IJE Advance Access published online on December 3, 2007
International Journal of Epidemiology, doi:10.1093/ije/dym240
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An exploratory method for estimating the changing speed of epidemic waves from historical data
1Department of Geography, University of Cambridge, Cambridge CB2 3EN, UK.
2Institute for Advanced Studies, University of Bristol, BS8 1UR, Bristol, UK.
3School of Geography, University of Nottingham, Nottingham NG7 2RD, UK.
*Corresponding author. Institute for Advanced Studies, University of Bristol, Royal Fort House, Bristol BS8 1UR, UK. E-mail: p.haggett{at}bristol.ac.uk
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Abstract |
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Background Historical data are necessary to establish long-term trends in disease incidence but pose analytical problems since their accuracy and reliability may be poorly specified.
Methods A robust measure of the spatial velocity, R0A, of epidemic waves from space-time series is proposed using binary data. The method was applied to the historical records of influenza morbidity for the island of Iceland over a 61-year period of influenza seasons from 1915–16 to 1975–76.
Results The onset of influenza waves tended to speed up over the period studied and the three pandemic waves associated with viral shifts in influenza A [Spanish influenza H1N1 (1918–19), Asian influenza H2N2 (1957–58) and Hong Kong influenza H3N2 (1968–69)] spread more rapidly around the island and struck earlier in the influenza season than did inter-pandemic waves, even when the latter were equally intensive as measured by total number of cases and case incidence.
Discussion The potential for using R0A in a real-time context is explored using French influenza data.
Conclusions The new measure of wave velocity appears to be applicable to those historical time series where breakdown into regional or local areas is available. The study is being extended to (i) other countries where similar influenza time series are available and (ii) to other diseases within Iceland.
Keywords Time series, exploratory data analysis, epidemic velocity, influenza, pandemics, Iceland
Accepted 24 October 2007
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Background |
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In epidemiology, long-term morbidity series are of interest where there is a need to establish historical trends in disease incidence. But the longer the series, the less likely is the sensitivity and specificity of disease reporting to have remained constant over time. Over the long term, morbidity recording is likely to be biased with the degree of bias often unknown and non-constant. Even over the short term, it is not clear whether during a period of high-disease incidence (e.g. the peak of an influenza pandemic), cases are (i) over reported because of assumptions of the prevailing cause or (ii) under reported because reporting physicians are overwhelmed by more pressing clinical demands on time. Historical evidence points in both directions.
Epidemiology is not alone in trying to establish trends from data of uncertain accuracy; economists, in particular, have been plagued by trying to establish economic trends from historically variable sources.1 Among the many methods proposed to overcome this problem, the work of John Tukey has been particularly important in establishing robust methods [what he termed exploratory data analysis (EDA)] to work alongside classical statistical models.2
In this article we propose, in the Tukey tradition, a robust method for exploring the speed of epidemic waves from long-term morbidity data. The spatial velocity of epidemic waves across human populations has attracted considerable theoretical interest3–5 and is a matter of practical public-health concern.6,7 The faster a wave of infection strikes a susceptible population, the less is the time available for implementing protective responses (e.g. isolation and vaccination). Velocity is thus particularly critical in the case of new diseases (e.g. SARS) or old diseases with changing characteristics (e.g. pandemic outbreaks of influenza where genetic shifts in the causative A virus mean that an infection is introduced into a population with little or no resistance to the new strain8,9). Here, we explore the second case.
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Methods |
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Sources of influenza morbidity data
Unusually among western countries, Iceland has since 1895 required direct notification of influenza cases by physicians.10 This concern for data collection stems from the island's early history that was marked by disastrous externally introduced epidemics, including the 1843 influenza outbreak that, although lasting only 2 months, doubled the expected death rate for the year.11 Annual totals and other summary data have been published in that country's annual public health reports (Heilbrigðissk
rslur) since that date with, for influenza, national monthly time series available from 1913 (Figure 1a). Physicians’ commentaries on the local course of the disease are still kept in manuscript form in the National Archives in Reykjavík. Of exceptional interest is the 61-year period spanning the middle of the 20th century from 1915 (Figure 1b). For these years, monthly data are broken down to a local level for some 50 medical districts (Figure 1c). This allows mapping of ways in which the disease spread around the island (Figure 1d), along with the calculation of general estimates of its changing spatial and temporal velocity over the period.
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The intricate spatial infrastructure of medical districts allows the local spread of the disease to be monitored across the island. This spatial network is, however, unstable over time as district boundaries have been modified to reflect changes in population and medical provision. Major changes to the 1907 medical district boundaries, current at the beginning of our study period, occurred in 1932, 1945 and 1955. We have therefore developed a standardized set of districts, based upon the 1945 configuration, which preserves spatial continuity over time (Figure 1c).
Methods for estimating velocity
The velocity of an epidemic wave through a population can be measured in different ways.12,13 Here we propose a robust method that uses (i) the binary (presence/absence) of a reported disease rather than the actual number of reported cases, (ii) the spatial extent of the outbreak in terms of infected districts and (iii) the time taken in months from the start of an outbreak for a disease to reach each district.14
We term the month in which an influenza case is first recorded in a given district the leading edge (LE) of the outbreak in that district, and the last month of record as the following edge (FE). Standard statistical analysis of the distribution of the two edges enables us to define a time-weighted arithmetic mean, 
and 
for each edge. In this study, the first month of the influenza season (September) is coded as t = 1. The subsequent months of the season are then coded serially as t = 2, t = 3, ... t = T, where T is the number of monthly periods from the beginning to the end of the season (i.e. T = 12 in our yearly cycle). For the LE, the equation is
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nt. The time-weighted mean is a useful measure of the velocity of the wave in terms of average time to district infection. A similar equation can be written for FE, and higher order moments can also be specified. To allow comparison between diseases with different wave characteristics, we convert these time-weighted means to a velocity ratio, V, (0 
V 1),
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The basic reproduction number (or rate or ratio), R0, is one of the most useful parameters used to characterize mathematically infectious disease processes. R0 is defined as the ratio between an infection rate (β) and a recovery rate (
):
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In terms of cases, R0 is the average number of secondary infections produced when one infected individual is introduced into a host virgin population. Methods for estimating the basic reproduction number for infectious diseases are given by Dietz,15 and an example of their use appears in the work by Watts.16
If we consider the two-phase shifts from susceptible to infective status (S 
I) and from infective to recovered status (I R) when an infectious disease arrives in an area, this raises the prospect of defining a spatial version of R0. In spatial terms, A, the spatial reproduction number, R0A, is the average number of secondary districts produced from one infected district in a virgin area. In a given study area, the integral SA (the proportion of the study area at risk of infection) is given by:
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The recovered areas integral, RA, is
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All three integrals are dimensionless numbers with values in the range (0, 1).
The integral SA has parallels to β in that a small value indicates a very rapid spread while the integral RA has parallels to in that a small value indicates a very rapid recovery. Since both terms are inversely related to their power, we suggest that their complement might be substituted in estimating a spatial version of R0, namely
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Such a spatial reproduction number would measure the propensity of an infected district to spawn other infected districts in later time periods. In effect it provides an indicator of the tendency of an infected district to produce secondaries. Values over unity imply a tendency to spread and calibrate the velocity of such spread (the larger the value, the greater the rate of spread).
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Results |
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Over the whole 61 years studied, Iceland's doctors reported 530 276 cases of influenza, half of them from Reykjavík and the immediate surrounding areas (Figure 1d). Although reported cases are likely to be under estimates, the broad shape of outbreaks in both space and time is readily discernible. The distribution throughout the year shows clear peaks in March–April with the low periods in August–September, a pattern typical of many northern latitude countries. For Iceland we have thus used, as our temporal unit, an influenza season running from September 1 to August 31 of the following year.
For Iceland as a whole, the time series of both edges is shown in Figure 2a. The equations used to estimate their values are given in the ‘Methods’ section of the article [equation (1)]. Despite marked year to year variation, the average trend shown by the linear regression line for the LE is distinctly upward implying that waves have speeded up over time, i.e. influenza waves moved around the island faster at the end than at the beginning of the study period. In contrast the position of the FE when influenza incidence ceased in any influenza season has remained essentially unchanged. This implies that the duration of reported influenza incidence grew slowly longer, from around 2.5 months in 1915–16 to nearly 4.0 months in 1975–76 (Figure 2b).
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Three of the 61 seasons studied were associated with pandemics of influenza A (the Spanish, Asian and Hong Kong pandemics). Figure 3 uses data on the spatial extent of influenza in each season to plot the position of the pandemic front with reference to the three seasons which immediately preceded or followed it. In the Spanish and Asian pandemics the front stands out clearly but in the third (Figure 3c) the Hong Kong front appears to spread over two seasons, supporting Viboud's17 concept that it was a ‘smoldering’ pandemic.
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Although pandemic years had large numbers of influenza cases, they were not the largest recorded over the period. The 1937–38 season had the largest number of cases (21 977) and the highest monthly rate and, as Figure 1a shows, monthly case numbers in several inter-pandemic years exceeded those with pandemics. In Table 1 we divided the 61 seasons into three groups: (a) pandemic (3), (b) high-intensity inter-pandemic (24), with case rates greater than the lowest pandemic season and (c) low-intensity inter-pandemic (34), with case rates lower than the lowest pandemic season. The average velocity of the LEs for the three groups is equal to (a) 2.83 months, (b) 5.53 months and (c) 6.03 months, respectively. This suggests that pandemic seasons have higher velocities than inter-pandemic years, and that this higher velocity is maintained even compared with inter-pandemic influenza seasons of similar intensity levels as pandemic seasons. Values for R0A were calculated for each influenza season (Figure 3d) and for the three categories of influenza season, (a), (b) and (c), defined above. The same differentials were observed (Table 1).
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Discussion |
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The method described may be used with surveillance data in a real-time situation and, since it may be used to compare epidemic speeds from one country to another, may have an early warning potential. The parameters of the model given in equations (1)–(7)
may be calculated for each month from the start of an outbreak and progressively re-calculated in each successive month (as fresh data becomes available) through to the end of the outbreak. An historical example of this method is shown in Figure 4. Here monthly influenza data from France over the 103 years from 1887–88 to 1998–99 were studied, the 5 shift seasons identified and these compared with the 2 years that preceded and followed them. Figure 4 shows the spatial reproductive numbers (R0A) measured in real time from the start of each influenza season. The contrasting behaviour of R0A in shift and non-shift years suggest there may be a capability for the method to pick up unusual wave characteristics early in the course of an outbreak and alert other yet unaffected areas. Influenza and monthly data have been used here for ease of illustration; in principle the methods proposed could be used with a wide range of infectious diseases and for shorter (i.e. weekly or daily) reporting periods.
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Conclusions |
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The new measure of wave velocity applied to a 61-year run of influenza morbidity records shows that for Iceland (i) the onset of waves tended to speed up over the period 1914–75 and (ii) waves in three viral shift (pandemic) seasons spread faster than did other equally large waves in non-shift (inter-pandemic) seasons. In principle, the proposed measure of wave velocity should be applicable to other epidemiological time series where breakdown into regional or local areas is available. The study is being extended to (i) other countries where similar influenza time series are available (notably France) and (ii) to other diseases within Iceland. If our findings are confirmed elsewhere, this may have wider implications for public health measures. It would suggest that any new influenza pandemic in the 21st century, whether emerging from avian influenza or other sources, is likely to appear earlier than normal in the influenza season and to spread spatially more rapidly than normal. Both conclusions underscore the role of surveillance and early virus watch identification.
Key Messages
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Acknowledgements |
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This work on Icelandic records has been supported by a History of Medicine grant from the Wellcome Trust.
Conflicts of interest: None declared.
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