Longini Koopman model: In epidemiology the model for primary and secondary infection, based on the classification of the extra-binomial variation in an infection rate which might arise due to 'clustering' of the infected individual within the households. The assumptions underlying the model are given as follows:

* a person might become infected at most once during the course of epidemic;

* all persons are members of the closed 'community'. Additionally each person belongs to the single 'household'. A household might comprise of one or several individuals;

* the sources of infection from community are distributed homogeneously through- out the community. Household members merge at random within the household;

* each person could be infected either from within household or from the community of it.

The probability that the person is infected from community is not dependent on the number of infected members in his or her household;

The probability that exactly the k additional individuals will become infected for the household with s initial susceptible and j  is the initial infections is
 


where B is probability that the susceptible individual is not infected from community during the course of the infection, and Q is the probability that the susceptible person escape infection from the single infected household member.