Measuring Spatial Dependence for Infectious Disease Epidemiology

Abstract

Global spatial clustering is the tendency of points, here cases of infectious disease, to occur closer together than expected by chance. The extent of global clustering can provide a window into the spatial scale of disease transmission, thereby providing insights into the mechanism of spread, and informing optimal surveillance and control. Here the authors present an interpretable measure of spatial clustering, τ, which can be understood as a measure of relative risk. When biological or temporal information can be used to identify sets of potentially linked and likely unlinked cases, this measure can be estimated without knowledge of the underlying population distribution. The greater our ability to distinguish closely related (i.e., separated by few generations of transmission) from more distantly related cases, the more closely τ will track the true scale of transmission. The authors illustrate this approach using examples from the analyses of HIV, dengue and measles, and provide an R package implementing the methods described. The statistic presented, and measures of global clustering in general, can be powerful tools for analysis of spatially resolved data on infectious diseases.

Conflict of interest statement

Competing Interests: The authors have declared that no competing interests exist.

Figures

Fig 1

The transmission kernel determines the spatial area where directly transmitted cases may be found (area of elevated risk, red circle in (B)). In addition, due to branching in the transmission chain there may be a larger area of elevated prevalence from more distally related cases (area of elevated prevalence, blue circle in (A) and (B)). (C) Example transmission chains. Each color represents a transmission chain of a different strain. The strains are divided into two serotypes, with the blue and red circles representing one serotype and the green and yellow squares a second serotype. (D) Increasing ability to discriminate between different transmission chains will increase the power of the τ-statistic to identify the spatial extent of elevated prevalence. Where case-pairs are known, we can identify the area of elevated risk (black line).

Fig 2. Overview of simulated epidemics of 100 separate transmission chains.

Each transmission chain consisted of cases caused by a single strain, with all strains divided into four serotypes. (A) Distribution of underlying population for simulated data. The black square represents the area of analysis to avoid edge effects from the simulation process. The two black dots represent two ‘surveillance hospitals’ (relevant for the spatially-biased observation scenario set out in Fig 3). (B) Using disease type information (blue) allow the τ-statistic to correctly show no spatial clustering of disease in a spatially clustered population, whereas the pair correlation function would falsely indicate clustering in disease cases. (C) τ-statistic results from simulated epidemics where the cases infected individuals between 0 and 100m away. Each case infected one individual (effective reproductive number of one). (D) As (C) but each case infects two individuals (effective reproductive number of two).

Fig 3. Performance of the τ-statistic under different observation scenarios: complete observation (blue solid line), spatially random observation (dashed blue line) and spatially biased observation (purple line).

For the spatially random observation all cases had a 1% probability of being observed. For the partially observed observation, the probability of observation was 0.1xexp(-d), where d was the distance (in km) to the closest ‘surveillance hospital’ in the area (marked by two black dots in Fig 2A).

Fig 4

(A) τ-statistic results from 1,912 dengue cases that presented at a Bangkok hospital between 1995 and 1999. The solid line represents the probability of cases occurring within the same month being of the same serotype, relative to any two cases being of the same serotype and occurring within the same month. The dashed line is the same analysis but with a temporal lag of four months and the dotted line are the results using a temporal lag of 15 months. Confidence intervals for the temporal lags of four months and 15 months are presented in S1 Fig. (B) Smoothed τ-statistic of HIV incident to HIV prevalent individuals in Rakai Uganda. The τ-statistic is calculated within households (i.e, at 0m) and in a sliding 250m window up to the average distance between community households, 3.3km. (C) τ-statistic results from 188 measles cases from Hagelloch, Germany in 1861–1862. The τ-statistic is calculated using a sliding 50m window and compares the risk of observing cases at that distance range for individuals sick within two weeks of each other to all time.