Estimating the reproductive number in the presence of spatial heterogeneity of transmission patterns

Abstract

Background: Estimates of parameters for disease transmission in large-scale infectious disease outbreaks are often obtained to represent large groups of people, providing an average over a potentially very diverse area. For control measures to be more effective, a measure of the heterogeneity of the parameters is desirable.

Methods: We propose a novel extension of a network-based approach to estimating the reproductive number. With this we can incorporate spatial and/or demographic information through a similarity matrix. We apply this to the 2009 Influenza pandemic in South Africa to understand the spatial variability across provinces. We explore the use of five similarity matrices to illustrate their impact on the subsequent epidemic parameter estimates.

Results: When treating South Africa as a single entity with homogeneous transmission characteristics across the country, the basic reproductive number, R0, (and imputation range) is 1.33 (1.31, 1.36). When fitting a new model for each province with no inter-province connections this estimate varies little (1.23-1.37). Using the proposed method with any of the four similarity measures yields an overall R0 that varies little across the four new models (1.33 to 1.34). However, when allowed to vary across provinces, the estimated R0 is greater than one consistently in only two of the nine provinces, the most densely populated provinces of Gauteng and Western Cape.

Conclusions: Our results suggest that the spatial heterogeneity of influenza transmission was compelling in South Africa during the 2009 pandemic. This variability makes a qualitative difference in our understanding of the epidemic. While the cause of this fluctuation might be partially due to reporting differences, there is substantial evidence to warrant further investigation.

Figure 1

Imputed epidemic curves. Gray shading indicates the variability in the imputed data. The dashed line indicates the observed onset data.

Figure 2

Epidemic curves for each of the nine provinces. The shaded area indicates the variability in the imputed values.

Figure 3

Estimates of R_t using the transmission matrices, as described in the text. The estimates shown represent the average of the R_t estimates obtained across the 500 imputed epidemics. Days when no cases were reported have a R_t of 0, though we smooth through this for the purpose of visual presentation.

Figure 4

Estimated R_t by province. The line types for each plot are the same as those used in the previous figure. Days when no cases were reported have a R_t of 0, though we smooth through this for the purpose of visual presentation.

Figure 5

The relationship between characteristics of each of the provinces and the outbreak. Lines drawn reflect the least squares regression line for the relationship between the two variables. The first panel shows the relationship between the population size and the size of the outbreak in each province. The second panel describes the relationship between the population size and R₀. The third panel illustrates the relationship between the land area and R₀ obtained for each of the transmission matrices. The final panel plots the relationship between population density and R₀ for each transmission matrix. Line types follow the legend in Figure 3.