Reed-Frost Chain Binomial Models

A classic model of infectious disease transmission was developed during the 1930s by Lowell J. Reed and Wade Hamptom Frost of Johns Hopkins. Because the model is simple to explain and provides valuable insights, we will discuss it at this time. In the classical Reed-Frost model, we assume a fixed population of size N . At each time, there are a certain number of cases of disease, C, and a certain number of susceptibles, S. We assume each case is infectious for a fixed length of time, and ignore the latent period; when individuals recover, we assume that they are immune to further infection. During the infectious period of each case, we assume that susceptibles may be infected, so that the disease may propagate further. This constitutes an idealized, or abstract, model, exhibiting some features of an epidemic system. Because we assume a fixed length infectious period and neglect the latent period, the generations of infection stay separate. At the beginning, we have only the generation of cases that starts the disease transmission. After the recovery of this generation, the new cases that resulted from transmission constitute the second generation of cases. These, in turn, recover, but may give rise to a third generation. Let C1 be the number of cases in the first generation, and S1 be the number of susceptibles that the first generation may place at risk of new infection. Similarly, let C2 be the number of cases in the second generation, and S2 the number of susceptibles present for the second generation of cases to potentially expose to disease; in general, the number of cases at generation t is denoted Ct and there are St susceptibles at that time. The basic Reed-Frost model assumes homogeneity of risk of infection throughout the population. In particular, we assume that each susceptible has a risk p of being infected by any of the infectives in the population. In a more realistic model, we might assume that