Since the high economic growth period in Japan private motorcars have proliferated in the local cities, where they have become necessities of life for most dwellers today. This paper considers the present state of proliferation by applying some statistical methods to a set of cross-sectional data, an array of transportation and socio-economic variables in 1990 for each of eighty-four medium-sized cities located outside major metropolitan areas (Table 1). In order to solve the problem, the author studies the systematic relationship among the variables or the transportation system. The results may be summarized as follows:
First, the eighteen variables on attributes of household, proliferation rate of private vehicle, modal choice in commuting, urban form, road environment and public transit shown in Table 2 were defined as indicators of the transportation system. Using exploratory factor analysis, they were grouped and simplified into five common factors which can be used as sorts of latent variables. The results of the factor analysis are given in Table 3. Of five factors extracted, Factor 5 was not identified even after a promax oblique rotation. Factor 1 was identified as household car ownership, Factor 2 as private traffic generation and traffic restraint, Factor 3 as public traffic generation, Factor 4 as compact car (Kei-jidosha) ownership in suburbs. These four factors correspond to essential elements of the above-mentioned transportation system.
Second, for the respective group of key variables comprising each factor, the causal sequence in their internal correlations was examined by means of path analysis to clarify a property of the element. The following became clear after the investigation of the four arrow diagrams in Figure 1 to Figure 4 that show the results of the analysis: (1) The level of household car ownership is influenced by the number of commuters in the household and the family income. In particular the income level has an effect on the proliferation rate of passenger cars. (2) The incidence of traffic accidents is influenced by the model choice of motorcycles in commuting and the level of traffic congestion. This causal relationship is consistent with empirical facts. (3) The level of proliferation of both bus and taxis is influenced by the D.I.D.'s population density. It was proved that the urban form affects the level of public transit service. (4) The level of household compact car ownership is influenced by the proportion of the D.I.D.'s area to the city area. That is, the larger the proportion is, the higher becomes the level.
Third, the relationship among elements of the transportation system was illustrated by hypothesizing a causal model for latent variables derived from the factors and then testing it through the multiple indicator method. The results in Table 4 and Figure 5 were obtained empirically, and indicate that three latent variables derived from the first three factors have significant relationships causally. The model represents a link in the chain of the causal cycle in which proliferation of private motorcars causes loss of public transit passengers.
Finally, the first latent variable scores for the eighty-four cities were estimated and examined. This latent variable, household car ownership, is a key exogenous one which precedes causally. Figure 6 shows the highest scores to be located in the northern Kanto, the Hokuriku, and the Tokai Districts, the lower ones generally in Northeastern and Western Japan. Furthermore, cities with higher scores are those in which the secondary activities in the economy are of great importance and the increase in population is remarkable (Table 5).

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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:
L_i(Y_{i, t=0}, …, Y_{i, t=T}|β°_i, δ_i)=Π_t1/√2πVar[Y_{i, t+1}]·exp{-1/2Var[Y_{i, t+1}](Y_{i, t+1}-E[Y_{i, t+1}])}, E[Y_{i, t+1}]=β°_i/M_iX_{i, t}Σ_jm_ijY_{j, t}, Var[Y_{i, t+1}]=β°_i/M_iX_{i, t}Σ_jm_ijY_{j, t}(1-β°_i/M_iΣ_jm_ijY_{j, t}), X_{i, t}=δ_i-Σⁱ_s=0Y_{i, s}, where M_i=Σ_jm_ij; L_i denotes the likelihood of the model for region i; X_{i, t} denotes the estimated rate of susceptibles in region i at week t; Y_{i, t} denotes the reported rate of infectives in region i at time t; m_ij 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; δ_i 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 Σ_jm_ijY_{j, t}/M_i. The probability of a susceptible in region i infected at time t is, therefore, β°_iΣ_jm_ijY_{j, t}/M_i.
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 T_ij correspond to m_ij as follows: m_ij=T_ij+T_ji (i≠j), m_ii=T_ii.
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).

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This paper examines regional difference in distance-decay parameters of a doubly-constrained gravity model, using a data set of the 1985 inter-municipal migration in Japan. The parameters are estimated for each of 28 prefectures. The model provides reasonably good representation of migratory flows for all the prefectures (Table 3).
The distance-decay parameters are distributed roughly in such a way that the prefectures of central Japan have more-negative estimates and those of peripheral Japan have less-negative estimates (Fig. 2) as if a nationwide map-pattern effect were operating. Simple correlation analyses reveal that the distance-decay parameter is significantly correlated with demographic variables relavant to migration volume and population size as well as socio-economic variabels, but rarely correlated with map-pattern variables (Table 4).
A stepwise multiple-regression analysis is employed to regress the distance-decay parameter on only those significant at the 0.10 level or less, among the 46 independent variables. As a result, 56% of the total variance is accounted for by 4 variables (Table 5): population density, growth rate of manufacturing workers, physical-structure index (Griffith and Jones, 1980) capturing a prefecture-specific map-pattern effect and out-migration volume from prefectures.
Taking into account their signs of standardized partial regression coefficients, it is concluded that the distance-decay parameter is more-negative in the prefectures which are high-densely inhabited, which suggests, in a sense, that urbanization has reached an advanced stage, where municipalities are well connected, where industrial locations have recently become stagnant and where large migratory outflows from prefectures do not take place, and vice versa.

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