We propose a stochastic, branching-process model of early events in vivo in human or simian immunodeficiency virus (HIV or SIV) infection and study the influence that the time of appearance of virus-specific antibodies or cytotoxic cells, or of administration of antiretroviral drugs, has on the probability of progression to a chronic infection. In some biological scenarios, our model predicts that a few days' delay in response or intervention would make little difference, while in others it would be highly deleterious. We show that prophylactic efficacy does not require perfect efficiency at neutralizing infectious virus. Data from a trial of PMPA, a potent antiretroviral drug, as post-exposure therapy for SIV infection in macaques, reported by C.-C. Tsai, P. Emau, K.E. Follis, T.W. Beck, R. E. Beneveniste, N. Bischofberger, J.D. Lifson, W.R. Morton (J. Virol. 72 (1998) 4265), provides a test of the model. We show that their observations are consistent with a branching-process without invoking supplementary viral- or host-variability. Finally, most animal trials of antiviral drugs or vaccines use very high viral inoculums; our model demonstrates that in such experiments we risk greatly underestimating the efficacy of these agents.