ABSTRACT
In the following lab, optimal control is used to find an optimal chemotherapy strategy in the treatment of the human immunodeficiency virus (HIV). Unlike the last lab, where the dynamics of a population affected by an epidemic were considered, this problem studies the immune system of an individual. A great deal of research has been conducted on the effect of chemotherapy
on the HIV virus. For example, in [105], Kirschner et. al. study the effects of chemotherapy on reducing viral production, which is most applicable to drugs such as protease inhibitors. Here, we consider the chemotherapy of reverse transcription inhibitors, such as AZT, which affects the “infectivity” of the virus. These drugs interrupt key stages of the infection process during the life cycle of HIV within a host cell. Butler, Kirschner, and Lenhart created a model for this type of interaction and used optimal control to develop treatment strategies in [25]. This lab is based on their work. It is assumed the treatment acts to reduce the infectivity of the virus for a
finite time, until drug resistance occurs. The measure of benefit of chemotherapy treatment is based solely on the increase of the CD4+T cell count. Thus, the model used describes the interaction of the immune system with HIV. Let T (t) and Ti(t) be the concentration of uninfected and infected CD4+T cells, respectively, and let V (t) be the concentration of free virus particles. In this instance, concentration refers to the population count per unit volume. Let
s
1 + V (t)
be the source term from the thymus, representing the rate of generation of new CD4+T cells. Let r be the growth rate of T cells per day. This growth is assumed to be logistic, with a maximum level of Tmax. Let kV (t)T (t) be the rate that free virus cells infect T cells. Let m1, m2, m3 be the natural death rates of uninfected CD4+T cells (T ), infected CD4+T cells (Ti), and free virus particles (V ), respectively. Once infection of a T cell occurs, replication of the virus is initiated and an average of N virus particles are produced before the host cell dies. The control, u(t), is the strength of the chemotherapy, where u(t) = 0 is
maximum therapy and u(t) = 1 is no therapy. We note that maximum therapy u = 0 is probably unrealistic to achieve; a more realistic positive lower bound would be better. We leave the problem as originally stated, though. A flow chart is given in Figure 14.1. We wish to maximize the number of uninfected
Models
the “cost” of the chemotherapy to the body. The fixed time frame simulates the period before drug resistance occurs. Letting A ≥ 0 be the cost, or weight, parameter, the problem is
max u
AT (t)− (1− u(t))2 dt
subject to T ′(t) = s
1 + V (t) −m1T (t) + rT (t)
[ 1− T (t) + Ti(t)
Tmax
] − u(t)kV (t)T (t),
T ′i (t) = u(t)kV (t)T (t)−m2Ti(t), V ′(t) = Nm2Ti(t)−m3V (t), T (0) = T0 > 0, Ti(0) = Ti0 > 0, V (0) = V0 > 0, 0 ≤ u(t) ≤ 1.
