# Smallpox and Season: Reanalysis of Historical Data

Authors: Hiroshi Nishiura; Tomoko Kashiwagi
Journal: Interdisciplinary Perspectives on Infectious Diseases
 (2009)

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## Abstract

Seasonal variation in smallpox transmission is one of the most pressing ecological questions and is relevant to bioterrorism preparedness. The present study reanalyzed 7 historical datasets which recorded monthly cases or deaths. In addition to time series analyses of reported data, an estimation and spectral analysis of the effective reproduction number at calendar timet, R(t), were made. Meteorological variables were extracted from a report in India from 1890–1921 and compared with smallpox mortality as well as R(t). Annual cycles of smallpox transmission were clearly shown not only in monthly reports but also in the estimates of R(t). Even short-term epidemic data clearly exhibited an annual peak every January. Both mortality and R(t) revealed significant negative association (P<.01) and correlation (P<.01), respectively, with humidity. These findings suggest that smallpox transmission greatly varies with season and is most likely enhanced by dry weather.


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## Body

## 1. Introduction

Smallpox is the only disease to have been eradicated worldwide [1]. Despite
the success story of vaccination and other public health interventions, the
number of susceptible individuals has grown to date following cessation of
routine vaccination, and the threat of bioterrorist attack has led to debates
on countermeasures in such an event [2]. Various mathematical studies have been
conducted as part of a preparedness program, including large-scale simulation of
a bioterrorist attack and the public health countermeasures against it [3–6]. Theoretical
studies on the spread of smallpox include not only simulations but also
quantitative analysis of historical data [7–10]. A
statistical modeling study suggests that a small outbreak could be contained only
implementing contact tracing and isolation [11]. Moreover, those who underwent
vaccination in the past are believed to be still protected against severe and
fatal manifestations of smallpox even today [7, 12].Studies on smallpox control have progressed in parallel with the
development of epidemiological and statistical methods, and because of the eradication
before maturation of biostatistics, many questions have remained in regards to
the details of the epidemiology. Seasonal variation in smallpox transmission is
one of the most pressing ecological questions playing a key role in determining
the transmission dynamics, should a future outbreak occur following the deliberate release [1, 4]. For example, clarification of the seasonal
preference of variola virus is crucial for identifying and forecasting the
disease risk using ecological data [13]. Although seasonal occurrence of
smallpox was documented early on among directly transmitted infectious diseases
[14, 15], and whereas the disease is believed to be one of the “winter diseases”
in industrialized countries, even the presence of seasonality has not been investigated
in detail.The best available evidence stems from a series of studies by Sir Leonard
Rogers (1868–1962) [16], who
conducted epidemiologic surveys of smallpox outbreaks in India
over a long period of time [17–19]. He also conducted
a similar survey in England and Wales [20]. By analyzing the monthly mortality data from the late 19th to the early
20th century in these countries, Rogers argued that the smallpox epidemic in India is relatively
uniform (i.e., not apparently cyclical) compared to that in England and Wales [17, 19, 21]. Further, he
descriptively and implicitly suggested that there is a negative correlation
between humidity and smallpox mortality, but there was little association
between smallpox and rainfall [17, 18]. This effort was followed by Russell and
Sundararajan [22] who supported the notion that a dry environment offers
favorable conditions for smallpox transmission. These consistent findings have also
been reported during the Smallpox Eradication Program (SEP), where a peak of smallpox
incidence occurred from December to January in the Northern hemisphere (e.g., Indonesia and Bangladesh)
and from May to June in the Southern hemisphere (e.g., Brazil) [23–26]. However, the
observed data during the SEP were greatly modified by intensive immunizations,
and perhaps because of this, epidemics in other locations were not suggested to
be seasonal [26–28], leaving this issue yet to be clarified.Despite the rigorous efforts before the global eradication, later
progress on this issue was unfortunately subtle. Upham once revisited Rogers's dataset from India,
anthropologically discussing potential reasons why the American Southwest was less infested by smallpox
[29]. A time series technique was applied to historical data in Finland and England  [30–32], showing that
periodicity is mainly regulated by the susceptible fraction of a population in
question [33]. However, despite the analyses on the impact of vaccination and
migration on periodicity, seasonal patterns of transmission have not been
explicitly studied, mainly because of a lack of data precision. In a historical
study examining smallpox in England from the 16th to 17th centuries, the time referred to as the “little ice age,”
it has been documented that long-term climatic changes did little to the
smallpox transmission [34], but this conclusion was drawn without
quantitatively and explicitly analyzing the data. Instead, the quality of time
series data and its impact on seasonality were discussed in relation to social
backgrounds of smallpox control [35, 36], but again no rigorous statistical analyses
were made using observed data.Accordingly, several lingering questions remain. Is smallpox transmission
really seasonal? If so, is the seasonality associated with humidity? Clarification
of these points will not only enhance our understanding of the pattern of smallpox
transmission, but also will be crucial for identifying the seasonal preference
of variola virus with some implications for bioterrorism preparedness plans.
The present study is aimed at examining the presence of seasonality and clarifying
the relationships between smallpox and climate. We reanalyzed various
historical datasets, suggesting a new method for the analysis of time series.


## 2. Materials and Methods

### 2.1. Data Source: Historical Records

Seven temporal distributions of smallpox at different times and locations
were extracted from historical literature. This literature review was based on
references collected by tracking all the references given in the relevant
publications and repeating this task until we could find no further references;
the details are given elsewhere [37, 38]. Figure 1 shows the time series data
by location with a monthly reporting interval. Chronologically, epidemic
records for The Hague (1755–1773), Berlin (1758–1774), Zurich (1870–1887), the entire
Netherlands (1870–1873), Northwest
Frontier province in India (1890–1921), Shanghai
(1900–1913), and Bombay
(1902–1907) provide monthly
data of smallpox with time and were used for further analysis [17, 39–45]. The first
two records contain data before the introduction of vaccination. Except for Zurich, which documents the
monthly number of cases, the remaining datasets record only monthly deaths.
Death data are given as the absolute number of deaths, except where indicated.
With regard to the magnitude of the epidemics, the annual averages of the
disaster size were 10.1 deaths (The Hague), 32.9 deaths (Berlin), 9.9 cases
(Zurich), 428.6 deaths (the entire Netherlands), 5.28 deaths per 100 000
(Northwest Frontier province in India), 21.5 deaths (Shanghai), and 2.45 deaths
per 100 000 (Bombay). By examining another historical record of the smallpox
epidemic in Tokyo, it was found that the mean (and the standard deviation) and the median (25–75% quartile)
time from onset to death were 29.1 (13.8) and 26.0 (19.0–37.0) days,
respectively [46]. Thus, it is reasonable to assume that the relative frequency
of death with time represents that of onset accompanied by approximately a 1
month delay. Moreover, the infection may have happened approximately half a
month before the onset [9]. Meteorological variables with time were given only
in Rogers’s observations [17], which contained the monthly rainfall
(inch) and the absolute humidity.Temporal distributions of smallpox. Temporal patterns of smallpox are
shown, which were extracted from historical records in (a) The Hague, The Netherlands, 1755–1773,
(b) Berlin, Germany, from 1758–1774, (c) Zurich, Switzerland, 1870–1887, (d) the Entire Netherlands,
1870–1873, (e) Northwest Frontier province, India, 1890–1921, (f) Shanghai, China, 1900–1913,
and (g) Bombay, India, 1902–1907. Death data are shown in (a), (b), (d), and (f). Cases are shown in (c). Mortality (i.e., deaths per
100 000) data are shown in (e) and (g). See [17, 39–45] for original
data.
(a)(b)(c)(d)(e)(f)(g)


### 2.2. Time Series Analysis

First, the presence of seasonality was examined for all 7 datasets using
spectral density analysis. Spectral analysis is based on the idea of a
theoretical power-spectrum, which partitions the total variance of the series
among sinusoidal components [47]. In other words, spectral density decomposes a
time series function into a sum of sines and cosines. The density plot (i.e.,
correlogram) was graphically plotted to determine if a sharp peak at a period
of 12 months exists, corresponding to an annual cycle (i.e., seasonality).


### 2.3. Estimation of the Effective Reproduction Number

Second, seasonality that was evaluated using the effective reproduction
number,R(t), defined as the actual average number of secondary cases per
primary case at calendar time t. R(t)
shows a time-dependent variation with a decline in susceptible individuals
(intrinsic factors) and with the implementation of control measures (extrinsic
factors). If R(t)<1, it suggests that the epidemic is in decline (vice verca, if R(t)>1). This approach was employed to clearly show the seasonal patterns of
transmission and to partly address the issue of dependence among cases, that
is, statistically, the observation of an infected individual is not independent
of other infected individuals, since the disease is transmitted directly from
human to human.The following approximation was made to derive estimates ofR(t).
Supposing that the number of new infections at calendar time t is I(t), the transmission dynamics are
described by the renewal equation using R(t) [48, 49]:(1)I(t)=R(t)∫0∞I(t−τ)ω(τ)dτ, where ω(τ) is the probability density
function of the generation time. The right-hand side of (1) represents
secondary transmissions at calendar time t,
which are determined by the number of those who were infected at time t−τ, I(t−τ), and the magnitude of
secondary transmissions at time t, R(t).
Since the data in the present study were recorded only monthly, the equation
has to be simplified to comply with discrete points of time data. From the
beginning of the history of mathematical modeling of smallpox in the late 19th
century [50], cases tended to be modeled by generation, the idea of which is
applied as follows. Given the number of cases in generation i, Ii, the expected number of cases in generation i+1,
E(Ii+1) is given by(2)E(Ii+1)=RiIi, where Ri is the effective reproduction number of generation i [51]. That is, the reproduction number
is simply given by ratio of successive generations of cases, which was
implicitly understood in history by a pioneering epidemiologist, Clare Oswald
Stallybrass (1881—1951) who
applied the theory to analyze the seasonality of various infectious diseases
[52, 53]. Since the mean generation time of smallpox is approximately 15 days
(i.e., half a month) [50, 54], monthly data contains exactly two generations. Let
us consider three successive generations, i, i+1, and i+2. Given the reproduction numbers Ri and Ri+1, we get(3)E(Ii+1)=RiIi,E(Ii+2)=Ri+1Ii+1. Considering that the generations i and i+1 are grouped together and reported in
the same month j, the reproduction
number cannot be estimated by generation i.
Instead, by assuming that the reproduction numbers in the successive
generations are identical, that is, Ri=Ri+1(:=Rj), (3) can be rearranged as	(4)E(Ii+1)=RjIi,E(Ii+2)=RjIi+1. The expected number of cases in the next generation i+3 is given
by product of Ii+2 and the
reproduction number in the next month j+1, Rj+1, that is,(5)E(Ii+3)=Rj+1Ii+2. Given (4) and (5), the number
of cases in month j+1, Cj+1(:=Ii+2+Ii+3)
is given using Cj(:=Ii+Ii+1), that is,(6)E(Cj+1)=E(Ii+2+Ii+3)=(1+Rj+1)Ii+2=(1+Rj+1)RjIi+1=(1+Rj+1)Rj2Ii=(1+Rj+1)Rj2Cj1+Rj. We assume that the expected
values are sufficient to characterize Poisson distributions. This assumption
indicates that the conditional distribution of the reported number of cases in
month j+1, Cj+1, given Cj is given by(7)Cj+1∣Cj∼Poisson[(1+Rj+1)Rj21+RjCj]. Thus, for the observation of
cases (or deaths with a 1 month lag) from month 0 to N, the likelihood of estimating Rj is given by(8)L=constant×∏j=0N−1[(1+Rj+1)Rj21+RjCj]Cj+1×exp⁡[−(1+Rj+1)Rj21+RjCj]. By minimizing the negative
logarithm of (8), the maximum likelihood estimates of the monthly-approximated
reproduction numbers, Rj were obtained.


### 2.4. Multivariate Modeling

Third, to identify the characteristic factors of seasonal variation in smallpox
transmissions, the relationships between meteorological variables (i.e.,
rainfall and humidity) and incidence (mortality) as well as the effective
reproduction number were examined. To examine the influence of seasonal
variables on the temporal trend of smallpox, we employed one of the generalized
linear models with the construction of a Poisson regression model incorporating
monthly and yearly terms [55]:(9)E(Cj)=exp{α+β1(year)+β2[sin⁡(2π×month12)]+β3[cos⁡(2π×month12)]}, where E(Cj) is the expected number of cases (deaths) in month j, α is
a constant, and βi are the regression coefficients for year
or month. The relationship was investigated using both univariate and multivariate
models. In the multivariate model, the year of occurrence was controlled for,
but the sine and cosine of the month were not included due to colinearity with
rainfall. The mortality rate ratios (MRR) for the occurrence of smallpox death
were used to evaluate the impact of each meteorological variable on smallpox.With regard to the relationship between meteorological variables andR(t),
multiple linear regression analysis was employed to determine factors contributing
to R(t). Because of the obvious cyclical nature of the observed data
yielding an autocorrelation in the linear regression analysis (Durbin-Watson =
0.23), the monthly periodic terms (as shown in (9)) were added to the list of
independent variables. The level of statistical significance was set at α=0.05. All statistical data were analyzed
using the statistical software JMP version 7.0 (SAS Institute Inc., Cary, NC, USA).


## 2.1. Data Source: Historical Records

Seven temporal distributions of smallpox at different times and locations
were extracted from historical literature. This literature review was based on
references collected by tracking all the references given in the relevant
publications and repeating this task until we could find no further references;
the details are given elsewhere [37, 38]. Figure 1 shows the time series data
by location with a monthly reporting interval. Chronologically, epidemic
records for The Hague (1755–1773), Berlin (1758–1774), Zurich (1870–1887), the entire
Netherlands (1870–1873), Northwest
Frontier province in India (1890–1921), Shanghai
(1900–1913), and Bombay
(1902–1907) provide monthly
data of smallpox with time and were used for further analysis [17, 39–45]. The first
two records contain data before the introduction of vaccination. Except for Zurich, which documents the
monthly number of cases, the remaining datasets record only monthly deaths.
Death data are given as the absolute number of deaths, except where indicated.
With regard to the magnitude of the epidemics, the annual averages of the
disaster size were 10.1 deaths (The Hague), 32.9 deaths (Berlin), 9.9 cases
(Zurich), 428.6 deaths (the entire Netherlands), 5.28 deaths per 100 000
(Northwest Frontier province in India), 21.5 deaths (Shanghai), and 2.45 deaths
per 100 000 (Bombay). By examining another historical record of the smallpox
epidemic in Tokyo, it was found that the mean (and the standard deviation) and the median (25–75% quartile)
time from onset to death were 29.1 (13.8) and 26.0 (19.0–37.0) days,
respectively [46]. Thus, it is reasonable to assume that the relative frequency
of death with time represents that of onset accompanied by approximately a 1
month delay. Moreover, the infection may have happened approximately half a
month before the onset [9]. Meteorological variables with time were given only
in Rogers’s observations [17], which contained the monthly rainfall
(inch) and the absolute humidity.Temporal distributions of smallpox. Temporal patterns of smallpox are
shown, which were extracted from historical records in (a) The Hague, The Netherlands, 1755–1773,
(b) Berlin, Germany, from 1758–1774, (c) Zurich, Switzerland, 1870–1887, (d) the Entire Netherlands,
1870–1873, (e) Northwest Frontier province, India, 1890–1921, (f) Shanghai, China, 1900–1913,
and (g) Bombay, India, 1902–1907. Death data are shown in (a), (b), (d), and (f). Cases are shown in (c). Mortality (i.e., deaths per
100 000) data are shown in (e) and (g). See [17, 39–45] for original
data.
(a)(b)(c)(d)(e)(f)(g)


## 2.2. Time Series Analysis

First, the presence of seasonality was examined for all 7 datasets using
spectral density analysis. Spectral analysis is based on the idea of a
theoretical power-spectrum, which partitions the total variance of the series
among sinusoidal components [47]. In other words, spectral density decomposes a
time series function into a sum of sines and cosines. The density plot (i.e.,
correlogram) was graphically plotted to determine if a sharp peak at a period
of 12 months exists, corresponding to an annual cycle (i.e., seasonality).


## 2.3. Estimation of the Effective Reproduction Number

Second, seasonality that was evaluated using the effective reproduction
number,R(t), defined as the actual average number of secondary cases per
primary case at calendar time t. R(t)
shows a time-dependent variation with a decline in susceptible individuals
(intrinsic factors) and with the implementation of control measures (extrinsic
factors). If R(t)<1, it suggests that the epidemic is in decline (vice verca, if R(t)>1). This approach was employed to clearly show the seasonal patterns of
transmission and to partly address the issue of dependence among cases, that
is, statistically, the observation of an infected individual is not independent
of other infected individuals, since the disease is transmitted directly from
human to human.The following approximation was made to derive estimates ofR(t).
Supposing that the number of new infections at calendar time t is I(t), the transmission dynamics are
described by the renewal equation using R(t) [48, 49]:(1)I(t)=R(t)∫0∞I(t−τ)ω(τ)dτ, where ω(τ) is the probability density
function of the generation time. The right-hand side of (1) represents
secondary transmissions at calendar time t,
which are determined by the number of those who were infected at time t−τ, I(t−τ), and the magnitude of
secondary transmissions at time t, R(t).
Since the data in the present study were recorded only monthly, the equation
has to be simplified to comply with discrete points of time data. From the
beginning of the history of mathematical modeling of smallpox in the late 19th
century [50], cases tended to be modeled by generation, the idea of which is
applied as follows. Given the number of cases in generation i, Ii, the expected number of cases in generation i+1,
E(Ii+1) is given by(2)E(Ii+1)=RiIi, where Ri is the effective reproduction number of generation i [51]. That is, the reproduction number
is simply given by ratio of successive generations of cases, which was
implicitly understood in history by a pioneering epidemiologist, Clare Oswald
Stallybrass (1881—1951) who
applied the theory to analyze the seasonality of various infectious diseases
[52, 53]. Since the mean generation time of smallpox is approximately 15 days
(i.e., half a month) [50, 54], monthly data contains exactly two generations. Let
us consider three successive generations, i, i+1, and i+2. Given the reproduction numbers Ri and Ri+1, we get(3)E(Ii+1)=RiIi,E(Ii+2)=Ri+1Ii+1. Considering that the generations i and i+1 are grouped together and reported in
the same month j, the reproduction
number cannot be estimated by generation i.
Instead, by assuming that the reproduction numbers in the successive
generations are identical, that is, Ri=Ri+1(:=Rj), (3) can be rearranged as	(4)E(Ii+1)=RjIi,E(Ii+2)=RjIi+1. The expected number of cases in the next generation i+3 is given
by product of Ii+2 and the
reproduction number in the next month j+1, Rj+1, that is,(5)E(Ii+3)=Rj+1Ii+2. Given (4) and (5), the number
of cases in month j+1, Cj+1(:=Ii+2+Ii+3)
is given using Cj(:=Ii+Ii+1), that is,(6)E(Cj+1)=E(Ii+2+Ii+3)=(1+Rj+1)Ii+2=(1+Rj+1)RjIi+1=(1+Rj+1)Rj2Ii=(1+Rj+1)Rj2Cj1+Rj. We assume that the expected
values are sufficient to characterize Poisson distributions. This assumption
indicates that the conditional distribution of the reported number of cases in
month j+1, Cj+1, given Cj is given by(7)Cj+1∣Cj∼Poisson[(1+Rj+1)Rj21+RjCj]. Thus, for the observation of
cases (or deaths with a 1 month lag) from month 0 to N, the likelihood of estimating Rj is given by(8)L=constant×∏j=0N−1[(1+Rj+1)Rj21+RjCj]Cj+1×exp⁡[−(1+Rj+1)Rj21+RjCj]. By minimizing the negative
logarithm of (8), the maximum likelihood estimates of the monthly-approximated
reproduction numbers, Rj were obtained.


## 2.4. Multivariate Modeling

Third, to identify the characteristic factors of seasonal variation in smallpox
transmissions, the relationships between meteorological variables (i.e.,
rainfall and humidity) and incidence (mortality) as well as the effective
reproduction number were examined. To examine the influence of seasonal
variables on the temporal trend of smallpox, we employed one of the generalized
linear models with the construction of a Poisson regression model incorporating
monthly and yearly terms [55]:(9)E(Cj)=exp{α+β1(year)+β2[sin⁡(2π×month12)]+β3[cos⁡(2π×month12)]}, where E(Cj) is the expected number of cases (deaths) in month j, α is
a constant, and βi are the regression coefficients for year
or month. The relationship was investigated using both univariate and multivariate
models. In the multivariate model, the year of occurrence was controlled for,
but the sine and cosine of the month were not included due to colinearity with
rainfall. The mortality rate ratios (MRR) for the occurrence of smallpox death
were used to evaluate the impact of each meteorological variable on smallpox.With regard to the relationship between meteorological variables andR(t),
multiple linear regression analysis was employed to determine factors contributing
to R(t). Because of the obvious cyclical nature of the observed data
yielding an autocorrelation in the linear regression analysis (Durbin-Watson =
0.23), the monthly periodic terms (as shown in (9)) were added to the list of
independent variables. The level of statistical significance was set at α=0.05. All statistical data were analyzed
using the statistical software JMP version 7.0 (SAS Institute Inc., Cary, NC, USA).


## 3. Results

### 3.1. Temporal Distribution and Spectral Density

The spectral densities are shown in Figure2 which can be reasonably interpreted
by comparatively examining the temporal distributions (Figure 1). With
regard to the data collected from The Hague and Berlin, the observations of
which were made before the introduction of vaccinations, periodic epidemics (i.e.,
super-annual cycles) are apparent where the interepidemic period ranges from 3
to 5 years (see Figures 1(a) and 1(b)). However, the annual cycle is not seen,
and thus, the spectral densities do not show a clear seasonal pattern (Figures
2(a) and 2(b)). On the contrary, the time series data in Zurich and Shanghai
clearly revealed a peak at 12 months (Figures 2(c) and 2(f)). The entire Netherlands
data covers a relatively short period of time compared to the other datasets (Figure 1(d)) with unclear seasonal and periodic frequencies in the spectral
diagram (Figure 2(d)). Although a small peak is seen at 12 months for the
data in the Northwest Frontier province in India
(Figure 2(e)), the
density plot exhibits a multimodal pattern, reflecting an irregular temporal
distribution (Figure 1(e)). In the Bombay
data, the annual cycle is most clearly highlighted in the temporal distribution
(Figure 1(g)), which is also reflected in the spectral density (Figure
2(g)).The spectral density plots for smallpox occurrences. (a)–(f) correspond to locations as chronologically
ordered in Figure 1. A sharp peak at a period of 12 months corresponds to the annual
cycle (seasonality), while other longer peaks may reflect a super-annual cycle
(periodicity). No adjustment was made in drawing the plots.
(a)(b)(c)(d)(e)(f)(g)


### 3.2. Effective Reproduction Number

Figure3 plots estimates of the effective reproduction number as a function of calendar time. The
vertical broken lines represent January in every year, while a horizontal
dashed line is a reference value yielding R(t)=1, that is, the threshold condition
of an epidemic. R(t) tends to increase during the winter
season for three early records (Figures 3(a), 3(b),
and 3(c)), but the
annual cycles are not seen. However, the short-term epidemic data for the
entire Netherlands clearly shows that three peaks of R(t) coincide in every January with
estimates above unity (Figure 3(d)). A similar pattern is observed in Shanghai and Bombay
(Figures 3(f) and 3(g)). Figure 4
shows the spectral density plots of R(t)
for the entire Netherlands and Northwest Frontier province in India. Although spectral densities
of death and mortality (Figures 2(d) and 2(e)) did not exhibit a clear
annual cycle, the obvious peak at 12 months is seen for both datasets in terms
of R(t) (Figures 4(a) and 4(b)). That is, seasonal patterns of
smallpox transmission were reasonably shown with the use of R(t)
even for the short- and long-term time series.Estimates of the effective reproduction numbers with time,R(t).
(a)–(f) correspond to locations as chronologically
ordered in Figure 1. The horizontal dashed line indicates where the
reproduction number is unity. The vertical broken lines represent every
January. R(t) cannot be estimated where the observed number of cases (or
deaths) was 0 and is not shown for such time points.
(a)(b)(c)(d)(e)(f)(g)The spectral density plots
forR of smallpox. The spectral density plots for the
estimated reproduction numbers of smallpox in (a) the entire Netherlands, 1870–1873 and (b)
Northwest frontier province, India, 1890–1921. Compared
with Figures 2(d) and 2(e), the spectral densities clearly indicate a sharp
peak at a period of 12 months.
(a)(b)


### 3.3. Factors Characterizing Seasonal Variation

Table1
shows the output of univariate and multivariate models for explaining smallpox mortality
in India
using meteorological variables. In both models, rainfall was not significantly
associated with smallpox mortality. However, significant negative association
was found for humidity (adjusted MRR = 0.387 (95% confidence interval (CI):
0.311, 0.481), P<.01). Table 2 summarizes the relationship between
the effective reproduction number and meteorological variables using a multiple
linear regression model. On a whole, the model showed a weak predictive
ability. However, humidity was again identified as an explanatory variable
which significantly reduces the effective reproduction number (P<.01). No significant correlation was found between R(t) and rainfall.Table 1
Monthly weather patterns and smallpox mortality in Northwest frontier province, India, from 1890–1921.
Meteorological elementUnivariate modelMultivariate model¶MRR‡(95% CI†)PMRR‡(95% CI†)PRainfall (inch)0.979 (0.952, 1.003).100.991 (0.964, 1.018).53Absolute humidity0.384 (0.309, 0.477)<.010.387 (0.311, 0.481)<.01‡MRR,
mortality rate ratio, reflects change in risk of smallpox death per unit (or
absolute value) in the meteorological variable in question. †CI,
confidence interval. ¶The
multivariate model was also adjusted for the calendar year.Table 2
Monthly weather patterns and the effective reproduction number of smallpox in Northwest frontier
province, India, from 1890–1921.
Variableβ†S.E.‡t-RatioPIntercept1.5790.13911.37<.01Rainfall (inch)0.0080.0150.51.61Absolute humidity−1.1690.319−3.66<.01Cosine−0.1810.078−2.32.02Sine−0.2110.064−3.29<.01†Parameter coefficient; ‡Standard error; r2=0.057 (F-ratio = 5.67, P<.01), dependent variable = effective
reproduction number, Durbin-Watson = 2.600.


## 3.1. Temporal Distribution and Spectral Density

The spectral densities are shown in Figure2 which can be reasonably interpreted
by comparatively examining the temporal distributions (Figure 1). With
regard to the data collected from The Hague and Berlin, the observations of
which were made before the introduction of vaccinations, periodic epidemics (i.e.,
super-annual cycles) are apparent where the interepidemic period ranges from 3
to 5 years (see Figures 1(a) and 1(b)). However, the annual cycle is not seen,
and thus, the spectral densities do not show a clear seasonal pattern (Figures
2(a) and 2(b)). On the contrary, the time series data in Zurich and Shanghai
clearly revealed a peak at 12 months (Figures 2(c) and 2(f)). The entire Netherlands
data covers a relatively short period of time compared to the other datasets (Figure 1(d)) with unclear seasonal and periodic frequencies in the spectral
diagram (Figure 2(d)). Although a small peak is seen at 12 months for the
data in the Northwest Frontier province in India
(Figure 2(e)), the
density plot exhibits a multimodal pattern, reflecting an irregular temporal
distribution (Figure 1(e)). In the Bombay
data, the annual cycle is most clearly highlighted in the temporal distribution
(Figure 1(g)), which is also reflected in the spectral density (Figure
2(g)).The spectral density plots for smallpox occurrences. (a)–(f) correspond to locations as chronologically
ordered in Figure 1. A sharp peak at a period of 12 months corresponds to the annual
cycle (seasonality), while other longer peaks may reflect a super-annual cycle
(periodicity). No adjustment was made in drawing the plots.
(a)(b)(c)(d)(e)(f)(g)


## 3.2. Effective Reproduction Number

Figure3 plots estimates of the effective reproduction number as a function of calendar time. The
vertical broken lines represent January in every year, while a horizontal
dashed line is a reference value yielding R(t)=1, that is, the threshold condition
of an epidemic. R(t) tends to increase during the winter
season for three early records (Figures 3(a), 3(b),
and 3(c)), but the
annual cycles are not seen. However, the short-term epidemic data for the
entire Netherlands clearly shows that three peaks of R(t) coincide in every January with
estimates above unity (Figure 3(d)). A similar pattern is observed in Shanghai and Bombay
(Figures 3(f) and 3(g)). Figure 4
shows the spectral density plots of R(t)
for the entire Netherlands and Northwest Frontier province in India. Although spectral densities
of death and mortality (Figures 2(d) and 2(e)) did not exhibit a clear
annual cycle, the obvious peak at 12 months is seen for both datasets in terms
of R(t) (Figures 4(a) and 4(b)). That is, seasonal patterns of
smallpox transmission were reasonably shown with the use of R(t)
even for the short- and long-term time series.Estimates of the effective reproduction numbers with time,R(t).
(a)–(f) correspond to locations as chronologically
ordered in Figure 1. The horizontal dashed line indicates where the
reproduction number is unity. The vertical broken lines represent every
January. R(t) cannot be estimated where the observed number of cases (or
deaths) was 0 and is not shown for such time points.
(a)(b)(c)(d)(e)(f)(g)The spectral density plots
forR of smallpox. The spectral density plots for the
estimated reproduction numbers of smallpox in (a) the entire Netherlands, 1870–1873 and (b)
Northwest frontier province, India, 1890–1921. Compared
with Figures 2(d) and 2(e), the spectral densities clearly indicate a sharp
peak at a period of 12 months.
(a)(b)


## 3.3. Factors Characterizing Seasonal Variation

Table1
shows the output of univariate and multivariate models for explaining smallpox mortality
in India
using meteorological variables. In both models, rainfall was not significantly
associated with smallpox mortality. However, significant negative association
was found for humidity (adjusted MRR = 0.387 (95% confidence interval (CI):
0.311, 0.481), P<.01). Table 2 summarizes the relationship between
the effective reproduction number and meteorological variables using a multiple
linear regression model. On a whole, the model showed a weak predictive
ability. However, humidity was again identified as an explanatory variable
which significantly reduces the effective reproduction number (P<.01). No significant correlation was found between R(t) and rainfall.Table 1
Monthly weather patterns and smallpox mortality in Northwest frontier province, India, from 1890–1921.
Meteorological elementUnivariate modelMultivariate model¶MRR‡(95% CI†)PMRR‡(95% CI†)PRainfall (inch)0.979 (0.952, 1.003).100.991 (0.964, 1.018).53Absolute humidity0.384 (0.309, 0.477)<.010.387 (0.311, 0.481)<.01‡MRR,
mortality rate ratio, reflects change in risk of smallpox death per unit (or
absolute value) in the meteorological variable in question. †CI,
confidence interval. ¶The
multivariate model was also adjusted for the calendar year.Table 2
Monthly weather patterns and the effective reproduction number of smallpox in Northwest frontier
province, India, from 1890–1921.
Variableβ†S.E.‡t-RatioPIntercept1.5790.13911.37<.01Rainfall (inch)0.0080.0150.51.61Absolute humidity−1.1690.319−3.66<.01Cosine−0.1810.078−2.32.02Sine−0.2110.064−3.29<.01†Parameter coefficient; ‡Standard error; r2=0.057 (F-ratio = 5.67, P<.01), dependent variable = effective
reproduction number, Durbin-Watson = 2.600.


## 4. Discussion

The present study
reanalyzed historical records of smallpox to examine the presence of
seasonality and to partly clarify the characteristic factors. Although 18th century
data did not show an apparent annual cycle, the remaining records reasonably
showed seasonal variations either in the monthly observation or the
reproduction number. In particular, even the short-term epidemic data for the
entire Netherlands
clearly revealed peaks of transmission every January. Although several
important meteorological variables were missing (e.g., temperature and
atmospheric pressure), Rogers’s observation permitted investigations of a few variables as underlying factors
characterizing the seasonality. Analyzing the meteorological data in India, both smallpox
mortality and the reproduction number yielded significant negative association
and correlation with humidity. Rainfall did not appear to be a useful predictor
of seasonality.One important
message drawn from this exercise is that smallpox transmission is confirmed as seasonal
and this is most likely associated with dry weather. This finding is consistent
with implicit suggestions which have accumulated in the historical literature
[1, 17, 19]. Whereas the data from The Hague and
Berlin did not offer the relevant interpretations, their periodic peaks were also observed
during the winter seasons. Assuming that these records captured mainly the
large periodic outbreaks alone, it is plausible that the old data were
accompanied by underreporting during less intensive years, and thus, did not
precisely contain subtle seasonal fluctuations. Given that the seasonal force
of infection was obvious even in the short-term epidemic data from the entire Netherlands,
not only endemic but also epidemic smallpox would greatly vary with the season
and most likely would be enhanced by dry weather. Historically, virologists
attempted to attribute the annual cycle to the seasonal preference of the variola
virus [56–58]. To date, it
is known that the variola virus could survive in an infective state under
different conditions of temperature and humidity [56, 57]. However, as temperature
and humidity rise above 30°C and 55%, respectively, the virus is known to
immediately lose infectivity [57]. Such a virological explanation supports the epidemiologic
findings from this present study and reasonably explains the seasonal
preference of the virus as a factor behind the seasonality of outbreaks. The
above-mentioned point implies that we cannot ignore the seasonality even in the
event of a short-term reintroduction of variola virus due to bioterrorist
attack.A technical
development in analyzing seasonal data is also notable. Since the observation
of an infected individual is not independent of other infectious individuals,
direct application of the generalized linear model has not been
appropriate to
date. One approach to resolve this issue is to employ a Bayesian method,
explicitly dealing with dependence in a Poisson regression model [59], which is,
however, computationally complicated for nonspecialists. As an alternative, the
present study suggested the use of R(t). R(t) reasonably reflects time-dependent
changes in the susceptible fraction of the population in question and other various
factors determining the transmission (including seasonality) [60, 61]. In
particular, it should be noted that R(t) does not reflect onset or death but
can represent an infection event with time, proving its potential as a marker
to model seasonal and periodic transmission cycles. In addition, quantitative
assessment of R(t) is theoretically important, because the amplitude of seasonal
forces of infection characterizes the length of the interepidemic period
[33, 62, 63]. A continued super-annual cycle mathematically requires seasonally
varying forces of infection, which determines the season-specific threshold
condition [64] and evolutionary dynamics of a disease [65, 66]. To the best of
our knowledge, the present study is the first to suggest a quantitative method to
reasonably extract the amplitude using the notation of R(t) and extending the previous
efforts of Stallybrass [53].Most infectious
diseases show characteristic seasonal variations in incidence. The present
study confirms that the transmission of smallpox does vary with season.
However, compared to the seasonal ecology of insects in vector-borne diseases,
seasonal factors for directly transmitted diseases are far more complex, and
thus, questions remain as to what exactly are the factors behind the
seasonality of smallpox. At least, experimental evidence supports the role of dry weather in the dynamics of influenza [67, 68]; a recent study found that low (dry) relative humidity in the range of 20 to 30% produced the spread of the influenza virus faster than at relative humidity in higher percentages. In fact, at a humidity of 80% or above, the study found no spread of the flu [68]. Since there are also various social factors which vary
with the season, the seasonal preference of pathogens cannot be captured
without sufficiently highlighting the time-varying human contact patterns, and
thus, further analyses (e.g., reanalysis of small-scale outbreaks where we can
adjust the contact frequency) are needed. We hope that the present study
enhances the similar reanalysis of historical data, triggering an interest in
investigating the relationship between the transmission of directly transmitted
infectious diseases and climatic changes.


## 5. Conclusions

Seven historical datasets of smallpox were reanalyzed to
examine the presence of seasonality and to identify the characteristic factors.
Annual cycles were clearly shown not only in the monthly reports but also in
the estimates of the effective reproduction number. Even for a short-term
epidemic, the transmission of smallpox would most likely be enhanced by dry
weather.



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*Source: 591935-2009-01-04.xml*