Real-time forecasting of an epidemic using a discrete time stochastic model: a case study of pandemic influenza (H1N1-2009)
© Nishiura; licensee BioMed Central Ltd. 2011
Received: 9 December 2010
Accepted: 16 February 2011
Published: 16 February 2011
Real-time forecasting of epidemics, especially those based on a likelihood-based approach, is understudied. This study aimed to develop a simple method that can be used for the real-time epidemic forecasting.
A discrete time stochastic model, accounting for demographic stochasticity and conditional measurement, was developed and applied as a case study to the weekly incidence of pandemic influenza (H1N1-2009) in Japan. By imposing a branching process approximation and by assuming the linear growth of cases within each reporting interval, the epidemic curve is predicted using only two parameters. The uncertainty bounds of the forecasts are computed using chains of conditional offspring distributions.
The quality of the forecasts made before the epidemic peak appears largely to depend on obtaining valid parameter estimates. The forecasts of both weekly incidence and final epidemic size greatly improved at and after the epidemic peak with all the observed data points falling within the uncertainty bounds.
Real-time forecasting using the discrete time stochastic model with its simple computation of the uncertainty bounds was successful. Because of the simplistic model structure, the proposed model has the potential to additionally account for various types of heterogeneity, time-dependent transmission dynamics and epidemiological details. The impact of such complexities on forecasting should be explored when the data become available as part of the disease surveillance.
Mathematical models in population biology and epidemiology have greatly progressed during the past few decades, supporting the argument for the relevance of theoretical models to the study of empirical observations [1, 2]. The transmission dynamics of infectious diseases have been well studied using modeling methods, facilitating our understanding of the mechanisms of disease spread [3–5], allowing the optimization of infectious disease control, and influencing public health policymaking [4, 6]. Of the various diseases that have been studied, the transmission dynamics of influenza have attracted much scientific interest, and from the beginning of the 2009 pandemic, mathematical modeling has progressed our understanding of the epidemiological dynamics of influenza (H1N1-2009) . Among the various applications of mathematical models to infectious disease epidemiology, future prediction is an area that has been understudied and methods for real-time and long-term prediction in large populations have yet to be sought [8–10]. A vast amount of past quantitative modeling effort has been devoted to the inverse problem methodology which focuses on statistical estimations of model parameters and in which the process of model building imposes strong assumptions about the underlying transmission dynamics [11, 12].
Prediction has two components: forecasting and projection . A forecast is a quantitative attempt to predict what will happen in the future, while a projection is an attempt to describe what would happen under certain assumptions and hypotheses. Given the many studies that have examined 'what if' scenarios of an influenza pandemic using a number of plausible parameter settings [14, 15], in a sense, one could regard the projection of influenza as having been widely studied. However, except for the monitoring and detection of outbreaks based on time series surveillance data , quantitative methods for forecasting have yet to be fully established. Although the real-time estimation of model parameters has been proposed with, for example, the aim of assessing the effectiveness of certain control measures in real-time , as mentioned above, such studies tended to focus on parameter estimation and quantitative forecasting has been understudied. During the course of an epidemic, it may be important to forecast the future course of the epidemic in real-time.
To date, three different approaches have been proposed for the real-time forecasting of influenza. The first employs a parsimonious, but flexible, power-law logistic equation to directly fit the parametric model (the analytical solution) to epidemic curves [18, 19]. Despite the omission of the so-called "dependent happening", defined as an epidemiological phenomenon in which the risk of infection in one individual depends on the risk in other individuals in the same population unit, and the use of a simplistic minimization of the sum of squared errors, an SIR (susceptible-infected-recovered) epidemic model is known to be approximated by a family of logistic equations [20, 21], and the flexible power-law logistic equation has been shown to yield reasonable fits to empirical data of H1N1-2009 . A second approach employs a deterministic compartmental model to describe epidemic curves of pandemics that occurred during the 20th century . This model has been shown to yield very good fits to the data, although the fitting procedure using the deterministic model requires the estimation of a total of nine parameters and computing the uncertainty bounds of forecasts is complex. One can, of course, reduce the complexity by reducing the number of unknown parameters before implementing the forecasting. The third, a hybrid stochastic epidemic model that employs a Bayesian method, was applied to H1N1-2009 in Singapore . Although the Bayesian method yields reasonable uncertainty bounds of forecasts through the posterior distribution, a likelihood-based approach to improve our analytical understanding has yet to be considered. Accordingly, a simple likelihood-based model for forecasting that permits us to compute the prediction interval (the interval in which future observations will fall with a certain probability), is called for.
The aims of the present study are; (i) to develop a simple and practical approach to the real-time forecasting of an epidemic, and (ii) to apply the proposed method to a case study of pandemic influenza (H1N1-2009) in Japan. Here the empirical data for H1N1-2009 in Japan and technical problems of forecasting epidemics are described and a discrete time stochastic model that is analogous to an SIR epidemic model is derived. By imposing a branching process approximation to adhere to discrete time data, a simple method for computing the 95% prediction interval is proposed.
Description of the data
The aim of the present study is to forecast the future weekly incidence during the course of the epidemic. The four arrows in Figure 1 indicate the weeks of prediction (weeks 42, 45, 48 and 51 in 2009) that were selected to compare the validity of forecasting. These weeks were chosen for comparison because they are close to the peak and it is known that the forecasting of epidemics is of limited accuracy before the peak incidence is observed [18, 22, 23] and is likely to be greatly improved near the peak. The highest incidence was observed in week 48, so forecasts in weeks 42 and 45 represent those before the peak, in week 48 those at the peak and in week 51 those after the peak. To simplify the calculations that follow, the calendar weeks (week 27, 2009 to week 18, 2010) in which the data were collected have been set to match the actual weeks of the study (week 0 to 44).
Four major technical challenges for the real-time prediction should be noted. First, the observed epidemic curve represents only a single sample path (or a single stochastic realization) among all possible trajectories of the epidemic . This implies that the model should account for stochastic variations in the data [28, 29]. Second, because the virus is transmitted from host to host (human to human infection), an observation at time t depends on the previous series of observations up to time t-1 , reflecting the abovementioned dependent happening and statistically requiring conditional assessments. Third, any empirical data are reported and published at discrete time intervals, while, for the purpose of forecasting, ideal statistical data are continuous. The data in Figure 1 are based on weekly reporting which does not offer any information regarding the dynamics within each reporting interval. Fourth, the observed data usually involve reporting delays. Moreover, accounting for heterogeneity (spatial heterogeneity and social patterns of contact) and time-dependent epidemiological dynamics (seasonality of transmission, contact behaviors and public health interventions) is ideally required to give detailed insights into the epidemiological dynamics. Because the data in Figure 1 describe a single temporal distribution of the epidemic curve for an entire population of Japan, it does not have the information necessary to explicitly address these heterogeneities.
Chain binomial model
Because the problems of delay and heterogeneity cannot be explicitly addressed without additional epidemiological information, data in Figure 1 are regarded as the weekly number of new infections (without any delay) generated by a homogeneously mixing population. It is also assumed that no intervention took place. These theoretical simplifications do not permit the interpretation of the model parameters explicitly in practical terms, and so the details of actual dynamics have, for now, been ignored. Rather, the focus is on the predictive performance of the simple model. The estimated parameters do retain practical interpretations for a hypothetical population in which the data generating process used for Figure 1 exactly follows the theoretical assumptions that are made.
Detailed properties of the Reed-Frost model are reviewed elsewhere . Assuming that λ k+1 = exp(-βC k) and that the reporting interval is close to the infectious period of the disease of interest, the Reed-Frost model has been shown to be comparable to an SIR epidemic model with certain assumptions [31, 34], and an extension of this type of Markov model has been applied to the real-time forecasting of influenza . Despite its usefulness, the Reed-Frost model is not readily analyzed for large S 0 (due to binomial arguments), and is mainly applicable to small populations. Although the issue of a large S 0 has been addressed for computing the final size (i.e. the total number, or the proportion, of infections throughout the course of an epidemic) by means of the so-called Sellke construction [35, 36], an approximate strategy is required for implementing real-time forecasting in a large population (see Barbour and Utev  for a detailed derivation of the approximation).
An approximate branching process
As mentioned above, the chain binomial model can be related to the SIR epidemic model with some adjustment of the generation time  (the time interval between infection of a primary case and infection of a secondary case caused by the primary case ), although the crudely reported weekly data sometimes include a few generations of cases within each reporting interval. For instance, a contact tracing of H1N1-2009 in the Netherlands estimated the mean generation time as T g = 2.7 days , implying that weekly data can include more than two generations of influenza cases. Therefore, a different approach by imposing a linear argument to the dynamics within each reporting interval has been used.
Assuming that T g is known (2.7 days), then the epidemic curve is governed by only two parameters, S 0 and R i. Thus, an SIR model with a constant generation time has been simplified to a branching process model that explicitly accounts for the practical interpretation of the observed weekly cumulative incidence C k.
Statistical estimation and computation of the uncertainty bounds
which is valid only in the case of a constant generation time.
because all possible chains for both weeks K+1 and K+2 have to be considered. The sums have to be calculated directly. Similarly, for a later week K+m, the sums of all possible chains in weeks K+1, K+2, ..., K+m-1 have to be computed. Although finding C U and C L for later chains requires a computer programming code, the chain Poisson model still remains computationally very simple. Alternatively, a negative binomially distributed offspring distribution  in which a dispersion parameter has to be jointly estimated could be used.
where n is the number of weeks of observation involving conditional expectation or prediction (n = 44 in the case study). MAE was chosen to measure the validity of forecasting, because (i) the scale does not directly influence the assessment of the predictions as a whole nor does it affect the comparative examination by week of prediction and (ii) the comparison is made against a single observed time series data set .
Estimates of parameters for the proposed model using weekly incidence data of influenza (H1N1-2009) in Japan
Week of prediction*
Initial reproduction number
Initially susceptible individuals (×105)
Total number of cases (×105)†
1.14 (0.88, 1.40)
113083 (0, 256710827)
26778 (25826, 27749)
1.18 (1.10, 1.28)
391 (218, 741)
573 (66, 1637)
1.15 (1.07, 1.21)
754 (0, 2225)
183 (105, 261)
1.15 (1.09, 1.20)
716 (540, 1104)
175 (100, 251)
1.13 (1.09, 1.18)
834 (664, 1149)
188 (101, 274)
The estimate of S 0 differed greatly depending upon the weeks of prediction. At week 15, S 0 was overestimated to the extent that it exceeded the actual population of Japan (approximately 1200×105). Although an advantage of the proposed stochastic model is its potential to estimate S 0 from incidence data, the estimates of S 0 before the epidemic peak appeared to be inaccurate. Based on the entire epidemic curve, S 0 was estimated to be 834×105, indicating that 69.5% of the Japanese population was initially susceptible. Given that the estimate agrees well with the result of serological surveillance , S 0 for the entire epidemic curve may be validly quantified even without the population data. Despite slight underestimations, the estimates of S 0 at and after the epidemic peak are close to the estimate based on week 44 with overlapping CIs.
Even in week 15, assuming that S 0 is known (set as 834×105 persons based on week 44), the epidemic curve described by R i alone qualitatively captured the observed epidemic curve (figure not shown). R i was estimated to be 1.16 (95% CI: 1.08, 1.23) and the MAE was reduced to 1.26, indicating that early forecasting is sensitive to variations in S 0 which is influenced by variations in the growth rate. When a constraint for the upper boundary of S 0 using the entire population size for Japan (say, 1200×105) was imposed for the prediction at week 15, R i and MAE were, estimated at 1.15 and 2.96, respectively (the original MAE at week 15 was 663 as shown in Table 1). Clearly, the validity of the prediction was greatly improved by using a constraint on the population size. Nevertheless, it should be noted that the use of a constrained S 0 imposes the arbitrary assumption that the entire population was initially susceptible and was fully involved in the transmission dynamics.
The present study has proposed a method for real-time forecasting based on crudely reported weekly incidence data, accounting for demographic stochasticity and conditional measurement and employing a simple discrete time stochastic model. The proposed model was constructed using a branching process approximation of a chain binomial model. In particular, realizing that the weekly incidence data of influenza C k is less interpretable than the incidence data of other diseases with longer generation times (e.g. measles), the iterative model was parameterized by assuming exponential growth of cases within each reporting interval. Consequently, the parsimonious model resulted in a novel, yet fully tractable form. Although the proposed stochastic model is analogous to models with a series of chains, it can incorporate a more realistic distribution of the generation time and, given more detailed epidemiological information, has a broad range of extensions. Moreover, the chains of Poisson offspring distributions enable the computation of the 95% prediction intervals. It is known that a non-linear model does not allow simple computation of the prediction interval  and, although a more formal approach to computing the prediction interval should ideally account for future observations more explicitly (and to be strict, the prediction interval of the present study may better be referred to as the forecast region), the proposed approach is not very computationally demanding.
The biggest advantage of the proposed model is its potential to describe and predict the epidemic curve with interpretable parameters S 0 and R i under a homogeneous mixing assumption. In addition, the parameterization produces estimates that can be exploited to compute the final epidemic size. Nevertheless, as was observed in other attempts at real-time forecasting [18, 22, 23], the forecast appears to be very vulnerable to the timing of forecasting, especially during the early growth phase of an epidemic. Indeed, Figure 4 has captured the difficulty of early forecasting in terms of the MAE. Although, even at week 15, the qualitative behavior of forecasts is greatly improved by fixing S 0 or by imposing constraints for S 0 (and leaving only R i as a free parameter), the advantage of the proposed model is in its ability to estimate S 0 explicitly. Indeed, in practical settings it may be best to assume that S 0 is an unobserved variable. It should be noted that the results also imply that serological surveillance before and during an epidemic may be a great help in improving the forecasts .
Despite the omission of heterogeneity, when more precise data in time and structure becomes available, it can readily be incorporated into the proposed model. For example, the model can potentially be extended for age-dependent and spatially structured data like that used to compute the final epidemic size in a multi-host population . Such an extension could potentially begin to address the difficulty of real-time forecasting in the presence of a multimodal epidemic curve. That is, given that a few peaks in a single temporal distribution resulted from multiple epidemic curves in different spatial units , the spatial extension could capture different epidemic waves in different geographic areas . Another important future task is to allow the model to fully adhere to the data generating process. If the reporting delay and any time-dependent epidemiological information (e.g. data that are likely to inform a time-dependent covariate of the risk of infection) are known, the proposed model could potentially incorporate those aspects in the model-building strategy. The impact of such complexities on forecasting should be explored when the required information becomes available as part of the surveillance.
As was shown through the likelihood-based approach, the present study has demonstrated that real-time forecasting can rest on a simple discrete time stochastic model and has shown that the uncertainty bounds can reasonably be computed using the conditional offspring distributions. Despite the simplicity, the present study successfully offers a sound modeling strategy and a methodological avenue to implement real-time forecasting of an epidemic in the midst of its course.
Because real-time forecasting of epidemics has been understudied, in the present study a discrete time stochastic model, accounting for demographic stochasticity and conditional measurement was developed. The model permitted us to derive the uncertainty bounds using chains of conditional offspring distributions. The proposed method was applied to the weekly incidence of pandemic influenza (H1N1-2009) in Japan. The validity of forecasts made before the epidemic peak appeared, largely to depend on obtaining good parameter estimates, and the forecasts of both weekly incidence and final epidemic size greatly improved at and after the peak with all the observed data points falling within the uncertainty bounds. Because the structure of the proposed model is simple, it has the potential to additionally account for heterogeneity, time-dependent transmission dynamics and epidemiological details when that information becomes available as part of the data generating process.
HN is supported by the Japan Science and Technology Agency PRESTO program.
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