Optimal Control of The HIV-Infection Dynamics

Transcription

1 Australian Journal of Basic and Applied Sciences, 5(9): , 2011 ISSN Optimal Control of The HIV-Infection Dynamics 1 H. Ghiasi zadeh, 1 H. Chahkandi nejad, 1 R. Jahani, 2 H. Roshan Nahad and 2 M. Mohammad Abadi 1 Islamic Azad University, Birjand Branch, Birjand, Iran. 2 Islamic Azad University, Gonabad Branch, Gonabad, Iran. Abstract: Since the issue of including the most optimal amount of medicine consumption in the program of illness control program is important in the control of HIV because of its high cost and high risk of side effects (poisoning), so applying a mathematical model and modern techniques is necessary in solving optimal control issue for prescribing optimal amount. In this article, we use an ordinary differential equation system which models interaction between HIV viruses and human body immune system. Optimal control illustrates percentage of medicine therapy on interaction between CD4+T cells (a type of white cells) and viruses. Our aim is to maximize the number of CD4+T and minimize the rate of virus load by determining amount of optimum consumption of medicine in each day. Mentioned model is a nonlinear model which generally solving these problems is difficult and there is no specific method for solving them. In this article, in addition to introducing AVK discretization method, it is used for approximate solving the mathematical model. Key words: Optimal Control, HIV-Infection Dynamics, AVK method, Ordinary Differential Equation System. INTRODUCTION Various medicine therapies have been used for achieving to an optimal solution for medicine therapy of HIV in people with HIV up to now. There is this question that what is the best medicine therapy for these patients with regard to this issue that therapeutic period is limited? An ordinary differential equation model which describes the interaction between HIV viruses and human body immune system. By applying a suitable optimal control on it a suitable medicine program can be provided (Ledzewicz Urszula, 2002; Fister K. Renee, 1998). During HIV infection strong homoral immune responses and also HIV specific cellular immune are created, in spite of that patient s antiviral immunes responses encounters sustainable immune defect and illness development and HIV causes failure of almost all the immune system elements. Its main cause is the pollution of CD4+T cells by HIV viruses which finally results in body immune system failure. CD4+Ts are responsible in coordination of immune system responses and HIV reproduces through polluting them and immune system activation and weaken immune system. The main aim of HIV is T cells with CD4 receptors i.e. CD4+T cells. In other words, existence of CD4 receptors is necessary for entering HIV into a cell. In fact activation of T cells prepares condition for HIV reproduction in them (RADISAVLJEVIC-GAJIC VERICA, 2009). Mathematical Model: At the beginning of 80s which AIDS became epidemic, many attempts were made for controlling this fatal illness by biologists and medical associations. Meanwhile mathematicians made a big contribution by making mathematical models which illustrate interaction among viruses and human body immune cells. When medical associations began their research on new medicines for preventing illness development in new therapeutical strategies, developed mathematical models were used in the experiments and these models were completed each year to provide better mathematical model. The most important elements in this model include three types of CD4+t cells: unpolluted CD4+T cells which their condensation is illustrated by X(t) variable, polluted CD4+T cells which their condensation is illustrated by Y(t) variable, active polluted CD4+T cells which their condensation is illustrated by Z(t) variable. V(t) shows the rate of condensation of free viruses and dynamic model of these four is modeled by following equations (Ledzewicz Urszula, 2002): µ (1- (1) µ µ (3) 2 Corresponding Author: Hadi chahkandi nejad, lecturer of Islamic azad university, birjand branch, Tell: hadi64.msc@gmail.com 1051

2 1µ µ 4 In these equations, expressions of µv, µz, µy, µx with negative signs indicate the rate of normal death of unpolluted CD4+T cells, latent polluted CD4+T cells, active polluted CD4+T cells and free cells so kx v (t)x(t) expression is deducted from two equations of viruses populations (4) and healthy cells ( 1) and added to latent polluted cells population equation (2). Ky is the rate which according to it latent polluted cells are converted to polluted cells. Parameter of r shows the rate of population growth in CD4+T cells and s parameter shows production rate of new CD4+T by thymus. N max indicates maximum population for CD4+T cells and control variable of u(t) shows amount of the effect of chemical medicines on system. So LuzZ(t) which indicates the number of produced viruses by polluted CD4+T cells in the rate of L is added to virus population equation (4) and deduced by a factor of u(t) which indicates effect of medicine during therapeutic period from viruses population equation (4),( Ledzewicz Urszula, 2002). Optimal Control of Model: This control indicates a percentage of effect of medicine therapy on interaction between CD4+T and HIV viruses which is illustrated by u(t) although there are many side effects in medicine therapy and in wrong prescription of the amount of medicine causes virus resistance toward medicine or lead to poisoning. So therapeutic period must be limited (Ledzewicz Urszula, 2002). In most of medicine therapies therapeutic period is less than 500 days (Denise Kirschner, 1997). For the aim of keeping the level of healthy CD4+T cells higher and keeping side effects of medicine therapy in low level, following target function is used for maximizing above dynamics: The integrand X indicates that one wants to maximize the number of uninfected CD4+T cells. At the same time, the second term minimizes the negative effects of the chemotherapy where B > 0 represents a desired "weight" on the benefit and cost (Ledzewicz Urszula, 2002; Fister K. Renee, 1998).. Mentioned model is a nonlinear model which solving these problems is difficult and there is no specific method for it. In this article in addition to introducing AVK discretization method, we convert mentioned model to a problem of linear programming and use it for solving mentioned model. AVK Method: In the following, we introduce the AVK method. The method is systematically described and will result in an approximate analytic solution for the strongly nonlinear ODEs. For linearization, we use from relation belows (K.P. Badakhshan, 2007) : 6 Suppose is not defined in s. we introduce weak differentiation for calculating differentiation off in s we have: lim 7 Given > 0, for all x (s, s + ), we define: lim, x s, s 8 We can extended above approach to n dimensional. Theorem 5.1: Consider the nonlinear smooth function f : [0, 1. Then the optimal solution of the following optimization problem is the function (x). 1052

3 .. 9 where s = (,,, ) [0, 1 is an arbitrary point and p(.) is a vector of the form ( (.), (.),..., (.)). Proof. See [Vaziri]. Now based on theorem (5.1) the following definition may be stated for nonsmooth functions. Definition 5.1: Let f : [0, 1 is a non-smooth function. The global weak differentiation with respect to x in the sense of space is defined as the p(.) the optimal solution of the minimization problem which is shown in (9). In the case that n = 1, we may obtain differentiation of f(x) on [0, 1] by partitioning interval [0, 1] to n subinterval of [xi 1, xi] such that for all i = 1, 2,..., n. Let be a point in [, ]. we show differentiation of f(x) at x = s by p(s). We may obtain values of p(s), by solving the following problem: where and, is an arbitrary point for all i = 1, 2,..., n. (note:we may define for all i = 1, 2,..., n) Using AVK Method In Solving HIV Mathematical Model: Since therapeutic period must be less than 500days, we can assume the integral span from 0 to 50 and with resolving of this integral you can find the efficient daily amount of drug and the level of CD4+T cells and viruses in every 10 days of treatment. With placing Equations (1)to (4) in (9) we can find: 1 µ 1 µ µ 1µ µ 1 11 Thus the problem is converted to a linear programming problem which is solved easily. We solve equation (11) in AVK method and get answers of the unknowns of the problem according to figures (1) to (5). In solving equation (11) we use following values for constants of the problem and primary conditions [Fister, 1998]: µ = 0.02/d Death rate of uninfected and latently infected CD4+T cells. µ Z = 0.24/d Death rate of actively infected cells. µ = 2.4/d Death rate of free virus. X = mm3/d Rate CD4+T cells become infected by free virus. Y = 0.003/d Rate Y cells convert to actively infected. r = 0.03/d Rate of growth for the CD4+T cells. N = 1200 Number of free virus produced by Z cells. = 1500/mm3 Maximum CD4+T cell level. s = 10/mm3 Source term for uninfected CD4+T. X 1000 / =0.01 / Applying AVK Method In Solving Optimal Control Model: We assume equation (5) as the objective function for solving optimal control model and assume equations (1) to (4) as objective functions constraints and solve them by using AVK method and its results are illustrated in figures (6) to (15). 1053

4 Fig. 1: T cells behavior in the absence of optimal control. Fig. 2: Y cells behavior in the absence of optimal control. Fig. 3: Z cells behavior in the absence of optimal control. 1054

5 Fig. 4: viral behavior in the absence of optimal control. Fig. 5: amount of medicine consumption in the absence of optimal control. Fig. 6: T cells behavior in the optimal control(b=30). 1055

6 Fig 7: Y cells behavior in the optimal control(b=30). Fig. 8: Z cells behavior in the optimal control(b=30). Fig. 9: viral behavior in the optimal control(b=30). 1056

7 Fig. 10: amount of medicine consumption in the optimal control (B=30). Fig. 11: T cells behavior in the optimal control(b=110). Fig. 12: Z cells behavior in the optimal control(b=110). 1057

8 Fig. 13: Z cells behavior in the optimal control(b=110). Fig. 14: viral behavior in the optimal control(b=110). Fig 15: amount of medicine consumption in the optimal control (B=110). Conclusions: We examined a mathematical model which shows interaction between HIV viruses and cells of human body immune system responses and used new method of AVK discretization for its approximate solution. The results of model solution indicates that in spite of medicine consumption and lack of optimal control on model, the rate of CD4+T decreased significantly and viruses population increased more although the rate of medicine consumption is more. This indicates that if medicine consumption be prescribed without optimal control, it may 1058

9 result in antigenic escape and viruses resistance against medicine. By applying optimal control on mathematical model and solving by AVK method shows that by selecting proper amount of B and consuming less medicine, the number of CD4+T cells will experience less decrease during therapeutic period and virus load will have less increase during therapeutic period. REFERENCES Ledzewicz Urszula, Schattler Heinz, On optimal controls for a general mathematical model for chemotherapy of HIV, ieee, & 0102/$ AACC. Fister, K., Renee, Lenhart Suzanne, Scott McNally Joseph, OPTIMIZING CHEMOTHERAPY IN AN HIV MODEL, Electronic Journal of Di_erential Equations,1998, ftp or RADISAVLJEVIC-GAJIC, VERICA, Optimal Control of HIV-Virus Dynamics, Annals of Biomedical Engineering, vol. 37, No. 6, June 2009 (_ 2009) pp: , DOI: /s Denise Kirschner, Suzanne Lenhart, Steve Serbin, Optimal control of the chemotherapy of HIV,springer, J. Math. Biol. (1997) 35: 775Ð792. Vaziri, A.M., A.V. Kamyad and S. Effatiand, M. Gachpazan, A parametric linearization approach for solving nonlinear programming problems, Submitted in Aligarh Statistics. Badakhshan, K.P., A.V. kamyad, A. Azemi, "Using AVK method to solve nonlinear problems with uncertain parameters," Applied Mathematics and Computation., 189: