\input amstex \documentstyle{amsppt} \loadmsbm \magnification=\magstephalf \hcorrection{1cm} \vcorrection{-6mm} \nologo \TagsOnRight \NoBlackBoxes \headline={\ifnum\pageno=1 \hfill\else% {\tenrm\ifodd\pageno\rightheadline \else \leftheadline\fi}\fi} \def\rightheadline{EJDE--1998/32\hfil Optimizing chemotherapy in an HIV model \hfil\folio} \def\leftheadline{\folio\hfil K.R. Fister, S. Lenhart, \& J.S. McNally \hfil EJDE--1998/32} \def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt % Electronic Journal of Differential Equations, Vol. {\eightbf 1998}(1998), No.~32, pp.~1--12.\hfil\break ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \hfill\break ftp 147.26.103.110 or 129.120.3.113 (login: ftp)\bigskip} } \topmatter \title Optimizing chemotherapy in an HIV model \endtitle \thanks {\it 1991 Mathematics Subject Classifications:} 34B15, 49K15, 92D30.\hfil\break\indent {\it Key words and phrases:} Chemotherapy, HIV, Optimal Control. \hfil\break\indent \copyright 1998 Southwest Texas State University and University of North Texas.\hfil\break\indent Submitted April 15, 1998. Published December 4, 1998. \endthanks \author K. Renee Fister\\ Suzanne Lenhart \\ Joseph Scott McNally\endauthor \address K. Renee Fister, Department of Mathematics and Statistics, Murray State University, Murray, KY 42071 USA \endaddress \email kfister\@math.mursuky.edu \endemail \address Suzanne Lenhart Department of Mathematics, University of Tennessee, Knoxville, TN 37996 USA \endaddress \email lenhart\@math.utk.edu \endemail \address Joseph Scott McNally Emory University, School of Medicine, Atlanta, GA 30322 USA \endaddress \email jmcnall\@learnlink.emory.edu \endemail \abstract We examine an ordinary differential system modeling the interaction of the HIV virus and the immune system of the human body. The optimal control represents a percentage effect the chemotherapy has on the interaction of the CD4$^+$T cells with the virus. We maximize the benefit based on the T cell count and minimize the systemic cost based on the percentage of chemotherapy given. Existence of an optimal control is proven, and the optimal control is uniquely characterized in terms of the solution of the optimality system, which is the state system coupled with the adjoint system. In addition, numerical examples are given for illustration. \endabstract \endtopmatter \document \def\op{\overline p} \def\oq{\overline q} \def\ou{\overline u} \def\ov{\overline v} \def\ow{\overline w} \def\oz{\overline z} \def\oy{\overline y} \def\oT{\overline T} \def\oV{\overline V} \def\olambda{\overline \lambda} \heading{1. Introduction} \endheading Various chemotherapies for patients with human immunodeficiency virus (HIV) are being examined to determine the optimal scheme for treatment. The questions of when and how treatment should be initiated given that treatment can only be continued for a finite interval are analyzed. An ordinary differential equation model which describes the interaction of HIV in the immune system is utilized, and optimal control of this ordinary differential equation model is explored. Also, the concept that the treatment control affects the ability of the virus to further infect the patient is assumed. The challenge of this disease is that the CD4$^+$T cells (CD4 positive T lymphocytes, a type of white blood cells) that HIV infects are the very ones that are necessary to ward off the invasion. The $CD4$ represents a protein marker on the surface of the CD4$^+$T cell. The T in the CD4$^+$T cell describes the connection to the thymus gland. After the initial CD4$^+$T cells flow from the bone marrow where they are produced, they arrive in the thymus gland where they mature. On their surfaces, they possess proteins that can bind to foreign substances, such as HIV. At these connectors, the HIV is fused into the host CD4$^+$T cell. Since HIV is a retrovirus, the RNA of the virus is converted into DNA inside the CD4$^+$T cell. Thus, the DNA of the virus is duplicated and the new virus particles bud from the CD4$^+$T cell. This process can proceed slowly and allow the CD4$^+$T cell to survive, or it can cause the host cell to erupt and die. The hallmark of the onset of AIDS (Acquired Immune Deficiency Syndrome) is the depletion of the CD4$^+$T cell population, which lowers the effectiveness of the immune system. The CD4$^+$T cells serve as command centers for the immune system and elicit responses from the CD8$^+$T cells and the B cells that can lead to the destruction of the virus. See the discussion in reference [10] and the survey paper by Kirschner [7] for background and references in HIV models. Let $T$ represent the concentration of the uninfected CD4$^+$T cells, and let $T^*$, $T^{**}$ denote the concentrations of latently infected and actively infected CD4$^+$T cells. Let $V$ denote the concentration of free infectious virus particles. Definitions and numerical data for the parameters can be found before the references. Using the model from [9], the assumption is made that the populations evolve as follows: $$ \gather {dT\over dt} = {s\over 1+V} -\mu_T T + rT \left( 1- {T+T^*+T^{**} \over T_{\max}} \right) - K_1 VT \tag1.1 \\ {dT^*\over dt} = K_1 VT -\mu_T T^* - K_2 T^* \tag1.2 \\ {dT^{**}\over dt} = K_2 T^* -\mu_b T^{**} \tag1.3 \\ {dV\over dt} = N\mu_b T^{**} -K_1 VT -\mu_v V \tag1.4 \endgather $$ with initial conditions $T(0)=T_0$, $T^*(0)=T_0^*$, $T^{**}(0)=T_0^{**}$, and $V(0)=V_0$ for infection by both infected cells and virus. In (1.1), $s/(1+V)$ is a source term from the thymus and represents the rate of generation of new CD4$^+$T cells. The T cells have a finite life span with a death rate $\mu_T$ per cell. In (1.2), latently infected T cells are assumed to have a natural death rate, $\mu_T$, even though other factors can change the natural death rate. In (1.1), $r$ is the coefficient of the growth rate of T cells, which is a logistic-type growth. This growth ensures that the T cells never grow larger than $T_{\max}$. In (1.1) and (1.2), the term $K_1VT$ models the rate that free virus infects CD4$^+$T cells. After a T cell becomes infected, it becomes a latently infected T cell. Hence the $K_1VT$ term is subtracted from (1.1) and added to (1.2). Equation (1.3) describes the actively infected CD4$^+$T cells. At the rate $K_2$, latently infected T cells become actively infected. The actively infected T cells manufacture virus and die at a rate per cell $\mu_b$. Equation (1.4) models the free virus population. An assumption is made that when an actively infected CD4$^+$T cell becomes stimulated by antigen exposure, replication of the virus begins. Further, $N$ viruses are formed before the host cell dies. Also, free virus is lost by connecting to CD4$^+$T cells at a rate $K_1$. The term $-\mu_vV$ takes into account loss of infectivity or removal from the body. See the references [1, 6] for control problems on similar models of HIV infection and see [5] for background on treatment strategies. Note that this paper contains existence and uniqueness results with proofs for the optimal control, and such results are not given in [1] or [6]. Here, an optimal chemotherapy treatment is considered with the control affecting the interaction term $K_1VT$. This control represents the percentage of effect the chemotherapy has on interaction of T cells with the virus. The control for the chemotherapy, $u(t)$, multiplies the parameter $K_1$ in equations (1.1) and (1.2). Therefore, the control class is chosen to be measurable functions defined on [$t_0$, $t_1$], with the condition $0\le u(t) \le 1$. The interval of treatment is necessary since a plausible assumption is made that chemotherapy only has a certain designated time for allowable treatment. After some finite time frame, HIV is able to build up resistance to the treatment due to its mutation ability. Also, chemotherapy has potentially hazardous side effects. Therefore, the length of treatment is restricted. For most HIV chemotherapy drugs, the length of treatment is less than 500 days. In addition, the optimal control will depend implicitly on the length of the treatment, and the general shape of the optimal control stays the same as the treatment interval changes. Hence, %for $t_0\le t\le t_1$ where $t_1-t_0 < 2$ years, the state system is $$ \gather {dT\over dt} = {s\over 1+V} -\mu_T T + rT \left( 1- {T+T^*+T^{**} \over T_{\max}} \right) -(1- u(t)) K_1 VT \tag1.5 \\ {dT^*\over dt} = (1-u(t)) K_1 VT -\mu_T T^* - K_2 T^* \tag1.6 \\ {dT^{**}\over dt} = K_2 T^* -\mu_b T^{**} \tag1.7 \\ {dV\over dt} = N\mu_b T^{**} -K_1 VT -\mu_v V \tag1.8 \endgather $$ with given initial values for $T$, $T^*$, $T^{**}$, and $V$ at $t_0$. The objective functional is defined as $$ J(u) = \int_{t_0}^{t_1} \left[ T(t) - {1\over2} B u^2 \right] dt. $$ The parameter $B$ represents the ``weight'' on the benefit and cost. The benefit based on the T cell count is being maximized and the systemic cost based on the percentage effect of the chemotherapy given is being minimized. If $u(t)=1$ represents maximal use of chemotherapy, then the maximal cost is represented by $u^2$. The goal is to characterize the optimal control $u^*$ satisfying $\displaystyle{\max_{0\le u\le 1} J(u) = J(u^*)}$. In section 2, a constraint on $T_{\max}$ is discussed, {\it{a priori}} estimates on the solutions are determined, and existence of an optimal control is investigated. In section 3, the aim is to seek to maximize the objective functional which is based on the benefit of the T cell count less the cost of the damage to the patient's body. The optimal control is characterized using Pontryagin's Maximum Principle. In section 4, uniqueness of the optimality system, which is the state system coupled with the adjoint system, is determined. In section 5, a numerical illustration for our problem is presented. \heading{2. Existence of an optimal control} \endheading There are certain parameter restrictions that are imposed to ensure that this model is realistic. For the death rate at $T_{\max}$ to be greater than the supply rate, an assumption is that $$ \mu_T T_{\max} > s.\tag2.5 $$ The steady state population size should be below $T_{\max}$ in order for the T cell population to expand when stimulated by the infection of HIV. Furthermore if the population ever gets near $T_{\max}$, it's growth should slow. See [9] for analysis of stability properties for this model. Upper bounds are needed for the existence of an optimal control and in the uniqueness proof of the optimality system. Using $T(t)0 \\ {d\hat V\over dt} = N\mu_b \hat T^{**} \qquad \hat V(t_0)=V_0 \endgathered $$ or $$ \pmatrix \hat T^*\\ \hat T^{**}\\ \hat V\endpmatrix ^{'} = \pmatrix 0& 0& {T_{\max}}K_1\\ K_2& 0& 0\\ 0& N\mu_b& 0 \endpmatrix \pmatrix \hat T^*\\ \hat T^{**}\\ \hat V\endpmatrix $$ Since this is a linear system in finite time with bounded coefficients, then the supersolutions $\hat T^*$, $\hat T^{**}$, $\hat V$ are uniformly bounded. To determine existence of an optimal control to our problem, we use a result from ([2, Thm 4.1, pg. 68-69]). \proclaim {Theorem 2.1} Under assumption (2.5), there exists an optimal control $u^{*}$ that maximizes the objective functional $J(u)$. \endproclaim To prove this theorem, the following conditions must be satisfied. \item{i)} The class of all initial conditions with a control $u$ such that $u$ is a Lebesgue-integrable function on $[t_0,t_1]$ with values in the admissible control set and such that the state system is satisfied is not empty. \item{ii)} The admissible control set is closed and convex. \item{iii)} The right hand side of the state system is continuous, is bounded above by a sum of the bounded control and the state, and can be written as a linear function of $u$ with coefficients depending on time and the state variables. \item{iv)} The integrand of the functional is concave on the admissible control set and is bounded above by $c_2 -c_1 |u|^\beta $, where $c_1 >0$, and $\beta>1$. First, an existence result in Lukes ([8], Theorem 9.2.1) for the state system (1.5)--(1.8) for bounded coefficients is invoked. The control set is closed and convex by definition. The right hand side of the state system (1.5)--(1.8) has at most linear growth since the state solutions are {\it{a priori}} bounded. In addition, the integrand in the functional, $T(t) -{1\over2} B u^2$, is concave on the admissible control set. To complete the existence of an optimal control, one uses that $T(t) -{1\over2} B u^2\le c_2 -c_1 |u|^\beta $, where $c_1 >0$, and $\beta>1$, since $T(t)0$} \\ 0,&\text{if $s\le0$}\,.\endcases $$ If $\lambda_2-\lambda_1<0$ for some $t$, then $u^*(t) \ne 0$. Hence, $0 \widetilde C_1 + \widetilde C_2$ and $t_1 < {1\over3\lambda} \ln( { \lambda-\widetilde C_1\over \widetilde C_2 } )$, then $p=\op$, $p^*=\op^*$, $p^{**}=\op^{**}$, $q=\oq$, $w=\ow$, $z=\oz$, $v=\ov$, $y=\oy$. \qed \enddemo Uniqueness for a small time interval is not unusual in such a nonlinear boundary value problem. The unique optimal control $u^{*}$ is characterized in terms of the unique solution of the optimality system. The optimal control $u^{*}$ gives an optimal chemotherapy strategy for the HIV positive patient. \heading{4. Numerical Illustration} \endheading Using two different weight factors of $B=110$ and $B=30$, numerical solutions for the uninfected $T$ cells, virus concentration, and the optimal treatment strategy have been generated. The optimality system is solved using an iterative method with Runge-Kutta four scheme. Starting with a guess for the adjoint variables, the state equations are solved forward in time. Then those state values are used to solve the adjoint equations backward in time, and the iterations continue until convergence. See [3] for background on such iterative algorithms. The optimal treatment schedule is considered in the following picture. One notices that when $B$ increases from 30 to 110 that the chemotherapy is given at the maximal level for a shorter period of time before decreasing in a continuous manner. Mathematically, this is due to the fact that the percentage chemotherapy given is inversely related to the weight factor, $B$. This says that if the systemic cost to the patient increases, then the patient receives the maximal chemotherapy for a shorter period of time. % \input epsf \epsfxsize=9cm $$\epsffile{fig1.eps} $$ The graphs for the concentration of the uninfected $T$ cells and the virus concentration are viewed on the next two pages. As the optimal chemotherapy is given, the $T$ cell concentration increases in a logistic way. However, for a higher systemic cost, the $T$ cell population increases at a slower rate. With both weight factors, the maximal chemotherapy is given for about the first thirty days. The $T$ cell concentrations for these different factors produce similar graphs for the first thirty days. However, after the first thirty days of treatment, the $T$ cell concentration for $B=110$ increases but at a slower rate than for the $T$ cell concentration for $B=30$. % \epsfxsize=9cm $$\epsffile{fig2.eps} $$ If the systemic cost is greater, the body can not ``fight off'' the virus as effectively. Indeed, this idea is supported in the virus concentration graph. Therefore, for the higher systemic cost, the optimal chemotherapy given produces a lower $T$ cell concentration and a higher virus concentration. % \epsfxsize=9cm $$\epsffile{fig3.eps} $$ \noindent The following values were used for the constants in numerical calculations. \smallskip \settabs 3 \columns \+ $\mu_T = 0.02$/d & Death rate of uninfected and latently infected CD4$^+$T cells. \cr \+ $\mu_b = 0.24$/d & Death rate of actively infected cells. \cr \+ $\mu_V = 2.4$/d & Death rate of free virus. \cr \+ $K_1 =0.000024$mm$^3$/d & Rate CD4$^+$T cells become infected by free virus. \cr \+ $K_2 = 0.003$/d & Rate $T^{*}$ cells convert to actively infected. \cr \+ $r = 0.03$/d & Rate of growth for the CD4$^+$T cells. \cr \+ $N = 1200$ & Number of free virus produced by $T^{**}$ cells. \cr \+ $T_{\max} = 1500$/mm$^3$ & Maximum CD4$^+$T cell level. \cr \+ $s = 10$/mm$^3$ & Source term for uninfected CD4$^+$T. \cr \Refs \ref \no 1 \by Butler, S., D. Kirschner, and S. Lenhart \paper Optimal Control of the Chemotherapy Affecting the Infectivity of HIV \jour Advances in Mathematical Population Dynamics-Molecules, Cells and Man \publ Editors: O. Arino, D. Axelrod, and M. Kimmel, World Scientific Press \publaddr Singapore \yr 1997 \pages 557-569 \endref \ref \no 2 \by Fleming W.H. and R.W. Rishel \book Deterministic and Stochastic Optimal Control \publ Springer-Verlag \publaddr New York \yr 1975 \endref \ref \no 3 \by Hackbusch, W. K.\paper A numerical method for solving parabolic equations with opposite orientations \jour Computing \vol 20 \yr 1978 \pages 229-240 \endref \ref \no 4 \by Kamien, M. I. and N. L. Schwartz \book Dynamic Optimization \publ North-Holland \publaddr Amsterdam \yr 1991 \endref \ref \no 5 \by Kirschner, D. and G.F. Webb \paper A Model for treatment strategy in the chemotherapy of AIDS \jour J. AIDS \publ Bulletin of Mathematical Biology \vol 58 \yr 1996 \pages 367-391 \endref \ref \no 6 \by Kirschner, D., S. Lenhart, S. Serbin \paper Optimal Control of the Chemotherapy of HIV \jour Journal of Mathematical Biology \vol 35 \yr 1997 \pages 775-792 \endref \ref\no 7\by Kirschner,D. \paper Using Mathematics to Understand HIV Immune Dynamics \jour AMS Notices \vol 43:2 \yr 1996 \pages 191-202 \endref \ref \no 8 \by Lukes, D. L. \paper Differential Equations: Classical to Controlled \jour Mathematics in Science and Engineering \vol 162 \publ Academic Press\publaddr New York \yr 1982 \endref \ref \no 9 \by Perelson, A., D. Kirschner, and R. DeBoer \paper The Dynamics of HIV infection of CD4$^+$T Cells \jour Mathematical Biosciences \vol 114 \yr 1993 \pages 81-125\endref \ref \no 10 \by Yeargers, E., R. Skonkwiler, and J. Herod \book An Introduction to the Mathematics of Biology \publ Birkhauser\publaddr Boston \yr 1996 \endref \endRefs \enddocument \bye