Another look at Gani’s carrier-borne epidemic with two stages of infection

This note is concerned with a carrier-borne epidemic in which the individuals initially susceptible pass through successive stages of infection and get eventually removed by repeated contacts with carriers. The model proposed here constitutes a generalized version of Gani’s model and is studied by a different method based on a direct conditional argument. First, the time-dependent behaviour of the conditional process of the susceptibles and the infectives in the successive stages, given any realization of the carrier process, is obtained using a simple martingale approach. Several particular processes for the carriers, together with different functions for their global contact rate with the other individuals, are then considered and analyzed. Combining these results leads easily to the building of the joint probability generating function of the numbers of susceptibles, infectives and carriers present at any time in the population.