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Re: Mathematical modeling of SARS: errata and updates

Dear Editor

We write to follow on from our eLetter published in 2003.[1]

As more information becomes available regarding the diagnosis and laboratory testing of SARS, the official number of laboratory confirmed SARS cases in Taiwan during the 2003 outbreak has been officially determined to be 346.[2]

The duration of the outbreak by onset date is February 25 to June 15. In order to take advantage of the newly-available data for modeling purpose, we use the up-dated cumulative case data of the 346 lab confirmed cases to fit the exponential curve with first-order autocorrelation in the error structure.[3] We divided the time duration of the outbreak into four time periods, the resulting estimated mean effective reproductive number of the observed time period R* of the curve-fitting and a chronology of the significant events related to the outbreak which occurred in Taiwan at the dividing point of each time period is given in the Table.

Table 1 Mean effective reproductive numbers R* for each of the four time periods with chronological events of relevance for the time periods.

The temporal fluctuation in the value of R* is further exhibited in figure 1 with similar trend as that of the mean effective reproduction number for various time intervals obtained in [1] using the probable cases.

Figure Histogram for mean effective basic reproduction number R* during the four time periods of SARS outbreak in Taiwan, 2003

The drastic decrease in the mean effective reproduction number after April 29 further confirms the fact that the turning point for the outbreak to subside had occurred around April 29 [4]. These result shows that the mathematical modeling methodology used here is inherently consistent, regardless of whether we use the cumulative probable case data as in [1], or the more restrictive but reliable cumulative laboratory confirmed case data.

References

1. Hsieh YH, Chen CWS. (2003) Re: Mathematical modeling of SARS: Cautious in all our movements [electonic response to the JECH Severe Acute Respiratory Syndrome Supplement] jech.com 2003 http://jech.bmjjournals.com/cgi/eletters/57/6/DC1#66

2. Center for Disease Control (Taiwan). Available at http://www.cdc.gov.tw/sarsen

3. Hsieh YH, Chen CWS. (2003) Severe Acute Respiratory Syndrome: Numbers don¡¦t tell the whole story. British Medical J 2003; 326: 1395-1396.

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Re: Re: Mathematical modeling of SARS: Cautious in all our movements

Dear Editor

Dr Nishiura [1] accentuated that caution must be exercised in using mathematical models to ascertain the recent SARS epidemics.

The key issue, as we believe, is to understand the model and its results for what they are and, more importantly, for what they are not. It is especially true with the basic reproductive number R0, or its variant the effective reproductive number at time t Rt, which has been estimated for the recent SARS outbreaks in Beijing, Hong Kong,, Toronto, Taiwan, and Singapore in several recent articles (e.g. [2-6]). R0, the average number of secondary infections caused by an infective person upon entering a totally susceptible population, is a useful tool to gauge the initial trend of an epidemic. It is also often misunderstood and misused. Indeed, a recent news feature in Nature [7] described the basic reproductive number R0 as "A measure of a disease's infectiousness" corresponds to how many people, on average, are infected by each patient in the absence of any control measures, which erroneously left out the important requirement that the patient must be an index case in that population, i.e. all possible contacts of that person are susceptible to infection.

The effective reproductive number at time t Rt =R0 x(t), where x is the susceptible proportion of population at time t, measures number of infections caused by a new case at time t.[3] It is more important as a mean to understand the progression of the epidemic, taking into consideration the control measures, behavior changes, and climate as they have all been proven to be important in the case of SARS. Moreover, one can approximate the average growth rate of an epidemic over a given time interval while the epidemic is underway from the cumulative case data. From which one could then estimate the "mean effective reproductive number of the observed time period" R*, i.e. the average number of secondary infections caused by one infective person during the observed time interval. The precise definition gives the public officials a clear chronology of progression (or cessation) of the epidemic, albeit retrospectively.

For illustration, we used the cumulative number of probable SARS cases in Taiwan by onset date from March 12 to June 15,[8] exponential curve fitting with first-order autocorrelation in the error structure,[9] and the period of SARS infectivity of 29.03 days (i.e. time from onset to death or discharge) estimated from [10] to obtain the mean effective reproductive numbers for the five distinct periods during March 12 - June 15 (Table 1). A chronology of relevant events of importance is given as a footnote of Table 1. Figure 1 paints a clear picture of slowly growing epidemic in the beginning, to the outbreak kindled by the admission of first SARS patient to Ho Ping Hospital, the site of first hospital cluster infections, on April 9. The peak period of infections (4/11-4/26) ended with the shutdown of Ho Ping Hospital on April 24. The series of hospital clusters in Taipei and subsequently in the southern port city of Kaohsiung finally subsided with the May 11 shutdown of Chang Gung Hospital in Kaohsiung, due to successful intervention efforts to stop nosocomial infections, the last of which occurred shortly before June 9 the onset date of the last hospital infection in Taiwan. The result clearly points to the important lesson from the outbreak in Taiwan shutdown of hospitals where cluster infections have occurred had been a crucial step in breaking the local chains of transmissions. The effect of quarantine measures, however, is less clear and requires further study, perhaps with mathematical modeling. Clearly, retrospective mathematical modeling is an important reference for public health policy makers intending to contain possible future outbreaks with the most effective intervention measures as long as we understand them for what they are and what they are not.

Table 1 Mean effective reproductive numbers R* for each of the five time periods with events of relevance during the time periods

3/18 – Implementation of Level A quarantine.
4/09 – Admission of first SARS patient to Ho Ping Hospital.
4/24 – Shutdown of Ho Ping Hospital.
4/28 – Implementation of Level B quarantine.
5/11 – Shutdown of Chang Gung Hospital.
6/15 – Onset date of the last hospital infection.

References

(1) Nishiura H. Mathematical modeling of SARS: cautious in all our movements. J Epidem Com Health 2003; In Press.

(2) Riley S, Fraser C, Donnelly C, Ghani AC, Abu-Raddad LJ, Hedley AJ, et al. Transmission dynamics of the etiological agents of SARS in Hong Kong: Impact of public health interventions. Science 2003; 300: 961-66 (20 June 2003) Published online 23 May 2003 (10.1126/science.1086478)

(3) Lipsitch, M, Cohen T, Cooper B, Robins JM, Ma S, James L, et al. Transmission dynamics and control of severe acute respiratory syndrome. Science 2003; 300: 1966-70 (20 June 2003) Published online 23 May 2003; 10.1126/science.1086616

(4) Zhou G, & Yan G. Severe Acute Respiratory Syndrome epidemics in Asia. Emerg Infect Dis 2003; 9(12), In Press.

(5) Hsieh YH, Chen CWS, & Hsu SB. The Severe Acute Respiratory Syndrome outbreak in Taiwan: Lessons to be learned. Emerg Infect Dis 2003; To Appear.

(6) Chowell G, Fenimore PW, Castillo-Garsow MA, & Castillo-Chavez C. SARS outbreaks in Ontario, Hong Kong, and Singapore: the role of diagnosis and isolation as a control mechanism. J Theoret Biol 2003; 224: 1-8.

(7) Pearson H, Clarke T, Abbott A, Knight J, & Cyranoski D. SARS: what have we learned? Nature 2003; 424(6945):121-6. Nature 424, 121: 126(2003) (10 July 2003).

(8) Center for Disease Control (Taiwan). Available at http://www.cdc.gov.tw/sarsen

(9) Hsieh YH,. & Chen CWS. Severe Acute Respiratory Syndrome: Numbers don¡¦t tell the whole story. British Medical J 2003; 326: 1395-1396.

(10) Donnelly C, Ghani AC, Leung GM, Hedley AJ, Fraser c, Riley S, et al. Epidemiological determinants of spread of causal agent of severe acute respiratory syndrome in Hong Kong. Lancet 2003; 361(9371): 1761-66. (May 24 2003) Available at http://image.thelancet.com/extras/-3art4453web.pdf.

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Re: Mathematical modeling of SARS: Cautious in all our movements

Dear Editor

Dr Bernard CK Choi and Dr Anita WP Pak recently developed a simple approximate mathematical model to predict the cumulative incidence and death.[1] Although it’s certainly easy to understand and to use as they stated, every users must be cautious about misunderstanding the real applications and evaluations for SARS epidemics. This problem originates in their too rough assumptions.

Firstly, since the model is based on simplification of an epidemic model, so called Kermack & McKendrick model,[2] the fate of an epidemic depends on threshold theorem: that is, R0 > 1 or not. They used a single value of the basic reproductive number, R0, throughout the epidemic. This, however, is not an accurate description of actual transmission dynamics because R0 is likely to decrease after the onset of an epidemic is detected and announced. Considering threshold host densities in case of simple circumstances,[3] R0 would be written as R0 = N/NT. Here N is the population size, and 1/NT is the proportionality constant whose value would be determined by all manner of biological, social, and environmental aspects of transmission. In the real SARS epidemic, the latter value might be largely affected by social reactions as well as environmental factors (including weather conditions as the authors mentioned). It is well known that the potentials for panic and social stigma are often much greater than the risk for the disease such as SARS. It is also notable that great variability in R0 should be a key to consider SARS epidemic. To say that a crude picture of SARS can be estimated with the model is quite different from saying that the model can be used to evaluate the success of interventions.

Secondly, as authors stated ‘may not’, homogenous mixing would not be a correct depiction of actual population interactions of SARS transmission. Although we are still in the face of so many unknowns including super-spreading events (SSEs), the small number of transmissions in most of the countries that experienced SARS occurrences suggests that the close contact required for transmission did not occur.[4] For instance, Japan has so far not experienced domestic transmission even though it had experienced the entrance of an SARS-CoV infected person, whereas one index case, not SSEs, caused an epidemic originated from Hospital in Toronto.[5] When we note that 76% of the infections in Singapore were acquired in a health-care facility,[6] it is evident SARS can easily be spread by close personal contact. Those facts were relatively known at that time when authors developed their model. Therefore, it is too optimistic to apply their assumption to the real data that every infected person will pass the disease to exactly R0 susceptible individuals.

I agree that user-friendly mathematical models must be developed and the mathematical models should be kept as simple as possible so that public health officers as well as students could predict and learn more effectively. Such models and recommendations for general usage, however, may send the wrong message to the local health officers and/or public. The models for the public must be based on clear knowledge, especially of epidemiological determinants and ecology of pathogens, when it comes to general usage. In order to avoid misleading, the mathematical epidemiologists cannot be too careful to describe their limitations of own models.

References

(1) Choi BCK, & Pak AWP. A simple approximate mathematical model to predict the number of SARS cases and deaths. J Epidem Com Health. 2003; In Press.

(2) Kermack WO, & McKendrick AG. Contributions to the mathematical theory of epidemics – 1927. R Stat Soc J 1927;115:700-721. (Reprinted in Bull Math Biol 1991;53:33-55).

(3) Anderson RM, & May RM. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press, 1992.

(4) Arita I, Kojima K, & Nakane M. Transmission of Severe Acute Respiratory Syndrome. Emerg Infect Dis 2003; 9: 1183-4.

(5) Dwosh HA, Hong HHL, Austgarden D, Herman S, & Schabas R. Identification and containment of an outbreak of SARS in a community hospital. CMAJ 2003; 168: 1415-20.

(6) Center for Disease Control and Prevention. Severe Acute Respiratory Syndrome-Singapore, 2003. MMWR 2003; 52: 405-11.

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Re: Epidemiology of SARS - the missing pathogen?

This is indeed a strange disease. The epidemiology suggests it to be of relatively low infectivity, but high severity.This in itself is odd, especially if the causative agent is a virus and the principal mode of spread by coughing/droplet.Also odd is the undoubted existence of "superspreaders", who can infect very many of their contacts - I can't think of any parallels to this in respiratory virology.

Perhaps the SARS virus obeys the usual rules of droplet-transmitted respiratory infections, and is of high infectivity. However, due to shared antigens, a proportion of the population has an acquired resistance to the new virus, having already been exposed to another, relatively innocuous, virus that provides immune protection. It is possible that the proportion of humanity immune or partially immune to SARS could be as high as, say, 95% if the second virus were a very common one, e.g. one of the coronaviruses that causes coryza.This would explain the seemingly low, unexpectedly so, infectivity of the SARS agent.

Maybe this also explains "superspreaders". Picture humanity divided into two categories:
1). Those who have met a common related coronavirus, and consequently have a degree of immunity to SARS, say for the sake of argument 95% of the population.
2). Those who have not met it, and have no immunity, 5%. If the defences of the first group are overwhelmed by exposure to a huge SARS virus inoculum, perhaps they would contract a modified form of the disease, quickly recruit their immune systems to produce antibodies to a recognised infectious agent, be likely to recover, not shed large amounts of virus, not be all that infectious.The second group would get the disease in an exuberant form, excrete quantities of infectious material, be likely to succumb before their immune system could meet the challenge.....the superspreaders.

I believe that a coherent model of the SARS epidemic could be constructed from the above theory. This of course would not necessarily lend it validity, but it may be worth looking at.