Project D02 - Modelling the HIV Project D02 - Modelling the HIV Infection
Mathematical models have proven valuable in understanding the dynamics of HIV-1 infection. In this project, you will use actual data to study a very simple HIV dynamic model.
  1. In 1994, potent antiretroviral drugs were being tested. The results were dramatic. The amount of virus measured in blood plasma fell rapidly once the drug was given. The data from a single patient are shown below. (These results were typical.)

    Days after treatment 0 2 7 11 16 20 26
    Virions per ml (in thousands) 99.3 53.1 21.9 6.21 4.19 2.03 0.897

    Plot this data.

  2. Because the data appeared to show that the virus concentration fell exponentially for a short while after the patient was placed on the drug, the following model was introduced:
    dV
    dt
    		 = P - cV,
    where P is an unknown function representing the rate of virus production, c is a constant, and V is the virus concentration. Assume that the drug causes P = 0. Explain what this means. Is this a reasonable assumption? What is the solution of the differential equation in this case?
  3. Plot lnV versus t and use linear regression to estimate c. Also estimate the half-life, t1/2, of the virus in the plasma. What value for V0 is predicted by linear regression? Compare this to the actual value.
  4. Recall that the data given above was from a single patient (Patient A). The following table gives values of V0 (in thousands) and t1/2 (in days) for another ten patients.

    Patient B C D E F G H I J K
    Initial concentration (V0) 193 80 41 121 88 175 185 130 70 101
    Half-life (t1/2) 2.3 2.6 3.3 2.5 2.1 1.3 1.5 2.4 2.3 1.7

    Use the data from patients A-K to obtain mean values for V0 and t1/2. Determine the value of c that corresponds to this value of t1/2. Use this c from this point on.

  5. A patient with an HIV-1 concentration of 156,000 virions per ml was given the drug. After how many days would you expect the concentration to drop to 1000 virions per ml?
  6. Two days after treatment began, a patient's HIV-1 concentration was 75,300 virions per ml. Six days later the concentration was 23,500 virions per ml. Explain why these results worried the researchers.
  7. Refer to Problem 2. Data suggest that for a period of time before therapy begins, the concentration of HIV-1 is in a steady state i. e. dV/dt = 0. How could one use this to determine the viral production rate before therapy. Discuss the pros and cons of this approach.
Reference: Perelson, A. S. and Nelson, P. W., Mathematical Analysis of HIV-1 Dynamics in Vivo, SIAM Review, 41 (1999), pp. 3-44.

File translated from TEX by TTH, version 1.98.
On 24 May 2000, 22:18.