Reed-Frost Model Research Articles

369: An Approximate Chain-Binomial Model for Simple Epidemics

The study of the spread of epidemics is an important feature for the understanding of the growth of infectives in a population. The difficulty in developing a proper mathematical model for epidemics is due to the fact that their intrinsic properties and biological behaviour are not very well known. However, when a large population is considered one can make the assumption that the growth of infectives is Poisson. Using this assumption, Bartlett [1959] has developed theoretical models for the study of epidemics. Studies of simple and general epidemics were also done by Weiss [1965] and Dietz ([1966], [1967]). When one considers an epidemic in a population of small size it is not reasonable to assume that the growth of infectives is Poisson. Epidemics such as measles often occur in groups of small size and it is, therefore, interesting to investigate epidemics of this nature. It is reported in Bailey [1966] that a chain-binomial model was used for the analysis of data from measles epidemics in Rhode Island. In this model, it was assumed that the chance of an infection in a unit interval of time is p (= 1 q) so that the process develops as a chain of binomial distributions. In the present paper, a modification to the Reed-Frost chain-binomial model has been made without allowing the removal of infectives so that the total extinction of susceptibles is inevitable. With this assumption, a relation for the generating function for the number of susceptibles in the rnth time interval is derived. In the special case, when the probability of infection is small, the expected value and the variance of the number of susceptibles in the nth interval are obtained and the results are found to be realistic.

Markov Chain Methods in Chain Binomial Epidemic Models

In this paper, Markov chain methods are applied to chain binomial models in epidemics. In both Greenwood and Reed-Frost chain binomial models, it is shown that the susceptibles and susceptibles together with infectives respectively form Markov chains. These chains are used to obtain probabilities for the duration time and the total number of cases in an epidemic. A study of chain binomial models as Markov chains imbedded in continuous time processes is made. A practical application of the effects of inoculation on an epidemic is carried out, and some numerical results for the mean duration times and mean numbers of cases given.