On transient effects in the HIV/AIDS epidemic

During the initially exponential spread of the human immunodeficiency virus (HIV—the causative agent of AIDS) the growth rate of the number of AIDS cases decreases from plus infinity to the growth rate of HIV infections. A sensitivity analysis shows that for all reasonable values of the parameters of the HIV epidemic (incubation period, initial doubling time, etc.) the effect of this positive transient becomes negligible when the annual number of AIDS cases reaches a few dozen. Necessary and sufficient conditions are given for the growth rate of the number of AIDS cases to be monotonically decreasing during the positive transient. A mildly pathological density function for the incubation period of AIDS provides an example of a growth rate of AIDS that does not decrease monotonically, even though HIV is spreading exponentially. A negative transient occurs when the growth rate of HIV begins to decrease. In this context a somewhat surprising result emerges under the assumption that the growth rate of HIV is non-increasing: the growth rate of AIDS is at all times larger than the growth rate of HIV. A logistic HIV epidemic illustrates this result, and implications for the growth of the HIV epidemic in the United States and Europe are discussed. In particular, it is shown that the positive transient must have passed by 1982 in the United States and by 1986 or 1987 for the five European countries with the largest caseloads.