Interdisciplinary approaches in epidemic studies--II: Four geographic models of the flow of contagious disease.

Straightforward adaptation of geographical and regional-science models to conceptual modeling of the epidemic spread of a contagious disease is achieved by: using analogies and isomorphisms, and using as a real example the observations made during a well studied epidemic of variola minor (the mild form of smallpox), a typical contagious disease. The adaptation of the Wilson model of planning for urban development includes a static view (network) of the structure and activities of the population and of organizations (diffusion agencies such as day schools), and the dynamic view (mechanism of epidemic spread) which includes the changes with time of elements of the network brough by flow of disease. Adaptation of the Brown model of spatial diffusion yields flows of disease occurring between micro-scale units (households) of social interactions but aggregations of these units do not participate explicitly in the flows. The occurrence of successive generations of affected households is specified as well as the occurrence of definite stages of the epidemic progression. Adaptation of the Alves-Morrill model of spatial diffusion yields a network of social groups and interdependencies; a simplified network of the mechanism of spread that shows flows of disease between individuals grouped into generations of infected individuals and generations of infective individuals; and a more realistic view which shows the flows of disease between real epidemiological units such as households and school classes. Adaptation of the Morrill-Manninen model of spatial diffusion concentrates on the mechanism and parameters having the epidemic spread as output. The interdependencies between the parameters and between each parameter and the epidemic spread are represented, including feedback processes. Brown's model seems to be the best for describing the epidemic spread of contagious disease while the Morrill-Manninen model is the most promising for investigating the detailed mechanism of the spread. Since these two models complement each other, their combined use is indicated.

An objective reconstruction of the chain of contagion.

North's clustering method, which is based on a much used ecological model, the nearest neighbor distance, was applied to the objective reconstruction of the chain of household-to-household transmission of variola minor (the mild form of smallpox). The discrete within-household outbreaks were considered as points which were ordered in a time sequence using a 10-40 day interval between introduction of the disease into a source household and a receptor household. The closer points in the plane were assumed to have a larger probability of being links of a chain of household-to-household spread of the disease. The five defining distances (Manhattan or city-block distance between presumptive source and receptor dwellings) were 100, 200, 300, 400 and 500 m. The subchain sets obtained with the five defining distances were compared with the subchains empirically reconstructed during the field study of the epidemic through direct investigation of personal contacts of the introductory cases with either introductory or subsequent cases from previously affected households. The criteria of fit of theoretical to empirical clusters were: (a) the number of clustered dwellings and of subchains , (b) number of dwellings in a subchain and (c) position of dwellings in a subchain . The defining distance closest to the empirical findings was 200 m, which fully agrees with the travelling habits of the study population. Less close but acceptable approximations were obtained with 100, 300, 400 and 500 m. The latter two distances gave identical results, as if a clustering ceiling had been reached. It seems that North's clustering model may be used for an objective reconstruction of the chain of contagious whose links are discrete within-household outbreaks.

Temporal-spatial-social parameters in the spread of contagious disease.

A conceptual theoretical model was built on the basis of prominent concepts of the generally accepted knowledge on the spread of contagious disease. Subsequently, the model was applied to a real epidemic of variola minor (the mild form of smallpox) and four phases of the epidemic were disclosed. The phases discriminated themselves through their relationships to invasion of certain city subdivisions and rural districts and particularly, through the type of social units involved and the type of persons introducing the disease into these units. Introduction of variola minor into day schools with further spread in classes passed across a threshold which led to a boosting of the number of households affected and of the area involved in the epidemic. The phase of maximal spread corresponded to this operation of day schools as diffusion agencies. The temporal-spatial-social correspondence suggests that phases of the epidemic did occur as a result of periodic variation of the mechanism of spread.

Concepts of diffusion theory and a graphic approach to the description of the epidemic flow of contagious disease.

Concepts used to analyze sociological, geographic, and economic processes were adapted to an examination of the diffusion of contagious disease. The example used in applying these concepts was an epidemic of variola minor which continued for 12 months in an area of 1,006 square kilometers centered on the city of Bragança Paulista, Sao Paulo State (Brazil). A graphic procedure is proposed that depicts aspects of the epidemic flow of person-to-person transmission. Spatial, temporal, and sociological characteristics of the epidemic flow are disclosed in sequential diagrams. They represent geographic areas as well as schools and agglomerates of households affected by the epidemic at a given time, the mode of diffusion, and the source of the infection. The procedure yielded indirect evidence of the role of school pupils as introducers of variola minor into households and school classes. All subdivisions of the city, six of the seven rural districts, and four of the five elementary schools were affected through hierarchical (between-areas) diffusion. Subsequently, there was neighborhood (within-area) diffusion, and this resulted in new interactions between areas.

Contour mapping of the temporal-spatial progression of a contagious disease.

A methodology for contour-map study of contagious-disease epidemics is presented. Its application is exemplified in a smallpox epidemic occurring in a small Brazilian town. Computer-controlled contour-mapping of dates of introduction of variola minor into 169 households and the coordinates of the affected dwellings did not show a single contour pattern, but a group of subareal patterns of within-household outbreaks. Introduction by adults and pre-school children were distributed throughout the whole city area. However, introduction by school children formed two groups of contours and of affected dwellings. Each group was included in a discrete area corresponding to the zone of pupil recruitment of the two schools enrolling 91% of the school-child introductory-cases. The latter were responsible for introduction of the disease into 45% of the city's affected households. Altogether, both zones practically covered the whole city area. In either zone, several patterns surrounded the corresponding school. Even though no time value was entered for any school, contour maps clearly evidenced the influence of those two schools on spread of the epidemic. An estimated rate of linear spread of variola minor was 1.35 m per day in a city subarea.

Variola minor in Braganca Paulista County, 1956: household aggregation of the disease and the influence of household size on the attack rate.

Household aggregation of cases, one possible characteristic of person-to-person transmitted disease, was formally tested in one epidemic of variola minor by using a pair statistic. A significant result was found for all households as well as for households grouped by the type of environment, or by the phase of the epidemic growth in time. Secondary attack rates, when related to household size (number of susceptibles) showed only a marginal trend in rural households but no trend in urban or semi-rural households.

Variola minor in Bragança Paulista county, 1956. Observations on the clinical course of variola minor and on pregnancy in women with the disease.

Occasional observations on the clinical course of 485 cases of variola minor composing an epidemic are reported. 17% of the cases showed a complication. Otitis was observed in one case. An erythematous rash limited to the upper chest, neck and head appeared, instead of the pock eruption, after the pre-eruptive phase, in a previously vaccinated case. Convulsions, drowsiness, stupor, delusions, dizziness or deafness were observed in 23 patients whose individual characteristics are also presented. One of these cases showed a definite neurologicl syndrome: encephalitis. Neurologic complications were mostly seen in patients with medium to severe variola minor. Neither abortion nor death were seen among 4 pregnant women who developed variola minor, even though one of them delivered, on term, a dead foetus, 4 months after developing variola minor.

Variola minor in Bragança Paulista County, 1956. Distribution of onset of cases in the two schools including most school children with variola.

Class attendance during illness was confirmed for numerous pupils of two schools. Successive generations whose median cases were separated by an interval consistent with the "serial interval" of variola minor were clearly found in the epidemic curve for II of the 29 classes with cases from both schools. Twenty two other classes had no case at all.

Variola minor in Bragança Paulista County, 1956. Respiratory manifestations in cases of variola and in household contacts of cases.

A respiratory complication was recorded for 62 of the 485 cases of variola minor occurring in Bragança Paulista County, Brazil, 1956. The respiratory complication preceded onset of the classical variola minor in 2 cases; lasted only the pre-eruptive phase in 38 cases; lasted only the eruptive phase in 9 cases; lasted both the pre-eruptive phases in 1 case and had an unrecorded relationship to the clinical course in 12 cases. Six of the 17 household contacts with a respiratory syndrome contemporaneous with variola in the household had previously suffered from variola.

Variola minor in Braganca Paulista county, 1956: a trend-surface analysis.

Trend-surface analysis (TSA), a form of polynomial regression used in geology, ecology and geography, was applied to analysis of the spread of an epidemic of variola minor in a small Brazilian city. Cubic surfaces gave a generalized map of the space-time distribution of the epidemic, allowing those parts of the city to be identified where variola minor was spreading rapidly or slowly. The epidemic spread relatively quickly over the core area of the city, especially to the peripherally located household dwellings in a northeast to south-west direction. The dwellings of adults and pre-school children introducing the disease into their households broadly followed the overall pattern. School child introductory cases from a southern-located school yielded a saddle-shaped contour pattern, centered about the school; but this pattern was not repeated for the school serving the northern half of the city, which showed a ridge-shaped pattern dipping toward the west. Cubic surfaces for the influence of certain household and individual characteristics were investigated, but showed only week trends. The nearest match to the bowl-shaped overall pattern of introductory dates was provided by the vaccination level of the households. From this application, it appears that TSA permits the identification of regional trends in an objective manner and gives a quantitative measure of the importance of these regional trends in terms of the overall variation in the spatial pattern.

Analysis of the incidence of variola minor in an outbreak by use of a mathematical model.

The incidence of variola minor during an outbreak was analyzed by use of mathematical model B proposed by Chelsky and Angulo. Several parameters were estimated. Among these, the propagating rate (ratio of receptor cases to source cases) seems to predict the outbreak decline better than inspection of the incidence curve. The estimated mean generation interval (18.6 days) supports the thesis that variola minor is not transmitted at onset of illness but, on the average, about 4 days later. The ratio of clinical to subclinical infections (about 1 to 1) approximates those ratios found in serological surveys of variola minor and variola major outbreaks.

Variola minor in Bragança Paulista County, 1956: overall description of the epidemic and of its study.

An overall description of the epidemic of variola minor (alastrim) affecting Bragança Paulista County (Brazil) in 1956 is given. A total of 484 cases were recorded for 210 households, one hotel and one boarding school. At least 95 per cent of the households with cases of the disease in the capital city were surveyed and 90 per cent of the households in the rural districts. An orphanage, an old folk's home, the County jail and 10 schools operating in the capital city and some other social groups without cases were also surveyed, as well as 125 households without cases but with one or more contacts with the disease. An overall attack rate of 1924 cases, 267 cases and 781 cases per 100,000 inhabitants was obtained, respectively, for the capital city, the rural environment and the whole County. Clinical, epidemiological, serological and environmental surveys were conducted. Evidence on identification of the epidemic disease is presented. Study of the epidemic was made at these levels: the disease itself (frequency and severity); the disease in the individual (association of characteristics of persons with occurrence and clinical severity of variola minor); the disease in social units (occurrence and spread in households and school classes); the disease in small communities (occurrence and spread in housing projects, city blocks and farms); the disease in two large communities with contrasting socio-economic characteristics (the capital city and the rural environment); and the disease in the County as a whole.

Three tests for randomness of attack of social groups during an epidemic.

It is suggested that investigation of the attack of social units can be more illuminating of an epidemic process than the usual study of secondary attack rates of individuals. Three tests for detecting non-random differences in the occurrence of a disease between two groups of social units are presented. In each method the number of units attacked (X) in one of two groups of units is considered, rather than the number of individuals attacked. The tests are: (a) to use the Monte Carlo method, computer simulation of the epidemic process, to obtain an empirical distribution to which the observed value of X is compared; (b) to obtain the exact distribution of X; (c) to standardize X and compare it to the standard normal distribution. The three approaches are compared using data obtained from an epidemic of variola minor in two schools and the differences between the results are trivial. However, for large samples the only feasible approach is the Monte Carlo method.