The effect of pair formation and variable infectivity on the spread of an infection without recovery

For the spread of an infection without recovery we compare the properties of a model in which contacts are assumed to be instantaneous and every contact is with a new partner to a model with an explicit description of partnership formation and duration. We investigate the dynamic behaviour at the beginning of the epidemic, the exponential growth phase and the endemic equilibrium in a population with a simple demographic process. Comparisons are made with the assumption that the basic reproduction ratio R0 is equal for both models. In a second step we introduce variable infectivity into both models and investigate how that interacts with the process of pair formation and partnership duration. The main results are: If partnership duration is taken into account, then (a) the prevalence may initially decrease, even if R0 > 1; (b) the exponential growth rate is lower than in a model without partnership duration; (c) the endemic equilibrium is higher than in a model without partnership duration. Furthermore we show that for a model with positive partnership duration for one value of R0 one might have more than one corresponding epidemic growth rate and endemic steady state. This implies that one cannot estimate R0 from the endemic level without further information on the partnership process. Finally we demonstrate that even for a very low value of per contact infectivity in the second stage of infection, as has been postulated for HIV-infection, transmission during the second stage may contribute significantly to the spread of the epidemic due to the long duration of the second stage. Therefore, reduction of the high infectivity in the short first stage of infection only affects R0 markedly if average partnership duration is short.