Time variations in the transmissibility of pandemic influenza in Prussia, Germany, from 1918–19
© Nishiura; licensee BioMed Central Ltd. 2007
Received: 23 April 2007
Accepted: 04 June 2007
Published: 04 June 2007
Time variations in transmission potential have rarely been examined with regard to pandemic influenza. This paper reanalyzes the temporal distribution of pandemic influenza in Prussia, Germany, from 1918–19 using the daily numbers of deaths, which totaled 8911 from 29 September 1918 to 1 February 1919, and the distribution of the time delay from onset to death in order to estimate the effective reproduction number, Rt, defined as the actual average number of secondary cases per primary case at a given time.
A discrete-time branching process was applied to back-calculated incidence data, assuming three different serial intervals (i.e. 1, 3 and 5 days). The estimated reproduction numbers exhibited a clear association between the estimates and choice of serial interval; i.e. the longer the assumed serial interval, the higher the reproduction number. Moreover, the estimated reproduction numbers did not decline monotonically with time, indicating that the patterns of secondary transmission varied with time. These tendencies are consistent with the differences in estimates of the reproduction number of pandemic influenza in recent studies; high estimates probably originate from a long serial interval and a model assumption about transmission rate that takes no account of time variation and is applied to the entire epidemic curve.
The present findings suggest that in order to offer robust assessments it is critically important to clarify in detail the natural history of a disease (e.g. including the serial interval) as well as heterogeneous patterns of transmission. In addition, given that human contact behavior probably influences transmissibility, individual countermeasures (e.g. household quarantine and mask-wearing) need to be explored to construct effective non-pharmaceutical interventions.
In the history of human influenza, Spanish flu (1918–20), caused by influenza A virus (H1N1), has resulted in the biggest disaster to date. The disease is believed to have killed 20–100 million individuals worldwide, having a considerable impact on public health not only in the past but also in the present . Although the detailed mechanisms of its pathogenesis have yet to be clarified, pandemic influenza is characterized by severe pulmonary pathology due to the highly virulent nature of the viral strain and the host immune response against it . Even though future pandemic strains could potentially be different from that of Spanish flu, the threat of recent avian influenza epidemics is causing widespread public concern. In order to plan effective countermeasures against a probable future pandemic, a comprehensive understanding of the epidemiology of Spanish flu is crucial in offering insight into control strategies and clarifying what and how we should prepare for such an event at the community and individual level. Nevertheless, various epidemiological questions regarding the 1918–20 pandemic remain to be answered .
One use of historical epidemiological data is in quantification of the transmission potential of a pandemic strain, which can help determine the intensity of interventions required to control an epidemic. The most important summary measure of transmission potential is the basic reproduction number, R0, defined as the average number of secondary cases arising from the introduction of a single primary case into an otherwise fully susceptible population . For example, one of the best known uses of R0 is in determining the critical coverage of immunization required to eradicate a disease in a randomly mixing population, pc, which can be derived using R0: pc > 1-1/R0 . Moreover, knowing the R0 is a prerequisite for designing public health measures against a potential pandemic using simulation techniques. To date, the R0 of Spanish flu has been estimated using epidemiological records in the UK [6, 7], USA [8–10], Switzerland , Brazil  and New Zealand , all of which suggested slightly different estimates. Whereas studies in the US and UK proposed an R0 ranging from 1.5–2.0 [6, 7, 9], other studies indicated that it could be closer to or greater than 3 [8, 10–13]. In addition, an ecological modeling study proposed that the R0 of seasonal influenza is in the order of 20 , generating a great deal of controversy in its interpretation.
Another problem with Spanish flu data is that only a few studies have assessed the time course of the pandemic. Although effective interventions against influenza may have been limited in the early 20th century, it is plausible that the contact frequency leading to infection varied considerably with time owing to the huge number of deaths and dissemination of information through local media (e.g. newspapers and posters). To shed light on this issue, it is important to evaluate time-dependent variations in the transmission potential. Explanation of the time course of an epidemic can be partly achieved by estimating the effective reproduction number, R(t), defined as the actual average number of secondary cases per primary case at time t (for t > 0) [15–17]. R(t) shows 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 and may be regarded as being 'under control' at time t (vice versa, if R(t) > 1).
This paper has two main purposes, the first of which is to examine one of the possible factors yielding the slightly different R0 estimates of pandemic influenza in recent studies. Specifically, this variation is examined in relation to the choice of a key model parameter (the serial interval) frequently derived from the literature. The second is to assess the transmissibility of pandemic influenza with time. The time course of a pandemic is likely to be influenced by heterogeneous patterns of transmission and human factors that modify the frequency of infectious contact with time. The latter aim is concerned with a common assumption in many influenza models, that the transmission rate is independent of time. Under this assumption, in a homogeneously mixing population, transmissibility with time has to be characterized only by the depletion of susceptible individuals due to infection, resulting in a monotonic decrease. However, this might not be true for Spanish flu, even though its social background (e.g. media reports and global alert) was rather different from that of severe acute respiratory syndrome (SARS) in 2002–03, for example, which accompanied huge behavioral changes. The daily number of deaths during the fall wave (from September 1918 – February 1919) and the relevant statistics in Prussia, Germany  (see [Additional file 1]), are used in the following analysis.
Temporal distribution of influenza
Time variations in the transmission potential
Estimates of R and the serial interval
Reported estimates of the basic reproduction number of pandemic influenza during the fall wave (2nd wave) from 1918–19
Serial interval (days)
Fitting of a time-independent system with the entire epidemic curve
San Francisco, USA
45 cities in the USA
UK (entire England and Wales) ‡
Sao Paulo, Brazil
83 cities in the UK
3.2 and 2.6
45 cities in the USA
Featherston Military Camp, New Zealand ¶
Simulated epidemic curve
This paper has examined time variations in the transmission potential of pandemic influenza in Prussia, Germany, from 1918–19. R(t) was estimated using a discrete-time branching process, allowing reasonable assessment of the impact of the serial interval. Whereas two different stochastic models have been proposed to quantify the time variations in transmission rate [23, 24], the present study showed that reasonable estimates of R(t) can be inferred using a far simpler method without assuming the number of susceptible individuals or further details of the disease dynamics. There were two important findings. First, R(t) depends on the assumed length of the serial interval; second, it varied with time and did not decline monotonically, reflecting underlying time variations in secondary transmission. In the Prussian epidemic, R(t) stayed close to 1 in the middle of the epidemic and then increased at a later stage.
In addition, the different recently reported R0 estimates for pandemic influenza were implicitly compared. Long serial intervals, estimates of which are often derived from the literature, seem to have yielded high estimates of R0, the relationship of which has been extensively investigated in previous studies by means of sensitivity analysis [8, 25], implying that a precise estimate of the serial interval is crucial for elucidating the finer details of R0 . This point has to be interpreted cautiously in relation to Table 1, since essentially there are two potential sources of variations in R0:
(A) Estimates of R0 will greatly vary according to model assumptions and the structure and type of data used to infer the relevant parameters .
(B) R0 can differ with time and place. That is, the transmission potential is generally influenced by various underlying social and biological conditions (e.g. contact patterns, differential susceptibility and pathogenic factors) [27, 28].
It should be noted that the present study examined only some of the factors related to (A) and did not explicitly test this hypothesis. Indeed, there are other plausible explanations for the variations in R0 in Table 1. For example, point (A) may be particularly true for the UK study, the small estimates of which may be attributable to the modeling assumption that fitted the model to three waves of the pandemic . Moreover, the New Zealand study is a good example of point (B) . This epidemic was observed in a community with closed contact (i.e. an army camp), which could result in high estimates of R0 even assuming a short serial interval. Thus, no definitive reason for the differences in R0 can be clarified unless each model is examined in relation to others, permitting explicit comparisons and robustness assessment . However, despite this, it is remarkable that differences in R(t) were obtained using the assumed serial interval lengths employed in the present study and that the differences in the R0 of pandemic influenza were also consistent with this well-known relationship (i.e. between R0 and the serial interval). The finding implies that it is critically important to clarify details of the natural history of a disease in order to offer robust assessments. In addition, further controversy concerning the R0 of seasonal influenza (= 20) needs to be addressed by exploring in detail the immune protection mechanisms of influenza .
The second finding of the present study concerns the time variations in secondary transmission. Although it is commonly assumed that a large epidemic only declines to extinction with depletion of susceptible individuals, this assumption leads to a monotonic decline in R(t). That is, in a homogeneously mixing population, R(t) is given by R0S(t)/S(0), where S(t) is the number of susceptible individuals at time t . Whereas the decline in R(t) in Prussia probably reflected a decline in susceptible individuals, the observed qualitative pattern (i.e. a non-monotonic decline in R(t)) is likely to have involved other factors not included in usual assumptions of homogeneously mixing models. The non-monotonic decline in R(t) could reflect (i) heterogeneous patterns of transmission and/or (ii) other time-dependent underlying factors. For example, two important factors need to be discussed with regard to heterogeneous transmission. The first, age-related heterogeneity in transmission was ignored in the present study. Whereas the case fatality of pandemic influenza varied with age (exhibiting a W-shaped curve not only for mortality but also for case fatality ), the present study assumed fixed and crude case fatality for the entire population. Thus, if the age-related transmission patterns yield time variations in age-specific incidence , the decline in R(t) could partly be attributable to age-related heterogeneity. Similarly, the time from onset to death may also vary by age-related factors. The second important factor is social heterogeneity in transmission (e.g. spatial spreading patterns). For example, considering realistic patterns of influenza spread in a location with urban and rural sub-regions, slow decline in incidence could originate from heterogeneous spatial spread between and within rural sub-regions. If some rural areas previously free from influenza are infested by a few cases at some point in time, such local spread could modify the overall epidemic curve. Since the present study assumed a closed population because detailed data were lacking, additional information (e.g. cases with time and place) is needed to elucidate the finer details.
With respect to (ii), other time-dependent underlying factors, it is likely that public health measures as well as human contact behaviors (including human migration) also influence the time course of an epidemic. From a very early study , it has been suggested that human behavioral changes (or differing transmission rates due to time-varying contact patterns) are observed during the course of an epidemic. If this is the case, the finding suggests that time-varying transmission potential is not only the case for SARS (i.e. recent epidemics accompanied by considerable media coverage) [15, 32, 33] but also for historical epidemics with a huge magnitude of disaster. Indeed, recent studies on Spanish flu in the US that employed rough assumptions implied that interventions had a considerable impact on the time trend [34, 35]. This also reasonably explains why high estimates of R0 are likely to originate from fitting an autonomous model to the entire epidemic curve. In practical terms, such a result implies that human behaviors could considerably influence transmissibility, and moreover, could potentially be a necessary countermeasure. Understanding the significant impact of human contact behaviors on the time course is therefore of importance . For example, non-pharmaceutical individual countermeasures are crucial for poor resource settings, especially in developing countries . In addition to community-based measures such as social distancing and area quarantine, it is also crucial to suggest what can be done at the individual level. In line with this, the effectiveness of individual countermeasures (e.g. household quarantine and mask wearing) needs to be further explored using additional data (i.e. of seasonal influenza) and models.
In summary, this paper showed the relationship between the R(t) and serial interval and assessed time variations in the transmissibility of pandemic influenza. The findings imply a need to detail the natural history of influenza as well as heterogeneous patterns of transmission, suggesting that robust assessment can only be made when population- and individual-based disease characteristics are clarified  and implying that further observations in clinical and public health practice are crucial. Given that individual human contact behaviors could influence the time variations in transmission potential, further understanding of the importance of individual-based countermeasures (e.g. household quarantine and mask wearing) could therefore offer hope for development of effective non-pharmaceutical interventions.
Medical officers in Prussia recorded the daily number of influenza deaths from 29 September 1918 to 1 February 1919 (Figure 1) ; a total of 8911 deaths were reported (see [Additional file 1]). Throughout the pandemic period in Germany, the largest number of deaths was seen in this fall wave . Prussia represents the northern part of present Germany and at the time of the pandemic was part of the Weimer Republic as a free state following World War I. The death data were collected from 28 different local districts surrounding the town of Arnsberg, which, at the time of the epidemic, had a population of approximately 2.5 million individuals (the mortality rate in this period being 0.36%). Although case fatality for the entire observation area was not documented, the numbers of cases and deaths during part of the fall wave were recorded for 25 districts. Among a total of 61,824 cases, 1609 deaths were observed, yielding a case fatality estimate of 2.60% (95% CI: 2.48, 2.73). For simplicity, the inflow of infected individuals migrating from other areas was ignored in the following analysis.
Back-calculation of the daily case onset
where p is the case fatality ratio, which is independent of time. Although the case fatality, p, was not taken into account in Figure 1, the following model reasonably cancels out the effect of p assuming that the conditional probability of death given infection is independent of time.
Estimation of the reproduction number
Equation (2) is a slightly different expression of a method proposed for SARS . The advantages of this model include: (i) we only need to know the time of onset of cases (i.e. the model does not require the total number of susceptible individuals or detailed contact information) and (ii) the time-dependent reproduction number can be reasonably estimated using a far simpler equation than other population dynamics models. Unfortunately, detailed information on the distribution of the serial interval, g(τ), is not available for pandemic influenza, and historical records often offer only an approximate mean length. Although a recent study estimated the serial interval from household transmission data of seasonal influenza , this is likely to have been considerably underestimated owing to the short interval from onset to secondary transmission within the households examined. Thus, the analyses conducted in the present study simplify the model using various mean lengths of the serial interval assumed in previous works. Supposing that we observed Ci cases in generation i, the expected number of cases in generation i+1, E(Ci+1) occurring a mean serial interval after onset of Ci is given by:E(Ci + 1) = C i R i
where Ri is the effective reproduction number in generation i. That is, cases in each generation, C1, C2, C3, ..., Cn are given by C0R0, C1R1, C2R2, ..., Cn-1Rn-1 and also by , respectively. By incorporating variations in the number of secondary transmissions generated by each case into the same generation (referred to as offspring distribution), the model can be formalized using a discrete-time branching process . The Poisson process is conventionally assumed to model the offspring distribution, representing stochasticity (i.e. randomness) in the transmission process. This assumption indicates that the conditional distribution of the number of cases in generation i+1 given Ci is given by:Ci + 1|C i ~ Poisson[C i R i ]
Since the Poisson distribution represents a one parameter power series distribution, the expected values and uncertainty bounds of Ri can be obtained for each generation. The 95% CI were derived from the profile likelihood. Since the length of the serial interval in previous studies ranged from 0.9 to 6 days [8, 10, 13], three different fixed-length serial intervals (i.e. 1, 3 and 5 days) were assumed for equation (5) with respect to the observed data. Although application of the Heaviside step function for the serial interval suffers some overlapping of cases in successive generations, this study ignored this and, rather, focused on the time variation in transmissibility using this simple assumption. For each length, the daily number of cases was grouped by the determined serial interval length. Whereas the choice of serial interval therefore affects estimates of Ri, it does not affect the ability to predict the temporal distribution of cases. It should be noted that this simple model assumes a homogeneous pattern of spread.
To assess the performance of the above-described estimation procedure, stochastic simulations were conducted. The simulations directly used the branching process model, the offspring distribution of which follows a Poisson distribution with expected values, Ri, estimated for each interval, i. Although the offspring distribution tends to exhibit a right-skewed shape (which was approximated by negative binomial distributions in recent studies [15, 22, 39]), it is difficult to extract additional information from the temporal distribution of cases only, so this paper focused on time variations in R(t) rather than individual heterogeneity. Each simulation was run with one index case at epidemic day 0. For the first two serial intervals, primary cases were set to generate 2.52 and 1.95 secondary cases deterministically in order to avoid immediate stochastic extinctions. Simulations were run 1000 times.
The author thanks Klaus Dietz for useful discussions. This study was supported by the Banyu Life Science Foundation International and the Japanese Ministry of Education, Science, Sports and Culture in the form of a Grant-in-Aid for Young Scientists (#18810024, 2006).
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