Predicting the spatial dynamics of rabies epidemics on heterogeneous landscapes

Abstract

Often as an epidemic spreads, the leading front is irregular, reflecting spatial variation in local transmission rates. We developed a methodology for quantifying spatial variation in rates of disease spread across heterogeneous landscapes. Based on data for epidemic raccoon rabies in Connecticut, we developed a stochastic spatial model of rabies spread through the state's 169 townships. We quantified spatial variation in transmission rates associated with human demography and key habitat features. We found that large rivers act as semipermeable barriers, leading to a 7-fold reduction in the local rates of propagation. By combining the spatial distribution of major rivers with long-distance dispersal we were able to account for the observed irregular pattern of disease spread across the state without recourse to direct assessment of host-pathogen populations.

Figure 1

The month when rabies was first observed in each township, O_i, relative to the month of the first reported case in Ridgefield Township (March, 1991) in western Connecticut. The letters are the first names of townships discussed in the text: Ridgefield (R), Enfield (e), Union (u), Putnam (p), Clinton (c), Bridgewater (b), and South Windsor (sw).

Figure 2

A stochastic model was used to simulate the heterogeneous spread of raccoon rabies on an irregular network, illustrated here on a simple array. An infected township, i, infects its adjacent neighbor, j, at a rate λ_i,j. In addition, a township, j, may become infected because of translocation of rabid raccoons at a rate μ_j. Heterogeneity was incorporated by allowing the local rates from the neighbors, {λ_i,j}, and the rate of translocation, {μ_j}, to be different, possibly unique in different models. Each algorithm for associating a set of rates with rivers or human population density defines a stochastic candidate model. The simulation algorithm involved six steps. (A) For each township, add the rates from all possible routes of infection. (B) Add the township rates to compute a total rate. (C) Draw a random number to determine the elapsed time. (D) Check to see if any of the edges had become infected in the elapsed interval. (E) If no edges were forced, select a random township to infect. (F) Infect the forced edge or the infected township, update the local rates, and repeat until each township becomes infected.

Figure 3

The data and output from the fitted model River. (a) The expected vs. observed date of first appearance. (b) The elements of the χ² statistic, (O_i − E_i)²/E_i, plotted at the centroid of each township. In red circles the first observed case of rabies was earlier than that predicted by the model; outliers are Putnam (p), South Windsor (p), Union (u), Clinton (c), Enfield (e), and Bridgewater (b). In green squares, the first observed case was later than the prediction. (c) The date of the first appearance vs. the date of the median case; by this criterion, Putnam clearly is an outlier. (d) The epidemic simulated without rivers and plotted by the associated delay. According to the models, a 7-fold slower rate across rivers is responsible for an 11-month delay in the expected appearance of rabies in the southeast corner of Connecticut. The expected delay would be 16 months without long-distance translocation.