A mathematical model is built for influenza or other similar disease epidemics in a multi-region setting. The model is an extended type of chain-binomial model applied to a large population (Cliff et al., 1981), taking into account interregional infection by interregional contacts of people. If the magnitude of the contact is presented by simple distance-decay spatial interaction or the most primitive gravity model, a conventional gravity-type epidemic model (Murray and Cliff, 1977; Thomas, 1988) is deduced.
Given the number of infectives and susceptibles, the chain-binomial model predicts the number of infectives in the next period with binomial probability distribution. Available data are, however, weekly cases per reporting clinic in each prefecture reported by the surveillance project, characterized by continuous variation; the data could be a surrogate index for rates of infection. The author modified the model to use rates of infectives and susceptibles, and used a normal approximation of binomial distribution. With the maximum-likelihood method, this model can be calibrated. The specification of the model is as follows:
Li(Yi, t=0, …, Yi, t=T|β°i, δi)=Πt1/√2πVar[Yi, t+1]·exp{-1/2Var[Yi, t+1](Yi, t+1-E[Yi, t+1])}, E[Yi, t+1]=β°i/MiXi, tΣjmijYj, t, Var[Yi, t+1]=β°i/MiXi, tΣjmijYj, t(1-β°i/MiΣjmijYj, t), Xi, t=δi-Σis=0Yi, s, where Mi=Σjmij; Li denotes the likelihood of the model for region i; Xi, t denotes the estimated rate of susceptibles in region i at week t; Yi, t denotes the reported rate of infectives in region i at time t; mij denotes the size of interregional contact with the people in regions j for the people in region i; β°i denotes the infection parameter in region i; δi denotes the parameter concerned with the rate of initial susceptibles in region i.
The model posits that the average number of people who come into contact with a susceptible in prefecture i is a constant, and that the average rate of infectives of the people is ΣjmijYj, t/Mi. The probability of a susceptible in region i infected at time t is, therefore, β°iΣjmijYj, t/Mi.
This model was applied to a weekly incidence of influenza in each prefecture, from the 41st week, 1988, to the 15th week, 1989, Japan, letting the size of interregional passenger flow Tij correspond to mij as follows: mij=Tij+Tji (i≠j), mii=Tii.
Goodness-of-fits (Table 1) of one-week-ahead forecasts were almost satisfactory except for prefectures whose epidemic curves were bi-modal (e.g., Hokkaido) or whose transition speed between epidemic breakout and peak was too high (e.g., Yamagata). The latter might be explained by a cluster of group infection (e.g., school classes) in an earlier phase of the epidemic (see Fig. 4).
