OPTIMAL CONTROL OF AN HIV MODEL

Transcription

1 H. R. ERFANIAN and M. H. NOORI SKANDARI/ TMCS Vol.2 No. (211) The ournal of Mathematics and Computer Science Available online at The ournal of Mathematics and Computer Science Vol.2 No. (211) OPTIMAL CONTROL OF AN HIV MODEL H. R. ERFANIAN a and M. H. NOORI SKANDARI b a Department of Mathematics & Statistics, University of Science and Culture, Tehran, Iran. b Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran. hadinoori3@yahoo.com Received: Auust 21, Revised: November 21 Online Publication: anuary 211 Abstract We consider an HIV model, based on optimal control, for identifyin the best treatment stratey in order to maximize the healthy cells by usin chemotherapies with minimum side effects. In this paper, a new approach is introduced which transform the constraints of problem to the interal constraints. By an approximation, we obtain a finite dimensional linear prorammin problem which ive us an approximate solution for oriinal problem. KeyWords: HIV Model, Optimal Control, Linear prorammin, Measure theory,chemotherapy. 1- Introduction One of the worst diseases in whole world is AIDS (Acquired Immunity Deficiency Syndrome). It is caused by the human immunodeficiency virus (HIV). There is still much work to be completed in the search for an anti-hiv vaccine. Most of the chemotherapies are aimed at killin or haltin the pathoen, but treatment which can boost the immune system can serve to help the body fiht infection on its own. The new treatments are aimed at reducin viral population and improvin the immune response. This brins 65

2 H. R. ERFANIAN and M. H. NOORI SKANDARI/ TMCS Vol.2 No. (211) new hope to the treatment of HIV infection, and we are explorin strateies for such treatments usin optimal control techniques. Once HIV enters the body, the human immune system tries to et rid of it.. The The invasion is reported to CD T cells CD is a protein marker in the surface of the T cell, and the letter T refers to thymus, the oran responsible for maturin these cells after they mirate from the bone marrow (where they are manufactured). The surface of CD T possesses a protein that can bind to forein substances such as HIV. The HIV needs a host in order to reproduce and the above mentioned protein provides shelter. The HIV virus is a retrovirus, the RNA of the virus is converted into DNA inside the. Thus, when infected bein to multiply to fiht CD T cells CD T cells this pathoen, they produce more virus (see [1], [3], [5], [6]). 2. Statement of the Model We consider a model which presented by Gumel et al.[3 ]describes the interaction of HIV and the immune system of the body. In this model, the variables are T, T i and V present the number of healthy CD T cells, infected CD T cells and free viruses respectively. The number of T cells in the person is affected by the rate of natural rowth of T cells, the rate of production of CD T cells due to the presence of the virus(note that if the virus ets into the body, T cells multiply themselves to the maximum level to resist the virus), the natural death rate and rate of infection of healthy cells due to the presence of the virus. The number of T i cells depend on the rate of production of infected cells from actively infected CD T cells, the rate at which the virus infects free cells, natural death, the action of anti-hiv cyto-toxic T lymphocyte cells and the viral lysis. Similarly, the population of the virus is determined by the production rate of viruses by actively infected T cells, the rate at which the virus enters into the rested T cells and the natural death rate. In this model, the variables are T, T i and V represent the number of healthy, infected CD T cells and free viruses, respectively. CD T cells 651

3 H. R. ERFANIAN and M. H. NOORI SKANDARI/ TMCS Vol.2 No. (211) dt() t s rt ( t ) V ( t ) 1T ( t ) kv (1 u1( t )) T( t ) V ( t ), dt dti () t kv (1 u1( t )) T( t ) V ( t ) rk v (1 u1( t )) T( t ) V ( t ) 2T i ( t ) dt (1) (2) k k T ( t ) V ( t ) rk k T ( t ) V ( t ), c v c v dv ( t ) N (1 L )(1 u2( t )) Ti ( t ) 3V ( t ) (1 ) ktt( t ) V ( t ), (3) dt with iven initial values for T, T i and V at t respectively by T, T i and Define the obective functional t f t ( u, u ) ( T ( t ) ( A ( u ( t )) A ( u ( t )) )) dt. () V. In other words, we are maximizin the benefit based on the healthy T cells count and minimizin the cost based on the percentae effect chemotherapy iven (i.e. u 1 and u 2 ). The parameters A1, A2 represent the weihts on the benefit and cost. * * The oal is to seek an optimal control pair ( u, u ) such that where U is the control set defined by 1 2 * * ( u, u ) max{ ( u, u ):( u, u ) U }, U { u ( u, u ): u measurable, u ( t ) 1, t [ t, t ] for i 1,2}. 1 2 i i f In the above model, parameters and constants, defined as follows: the value of functionin thymus, s rate of supply of CD T cells r rate of production of CD T cells due to the HIV, 1 natural death rate of CD T cells, kv rate of infection of activated CD T cells, 2 natural death rate of infected CD T cells, 3 kc natural death rate of free viruses, rate of Cyto-toxic T lymphocytes action viral lysis, N rate of production of HIV from actively infected CD T cells, kt rate of viral entry quiescent restin CD T cells, 652

4 H. R. ERFANIAN and M. H. NOORI SKANDARI/ TMCS Vol.2 No. (211) In the next section the problem is chaned to a problem in measure space, where we interface with a linear prorammin problem. In the followin we replace the problem by another one in which the maximum of the obective functional () is calculated over a set of positive Radon measures to be defined as follows. Some authors have used this approach in a variety of optimal control problems; we mention [2],[7],[8],[1] and the pioneerin work of Rubio ([9]) as well. Let A U, where [ t, t f ] and t, x [ x1( t ), x 2( t ), x 3( t )] or x [ T( t ), Ti ( t ), V ( t )] A is the traectory of the controlled system and A is a compact set of 3 R, t, u( t ) [ u1( t ), u2( t )] U is the control and U is a 2 compact set of R. we may rewrite optimization problem (1) -() as the followin reduced form: t f max 2 2 ( x, u ) ( x1( t ) ( A1( u1( t )) A2( u2( t )) )) dt (5) t s. t. x ( t ) ( t, x, u ), t (6) 3. Classical Control problems We shall say that a traectory-control pair p [ x (.), u (.)], is admissible if the followin conditions hold: i) x ( t ) A, t. ii) u ( t ) U, t. iii) The boundary conditions x ( t ) x and x ( t ) x x is unknown. iv) The pair p satisfies the differential equation (6) a.e. on a a b b is satisfied, where b Let B an open ball in R containin A. Let C ( B) be the space of all realvalued continuously differentiable functions on B which are bounded on B toether with their first derivatives. Let C ( B), and define function as follows: ( t, x, u) ( t, x ) ( t, x, u) ( t, x ) ( t, x, u) (7) x The function is in the space C ( ) of all real-valued continuous functions defined on the compact set. Thus we have t. 653

5 H. R. ERFANIAN and M. H. NOORI SKANDARI/ TMCS Vol.2 No. (211) ( t, x, u ) dt ( ( t, x ) x ( t, x )) dt x ( t, x ) dt ( t, x ) ( t, x ) (8), C( B) t b b a a O Let D ( ) be the space of infinitely differentiable real-valued functions with compact support in O. Define ( t, x, u) x ( t ) ( t, x, u) ( t ) 1,2,..., n D( ). (9) Then (see [9]) Put ( t, x, u ) 1,2,..., n D ( ). ( t, x, u) ( t ) ( t, x, u) (1) that is, a function which depends on the time variable only; then ( t, x, u ) ( t ),( t, x, u ) (11) If p is an admissible pair, the equality (1) with the choice (11) for the function implies that ( t, x, u ) dt a, C '( B ) (12) Where a is the interal of f over, independent of x and u. Now, the mappin p : F F ( t, x ( t ), u ( t )) dt, F C ( ). (13) Defines a positive, linear functional on C ( ). By the Riesz representation theorem, there exists a unique positive Radon measure on which ( F ) F ( t, x, u ) dt Fd ( F ), F C ( ) p. (1) 65

6 H. R. ERFANIAN and M. H. NOORI SKANDARI/ TMCS Vol.2 No. (211) Let M ( ) be the set of all positive Radon measure on. Define the positive Radon measure M ( ) such that maximizes the followin linear functional I( ) ( f) (15) subect to ( ), C( B), (16) ( ), 1,2,..., n D( ), (17) ( ) a, C1( ), (18) We define Q to be the set of all measures in M ( ) that satisfy equalities (16)-(18) than we can show that there exists an optimal measure in the Q for which ( f) ( f) for all Q (see [8]). This problem is an infinite-dimensional linear prorammin problem and all the functions in (15)-(18) are linear with respect to measure. We obtain the approximate solution of this problem by the solution of a finitedimensional linear proram. In this section, we are limitin constraint (15)-(18) as follows: i) We choose the function ( t, x ) x i r, r 1,2,..., M1. and i 1,2,..., n. Then we r1 have rx i i ( t, x, u). ii) We consider 2 r( t t ) ( ) sin[ a r t ], r 1,2,..., M 21, t 2 r( t t ) ( ) 1 cos[ a r t ], r M 21 1,...,2 M 21, t Where t tb ta, M 2 2 nm 21 We shall call h, h=1,2,, the sequence of functions of the type ( t, x, u) x ( t ) ( t, x, u) ( t ), r 1,2,..., M. r r r iii) For the third type of constraints, we choose 1 t s s otherwise. t ( 1) with [ a s t t, a st s ] s 1,2,..., L (see[9]). L L In the previous section, we have limited the number of constraints in the oriinal linear proram; the underlyin space is not, however, finitedimensional

7 H. R. ERFANIAN and M. H. NOORI SKANDARI/ TMCS Vol.2 No. (211) Definition: Let A be borel set and z a point in the space. The Radon measure z will be called atomic measure if 1 z A ( z)( A) ow. Now we approximate by a linear combination of atomic measure in followin proposition of [9]. Proposition: Let be a countable dense subset of. Given, a measure M ( ) can be found such that ( ) f, and ( ), 1,2,..., M, and the measure has the form k M k k ( z ), k 1 k where ( z ) is atomic measure, z, k, k 1,2,..., M. The infinite-dimensional linear prorammin problem in (15)-(18) can be approximated by the followin linear prorammin problem in which z, 1,2,..., N belons to a dense subset of. N max f ( z ), (19) 1 N i i s. t ( z ), i 1,2,..., M, (2) N N ( z ), h 1,2,..., M, (21) A ( z ) a, s 1,2,..., L, (22) s f, 1,2,..., N, where z ( t, y, x ), 1,2,..., N are constructed by dividin the sets, A, U into the number of equal subsets. By usin manner similar (see [9]) we can approximate the optimal pair [ x(.), u (.)] by considerin k such that if we set t ( k1, k ) then k u() t uk and the traectory x (.) will be obtained by the equation (6).. Computational Results We now present results from solvin model which assumed the parameters and initial values the same Gumel [3] as follows:

8 H. R. ERFANIAN and M. H. NOORI SKANDARI/ TMCS Vol.2 No. (211) s =1, r =.3, kv., N=1, kc.5, L=.25,.9,.2, kt.8, 12.1, with T () 1, Ti () 1 and V () 1. For more clearness, it is better to present these results throuh raphs. Fiures 1 and 2 show Healthy cells before and after treatment and Viral load before and after treatment, respectively. Fiures 3 and show the optimal control u1and u 2, respectively. In fact they show the best policy of drus treatment. Fi.1. Healthy cells before and after treatment treatment Fi.2.Viral load before and after Fi.3.The first control 5. Conclusion Fi.. The second contro In this paper an applicable and practical method for solvin nonlinear optimal control problems is presented that we developed it for best chemotherapy in treatment of HIV which this method is based on linear technique. It seems that by usin this method, we can obtain fine results by considerin a linear treatment of the nonlinear differential equations. Moreover, it is not necessary to impose any restriction on obective function of model. 657

9 H. R. ERFANIAN and M. H. NOORI SKANDARI/ TMCS Vol.2 No. (211) References [1] A.A.Eiu, An Efficient Treatment Stratey for HIV Patients Usin Optimal Control, African Institute for Mathematical Sciences (AIMS), Supervised by: Prof. Kailash C. Patidar,University of the Western Cape, South Africa,Submitted in partial fulfilment of a postraduate diploma at AIMS, 22 May 28. [2] H. R. Erfanian, A. V. Kamyad and S. Effati, The optimal method for solvin continuous linear and nonlinear prorams, Applied mathematical sciences, Vol. 3, No. 25, pp , 29. [3] A.B. Gumel, P.N. Shivakumar, B.M. Sahai, A mathematical model for the dynamics of HIV-1 durin the typical course of infection, in: Third World Conress of Nonlinear Analysts, vol. 7, pp , 21. [] A.Heydari, M.H.Farahi, A.A.Heydari, Chemotherapy in an HIV model by a pair of optimal control, Proceedins of the 7th WSEAS International Conference on Simulation, Modellin and Optimization, Beiin, China, September 15-17, 27. [5] H.R.oshi, Optimal control of an HIV immunoloy model, Optimal Control Applications and Methods, No.23, pp , 22. [6]. Karrakchou, M. Rachik, and S. Gourari. Optimal control and Infectioloy: Application to an HIV/AIDS Model. Applied Mathematics and Computation, 177:87 818, 26. [7]. A. V. Kamyad,. E. Rubio and D. A. Wilson, An optimal control problem for the multidimensional diffusion equation with a eneralized control variable, ournal of Optimization Theory and Applications, No.75(1), pp ,1992. [8] A. V. Kamyad,. E. Rubio, and D. A. Wilson, The optimal control of the multidimensional diffusion equation, ournal of Optimization theory and Application, No.7 (1991), pp ,1991. [9]. E. Rubio, Control and Optimization the Linear Treatment of Non-linear Problems, Manchester, U. K., Manchester University Press, [1] H. Zarei, A. V. Kamyad, S. Effati, Maximizin of Asymptomatic Stae of Fast Proressive HIV Infected Patient Usin Embeddin Method Intellient Control and Automation, No.1, pp.8-58,