Estimation for the infection curves for the spread of Severe Acute Respiratory
Syndrome (SARS) from a back-calculation approach

Ping Yan

Modelling and Projection Section
Centre for Infectious Disease Prevention and Control
Population and Public Health Branch
Health Canada

ABSTRACTBACKGROUND
Available data about the spread of the Severe Acute Respiratory Syndrome (SARS) often comewith two types: that by dates of report and that by dates of onset of symptoms. The latter is anapproximation of the epidemic curve. Statistical methods can suggest the most plausible
epidemic curve by the time of infection.
METHODS
Input data are SARS cases by dates of onset. Back-calculation methods are applied. The
incubation distributions used in the algorithm are three log-logistic models: (i) that with median= 4.18 days and a shape parameter 3.413 (95% quantile 9.9 days); (ii) that with a shorter
incubation distribution (median= 3.453 days; 95% quantile = 6.569 days); (iii) that with a longer
incubation distribution (median = 5.139 days; 95% quantile = 17.2 days.)
RESULTS
Three infection curves based on the above three incubation distribution assumptions have beenreconstructed separately for Singapore, Viet Nam, Hong Kong and Canada. Where available,
these infections curves are compared with the trend based on dates of report, along with
documented event history of public reactions.
CONCLUSIONSThe public knowledge (e.g. media) and actions(e.g. quarantine a large number of people) areoften driven by reported outbreaks, subject to a delay of approximately two weeks since the timeof infection. Second, it is important to be aware of the underline mechanisms that manifest thedata that are publically available. The trends by dates of report are distorted not only by a timedelay, but also by reporting patterns in different jurisdictions. The trends by dates of onset havecloser resemblance of the epidemic curves, but may be biassed by reporting delay. Finally, since
the beginning of the multiple country outbreaks SARS, some knowledge has been gained on thedistributions of time from the onset of symptoms to either death or recovery, and the incubation
distribution from the point of infection to the onset of symptoms. Jointly with a plausible
reconstructed infection process, then one may be able to estimate key parameters such asinfectivity.

INTRODUCTION
Data about the spread of the Severe Acute Respiratory Syndrome (SARS) often come with twotypes. One is that based on dates of report as represented by the far right compartment in Figure
1 (top). This is what the media and the public perceive and is the driving force for immediate
action upon notification of an outbreak. The other is based on dates of onset of symptoms as
represented by the far left compartment in Figure 1 (top). Reported data from Beijing (1) in
Figure 1 (bottom) provide a good illustration of the reporting mechanism.
Classified as aprobable case
Onset:
feeling
unwellReported bymedia

Under observation
Admission tohospital

ruled out

Classified as asuspected caseruled out

Dates of onset are retrospectively ascertained

Beijing
400

Newly reported probable cases

Newly reported suspected
Updated to probable from previous suspected
New cases as probable

300
200
1000-100

Ruled out from previous suspected

Figure 1. Illustration of mechanisms that manifest data as reported and anexample from Beijing

The time trends by dates of onset are approximations to the epidemic curves. These curves areavailable from a number of government websites and the World Health Organization website.
With some knowledge of the incubation period and the shape of its distribution, statisticalmethods can suggest the most plausible epidemiologic curve by the time of infection. Once one
can make a plausible reconstruction of the realized infection process, there will be two
immediate implications. The first is to compare the infection process with the observed trends,
especially with that by dates of report, since most of the public reactions are driven by reported
outbreaks. This will help to design more timely response measures based on lessons learned.
The second is to use reconstructed infection process, together with available data, as well as
knowledge of the incubation distribution, and knowledge of the “removal process”, characterized
by the distributions of time from the onset of symptoms to either death or recovery, to estimate
some of the most important epidemiology parameters, such as infectivity. All data used in thismanuscript are available from public domain. The World Health Organization gives thefollowing website, where one can download epidemiologic curves of SARS from selected
regions in the world as reported by May 2, 2003
http://www.who.int/csr/sarsepicurve/2003_05_02epicurve.pdf
The Singapore Ministry of Health and Health Canada also routinely publish respective
epidemiologic curves on their own websites:
http://app.moh.gov.sg/sar/sar01.asp
http://www.hc-sc.gc.ca/pphb-dgspsp/sars-sras/prof_e.html
In order to keep consistancy, only probable cases are considered.

METHODThe back-calculation methods used here are the EMS algorithm as proposed by Becker, Watson
and Carlin (2), originally developed in the HIV/AIDS context to assess the extent of the HIV
epidemic and to use the reconstruction as a basis for predicting AIDS incidence. Becker andBritton (3) also pointed out the usefulness of reconstructing the infection process for otherdiseases for the purpose of taking the advantage of the explicit expressions available for
maximum likelihood estimates of parameters when the infection process is fully observed.

The incubation distribution for the time the date of infection to the onset of symptoms must toused in back-calculation. The current consensus, based on empirical observations, seems toagree that the average incubation time is short: 2 – 10 days (3). There are three drawbacks.
The first is the representativeness, because in these studies, only a subset of cases that can beascertained to a single exposure event whereas for the majority of the cases, there are multiple
exposures points or with exposures that can not be easily defined and measured. The second isrelatively small number of cases that can be ascertained to a single exposure event. This implies
large uncertainty in terms of confidence limits. The third is that the dates of exposure are
retrospectively ascertained from diagnosed SARS patients. Data suffer from time-length bias.
For each patient, the observed incubation time is limited to an observation window no longer
than the length from the date of exposure to the date of analysis. This window makes one toover sample "shorter" incubation periods.

Based on observed data, there might be occasional longer observed incubation time and might beconsidered as “outliers”. As time goes by, what one characterizes as "outliers" today, may notlook at outliers any more. So the question is, what might be the percentage of SARS cases
which might have incubation period longer than 10 days. The proposed model used in the backcalculation is a log-logistic distribution. The choice is based on the following reasons:1.

Flexible: suitable to describe the distribution in a population consist of a mixture ofindividuals with short and long incubation periods and the capacity to accommodate“outliers”. Both log-logistic and log-normal distributions are particularly suitable for thisconsideration.

2.

Empirical: Sartwell (5, 6) found that log-normal distributions gave good descriptions of
the variation in incubation periods for a considerable number of well-known disease. It
is well known that the log-logistic distribution provides a good approximation to the lognormal distribution.

3.

Practical: the log-logistic distribution has a much simpler algebraic expression than thelog-normal distribution. It is very practical because the log-logistic distribution can beparametrized by any given two quantiles. For example, the standard form of a loglogistic distribution can be expressed by

Pr{ X > x} =1
1 + (λx ) β

where the inverse of scale parameter is the median: m = 8–1 . $ is a shape parameter
which, together with the median parameter m, defines other quantiles of the distribution.
Therefore, the log-logistic distribution can be re-parametrized from scale-shape (8,$) tomedian - 95% quantile: (m, t95 ). The conversion ism=t 95 =1λ 2.944438979  .
× exp
λβ1

95% quantile: 9.9 days95 % C.I. 6.57 – 17.21 days

median: 4.175 days
95% C.I. 3.45 – 5.139days

5101520

Number of days since exposure

Figure 2. Incubation distributions used for back-calculation.
The parametric models for the incubation distribution in the back-calculations are represented by
the three smooth curves in Figure 2. These are the cumulative probabilities from the time ofinfection to the time of onset measure in days. The solid middle line corresponds to a loglogistic distribution with m = 4.18 days and t95 = 9.9 days. The dotted line with a shorter
distribution has m = 3.45 days and t95 = 6.57 days. The dotted line with a longer distribution hasm = 5.14 days and t95 = 17.21 days. These three scenarios are compared with empiricallyestimated distribution, represented as step-function based on the 42 cases in Ontario asdocumented in (4).

RESULTS
For each of the following countries or regions, three reconstructed infection curves in lines arepresented, corresponding to the short, medium and long incubation distributions as illustrated inFigure 2. For comparison purposes, time trends by dates of onset and by dates of report arepresented in bars.
Viet Nam:
There has been no newly reported probable cases since April 14, 2003. Therefore, data
presented by dates of onset are assumed as accurate. No further reporting delay adjustment isneeded. The estimated infection curves using back-calculation show a clear pattern as often
described in mathematical models. The short, medium and long incubation distribution
assumptions do not seriously affect the interpretations. Comparing with the epidemic curve
represented by dates of onset, the latter not only suffer a slight delay in time, but also a slight
distortion in shape, due to the variance of the incubation distribution.

1086420Figure 3. Reconstructed infection curves compared with trend by dates of onset in Viet Nam.
Singapore:
Back-calculated infection curves of SARS are presented in Figure 4, together with trends bydates of onset and by dates of report. Trend by dates of report was compiled from daily briefings
from the Singapore Ministry of Health website:
http://app.moh.gov.sg/sar/sar03.asp
Documented transmission and linkages among these clusters are also available from theaforementioned website. Therefore, some major events related to public reactions are also
illustrated in Figure 5. The acrnyms are:
TTSH = Tan Tock Seng Hospital
SGH = Singapore General Hospital
PPWM = Pasir Panjang Wholesale Market

Total numbers under quarantine

PPWM quarantine

Discharge under quarantine

Contact under quarantine

3000

April 19-20: Notificationof an outbreak at
P.P.wholesale market.

2500
2000

April 6: Notification of
the outbreak outside
TTSH, in SGH
16141500
1000

By dates of report
By dates of onset
Lines: by dates of infection

500

12108May 18: a single
repo rted case who
travelled to Malaysia

6He felt u nwell o n May 54201614
12TTSH
clusterMar.24: The link case admitted
to SGH Ward 57 forgastrointestinal bleeding.

April 8: A taxi driver, presumed
index case for the PPWM
outbreak was likely infected.

Feb. 21-23:
Three index cases
infected in8 Hong Kong

10He became unwell (infectious) on
Apr.14.

64

SGH
cluster

2PPWM
clusterTh e single rep orted caseon May 1 8 who travelled
to Malaysia o n Apirl 25,May and May 5. Most
likely infected d uring theMay 1 trip.

0Mar.5-20: A link case stayed inTTSH for chronic kidney problems.

Figure 4. Singapore: Infection curves compared with trends by onset and report dates, alongwith document major event history on public notification and action.

In addition, Figure 4 illustrates how major actions are coincidence with the trend by dates ofreport, which took place immediately at the newly reported clusters of outbreaks. However, theinfection time often took place approximately two weeks before the public notification.
Hong Kong:Since Hong Kong is continuing reporting new SARS cases, one needs to adjust the numbers bydates of onset for reporting delays. The authors thanks Dr. Nigel Gay from the World HealthOrganization for providing the reporting delay adjusted numbers used for back-calculation.
The event history about the outbreak in the Amoy Gardens Complex is used in Figure 5 toillustrate the time lag between the infection time and the time when reported cases started to alert
public and drove the action.
Lines: by dates of infection

By dates of onset

By dates of report

120100The Amoy Gardens index case:
Symptomatic on Mar.14, visited
the Complex on Mar. 14 & 19.

Mar.31: Notification of
the outbreak at Amoy
Gardens Complex

806040200Figure 5. Hong Kong: infection curves compared with trends by onset and by report dates

Canada:The trend analysis for Canada has the same message as what we have seen for Singapore and forHong Kong. Figure 6 uses a single event to highlight the time lag between the earliest public
notice of the outbreak in a religious group, which was driven by the elevated reported cases onApril 14 and 15. The most plausible starting point of this infection is before March 30.
March 30: Earliest
onset among a religious group
April 14: Earliest public
notification of the15

outbreak in thereligious group10

March 25: SARS became areportable, virulent, communicabledisease under Ontario's HealthProtection and Promotion Act

Cumulative
number April
19 & 20 (Easterlong weekend)

50Figure 6. Canada: Infection curves compared with trends by onset and report dates, along withsome document event history on public notification.
Synthesized infection curves:
Figure 7 presents the infection curves reconstruction for different regions based on the medium
scenario of the incubation distribution, which is a log-logistic distribution with median = 4.18

days and 95% quantile = 9.9 days. All these regional outbreaks are linked to a single “superspreading event” that took place between Feb. 21 and Feb. 23, 2003 at the ninth floor of theMetropole Hotel in Hong Kong. In terms of infection dynamics over time, Figure 7
demonstrates a synchronized recurrent waves, not necessarily dampening, between the infection
curve in Hong Kong and the infection curve in Singapore. For all these four regions, therecurrent waves tend to share a common periodicity.

Hong Kong

100806040Viet Nam
20CanadaSingapore

0Figure 7. Infection curves reconstructed for different regions based on the medium scenario ofthe incubation distribution.
CONCLUSION
Back-calculation methods, widely used in the study of the HIV/AIDS epidemic in early 1990's,
have found use in the current fight against SARS. This analysis illustrates the following aspects:
1.

Public notification and reaction are driven by reports, which tend to be approximately
two weeks after the initiation of a local outbreak. Further research is necessary inidentifying ways to shorten this time lag, and to identifying ways for more efficient
control measures.
2.

For modellers who fit mathematical models to observed epidemic curves, it is important
to know the underline mechanisms that manifest the data, including reporting patterns
driven by local jurisdictions and occasional irregularities.

3.

Back-calculated infection curves provide additional information on the infection process,which are not directly observable. With this additional information, one may be able toestimate some important parameters in mathematical models, such as infectivity, which
may not be easily estimated only based on observed data.

The back-calculated epidemic curves provide empirical contribution to more advanced
mathematical modelling. Some open challenges exist. For example, data may exist throughcontact tracing such that one can construct a linked data as who infects whom. Therefore, it ispossible to match the reconstructed trends by dates of infection over calendar time with thatobserved by generation time. In classical mathematical models for infectious diseases, it is often
assumed that the initial spread of an outbreak in calendar time can be approximated by anexponential curve represented by i (t ) ∝ Ce . It is also often assumed that the initial spread byrtgeneration time can be approximated by a time-stationary branching process, defined by a mean
parameter R0 , and that the decline of the number of susceptibles in subsequent generations in theearly phase can be ignored. Under certain assumptions, such as the S-I-R model under random
mixing, the initial growth in calendar time and in generation time can be matched through r = R0
–1, and R0 is the basic reproductive number.
On the other hand, the transmission of SARS has been mainly confined to hospital settings,
within households of infected patients, and in some cases, within close-knit communities such asa specific religious group in Ontario and possibly crowded university dormitories in Beijing.
Such environments may contribute to the “super-spreading events”, characterized by an initialcase infecting many secondary cases and leaving each case in the second and third generations
infecting relatively few. Hence one may observe a non-stationary branching process in the very
initial phase in generation time. Therefore, if one uses the infection curve in calendar time toestimate such a parameter R0, it will be useful to discuss the meaning of this parameter in abranching process setting. Intuitively, it may correspond to a “pseudo-branching process” which
is time-stationary, with R0 being the average of the number of secondary cases from the initial“super-spreader” and the number of number secondary cases from cases in the next few
generations during this initial phase from the observed non-stationary branching process. In
other words, the initial growth in generation time described by the observed non-stationary
branching process generates the initial growth of the infection curve in calendar time, but thelatter can be mimicked by the time-stationary pseudo-branching process. I take notice that suchwork is underway, with references to two recently published articles by Lisptich, et al. (7) and byRiley, et al. (8). In their work, detailed data from Singapore and Hong Kong have been analyzed
with special treatments to the “super-spreading events”, without referencing the branchingprocess aspects. I hope that the back-calculated infection processes which are not in theobserved data may provide additional insight to their work and additional work needs to be done.

REFERENCES(1) The website of Ministry of Health, People’s Republic of China (in Chinese language):
http://www.moh.gov.cn/zhgl/yqfb/index.htm.
(2) Becker, N.G., Watson, L.F. and Carlin, J.B. (1991). A method of non-parametric backprojection and its application to AIDS data. Statistics in Medicine, 10, 1527-1542.
(3) Becker, N.G. and Britton, T. (1999) Statistical studies of disease incidence. Journal of Royal
Statistics Society (B), 61, 287–307.
(4) Epi-Update: Interim Report on the SARS outbreak in the Greater Toronto Area, Ontario,Canada, April 24, 2003. http://www.hc-sc.gc.ca/pphb-dgspsp/sars-sras/pef-dep/gta20030424_e.html. Health Canada.
(5) Sartwell, P.E. (1950). The distribution of incubation periods of infectious disease.
American Journal of Hygen, 51 , 308-318.
(6) Sartwell, P.E. (1966). The distribution of incubation period and the dynamics of infectious
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(7) Lipsitch, M, Cohen, T., Robins, J.M., Ma, S., et al. (2003) Transmission dynamics andcontrol of Severe Acute Respiratory Syndrome. Sciencexpress/www.sciencexpress.org/23
May 2003/Page 1/10.1126/science.1086616.
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