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Different Epidemic Curves for Severe Acute Respiratory Syndrome Reveal Similar Impacts of Control Measures

Different Epidemic Curves for Severe Acute Respiratory Syndrome Reveal Similar Impacts of Control... American Journal of EPIDEMIOLOGY Volume 160 Copyright © 2004 by The Johns Hopkins Bloomberg School of Public Health Number 6 Sponsored by the Society for Epidemiologic Research September 15, 2004 Published by Oxford University Press ORIGINAL CONTRIBUTIONS Different Epidemic Curves for Severe Acute Respiratory Syndrome Reveal Similar Impacts of Control Measures Jacco Wallinga and Peter Teunis From the National Institute for Public Health and the Environment, Bilthoven, the Netherlands. Received for publication October 27, 2003; accepted for publication March 29, 2004. Severe acute respiratory syndrome (SARS) has been the first severe contagious disease to emerge in the 21st century. The available epidemic curves for SARS show marked differences between the affected regions with respect to the total number of cases and epidemic duration, even for those regions in which outbreaks started almost simultaneously and similar control measures were implemented at the same time. The authors developed a likelihood-based estimation procedure that infers the temporal pattern of effective reproduction numbers from an observed epidemic curve. Precise estimates for the effective reproduction numbers were obtained by applying this estimation procedure to available data for SARS outbreaks that occurred in Hong Kong, Vietnam, Singapore, and Canada in 2003. The effective reproduction numbers revealed that epidemics in the various affected regions were characterized by markedly similar disease transmission potentials and similar levels of effectiveness of control measures. In controlling SARS outbreaks, timely alerts have been essential: Delaying the institution of control measures by 1 week would have nearly tripled the epidemic size and would have increased the expected epidemic duration by 4 weeks. disease outbreaks; estimation; infection; models, statistical; SARS virus; severe acute respiratory syndrome; statistics Abbreviations: CI, confidence interval; SARS, severe acute respiratory syndrome. Editor’s note: An invited commentary on this article throughout the world and seeded outbreaks in Vietnam, Singapore, and Canada (2–4). On March 12, 2003, the World appears on page 517, and the authors’ response appears on Health Organization issued a global alert. On March 15, the page 520. World Health Organization issued a second global alert and provided a case definition and a name, severe acute respira- On November 16, 2002, the first known case of atypical tory syndrome (SARS), for the new disease (1). By this time, pneumonia occurred in Guangdong Province in southern Hong Kong, Vietnam, Singapore, and Canada had instituted China (1). In late February 2003, the infection spread from general infectious disease control measures, such as quaran- Guangdong to Hong Kong. From there it spread by airline tine, isolation, and strict hygiene measures in hospitals (1). Correspondence to Dr. Jacco Wallinga, National Institute for Public Health and the Environment, Antonie van Leeuwenhoeklaan 9, 3721 MA Bilthoven, the Netherlands (e-mail: [email protected]). 509 Am J Epidemiol 2004;160:509–516 510 Wallinga and Teunis FIGURE 1. Epidemic curves (numbers of cases by date of symptom onset) for severe acute respiratory syndrome (SARS) outbreaks in a) Hong Kong, b) Vietnam, c) Singapore, and d ) Canada and the corresponding effective reproduction numbers (R ) (numbers of secondary infections generated per case, by date of symptom onset) for e) Hong Kong, f ) Vietnam, g) Singapore, and h) Canada, 2003. Markers (white spaces) show mean values; accompanying vertical lines show 95% confidence intervals. The vertical dashed line indicates the issuance of the first global alert against SARS on March 12, 2003; the horizontal solid line indicates the threshold value R = 1, above which an epidemic will spread and below which the epidemic is controlled. Days are counted from January 1, 2003, onwards. The etiologic agent of the disease, a coronavirus, was identi- the symptom onset date by t . For the SARS epidemics in fied on April 16, 2003 (5). Estimates of important epidemio- Hong Kong, Vietnam, and Singapore, we derived the dates logic measures such as the case-fatality rate and the of symptom onset from epidemic curves provided by the incubation period were reported in May 2003 (6–8). World Health Organization (9). For Canada, we used the epidemic curve provided by Health Canada (10). Despite the rapid progression of understanding of the new disease, differences between the epidemic curves (numbers of probable SARS cases by date of symptom onset) have Observed distribution of generation intervals been awaiting further explanation. For the affected regions, these epidemic curves have been publicly available from the The generation interval, denoted by τ, is the time from end of March 2003 onwards (9), and they reveal distinct symptom onset in a primary case to symptom onset in a temporal patterns in numbers of SARS cases. This is espe- secondary case. Sometimes this generation interval is called cially remarkable for the epidemics in Hong Kong, Vietnam, the serial interval (8) or generation time (11). The generation Singapore, and Canada, since these epidemics started almost intervals observed during the SARS outbreak in Singapore simultaneously in late February 2003 and similar control are well described by a Weibull distribution with a shape measures were put in place at almost the same time in March parameter α and a scale parameter β, with values corre- 2003 (figure 1, parts a–d). The question arises as to whether sponding to a mean generation interval of 8.4 days and a the affected regions differed in terms of transmission poten- standard deviation of 3.8 days (8). We denote this distribu- tial for SARS or effectiveness of control measures. In this tion by τ ∼ w(τ|α,β). paper, we interpret the observed epidemic curves with regard to disease transmission potential and effectiveness of control ESTIMATION OF REPRODUCTION NUMBERS measures, and we compare the epidemiologic profiles of SARS outbreaks in Hong Kong, Vietnam, Singapore, and Reproduction numbers Canada. The key epidemiologic variable that characterizes the transmission potential of a disease is the basic reproduction DATA number, R , which is defined as the expected number of secondary cases produced by a typical primary case in an Observed epidemic curves entirely susceptible population (12–15). When infection is The epidemic curve is the number of reported cases by spreading through a population, it is often more convenient date of symptom onset. For each reported case i, we denote to work with the effective reproduction number, R, which is Am J Epidemiol 2004;160:509–516 Analysis of Different SARS Epidemic Curves 511 defined as the actual average number of secondary cases per derivation of these equations is provided in Appendix 1. The primary case. The value of R is typically smaller than the estimation algorithm allows estimation of the effective value of R , and it reflects the impact of control measures and reproduction numbers for infectious diseases at a finer depletion of susceptible persons during the epidemic. If R temporal resolution under more general assumptions than exceeds 1, the number of cases will inevitably increase over was previously possible. time, and a large epidemic is possible. To stop an epidemic, R needs to be persistently reduced to a level below 1. Estima- Testing the estimation procedure with simulated data tion of R is a simple affair if there is information about who infected whom. Then it is possible to construct an infection To test how well the estimation procedure approximates network (11), wherein cases are connected if one person the underlying value of an effective reproduction number infected the other. Estimation of R involves simply counting during a typical SARS outbreak, we estimate the effective the number of secondary infections per case. reproduction numbers from simulated epidemic curves. We have constructed a stochastic, individual-based model that simulates epidemic processes with exactly specified proper- A likelihood-based estimation procedure ties. The model allows for a variable effective reproduction Often, estimation of the effective reproduction number is a number R as a function of symptom onset data t, and the complicated affair, because only the epidemic curve is model parameters are estimated from observations on the observed and there is no information about who infected SARS epidemic in Singapore (see Appendix 2). Applying whom. Recent analyses of closely monitored epidemics have the estimation procedure to simulated epidemic curves shown that it is possible to estimate the probability that one shows that most of the estimates are close to the actual repro- person has infected another if the spatial locations of the duction numbers and that a few estimates based on small infected persons are available (11, 16). However, when only outbreak sizes are below the actual values that are used in the times of symptom onset are available, most investigators simulation model. On average, the estimates tend to be lower resort to approximating R by assuming an exponential than the actual values but deviate less than 5 percent from the increase in the number of cases over time (8, 17) or by fitting actual reproduction numbers. If we account for the effects of a specific model that summarizes assumptions about the incomplete reporting and temporal change in generation epidemiology of the disease (7, 18, 19). Such assumptions interval, the estimates become only slightly less accurate, can be avoided by using a likelihood-based estimation proce- and on average they deviate less than 15 percent from the dure that infers “who infected whom” from the observed actual reproduction numbers (see Appendix 2). dates of symptom onset as provided by the epidemic curve. However, the computational burden of a straightforward RESULTS numerical evaluation of the likelihood appears to be enor- mous, since it requires consideration of all possible infection For Hong Kong, Vietnam, Singapore, and Canada, we networks, and even for a small outbreak of 50 cases have converted the epidemic curves into the time course of there are almost 7 × 10 possible infection networks (see effective reproduction numbers. The results are shown in Appendix 1). figure 1, parts e–h. These four large outbreaks were Here we show that it is possible to obtain likelihood-based sparked almost simultaneously by the same index patient. estimates of R while avoiding the computational problems if All regions have faced erratic “super-spread events” we use pairs of cases rather than the entire infection network. wherein cases produced more than 10 secondary infections. The relative likelihood p that case i has been infected by These “super-spread events” show up in parts e–h of figure ij case j, given their difference in time of symptom onset t – t , 1 as temporary increases in effective reproduction numbers i j can be expressed in terms of the probability distribution for around the symptom onset date of the index case for the the generation interval. This distribution for the generation “super-spread event.” In Hong Kong, Vietnam, and interval is available for many infectious diseases, and we Singapore, there were “super-spread events” marking the denote it by w(τ). The relative likelihood that case i has been start of the outbreak. In Hong Kong, Singapore, and infected by case j is then the likelihood that case i has been Canada, there were “super-spread events” after control infected by case j, normalized by the likelihood that case i measures were implemented. has been infected by any other case k: During the early phase of the SARS epidemic, before the first World Health Organization global alert was issued on p = wt() – t ⁄ wt() – t . March 12, 2003, the average effective reproduction numbers ij i j i k ik ≠ were markedly similar across the regions: Each case The effective reproduction number for case j is the sum over produced approximately three secondary infections (table 1). all cases i, weighted by the relative likelihood that case i has A value of R slightly higher than 3 is consistent with the been infected by case j: observed epidemics in all four regions. Around mid-March, control measures were implemented in all regions, and R = p . during this period the effective reproduction numbers j ij decreased sharply. For some regions, the effective reproduc- Note that we are not making any assumption regarding the tion numbers continued to decrease at a slow pace, distribution of numbers of secondary infections per case suggesting improvement of control measures while the (i.e., the p are independent in j). Additional detail on the epidemic was going on. After the first World Health Organi- ij Am J Epidemiol 2004;160:509–516 512 Wallinga and Teunis TABLE 1. Average daily effective reproduction number (R ) for cases of severe acute respiratory syndrome (SARS) with a symptom onset date before or after the issuance of the first global alert against SARS on March 12, 2003, for regions where infection was introduced in late February Hong Kong Vietnam Singapore Canada Symptom onset R 95% CI* R 95% CI R 95% CI R 95% CI Before alert 3.6 3.1, 4.2 2.4 1.8, 3.1 3.1 2.3, 4.0 2.7 1.8, 3.6 After alert 0.7 0.7, 0.8 0.3 0.1, 0.7 0.7 0.6, 0.9 1.0 0.9, 1.2 * CI, confidence interval. zation global alert, each case produced approximately 0.7 We have presented the relation between observed secondary infections (table 1). A value of R = 0.7 is consis- epidemic curves and inferred reproduction numbers from tent with all four regions, except Canada. By the beginning an infection-network perspective. We are certainly not the of July 2003, transmission of the SARS virus had been first researchers to do so: Infection networks have been stopped in all regions where the infection was introduced in used extensively in the area of sexually transmitted late February 2003 (1). diseases (20), and an infection-network perspective was adopted to analyze a closely observed foot-and-mouth To explore the range of epidemic curves that can result from the same epidemic process, we performed extensive epidemic in Great Britain (11). Our contribution is the deri- vation of likelihood-based estimates of effective reproduc- computer simulations using a model of epidemics with char- tion numbers, requiring only the observed time of symptom acteristics similar to those of SARS (see Appendix 2). The outcome of 10,000 simulations shows a highly variable onset for the observed cases. The use of a likelihood frame- work provides a set of powerful tools for inference, uncer- epidemic size and epidemic duration: The mean epidemic tainty analysis, and model selection (20); the use of only size is 685 cases (95 percent confidence interval (CI): 27, 2,446), and the mean epidemic duration is 98 days (95 time of symptom onset allows us to apply this method to routinely collected epidemic-curve data. However, estima- percent CI: 32, 187). All observed sizes and durations of tion of effective reproduction numbers from epidemic SARS epidemics are within this very wide range of possible outcomes resulting from chance alone. Additionally, we curves has an intrinsic limitation that should be kept in mind: The effective reproduction number contains entan- used computer simulations to explore the effect of the timing gled information about the transmission potential (i.e., the of implementation of control measures on the epidemic size and duration, in a setting that is typical for the SARS basic reproduction number) and the effectiveness of control measures. These two components can be disentangled only outbreaks. The simulation results show a high sensitivity to when we obtain additional information—for example, the timing of implementation of control. On average, a 1- week delay in implementation of control measures results in about the time of implementation of control measures. Moreover, individual contributions to the effective repro- a 2.6-fold increase in mean epidemic size and a 4-week duction number are entangled if their date of symptom extension of the mean epidemic duration. onset is smaller than the generation interval. This limitation can be overcome if more detailed information on who DISCUSSION infected whom is available (see Appendix 1). This study showed that there exists a direct relation For the SARS outbreak in Hong Kong, it is possible to between the epidemic curve and the time course of the repro- compare our results with previously published estimates. duction number R. This relation is determined by the distri- Our estimate of the average effective reproduction number bution of the generation intervals. The relation can be used to prior to the first global alert (R = 3.6, 95 percent CI: 3.1, 4.2) monitor the combined effects of transmission potential and is more precise than the estimate obtained by assuming an control measures during an epidemic. We have shown that exponential increase in the number of cases (R = 3.5, 95 the epidemic curves for SARS in Hong Kong, Vietnam, percent CI: 1.5, 7.7) (8) and more precise than the estimated Singapore, and Canada, though apparently different, are all lower bound excluding “super-spreading events” (R > 2.7) consistent with a single time course of the effective repro- (7). Our estimate of the average effective reproduction duction numbers for SARS. This apparent difference in number after the first global alert (R = 0.7, 95 percent CI: epidemic curves arises because chance effects, such as the 0.7, 0.8) is much higher than the previously estimated lower occurrence of a rare “super-spread event,” leave a lasting bound excluding “super-spread events” (R > 0.14) (7). This trace on the epidemic curve. In contrast, chance events mani- comparison illustrates that the estimation algorithm fest only as temporary increases in the reproduction number. presented here allows more precise estimation of the effec- Our analysis of the epidemic curves for SARS shows that tive reproduction numbers for infectious diseases under one should be cautious in taking a smaller epidemic size and more general assumptions than was previously possible. a shorter epidemic duration as proof of better infection The effectiveness of control measures against SARS can control. be estimated if it is assumed that the sudden decrease in the Am J Epidemiol 2004;160:509–516 Analysis of Different SARS Epidemic Curves 513 effective reproduction number for SARS around the time of 3. Poutanen SM, Low DE, Henry B, et al. Identification of severe acute respiratory syndrome in Canada. N Engl J Med 2003;348: the first global alert was indeed caused by the implementa- 1995–2005. tion of control measures. The decrease in the reproduction 4. Severe acute respiratory syndrome—Singapore, 2003. MMWR number from 3 to 0.7 then suggests that the control measures Morb Mortal Wkly Rep 2003;52:405–11. taken around this time prevented approximately 77 percent 5. Fouchier RAM, Kuiken T, Schutten M, et al. Aetiology: Koch’s (100 × (3 – 0.7)/3 percent ≈ 77 percent) of all potential postulates fulfilled for SARS virus. Nature 2003;423:240. secondary infections. This effectiveness suffices for eventu- 6. Donnelly CA, Ghani AC, Leung GM, et al. Epidemiological ally stopping infections in situations where each case causes determinants of spread of causal agent of severe acute respira- up to 4.3 (1/(1 – 0.77) ≈ 4.3) secondary infections. Recalling tory syndrome in Hong Kong. Lancet 2003;361:1761–6. that before implementation of control measures, each SARS 7. Riley S, Fraser C, Donnelly CA, et al. Transmission dynamics case produced, on average, approximately three secondary of the etiological agent of SARS in Hong Kong: impact of pub- infections, this effectiveness has been barely sufficient for lic health interventions. Science 2003;300:1961–6. the eventual interruption of the chain of human-to-human 8. Lipsitch M, Cohen T, Cooper B, et al. Transmission dynamics transmission of the SARS virus. and control of severe acute respiratory syndrome. Science 2003;300:1966–70. As the direct threat of a worldwide SARS epidemic has 9. World Health Organization. Epidemic curves—severe acute waned, the question arises as to how we can use the experi- respiratory syndrome (SARS). Geneva, Switzerland: World ence with SARS to improve infection control against new Health Organization, 2003. (World Wide Web URL: http:// infectious diseases. Our analysis of the epidemic curves for www.who.int/csr/sars/epicurve/epiindex/en/). (Last accessed SARS, as reported for Hong Kong, Vietnam, Singapore, October 24, 2003). and Canada, shows how crucial the rapid implementation 10. Health Canada. Epidemic curve of a SARS outbreak in Canada, of control measures has been in limiting the impact of the February 23 to 2 July, 2003 (N = 250). Ottawa, Ontario, Can- epidemics, both in terms of preventing more casualties and ada: Health Canada, 2003. (World Wide Web URL: http:// in terms of shortening the period during which stringent www.hc-sc.gc.ca/pphb-dgspsp/sars-sras/pdf-ec/ec_20030808. infection control measures were in place. A first lesson pdf). (Last accessed October 24, 2003). from the several SARS epidemics is that a timely alert 11. Haydon DT, Chase-Topping M, Shaw DJ, et al. The construc- against a new infectious disease is most essential. Our anal- tion and analysis of epidemic trees with reference to the 2001 UK foot-and-mouth outbreak. 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Am J Epidemiol 2004;160:509–516 514 Wallinga and Teunis APPENDIX 1 Estimation Procedure Infection networks An outbreak of an infectious disease can be described as a directed network in which the nodes represent cases and the directed edges between the nodes represent transmission of infection between cases. We consider an outbreak of n reported cases, of which q cases have contracted infection from outside the population. This leaves n – q persons whose primary case is among the reported cases. We label the cases by an index i ∈{1, …, n}. Because each case has exactly one primary case, each node in the infection network must have exactly one incoming edge. Because a case patient cannot have infected himself or herself, there cannot be any edges from a node to itself. The structure of a network that satisfies these constraints can be uniquely represented by a vector v, of which the ith element v(i) denotes the label of the primary case that has infected the case with label i. We use v(i) = 0 to refer to sources of infection outside the population. We denote the entire set of all infection n–1 networks that satisfy the above constraints by V. The number of different network structures in V is (n – q) , since, for any of the n – q nonimported cases, there are n – 1 possible primary cases. Note that the set V includes network structures with cycles and that such structures cannot represent transmission between cases. Likelihood inference for infection networks We use a probability model to infer the likelihood that a specific infection network v underlies the observed epidemic curve t. The probability model is built on the assumption that transmission of infection occurred only among the reported n cases. A key element in this model is the probability density function for the generation interval, w(τ|θ). Here, τ is the generation interval and θ is a vector of parameters that specify the probability distribution. We require w(τ|θ) = 0 for τ < 0. All infection networks with cycles have at least one negative generation interval, and these networks are assigned zero probability by this requirement. In the absence of an observed epidemic curve, each infection network is considered equally likely. This is equivalent to requiring independence between unobserved transmission events from case j to case i and from case j to any other case k. Henceforth we will refer to this requirement as the “independence condition.” In Appendix 2, we simulate epidemic processes that do not meet the above-mentioned technical conditions, and we use these simulations to test for the robustness of the likelihood-based estimation procedure. Likelihood functions The probability of observing epidemic curve t, given the parameters θ for the generation interval and v for the infection network, is in = –q L() v, θ t = wt() – t θ . (A1) i vi() i = 1 Because we are interested in the likelihood of sets of infection networks, we sum the likelihood over networks in a set. This requires a “weight function” c(v|θ) for each infection network. The independence condition implies that c(v|θ) is a constant, denoted c. The integrated likelihood over the set of all networks is therefore in = –q in = –q jn = LV() , θ t =cw() t – t θ =cw()t – t θ . (A2) i vi() i j ∑ ∏ ∏ ∑ i = 1 i = 1 j = 1,ji ≠ The integrated likelihood over the set of all infection networks in which case k has been infected by case l is in = –q jn = LV() , θ t = cw() t – t θ wt() – t θ . (A3) () kl , k l i j ∏ ∑ i = 1,ik ≠ j = 1,ji ≠ Estimation The relative likelihood that case k has been infected by case l is in = –q jn = cw() t – t θ wt() – t θ k l i j ∏ ∑ wt() – t θ LV() , θ t i = 1,ik ≠ j = 1,ji ≠ k l () kl , -- -------- -------- ------- -------- ------- -------- -- - p == ------- ------- -------- ------- - - ----- -------- ------- -------- -------- ------- -------- ------- -------- ------- -------- -------- ------- -= . (A4) () kl , LV() , θ t in = –q jn = mn = wt() – t θ cw()t – t θ k m i j ∑ ∏ ∑ m = 1,mk ≠ i = 1 j = 1,ji ≠ Am J Epidemiol 2004;160:509–516 Analysis of Different SARS Epidemic Curves 515 The relative likelihood of case l’s infecting case m is inde- first global alert on March 12, 2003, and to a value of R = 0.7 pendent of the relative likelihood of case l’s infecting any for cases with a symptom onset date on or after March 12, other case k (by the independence condition). The distribu- 2003. The model draws for each case the number of tion of the effective reproduction number for case l is secondary infections from a negative binomial distribution, which is determined by the mean R and the shape parameter 2 2 k = (R + R )/σ . The model uses values of k = 0.18 for cases kn = –q t t t t t with a symptom onset date before March 12, 2003, and k = R ∼ Bernoulli[] p (A5) l () kl , 0.08 for cases with a symptom onset date on or after March k = 1 12, 2003; these values correspond to the distribution of the number of secondary infections per case as observed during with an expected value the severe acute respiratory syndrome (SARS) outbreak in Singapore (4). The model draws for each new infection the kn = –q generation interval from a Weibull distribution with a mean ER() = p . (A6) l () kl , and standard deviation of 8.4 days and 3.8 days, respectively k = 1 (8). Each simulation is started by one index case that produces at least 10 secondary cases, which corresponds to a The average daily reproduction number R is calculated as SARS outbreak that is started by a so-called “super-spread the arithmetic mean over R for all of those cases l who show event,” to ensure that the epidemic takes off. the first symptoms of illness on day t. Accuracy of the estimation procedure Simultaneous estimation of parameters v and θ We test for accuracy of the proposed estimation procedure When we have observed transmission of infection between by first generating 20 epidemic curves using the stochastic some pairs of cases, it is possible to infer both infection simulation model and then estimating the time course of the network (v) and generation interval (θ) simultaneously. We reproduction number for the simulated epidemic curves using must consider the likelihood the proposed method. The estimated reproduction numbers tend to be close to the actual values used in the simulation model, except for simulated epidemic curves with a small LV() , θ t = (A7) () kl , number of cases in which the estimated reproduction numbers are well below the actual values (the actual values of the in = –q jn = reproduction numbers for the simulated epidemic curves were cv() θ wt() – t θcw()t – t θ , k l i j ∏ ∏ ∑ 3 and 0.7 for cases with symptom onset data before and after i = 1,ik ≠ j = 1,ji ≠ () kl , March 12, 2003, respectively; the average estimated values were 3.09 and 0.68). where (k,l) denotes all pairs of cases for which any case k is known to have been infected by another case l. Estimates for The effect of incomplete reporting both generation interval (θ) and infection network (v) are obtained by maximizing this equation using the expectation- The estimation procedure supposes that all infected maximization algorithm (21). Note that the independence persons will show overt clinical symptoms and that all condition applies only to the unobserved transmission cases will be reported, an assumption that might not be events. correct for an infectious disease like SARS. To test for the impact of incomplete reporting on the estimated effective reproduction number, we modified the model such that each infected individual would have a probability of his or APPENDIX 2 her case’s being reported of 0.5. The resulting estimates were only slightly less accurate than they were with Testing the Estimation Procedure with Simulated Data complete reporting (the actual values of the reproduction numbers for the simulated epidemic curves were 3 and 0.7 for cases with symptom onset data before and after March A stochastic simulation model 12, 2003, respectively; the average estimated values were 2.69 and 0.70). We have constructed a stochastic, individual-based model to generate infection networks that result from epidemic processes with exactly specified properties. We use such The effect of changes in generation interval simulated infection networks for testing the estimation procedure and for exploring the expected distribution of The estimation procedure presented here supposes that the epidemic size and epidemic duration. The model allows for a distribution of generation intervals does not change over time. variable effective reproduction number R as a function of However, for the SARS epidemic in Singapore, there was a symptom onset date t. In the simulations described here, we tendency for the generation interval to decrease after control set the effective reproduction number to a value of R = 3 for was implemented (8). Cases with a symptom onset date before cases with a symptom onset date before the issuance of the March 12, 2003, had a mean generation interval of 10.0 days Am J Epidemiol 2004;160:509–516 516 Wallinga and Teunis and a standard deviation of 2.8 days, whereas cases with a eterized according to the Singapore data. The estimates were, symptom onset date on or after March 12, 2003, must have had on average, slightly lower than they were with use of the same a mean generation interval of 8.2 days and a standard deviation generation interval throughout the epidemic (the actual values of 3.9 days (8). To test for the impact of this temporal change of the reproduction numbers of the simulated epidemic curves in the duration of the generation interval, we modified the were 3 and 0.7 for cases with symptom onset data before and model by using two different Weibull distributions, one before after March 12, 2003, respectively; the average estimated March 12 and the other one on or after March 12, each param- values were 2.60 and 0.66 over 20 simulations). Am J Epidemiol 2004;160:509–516 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png American Journal of Epidemiology Pubmed Central

Different Epidemic Curves for Severe Acute Respiratory Syndrome Reveal Similar Impacts of Control Measures

Wallinga, Jacco; Teunis, Peter
American Journal of Epidemiology , Volume 160 (6) – Sep 15, 2004
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American Journal of EPIDEMIOLOGY Volume 160 Copyright © 2004 by The Johns Hopkins Bloomberg School of Public Health Number 6 Sponsored by the Society for Epidemiologic Research September 15, 2004 Published by Oxford University Press ORIGINAL CONTRIBUTIONS Different Epidemic Curves for Severe Acute Respiratory Syndrome Reveal Similar Impacts of Control Measures Jacco Wallinga and Peter Teunis From the National Institute for Public Health and the Environment, Bilthoven, the Netherlands. Received for publication October 27, 2003; accepted for publication March 29, 2004. Severe acute respiratory syndrome (SARS) has been the first severe contagious disease to emerge in the 21st century. The available epidemic curves for SARS show marked differences between the affected regions with respect to the total number of cases and epidemic duration, even for those regions in which outbreaks started almost simultaneously and similar control measures were implemented at the same time. The authors developed a likelihood-based estimation procedure that infers the temporal pattern of effective reproduction numbers from an observed epidemic curve. Precise estimates for the effective reproduction numbers were obtained by applying this estimation procedure to available data for SARS outbreaks that occurred in Hong Kong, Vietnam, Singapore, and Canada in 2003. The effective reproduction numbers revealed that epidemics in the various affected regions were characterized by markedly similar disease transmission potentials and similar levels of effectiveness of control measures. In controlling SARS outbreaks, timely alerts have been essential: Delaying the institution of control measures by 1 week would have nearly tripled the epidemic size and would have increased the expected epidemic duration by 4 weeks. disease outbreaks; estimation; infection; models, statistical; SARS virus; severe acute respiratory syndrome; statistics Abbreviations: CI, confidence interval; SARS, severe acute respiratory syndrome. Editor’s note: An invited commentary on this article throughout the world and seeded outbreaks in Vietnam, Singapore, and Canada (2–4). On March 12, 2003, the World appears on page 517, and the authors’ response appears on Health Organization issued a global alert. On March 15, the page 520. World Health Organization issued a second global alert and provided a case definition and a name, severe acute respira- On November 16, 2002, the first known case of atypical tory syndrome (SARS), for the new disease (1). By this time, pneumonia occurred in Guangdong Province in southern Hong Kong, Vietnam, Singapore, and Canada had instituted China (1). In late February 2003, the infection spread from general infectious disease control measures, such as quaran- Guangdong to Hong Kong. From there it spread by airline tine, isolation, and strict hygiene measures in hospitals (1). Correspondence to Dr. Jacco Wallinga, National Institute for Public Health and the Environment, Antonie van Leeuwenhoeklaan 9, 3721 MA Bilthoven, the Netherlands (e-mail: [email protected]). 509 Am J Epidemiol 2004;160:509–516 510 Wallinga and Teunis FIGURE 1. Epidemic curves (numbers of cases by date of symptom onset) for severe acute respiratory syndrome (SARS) outbreaks in a) Hong Kong, b) Vietnam, c) Singapore, and d ) Canada and the corresponding effective reproduction numbers (R ) (numbers of secondary infections generated per case, by date of symptom onset) for e) Hong Kong, f ) Vietnam, g) Singapore, and h) Canada, 2003. Markers (white spaces) show mean values; accompanying vertical lines show 95% confidence intervals. The vertical dashed line indicates the issuance of the first global alert against SARS on March 12, 2003; the horizontal solid line indicates the threshold value R = 1, above which an epidemic will spread and below which the epidemic is controlled. Days are counted from January 1, 2003, onwards. The etiologic agent of the disease, a coronavirus, was identi- the symptom onset date by t . For the SARS epidemics in fied on April 16, 2003 (5). Estimates of important epidemio- Hong Kong, Vietnam, and Singapore, we derived the dates logic measures such as the case-fatality rate and the of symptom onset from epidemic curves provided by the incubation period were reported in May 2003 (6–8). World Health Organization (9). For Canada, we used the epidemic curve provided by Health Canada (10). Despite the rapid progression of understanding of the new disease, differences between the epidemic curves (numbers of probable SARS cases by date of symptom onset) have Observed distribution of generation intervals been awaiting further explanation. For the affected regions, these epidemic curves have been publicly available from the The generation interval, denoted by τ, is the time from end of March 2003 onwards (9), and they reveal distinct symptom onset in a primary case to symptom onset in a temporal patterns in numbers of SARS cases. This is espe- secondary case. Sometimes this generation interval is called cially remarkable for the epidemics in Hong Kong, Vietnam, the serial interval (8) or generation time (11). The generation Singapore, and Canada, since these epidemics started almost intervals observed during the SARS outbreak in Singapore simultaneously in late February 2003 and similar control are well described by a Weibull distribution with a shape measures were put in place at almost the same time in March parameter α and a scale parameter β, with values corre- 2003 (figure 1, parts a–d). The question arises as to whether sponding to a mean generation interval of 8.4 days and a the affected regions differed in terms of transmission poten- standard deviation of 3.8 days (8). We denote this distribu- tial for SARS or effectiveness of control measures. In this tion by τ ∼ w(τ|α,β). paper, we interpret the observed epidemic curves with regard to disease transmission potential and effectiveness of control ESTIMATION OF REPRODUCTION NUMBERS measures, and we compare the epidemiologic profiles of SARS outbreaks in Hong Kong, Vietnam, Singapore, and Reproduction numbers Canada. The key epidemiologic variable that characterizes the transmission potential of a disease is the basic reproduction DATA number, R , which is defined as the expected number of secondary cases produced by a typical primary case in an Observed epidemic curves entirely susceptible population (12–15). When infection is The epidemic curve is the number of reported cases by spreading through a population, it is often more convenient date of symptom onset. For each reported case i, we denote to work with the effective reproduction number, R, which is Am J Epidemiol 2004;160:509–516 Analysis of Different SARS Epidemic Curves 511 defined as the actual average number of secondary cases per derivation of these equations is provided in Appendix 1. The primary case. The value of R is typically smaller than the estimation algorithm allows estimation of the effective value of R , and it reflects the impact of control measures and reproduction numbers for infectious diseases at a finer depletion of susceptible persons during the epidemic. If R temporal resolution under more general assumptions than exceeds 1, the number of cases will inevitably increase over was previously possible. time, and a large epidemic is possible. To stop an epidemic, R needs to be persistently reduced to a level below 1. Estima- Testing the estimation procedure with simulated data tion of R is a simple affair if there is information about who infected whom. Then it is possible to construct an infection To test how well the estimation procedure approximates network (11), wherein cases are connected if one person the underlying value of an effective reproduction number infected the other. Estimation of R involves simply counting during a typical SARS outbreak, we estimate the effective the number of secondary infections per case. reproduction numbers from simulated epidemic curves. We have constructed a stochastic, individual-based model that simulates epidemic processes with exactly specified proper- A likelihood-based estimation procedure ties. The model allows for a variable effective reproduction Often, estimation of the effective reproduction number is a number R as a function of symptom onset data t, and the complicated affair, because only the epidemic curve is model parameters are estimated from observations on the observed and there is no information about who infected SARS epidemic in Singapore (see Appendix 2). Applying whom. Recent analyses of closely monitored epidemics have the estimation procedure to simulated epidemic curves shown that it is possible to estimate the probability that one shows that most of the estimates are close to the actual repro- person has infected another if the spatial locations of the duction numbers and that a few estimates based on small infected persons are available (11, 16). However, when only outbreak sizes are below the actual values that are used in the times of symptom onset are available, most investigators simulation model. On average, the estimates tend to be lower resort to approximating R by assuming an exponential than the actual values but deviate less than 5 percent from the increase in the number of cases over time (8, 17) or by fitting actual reproduction numbers. If we account for the effects of a specific model that summarizes assumptions about the incomplete reporting and temporal change in generation epidemiology of the disease (7, 18, 19). Such assumptions interval, the estimates become only slightly less accurate, can be avoided by using a likelihood-based estimation proce- and on average they deviate less than 15 percent from the dure that infers “who infected whom” from the observed actual reproduction numbers (see Appendix 2). dates of symptom onset as provided by the epidemic curve. However, the computational burden of a straightforward RESULTS numerical evaluation of the likelihood appears to be enor- mous, since it requires consideration of all possible infection For Hong Kong, Vietnam, Singapore, and Canada, we networks, and even for a small outbreak of 50 cases have converted the epidemic curves into the time course of there are almost 7 × 10 possible infection networks (see effective reproduction numbers. The results are shown in Appendix 1). figure 1, parts e–h. These four large outbreaks were Here we show that it is possible to obtain likelihood-based sparked almost simultaneously by the same index patient. estimates of R while avoiding the computational problems if All regions have faced erratic “super-spread events” we use pairs of cases rather than the entire infection network. wherein cases produced more than 10 secondary infections. The relative likelihood p that case i has been infected by These “super-spread events” show up in parts e–h of figure ij case j, given their difference in time of symptom onset t – t , 1 as temporary increases in effective reproduction numbers i j can be expressed in terms of the probability distribution for around the symptom onset date of the index case for the the generation interval. This distribution for the generation “super-spread event.” In Hong Kong, Vietnam, and interval is available for many infectious diseases, and we Singapore, there were “super-spread events” marking the denote it by w(τ). The relative likelihood that case i has been start of the outbreak. In Hong Kong, Singapore, and infected by case j is then the likelihood that case i has been Canada, there were “super-spread events” after control infected by case j, normalized by the likelihood that case i measures were implemented. has been infected by any other case k: During the early phase of the SARS epidemic, before the first World Health Organization global alert was issued on p = wt() – t ⁄ wt() – t . March 12, 2003, the average effective reproduction numbers ij i j i k ik ≠ were markedly similar across the regions: Each case The effective reproduction number for case j is the sum over produced approximately three secondary infections (table 1). all cases i, weighted by the relative likelihood that case i has A value of R slightly higher than 3 is consistent with the been infected by case j: observed epidemics in all four regions. Around mid-March, control measures were implemented in all regions, and R = p . during this period the effective reproduction numbers j ij decreased sharply. For some regions, the effective reproduc- Note that we are not making any assumption regarding the tion numbers continued to decrease at a slow pace, distribution of numbers of secondary infections per case suggesting improvement of control measures while the (i.e., the p are independent in j). Additional detail on the epidemic was going on. After the first World Health Organi- ij Am J Epidemiol 2004;160:509–516 512 Wallinga and Teunis TABLE 1. Average daily effective reproduction number (R ) for cases of severe acute respiratory syndrome (SARS) with a symptom onset date before or after the issuance of the first global alert against SARS on March 12, 2003, for regions where infection was introduced in late February Hong Kong Vietnam Singapore Canada Symptom onset R 95% CI* R 95% CI R 95% CI R 95% CI Before alert 3.6 3.1, 4.2 2.4 1.8, 3.1 3.1 2.3, 4.0 2.7 1.8, 3.6 After alert 0.7 0.7, 0.8 0.3 0.1, 0.7 0.7 0.6, 0.9 1.0 0.9, 1.2 * CI, confidence interval. zation global alert, each case produced approximately 0.7 We have presented the relation between observed secondary infections (table 1). A value of R = 0.7 is consis- epidemic curves and inferred reproduction numbers from tent with all four regions, except Canada. By the beginning an infection-network perspective. We are certainly not the of July 2003, transmission of the SARS virus had been first researchers to do so: Infection networks have been stopped in all regions where the infection was introduced in used extensively in the area of sexually transmitted late February 2003 (1). diseases (20), and an infection-network perspective was adopted to analyze a closely observed foot-and-mouth To explore the range of epidemic curves that can result from the same epidemic process, we performed extensive epidemic in Great Britain (11). Our contribution is the deri- vation of likelihood-based estimates of effective reproduc- computer simulations using a model of epidemics with char- tion numbers, requiring only the observed time of symptom acteristics similar to those of SARS (see Appendix 2). The outcome of 10,000 simulations shows a highly variable onset for the observed cases. The use of a likelihood frame- work provides a set of powerful tools for inference, uncer- epidemic size and epidemic duration: The mean epidemic tainty analysis, and model selection (20); the use of only size is 685 cases (95 percent confidence interval (CI): 27, 2,446), and the mean epidemic duration is 98 days (95 time of symptom onset allows us to apply this method to routinely collected epidemic-curve data. However, estima- percent CI: 32, 187). All observed sizes and durations of tion of effective reproduction numbers from epidemic SARS epidemics are within this very wide range of possible outcomes resulting from chance alone. Additionally, we curves has an intrinsic limitation that should be kept in mind: The effective reproduction number contains entan- used computer simulations to explore the effect of the timing gled information about the transmission potential (i.e., the of implementation of control measures on the epidemic size and duration, in a setting that is typical for the SARS basic reproduction number) and the effectiveness of control measures. These two components can be disentangled only outbreaks. The simulation results show a high sensitivity to when we obtain additional information—for example, the timing of implementation of control. On average, a 1- week delay in implementation of control measures results in about the time of implementation of control measures. Moreover, individual contributions to the effective repro- a 2.6-fold increase in mean epidemic size and a 4-week duction number are entangled if their date of symptom extension of the mean epidemic duration. onset is smaller than the generation interval. This limitation can be overcome if more detailed information on who DISCUSSION infected whom is available (see Appendix 1). This study showed that there exists a direct relation For the SARS outbreak in Hong Kong, it is possible to between the epidemic curve and the time course of the repro- compare our results with previously published estimates. duction number R. This relation is determined by the distri- Our estimate of the average effective reproduction number bution of the generation intervals. The relation can be used to prior to the first global alert (R = 3.6, 95 percent CI: 3.1, 4.2) monitor the combined effects of transmission potential and is more precise than the estimate obtained by assuming an control measures during an epidemic. We have shown that exponential increase in the number of cases (R = 3.5, 95 the epidemic curves for SARS in Hong Kong, Vietnam, percent CI: 1.5, 7.7) (8) and more precise than the estimated Singapore, and Canada, though apparently different, are all lower bound excluding “super-spreading events” (R > 2.7) consistent with a single time course of the effective repro- (7). Our estimate of the average effective reproduction duction numbers for SARS. This apparent difference in number after the first global alert (R = 0.7, 95 percent CI: epidemic curves arises because chance effects, such as the 0.7, 0.8) is much higher than the previously estimated lower occurrence of a rare “super-spread event,” leave a lasting bound excluding “super-spread events” (R > 0.14) (7). This trace on the epidemic curve. In contrast, chance events mani- comparison illustrates that the estimation algorithm fest only as temporary increases in the reproduction number. presented here allows more precise estimation of the effec- Our analysis of the epidemic curves for SARS shows that tive reproduction numbers for infectious diseases under one should be cautious in taking a smaller epidemic size and more general assumptions than was previously possible. a shorter epidemic duration as proof of better infection The effectiveness of control measures against SARS can control. be estimated if it is assumed that the sudden decrease in the Am J Epidemiol 2004;160:509–516 Analysis of Different SARS Epidemic Curves 513 effective reproduction number for SARS around the time of 3. Poutanen SM, Low DE, Henry B, et al. Identification of severe acute respiratory syndrome in Canada. N Engl J Med 2003;348: the first global alert was indeed caused by the implementa- 1995–2005. tion of control measures. 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Transmission dynamics case produced, on average, approximately three secondary of the etiological agent of SARS in Hong Kong: impact of pub- infections, this effectiveness has been barely sufficient for lic health interventions. Science 2003;300:1961–6. the eventual interruption of the chain of human-to-human 8. Lipsitch M, Cohen T, Cooper B, et al. Transmission dynamics transmission of the SARS virus. and control of severe acute respiratory syndrome. Science 2003;300:1966–70. As the direct threat of a worldwide SARS epidemic has 9. World Health Organization. Epidemic curves—severe acute waned, the question arises as to how we can use the experi- respiratory syndrome (SARS). Geneva, Switzerland: World ence with SARS to improve infection control against new Health Organization, 2003. (World Wide Web URL: http:// infectious diseases. Our analysis of the epidemic curves for www.who.int/csr/sars/epicurve/epiindex/en/). 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Am J Epidemiol 2004;160:509–516 514 Wallinga and Teunis APPENDIX 1 Estimation Procedure Infection networks An outbreak of an infectious disease can be described as a directed network in which the nodes represent cases and the directed edges between the nodes represent transmission of infection between cases. We consider an outbreak of n reported cases, of which q cases have contracted infection from outside the population. This leaves n – q persons whose primary case is among the reported cases. We label the cases by an index i ∈{1, …, n}. Because each case has exactly one primary case, each node in the infection network must have exactly one incoming edge. Because a case patient cannot have infected himself or herself, there cannot be any edges from a node to itself. The structure of a network that satisfies these constraints can be uniquely represented by a vector v, of which the ith element v(i) denotes the label of the primary case that has infected the case with label i. We use v(i) = 0 to refer to sources of infection outside the population. We denote the entire set of all infection n–1 networks that satisfy the above constraints by V. The number of different network structures in V is (n – q) , since, for any of the n – q nonimported cases, there are n – 1 possible primary cases. Note that the set V includes network structures with cycles and that such structures cannot represent transmission between cases. Likelihood inference for infection networks We use a probability model to infer the likelihood that a specific infection network v underlies the observed epidemic curve t. The probability model is built on the assumption that transmission of infection occurred only among the reported n cases. A key element in this model is the probability density function for the generation interval, w(τ|θ). Here, τ is the generation interval and θ is a vector of parameters that specify the probability distribution. We require w(τ|θ) = 0 for τ < 0. All infection networks with cycles have at least one negative generation interval, and these networks are assigned zero probability by this requirement. In the absence of an observed epidemic curve, each infection network is considered equally likely. This is equivalent to requiring independence between unobserved transmission events from case j to case i and from case j to any other case k. Henceforth we will refer to this requirement as the “independence condition.” In Appendix 2, we simulate epidemic processes that do not meet the above-mentioned technical conditions, and we use these simulations to test for the robustness of the likelihood-based estimation procedure. Likelihood functions The probability of observing epidemic curve t, given the parameters θ for the generation interval and v for the infection network, is in = –q L() v, θ t = wt() – t θ . (A1) i vi() i = 1 Because we are interested in the likelihood of sets of infection networks, we sum the likelihood over networks in a set. This requires a “weight function” c(v|θ) for each infection network. The independence condition implies that c(v|θ) is a constant, denoted c. The integrated likelihood over the set of all networks is therefore in = –q in = –q jn = LV() , θ t =cw() t – t θ =cw()t – t θ . (A2) i vi() i j ∑ ∏ ∏ ∑ i = 1 i = 1 j = 1,ji ≠ The integrated likelihood over the set of all infection networks in which case k has been infected by case l is in = –q jn = LV() , θ t = cw() t – t θ wt() – t θ . (A3) () kl , k l i j ∏ ∑ i = 1,ik ≠ j = 1,ji ≠ Estimation The relative likelihood that case k has been infected by case l is in = –q jn = cw() t – t θ wt() – t θ k l i j ∏ ∑ wt() – t θ LV() , θ t i = 1,ik ≠ j = 1,ji ≠ k l () kl , -- -------- -------- ------- -------- ------- -------- -- - p == ------- ------- -------- ------- - - ----- -------- ------- -------- -------- ------- -------- ------- -------- ------- -------- -------- ------- -= . (A4) () kl , LV() , θ t in = –q jn = mn = wt() – t θ cw()t – t θ k m i j ∑ ∏ ∑ m = 1,mk ≠ i = 1 j = 1,ji ≠ Am J Epidemiol 2004;160:509–516 Analysis of Different SARS Epidemic Curves 515 The relative likelihood of case l’s infecting case m is inde- first global alert on March 12, 2003, and to a value of R = 0.7 pendent of the relative likelihood of case l’s infecting any for cases with a symptom onset date on or after March 12, other case k (by the independence condition). The distribu- 2003. The model draws for each case the number of tion of the effective reproduction number for case l is secondary infections from a negative binomial distribution, which is determined by the mean R and the shape parameter 2 2 k = (R + R )/σ . The model uses values of k = 0.18 for cases kn = –q t t t t t with a symptom onset date before March 12, 2003, and k = R ∼ Bernoulli[] p (A5) l () kl , 0.08 for cases with a symptom onset date on or after March k = 1 12, 2003; these values correspond to the distribution of the number of secondary infections per case as observed during with an expected value the severe acute respiratory syndrome (SARS) outbreak in Singapore (4). The model draws for each new infection the kn = –q generation interval from a Weibull distribution with a mean ER() = p . (A6) l () kl , and standard deviation of 8.4 days and 3.8 days, respectively k = 1 (8). Each simulation is started by one index case that produces at least 10 secondary cases, which corresponds to a The average daily reproduction number R is calculated as SARS outbreak that is started by a so-called “super-spread the arithmetic mean over R for all of those cases l who show event,” to ensure that the epidemic takes off. the first symptoms of illness on day t. Accuracy of the estimation procedure Simultaneous estimation of parameters v and θ We test for accuracy of the proposed estimation procedure When we have observed transmission of infection between by first generating 20 epidemic curves using the stochastic some pairs of cases, it is possible to infer both infection simulation model and then estimating the time course of the network (v) and generation interval (θ) simultaneously. We reproduction number for the simulated epidemic curves using must consider the likelihood the proposed method. The estimated reproduction numbers tend to be close to the actual values used in the simulation model, except for simulated epidemic curves with a small LV() , θ t = (A7) () kl , number of cases in which the estimated reproduction numbers are well below the actual values (the actual values of the in = –q jn = reproduction numbers for the simulated epidemic curves were cv() θ wt() – t θcw()t – t θ , k l i j ∏ ∏ ∑ 3 and 0.7 for cases with symptom onset data before and after i = 1,ik ≠ j = 1,ji ≠ () kl , March 12, 2003, respectively; the average estimated values were 3.09 and 0.68). where (k,l) denotes all pairs of cases for which any case k is known to have been infected by another case l. Estimates for The effect of incomplete reporting both generation interval (θ) and infection network (v) are obtained by maximizing this equation using the expectation- The estimation procedure supposes that all infected maximization algorithm (21). Note that the independence persons will show overt clinical symptoms and that all condition applies only to the unobserved transmission cases will be reported, an assumption that might not be events. correct for an infectious disease like SARS. To test for the impact of incomplete reporting on the estimated effective reproduction number, we modified the model such that each infected individual would have a probability of his or APPENDIX 2 her case’s being reported of 0.5. The resulting estimates were only slightly less accurate than they were with Testing the Estimation Procedure with Simulated Data complete reporting (the actual values of the reproduction numbers for the simulated epidemic curves were 3 and 0.7 for cases with symptom onset data before and after March A stochastic simulation model 12, 2003, respectively; the average estimated values were 2.69 and 0.70). We have constructed a stochastic, individual-based model to generate infection networks that result from epidemic processes with exactly specified properties. We use such The effect of changes in generation interval simulated infection networks for testing the estimation procedure and for exploring the expected distribution of The estimation procedure presented here supposes that the epidemic size and epidemic duration. The model allows for a distribution of generation intervals does not change over time. variable effective reproduction number R as a function of However, for the SARS epidemic in Singapore, there was a symptom onset date t. In the simulations described here, we tendency for the generation interval to decrease after control set the effective reproduction number to a value of R = 3 for was implemented (8). Cases with a symptom onset date before cases with a symptom onset date before the issuance of the March 12, 2003, had a mean generation interval of 10.0 days Am J Epidemiol 2004;160:509–516 516 Wallinga and Teunis and a standard deviation of 2.8 days, whereas cases with a eterized according to the Singapore data. The estimates were, symptom onset date on or after March 12, 2003, must have had on average, slightly lower than they were with use of the same a mean generation interval of 8.2 days and a standard deviation generation interval throughout the epidemic (the actual values of 3.9 days (8). To test for the impact of this temporal change of the reproduction numbers of the simulated epidemic curves in the duration of the generation interval, we modified the were 3 and 0.7 for cases with symptom onset data before and model by using two different Weibull distributions, one before after March 12, 2003, respectively; the average estimated March 12 and the other one on or after March 12, each param- values were 2.60 and 0.66 over 20 simulations). Am J Epidemiol 2004;160:509–516

Journal

American Journal of Epidemiology – Pubmed Central

Published: Sep 15, 2004

References