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 - δT* (1)
where δ 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 above:
dT/dt = P - cV (2)
, but now the production of the virus can be modeled by P(t)=δT*(t). Here N is the total number of virions produced by an infected T-cell during its lifetime. Since 1/δ is the length of its lifetime, NδT* (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 (1) and (2) are the two coupled equations for the two variables T*(t) and V(t).
A drug therapy using RT inhibitors blocks infection, leading to k≈0. Setting k=0 in (1), solve for T*(t). Substitute it into (2) and solve for V(t).