In this work, a mathematical model for the human immunodeficiency virus (HIV) infection of CD4+ T-cells by using the optimal perturbation iteration method (OPIM) is analyzed. Optimization and classical perturbation techniques are combined to build the new proposed method. The iteration algorithm for systems of nonlinear differential equations for this optimal perturbation iteration technique is constructed for the first time. A test problem has been solved and some plots are given to show the reliability and efficiency of the proposed method. Obtained results exhibit the effectiveness and accuracy of the semi-analytical technique.Publisher's Versio

Deniz, Sinan

TWMS Journal of Applied and Engineering Mathematics

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Isik University Academic Open Access

TWMS J. App. and Eng. Math. V.11, N.1, 2021, pp. 273-281SEMI-ANALYTICAL APPROACH FOR SOLVING A MODEL FOR HIVINFECTION OF CD4+ T-CELLSS. DENİZ, §Abstract. In this work, a mathematical model for the human immunodeficiency virus (HIV)infection of CD4+ T-cells by using the optimal perturbation iteration method (OPIM) is an-alyzed. Optimization and classical perturbation techniques are combined to build the newproposed method. The iteration algorithm for systems of nonlinear differential equations forthis optimal perturbation iteration technique is constructed for the first time. A test problemhas been solved and some plots are given to show the reliability and efficiency of the pro-posed method. Obtained results exhibit the effectiveness and accuracy of the semi-analyticaltechnique.Keywords: HIV infection model, optimal perturbation iteration method, semi-analytical solu-tion.AMS Subject Classification: 81Q15,78A701. IntroductionCD4, which means a cluster of differentiation antigen 4, is actually a glyco-protein locatedin the outer region of immune cells such as T assistant cells, dendritic cells, and macrophages.They were invented in 1970 and were firstly known as leu-3 and T4 before being they are namedas CD4 in 1984. CD4+ T assistant cells are blood white cells which are usually known as themain part of the body immune system. They are often referred to as T-helper cells, CD4 cellsor T4 cells. The reason for saying ”helper cells” is that their main acts is to deliver signalsto alternative types of human immune cells, including CD8 terminator cells. After then theydemolish the poisonous particle. If those cells become consumed, for instance in untreated thehuman immunodeficiency virus (HIV) infection, or following immune suppression prior to atransplant, the body can be counted as defenceless to a wide range of contaminations that itwould in other way have been able to be forced to be defeated.Analyzing the mathematical models of some constructive situations enables us to discoverthe distinct dynamics of these kinds of situations. Many scientists have studied a number ofmodels to see different aspects of HIV infection. By doing so, they can also predict the spreadof this deadly ailment. To investigate the model, many different techniques have been usedsuch as variational iteration method [16], Laplace Adomian decomposition method [19], themodified variational iteration method [15], the homotopy analysis method [11], etc. In additionDepartment of Mathematics, Faculty of Art and Sciences, Manisa Celal Bayar University, 45140,Manisa, Turkey.e-mail: sinandeniz01@gmail.com; ORCID: https://orcid.org/0000-0002-8884-3680.§ Manuscript received: February 23, 2019; accepted: May 4, 2019.TWMS Journal of Applied and Engineering Mathematics, Vol.11, No.1; © Işık University, Department ofMathematics, 2021; all rights reserved.273274 TWMS J. APP. AND ENG. MATH. V.11, N.1, 2021Table 1. List of variables-parameters used in this work ([16]).Parameters-variables MeaningH Uninfected CD4+ T-cell concentrationI Unhealthy CD4 + T-cellsV Free HIV virus at time ts Source term for healthy CD4+ T- cellsµH Natural death rateg Growth rate of healthy CD4 + T-cellsHmax Maximum possibility of presence of number of T-cellsk Rate at which healthy cells become infectedk′ Rate of infectious cells to become actively infectedK Number of viral particlesµI Blanket death rates for infected cellsµV Loss rate for virus.τ The amount of delayto those papers, Perelson [17] presented a comprehensive work on the mathematical model ofthis infection involving a delay term in the differential equation. His team also investigateddifferent dynamics of HIV infection [20]. In [12], the authors also discussed the usefulness ofthe delay term in ordinary differential equations for population dynamics and studied on thestability analysis for the considered models. In [18], Nowak studied the injurious effects of longincubation period of virus infection and introduced a simple model for this disease.The motivation of this work is to improve the implementation of the semi-analytic optimalperturbation iteration method (OPIM) to investigate a model for HIV infection of CD4+ T cells:dH(t)dt= s− µHH(t) + gH(t)(1 − H(t) + I(t)Hmax)− kH(t)V (t)dI(t)dt= k′H(t− τ)V (t− τ) − µII(t) (1)dV (t)dt= KµII(t) − kH(t)V (t) − µV V (t).The above equation is the mathematical description of human health related phenomena, par-ticularly about the presence, decrease or increase in HIV. Figure 1 represents flow diagram forthe Eq. (1).The optimal perturbation iteration method has been developed by using the idea of pertur-bation iteration method (PIM) [5, 9]. Many studies show that approximate PIM solutions canbe healed by inserting the convergence control-parameter p to PIM algorithms. One can easilyadjust the convergence with the aid of the parameter p. OPIM has been applied to many kindsof differential equations [10, 8, 3, 7, 4, 6].This study is organized as follows: In the following part, section 2, the optimal perturbation-iteration algorithm is described and also improved for the solution of systems of differentialequations for the first time. In section 3, OPIM is implemented to a test problem to get theapproximate solutions. Finally, in Section 4, the conclusion of this research paper is given.2. Optimal Perturbation-Iteration AlgorithmIn this part, a perturbation-iteration algorithm is presented by using only one correction termin the straightforward perturbation expansion and a correction term of nth-order derivativesfrom the Taylor expansion of a function.S.DENIZ: SEMI-ANALYTICAL APPROACH FOR SOLVING A MODEL FOR HIV ... 275Figure 1. Flow diagram for HIV modelLet us look at the following non-linear (or linear) system of differential equationsAk(ẋk, xj , ε, t) = 0; k = 1, 2, ...,K; j = 1, 2, ...,K (2)where k denotes the number of differential equations in the system. Obviously, the system ofequations can be written asA1 = A1(ẋ1, x1, x2, x3, ..., xK , ε, t) = 0A2 = A2(ẋ2, x1, x2, x3, ..., xK , ε, t) = 0...Ak = Ak(ẋk, x1, x2, x3, ..., xK , ε, t) = 0.(3)Let us consider an approximate solution of the systemxk,n+1 = xk,n + εxck,n (4)with one correction term in the perturbation expansion. The subscript letter n symbolizes thenth order iteration over the obtained approximate solution. One can approximate to the systemwith a Taylor series expansion asAk =M∑m=01m![(ddε)mAk]ε=0εm; k = 1, 2, ...,K (5)in the neighborhood of ε = 0 whereddε=∂ẋk,n+1∂ε∂∂ẋk,n+1+K∑j=1(∂ẋk,n+1∂ε∂∂ẋk,n+1)+∂∂ε(6)is defined for the (n+ 1)th iterative equationsAk(ẋk,n+1, xj,n+1, ε, t) = 0 (7)substituting equation (6) into (5), an iteration algorithm is attained;Ak =M∑m=01m!ẋck,n ∂∂ẋk,n+1 +K∑j=1(ẋcj,n∂∂ẋj,n+1)+∂∂εmHkε=0εm = 0 (8)which is a first-order ordinary differential equation and can be solved for the correction termsxck,n Then, using Eq.(3), the (n + 1)th iteration solution can be found. Iterations are stoppedafter a satisfactory approximate solution is obtained.276 TWMS J. APP. AND ENG. MATH. V.11, N.1, 2021In order to improve the approximate PIM solutions, a new approach to perturbation iterationalgorithms is proposed. Based on the idea of homotopy analysis method (HAM) [1, 21, 23], aconvergence-control parameter p is furnished into Eq. (4) and then establish new components,described byxk,1(t; p) = xk,0 + p0xck,0xk,2(t; p) = xk,1 + p1xck,1...xk,m+1(t; p) = xk,n + pmxck,m.(9)It is evident that when the convergence parameters pm are all equal to one, (9) is convertedto the classical PIM. As observed in [1, 21], the region of validity of these parameters pm canbe determined by drawing constant level curves for some arbitrary values of the solution. Thiscan be done by selecting a non-zero value of the mth order approximate solution, mapping itout against the parameter p and tracking the interval of p for which only a small change in thevalue is sought. Although this approach enables us to estimate a value for p, it is not a sufficientmethod to find the optimum value. To get a better and optimum value, the similar strategymentioned by Marinca et al [14, 2] is used. To achieve this, the approximate solution xm isinserted into the Eq.(2). Correspondingly, the general problem results in the following residual:ResA1 = A1(ẋ1,m, x1,m, x2,m, ..., xK,m, p0, ε, t) = 0ResA2 = A2(ẋ2, x1,m, x2,m, ..., xK,m, p1, ε, t) = 0...ResAk = Ak(ẋk, x1,m, x2,m, ..., xK,m, pm−1ε, t) = 0.(10)Apparently, when Res(t; pm) = 0 then the approximation xm(t; p) will be the desired exactsolution.In fact, one cannot encounter with this case for especially nonlinear equations, but onecan perform some calculation for the functionalJResA1(p1) =∫Ω(ResA1)2(t; p0)dtJResA2(p2) =∫Ω(ResA2)2(t; p1)dt...JResAk(pK) =∫Ω(ResAk)2(t; pK−1)dt(11)where Ω is the domain that one wishes to obtain the solution of the problem. Therefore, theconvergence-control parameters p can be optimally identified from the conditiondJResA1dp1= 0,dJResA2dp2= 0, · · · , dJResAKdpK= 0. (12)Both of these techniques have their drawbacks. Therefore, firstly the level curves are used todetermine the valid region of p, then use the residual to find its optimum value around thisregion. All of these processes are called the optimal perturbation iteration method (OPIM)since one can obtain the optimal convergence control parameter in this manner.3. Test ProblemIn this section, the system (1) by using OPIM algorithms is solved. To start the iterations,the initial conditions will be taken as H(−τ) = H0(t) = H(0) = 900, I0(t) = I(0) = 0,V (−τ) = V0(t) = V (0) = 0.005. With these values, one can follow the procedure for OPIMmentioned in Section 2 to solve the following system:S.DENIZ: SEMI-ANALYTICAL APPROACH FOR SOLVING A MODEL FOR HIV ... 277dH(t)dt= 18 − 0.05H(t) + 0.05H(t)(1 − H(t) + I(t)120)− 0.0028H(t)V (t)dI(t)dt= 0.0029H(t− τ)V (t− τ) − 0.27I(t) (13)dV (t)dt= 600I(t) − 0.0028H(t)V (t) − 3.5V (t).Perturbation parameter ε can be embedded to the above systemA1 =dH(t)dt−(18 − 0.05H(t) + 0.05εH(t)(1 − H(t) + I(t)120)− 0.0028εH(t)V (t))A2 =dI(t)dt− (0.0029εH(t− τ)V (t− τ) − 0.27I(t)) (14)A3 =dV (t)dt− (600I(t) − 0.0028εH(t)V (t) − 3.5V (t))Correspondingly, the Eq. (8) turns intoḢk + εḢkc+ 18 − 0.05Ḣk + 0.05εḢk(1 − Ḣk + İk120)−0.0028εḢkV̇k = 0İk + εİkc+ 0.0029εḢk(t− τ)V̇k(t− τ) − 0.27İk = 0 (15)V̇k + εV̇kc+ 600İk − 0.0028εḢkV̇k − 3.5V̇k = 0.One can continue the process till fifth - order approximation and can obtain the following residualfunctions as:ResH5 = Ḣ5 − 18 + 0.05Ḣ5 − 0.05Ḣ5(1 − Ḣ5 + İ5120)+ 0.0028εḢ5V̇5ResI5 = İ5 −(0.0029Ḣ5(t− τ)V̇5(t− τ) − 0.27İ5)(16)ResV5 = V̇5 −(600İ5 − 0.0028Ḣ5V̇5 − 3.5V̇5)Unknown parameters p0, p1, p2 can be obtained by using the idea described in Section 2 asdJResH5dp0= 0,dJResI5dp1= 0,dJResV5dp2= 0. (17)Graphics of absolute residual errors obtained by optimal perturbation iteration method for eachgroup are displayed in Figures 2-7 for fourth and fifth order approximate solutions respectively.For this problem, the optimal values of p0 = 0.996121, p1 = 0.070589 and p2 = 0.200365 areobtained by minimizing the residual error for fifth - order approximation. One can easily getthe approximate OPIM results by substituting these parameters in their places.278 TWMS J. APP. AND ENG. MATH. V.11, N.1, 2021Figure 2. Absolute error for fourth order OPIM solution of H(t)Figure 3. Absolute error for fourth order OPIM solution of I(t)Figure 4. Absolute error for fourth order OPIM solution of V (t)4. ConclusionsIn this research paper, the approximate solutions of the mathematical model of HIV infectionsis analyzed. Newly developed optimal perturbation iteration method has been improved to solvethe system of differential equations. Applying this technique to the model for HIV infection ofCD4+ T-cells reveals that this method is powerful and effective to get the approximate solutionsS.DENIZ: SEMI-ANALYTICAL APPROACH FOR SOLVING A MODEL FOR HIV ... 279Figure 5. Absolute error for fifth order OPIM solution of H(t)Figure 6. Absolute error for fifth order OPIM solution of I(t)Figure 7. Absolute error for fifth order OPIM solution of V (t)of these types of problems. Graphics for absolute residual errors also support this claim. Finally,one can conclude that OPIM provide a simple and easy way to control and adjust the convergenceregion and it is also applicable to the nonlinear biological systems.280 TWMS J. APP. AND ENG. MATH. V.11, N.1, 2021References[1] Abbasbandy, S. and Zakaria, F. S., (2008), Soliton solutions for the fifth-order KdV equation with the homo-topy analysis method. Nonlinear Dynamics, 51(1-2), 83-87.[2] Ali, J., Islam, S., Islam, S. and Zaman, G., (2010), The solution of multipoint boundary value problems bythe optimal homotopy asymptotic method. Computers & Mathematics with Applications, 59(6), 2000-2006.[3] Bildik, N. and Deniz, S., (2017), A new efficient method for solving delay differential equations and a com-parison with other methods. 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M. and Ansari, A. R., (2010), Some solutions of the linear and non-linear Klein–Gordon equations using the optimal homotopy asymptotic method. Applied Mathematics andComputation, 216(10), 2898-2909. (2010): 2898-2909.[14] Iqbal, S. and Javed, A., (2011), Application of optimal homotopy asymptotic method for the analytic solutionof singular Lane–Emden type equation. Applied Mathematics and Computation, 217(19), 7753-7761.[15] Merdan, M., Gökdoğan, A. and Yildirim, A., (2011), On the numerical solution of the model for HIV infectionof CD4+ T cells. Computers & Mathematics with Applications, 62(1), 118-123.[16] Nazir, A., Ahmed, N., Khan, U. and Mohyud-Din, S. T., (2018), Analytical approach to study a mathematicalmodel of CD4+ T-cells. International Journal of Biomathematics, 11 (03), 1-12.[17] Nelson, P. W. and Perelson, A. S., (2002), Mathematical analysis of delay differential equation models ofHIV-1 infection. Mathematical biosciences, 179(1), 73-94.[18] Nowak, M. A. and May, R. M., (1991), Mathematical biology of HIV infections: antigenic variation anddiversity threshold. Mathematical Biosciences, 106(1), 1-21.[19] Ongun, M. Y., (2011), The Laplace Adomian Decomposition Method for solving a model for HIV infectionof CD4+ T cells. Mathematical and Computer Modelling, 53(5-6), 597-603.[20] Perelson, A. S., Kirschner, D. E. and De Boer, R., (1993), Dynamics of HIV infection of CD4+ T cells.Mathematical biosciences, 114(1), 81-125.[21] Rashidi, M. M., Domairry, G. and Dinarvand, S., (2009), Approximate solutions for the Burger and regular-ized long wave equations by means of the homotopy analysis method. Communications in Nonlinear Scienceand Numerical Simulation, 14(3), 708-717.[22] Shah, R. A., Islam, S., Siddiqui, A. M. and Haroon, T. , (2011), Optimal homotopy asymptotic methodsolution of unsteady second grade fluid in wire coating analysis. Journal of the Korean Society for Industrialand Applied Mathematics, 15(3), 201-222.[23] Tari, H., Ganji, D. D. and Rostamian, M., (2007), Approximate solutions of K(2, 2), KdV and modified KdVequations by variational iteration method, homotopy perturbation method and homotopy analysis method.International Journal of Nonlinear Sciences and Numerical Simulation, 8(2), 203-210.S.DENIZ: SEMI-ANALYTICAL APPROACH FOR SOLVING A MODEL FOR HIV ... 281Sinan DENİZ has been a research assistant of Mathematics at Manisa Celal BayarUniversity since September 2012. He received his BsC degree in “Mathematics” fromFatih University, Istanbul, Turkey. He completed his master study in Applied Mathe-matics at Manisa Celal Bayar University in 2014 and his PhD in 2018 at Manisa CelalBayar University with a dissertation about optimization of analytical approximate so-lutions of ODEs and PDEs. His main research interests are: Nonlinear Ordinary andPartial Differential Equations, Asymptotic and Computational Methods for ODEsand PDEs, Semi-Analytical and Numerical Methods, Mathematical Modeling

Semi-analytical approach for solving a model for HIV infection of CD4+ T-cells

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Semi-analytical approach for solving a model for HIV infection of CD4+ T-cells

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