The lifespan of a human CD4+ T cell can be modeled by:
Pr(cell still alive at time t) = e-αt t ≥ 0
where t is measured by days.
The immune system of an HIV patient harbors a mixed population of healthy and sick (HIV-infected) cells. The lifespan of healthy cells obeys the model above. The lifespan of infected cells obeys a similar model, except that α is replaced by 5α. Assume that the fraction of healthy cells is p and the fraction of infected cells is 1 - p.
a. Find the probability that a randomly selected cell is still alive after τ days.
b. In order to enrich a blood sample for healthy cells, the sample is kept for τ days before further use. Derive a formula for the probability that a cell chosen at random from the enriched sample is infected (as a function of the constants α, p, and τ). Assume that the cultured cells do not proliferate during the enrichment phase.