ)0t(ty)υ1s(y,T(y))dy,KI(t)=1υB(υ)s(t,I(t))+υB(υ)Γ(υ)0t(ty)υ1s(y,I(y))dy,KV(t)=1υB(υ)s(t,V(t))+υB(υ)Γ(υ)0t(ty)υ1s(y,V(y))dy.

Lemma 1

Let M ⊂ H be bounded in this way, we can find n,r,l > 0 such that

T(t2)T(t1)nt2t1foreveryTM,I(t2)I(t1)rt2t1foreveryIM,V(t2)V(t1)lt2t1foreveryVM.
ThenK(M)¯is compact.

Proof

Let N=max{1υB(υ)+s(t,T(t))} and 0 ≤ T (t) ≤ P. For T (t) ∈ M, then we have the followings

KT(t)=1υB(υ)s(t,T(t))+υB(υ)Γ(υ)0t(ty)υ1s(y,T(y))dy1υB(υ)N+υB(υ)NtυΓ(υ+1).

Let R=max{1υB(υ)+s(t,I(t))} and 0 ≤ I (t) ≤ Q. For I (t) ∈ M

KI(t)=1υB(υ)s(t,I(t))+υB(υ)Γ(υ)0t(ty)υ1s(y,I(y))dy1υB(υ)R+υB(υ)RtυΓ(υ+1).

In a similar way, S=max{1υB(υ)+s(t,V(t))} and 0 ≤ V (t) ≤ S. For V (t) ∈ M

KV(t)=1υB(υ)s(t,V(t))+υB(υ)Γ(υ)0t(ty)υ1s(y,V(y))dy1υB(υ)S+υB(υ)StυΓ(υ+1).
From the Eqs. (7)(9), the function K is bounded.

Now, we will consider t1< t2 and T (t) ∈ M. For ɛ > 0, if |t2t1| < δ then we obtain

KT(t2)KT(t1)=1υB(υ)s(t2,T(t2))+υB(υ)Γ(υ)0t2(t2y)υ1s(y,T(y))dy1υB(υ)s(t1,T(t1))υB(υ)Γ(υ)0t1(t1y)υ1s(y,T(y))dy1υB(υ)s(t2,T(t2))s(t1,T(t1))+υB(υ)Γ(υ)0t2(t2y)υ1s(y,T(y))dyυB(υ)Γ(υ)0t1(t1y)υ1s(y,T(y))dy1υB(υ)s(t2,T(t2))s(t1,T(t1))+υPB(υ)Γ(υ){0t2(t2y)υ1dy0t1(t1y)υ1},
and then
0t2(t2y)υ1dy0t1(t1y)υ1dy=0t1{(t1y)υ1(t2y)υ1}dy+t1t2(t2y)υ1dy=2(t2t1)υυ.
Now, we will investigate the following:
s(t2,T(t2))s(t1,T(t1))T(t2)T(t1)(υ¯+ra+k(b+c))F1nt2t1At2t1.
Putting Eqs. (11) and (12) in Eq. (10), we have
KT(t2)KT(t1)1υB(υ)At2t1+υPB(υ)Γ(υ)2t2t1υυ1υB(υ)At2t1+2υPB(υ)Γ(υ+1)t2t1.
Let δ1=ε1υB(υ)A+2υPB(υ)Γ(υ+1) and then we find
KT(t2)KT(t1)<ε.

With the same rule step by step, we can obtain the following for other two equations. For each ɛ > 0, we can find δ2=ε1υB(υ)B+2υQB(υ)Γ(υ+1), δ3=ε1υB(υ)C+2υVB(υ)Γ(υ+1) such that

KI(t2)KI(t1)1υB(υ)Bt2t1+2υQB(υ)Γ(υ+1)t2t1
and
KV(t2)KV(t1)1υB(υ)Ct2t1+2υVB(υ)Γ(υ+1)t2t1.

Consequently,

KI(t2)KI(t1)<εandKV(t2)KV(t1)<ε
are satisfied. So T (M) is equicontinuous and using Arzelo-Ascoli Theorem, T(M)¯ is compact.

Theorem 2

S : [a,b] × [0,∞) → [0,∞) be a continuous function and S (t,.) increasing for each t in [a,b]. Let us assume that one can find v,w satisfying M (D)v ≤ S (t,v), M (D)w ≥ S (t,w) for 0 ≤ v(t) ≤ w(t) and a ≤ t ≤ b. Then our new equation has a positive solution.

Proof

We should consider the fixed point of the operator K. We know that the operator K : H → H is completely continuous. Let T1≤ T2, I1≤ I2 and V1≤ V2 then we get

KT1(t)1υB(υ)s(t,T1(t))+υB(υ)Γ(υ)0t(ty)υ1s(y,T1(y))dyKT2(t).

By a similar way, we obtain

KI1(t)1υB(υ)s(t,I1(t))+υB(υ)Γ(υ)0t(ty)υ1s(y,I1(y))dyKI2(t),
and
KV1(t)1υB(υ)s(t,V1(t))+υB(υ)Γ(υ)0t(ty)υ1s(y,V1(y))dyKV2(t).

Hence K is increasing operator. By the help of conjecture, we have Km ≥ m and Kn ≤ n. So, the operator K : 〈n,m〉 → 〈n,m〉 is compact and continuous from Lemma 1. In that case, H is a normal cone.

To obtain uniqueness of the solutions, we begin the following steps.

KT1(t)KT2(t)1υB(υ)s(t,T1(t))s(t,T2(t))+υB(υ)Γ(υ)0t(ty)υ1s(y,T1(y))s(y,T2(y))dy1υB(υ)F1T1(t)T2(t)+υB(υ)Γ(υ)F10t(ty)υ1T1(y)T2(y)dy,
which yields
KT1(t)KT2(t){1υB(υ)F1+υF1bυB(υ)Γ(υ+1)}T1(t)T2(t).

For the other components of the model, we find

KI1(t)KI2(t){1υB(υ)F2+υF2bυB(υ)Γ(υ+1)}I1(t)I2(t),
and
KV1(t)KV2(t){1υB(υ)F3+υF3cυB(υ)Γ(υ+1)}V1(t)V2(t).

Therefore, if the following conditions hold

1υB(υ)F1+υF1bυB(υ)Γ(υ+1)<1,1υB(υ)F2+υF2bυB(υ)Γ(υ+1)<1,1υB(υ)F3+υF3cυB(υ)Γ(υ+1)<1,
the mapping K is a contraction, which implies fixed point and thus the model has a unique positive solution.

5 Concluding remarks

In this work, we have examined the system response of the HIV infection model in [1] by modeling the AB fractional derivative in Caputo sense. The theoretical studies have shown that the solution of the discussed model in Eq. (4) is exist and unique under the AB fractional derivative.

References

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    Badr Saad T Alkahtani, Abdon Atangana, Ilknur Koca, (2016), A new nonlinear triadic model of predator rey based on derivative with non-local and non-singular kernel, Advances in Mechanical Engineering, Vol:8, No:11.

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    Abdon Atangana, Dumitru Baleanu, (2016), New fractional derivatives with non-local and non-singular kernel: theory and applications to heat transfer model, Thermal Science, Vol:20, No:2, 763–769.

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    Necati Özdemir, Mehmet Yavuz, (2017), Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Pade Approximation, Acta Physica Polonica A, Vol:132, 1050–1053.

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    Toufik Mekkaoui, Abdon Atangana, (2017), New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, European Physical Journal Plus, Vol:132, No:10.

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    Zakia Hammouch, Toufik Mekkaoui, (2018), Circuit design and simulation for the fractional-order chaotic behavior in a new dynamical system, Complex & Intelligent Systems, Vol:4, No:4, 251–260.

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    Jagdev Singh, Devendra Kumar, Zakia, Hammouch, Abdon Atangana, (2018), A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation, Vol:316, 504–515.

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    Mehmet Yavuz, Necati Özdemir, Haci Mehmet Baskonus, (2018), Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, The European Physical Journal Plus, Vol:133, No:6.

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    Asıf Yokuş, Sema Gülbahar, (2019), Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation, Applied Mathematics and Nonlinear Sciences, Vol:4, No:1, 35–42.

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    Sümeyra Uçar, Nihal Yılmaz Özgür, Beyza Billur İskender Eroğlu, (2019), Complex conformable derivative, Arabian Journal of Geosciences, Vol:12, No:6, 201.

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    Abdon Atangana, Ernestine Alabaraoye, (2013), Solving a system of fractional partial differential equations arising in the model of HIV infection of CD4+ cells and attractor one-dimensional Keller-Segel equations, Advances in Difference Equations, 2013:94.

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    Ana RM Carvalho, Carla MA Pinto, Dumitru Baleanu, (2018), HIV/HCV coinfection model: a fractional-order perspective for the effect of the HIV viral load, Advances in Difference Equations, 2018:2.

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    İlknur Koca, (2018), Analysis of rubella disease model with non-local and non-singular fractional derivatives, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), Vol:8, No:1, 17–25.

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    Ebenezer Bonyah, J.F. Gomez-Aguilar, Augustina Adu, (2018), Stability analysis and optimal control of a fractional human African trypanosomiasis model, Chaos, Solitons & Fractals, Vol:117, 150–160.

  • [25]

    V.F. Morales-Delgado, J.F. Gomez-Aguilar, M.A. Taneco-Hernandez, R.F. Escobar-Jimenez, V.H. Olivares-Peregrino, (2018), Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel, Journal of Nonlinear Sciences Applications, Vol:11, No:8, 994–1014.

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    Amin Jajarmi, Dumitru Baleanu, (2018), A new fractional analysis on the interaction of HIV with CD4+ T-cells, Chaos, Solitons & Fractals, Vol:113, 221–229.

  • [27]

    Sümeyra Uçar, Esmehan Uçar, Necati Özdemir, Zakia Hammouch, (2019), Mathematical analysis and numerical simulation for a smoking model with Atangana Baleanu derivative, Chaos, Solitons & Fractals, Vol:118, 300–306.

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    Esmehan Uçar, Necati Özdemir, Eren Altun, (2019), Fractional order model of immune cells influenced by cancer cells, Mathematical Modelling of Natural Phenomena, Vol:14, No:3, 12p.

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  • [1]

    Şuayip Yüzbaşı, (2016), An exponential collocation method for the solutions of the HIV infection model of CD4+ T cells, International Journal of Biomathematics, Vol:9, No:3, 1650036.

  • [2]

    Şuayip Yüzbaşı, Murat Karaçayır, (2017), An exponential Galerkin method for solutions of HIV infection model of CD4+ T-cells, Computational Biology and Chemistry, Vol:67, 205–212.

  • [3]

    Swarnali Sharma, G.P. Samanta, (2016), Analysis of the dynamics of a tumor immune system with chemotherapy and immunotherapy and quadratic optimal control, Differential Equations and Dynamical Systems, Vol:24, No:2, 149–171.

  • [4]

    K.B. Oldham, J. Spanier, (1974), The Fractional Calculus. New York: Academic Press.

  • [5]

    A.A. Kilbas, H.M. Srivastava, Juan J. Trujillo, (2006), Theory and applications of fractional differential equations, Amsterdam: Elsevier.

  • [6]

    Necati Özdemir, Derya Karadeniz, Beyza Billur İskender, (2009), Fractional optimal control problem of a distributed system in cylindrical coordinates, Physics Letters A, Vol:373, No:2, 221–226.

  • [7]

    Fırat Evirgen, (2011), Multistage adomian decomposition method for solving NLP problems over a nonlinear fractional dynamical system, Journal of Computational and Nonlinear Dynamics, Vol:6.

  • [8]

    Dumitru Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, (2012), Fractional calculus models and numerical methods. Series on Complexity, Nonlinearity and Chaos. Boston: World Scientific.

  • [9]

    Fırat Evirgen, (2016), Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), Vol:6, No:2, 75–83.

  • [10]

    Abdon Atangana, İknur Koca, (2016), On the new fractional derivative and application to nonlinear Baggs and Freedman model, Journal of Nonlinear Sciences and Applications, Vol:9, No:5, 2467–2480.

  • [11]

    Badr Saad T Alkahtani, Abdon Atangana, Ilknur Koca, (2016), A new nonlinear triadic model of predator rey based on derivative with non-local and non-singular kernel, Advances in Mechanical Engineering, Vol:8, No:11.

  • [12]

    Abdon Atangana, Dumitru Baleanu, (2016), New fractional derivatives with non-local and non-singular kernel: theory and applications to heat transfer model, Thermal Science, Vol:20, No:2, 763–769.

  • [13]

    Necati Özdemir, Mehmet Yavuz, (2017), Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Pade Approximation, Acta Physica Polonica A, Vol:132, 1050–1053.

  • [14]

    Toufik Mekkaoui, Abdon Atangana, (2017), New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, European Physical Journal Plus, Vol:132, No:10.

  • [15]

    Zakia Hammouch, Toufik Mekkaoui, (2018), Circuit design and simulation for the fractional-order chaotic behavior in a new dynamical system, Complex & Intelligent Systems, Vol:4, No:4, 251–260.

  • [16]

    Jagdev Singh, Devendra Kumar, Zakia, Hammouch, Abdon Atangana, (2018), A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation, Vol:316, 504–515.

  • [17]

    Mehmet Yavuz, Necati Özdemir, Haci Mehmet Baskonus, (2018), Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, The European Physical Journal Plus, Vol:133, No:6.

  • [18]

    Asıf Yokuş, (2018), Comparison of Caputo and conformable derivatives for time-fractional Kortewege Vries equation via the finite difference method, International Journal of Modern Physics B, Vol:32, No:29, 1850365.

  • [19]

    Asıf Yokuş, Sema Gülbahar, (2019), Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation, Applied Mathematics and Nonlinear Sciences, Vol:4, No:1, 35–42.

  • [20]

    Sümeyra Uçar, Nihal Yılmaz Özgür, Beyza Billur İskender Eroğlu, (2019), Complex conformable derivative, Arabian Journal of Geosciences, Vol:12, No:6, 201.

  • [21]

    Abdon Atangana, Ernestine Alabaraoye, (2013), Solving a system of fractional partial differential equations arising in the model of HIV infection of CD4+ cells and attractor one-dimensional Keller-Segel equations, Advances in Difference Equations, 2013:94.

  • [22]

    Ana RM Carvalho, Carla MA Pinto, Dumitru Baleanu, (2018), HIV/HCV coinfection model: a fractional-order perspective for the effect of the HIV viral load, Advances in Difference Equations, 2018:2.

  • [23]

    İlknur Koca, (2018), Analysis of rubella disease model with non-local and non-singular fractional derivatives, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), Vol:8, No:1, 17–25.

  • [24]

    Ebenezer Bonyah, J.F. Gomez-Aguilar, Augustina Adu, (2018), Stability analysis and optimal control of a fractional human African trypanosomiasis model, Chaos, Solitons & Fractals, Vol:117, 150–160.

  • [25]

    V.F. Morales-Delgado, J.F. Gomez-Aguilar, M.A. Taneco-Hernandez, R.F. Escobar-Jimenez, V.H. Olivares-Peregrino, (2018), Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel, Journal of Nonlinear Sciences Applications, Vol:11, No:8, 994–1014.

  • [26]

    Amin Jajarmi, Dumitru Baleanu, (2018), A new fractional analysis on the interaction of HIV with CD4+ T-cells, Chaos, Solitons & Fractals, Vol:113, 221–229.

  • [27]

    Sümeyra Uçar, Esmehan Uçar, Necati Özdemir, Zakia Hammouch, (2019), Mathematical analysis and numerical simulation for a smoking model with Atangana Baleanu derivative, Chaos, Solitons & Fractals, Vol:118, 300–306.

  • [28]

    Esmehan Uçar, Necati Özdemir, Eren Altun, (2019), Fractional order model of immune cells influenced by cancer cells, Mathematical Modelling of Natural Phenomena, Vol:14, No:3, 12p.

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