Threshold behaviour for a chain-binomial S-I-S infectious disease

The present paper is concerned with a chain-binomial deterministic model for an infectious disease of the S-I-S type. The model extends the Cooke et al. (1977) and Longini (1980) model in the sense that it accounts for the distribution of the number of contacts made by each susceptible during an infectious period and for the probabilities of infection at the different contacts with infectives. The aim of the work is to investigate under which conditions the disease becomes endemic or not. Some partial results are first derived, but a complete analysis of the threshold behaviour seems very intricate. A more detailed discussion is then presented for the specific case where at least K + 1 contacts with infectives, K greater than or equal to 1 fixed, are required to make a susceptible infected.