A discrete branching process model for the spread of HIV via steady sexual partnerships

The transmission of HIV in a monogamous heterosexual population structured by the ordinal number of the current partnership is considered. The sexual carreer of a man (woman) is thought to be a succession of k( m) partnerships, and a multitype Galton-Watson process is defined, in which the objects are infections and the types are related to the ordinal number of the partnership during which a person has acquired the infection. Contrary to multitype models in which the types are not age-related in some sense, this process contains at least two singular types, namely infections acquired in the last partnership of a man or a woman. The criticality parameter of this branching process is the epidemic threshold parameter R(0). In the case k= m an epidemic is impossible, however large k may be, if the difference between the ordinal numbers of the partners in a pair is never > 1. When the frequency of pairs in which this difference is >or= 2 increases, then R(0) increases. This is demonstrated for the cases k= m=3 and k=4, m=3. The formulae obtained show also the joint influence of the mixing pattern and of variable infectivity. The result for the case of uniform mixing implies that a formula of May and Anderson (1987) is an approximation for k and m large.