Traveling waves and spatial hierarchies in measles epidemics
Reviewed 09/29/11
Traveling waves have been predicted for many natural systems driven primarily by some sort of stimulus and impediment; for example host-prey, pathogen-host, invasion, etc. However the empirical evidence for this type of dynamic is traditionally limited due to a lack of adequate spatial and temporal resolution of the ecological process. The patterns of the measles epidemics in England during the latter half of the 20th century provide the kind of data that is necessary for traveling waves analyses.
Annual changes in birthrates, seasonal changes in the transmission of the virus via children attending school, and the large-scale introduction of a vaccine in 1968 make this disease system highly non-stationary however. For this reason, traditional Fourier analysis methods were abandoned in favor of local wavelet power spectrum (LWPS) which is more equipped to handle local changes in periodicity over time and space. Some drawbacks of this method are the requirements of a large amount of data and the inability to handle the analysis of the full time series at once (a restriction in period analysis was required for the three major British cities phase angle analysis).
The authors present results showing that two peaks, a roughly annual and biannual, of disease could be discerned for England and Wales during the pre-vaccination era. After the introduction on the vaccine we see a gradual increase in the periodicity of the longer-term epidemic, formerly biannual, that coincides with a steady rise in immune potential hosts. The most intriguing dynamics result from a closer analysis of the spatial patterns of disease centered around the density of human population centers, ie cities verses rural disease dynamics. There is a clear increase in the phase differences between population centers the farther a town is from London or Manchester, two major cities located in the east and northwest of the country, respectively. A smaller, yet still significant, trend can be shown for increasing population sizes as well; as the city gets larger the phase difference between it and the major urban centers decreases.
These facts lead to some important conclusions, mainly that small towns tend to have a lag in epidemic because they are not large enough for the disease to remain endemic on a seasonal or yearly basis, they require an infection spark to drive high incidence levels of the disease. These sparks are brought in from nearby large cities which themselves are still prone to periodicity in the disease dynamics. Analysis of multiple cities, including London, Manchester, Norwich, and Cambridge, shows that the closer larger cities are to one another the more in sync the phase angles of their disease epidemics are. The biennial disease epidemic in Norwich, which is relatively more isolated, was close to a year off of the disease peaks in both Cambridge and London, which had similar phase alignment.
The phase difference between Manchester and its surrounding suburbs was not as clear of a trend as that for London and its surrounding towns, leading to the conclusion that the presence of multiple large urban centers in a small section of the country complicates disease dynamics in that area. Small towns around Manchester were influenced by the disease dynamics in this urban hub, but also received signals from other urban centers in the area such as Liverpool and Sheffield. These trends of disease resurgence in small population centers via spark introductions from large population centers has a strangely rescue-effect-like mentality borrowed from colonization-extinction dynamics in metapopulation studies. The ability to handle non-stationarity in temporal dynamics and environmental heterogeneity of host populations may have increasing value to ecological modeling in the future.
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