Limit theorems for the generalized size of epidemic in a Markov model with immunization

Abstract

We suggest a new Markov model for the spread of an epidemic in a closed population in which, in addition to conventional transitions interpeted as the process of infection of susceptibles and removing of infectives there exist also transitions related to immunization of susceptibles. In the well-known model with natural immunization the probability of the latter transition is proportional to the number of possible contacts of susceptibles and infectives, whereas in the model in question this probability is proportional only to the number of infectives; that is, the probability of immunization increases with increasing the number of infectives. For the model introduced we describe the class of limit laws for the generalized size of an epidemic (the sum of the number of infected and the number of immunized ones by the time the epidemic is finished) for various relations between the input parameters of the model under the assumption that the initial numbers of susceptibles and infectives tend to infinity and the parameters of the model are varying (the ‘triangular arrays’)