Cells that are susceptible to HIV infection are called T(target) cells. Let T(t) be the population of uninfected T-cells, T*(t) that of the infected T-cells, and V(t) the population of the HIV virus. A model for the rate of change of the infected T-cells is
dT*/dt = kVT - gT*, (eq1)
where g is the rate of clearance of infected cells by the body, and k is the rate constant for the infection of the T-cells by the virus. The equation for the virus is the same as
dV/dt = P-cV, (eq2)
but now the production of the virus can be modeled by
P(t) = NgT*(t).
Here N is the total number of virions produced by an infected T-cell during its lifetime. Since 1/g is the length of its lifetime, NgT*(t) is the total rate of production of V(t).
At least during the initial stages of infection, T can be treated as an approximate constant. Equations (eq1) and (eq2) are the two coupled equations for the two variables T*(t) and V(t).
A drug therapy using RT (reverse transcriptase) inhibitors blocks infection, leading to k~=0. Setting K = 0 in (eq 1), solve for T*(t). Substitute it into (eq 2) and solve for V(t). Show that the solution is
V(t) = [V(0)/(c-g)][ce^(gt) - ge^(-ct)].
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