We studied the severe acute respiratory syndrome (SARS) outbreak in Taiwan, using the daily case-reporting data from May 5 to June 4 to learn how it had spread so rapidly. Our results indicate that most SARS-infected persons had symptoms and were admitted before their infections were reclassified as probable cases. This finding could indicate efficient admission, slow reclassification process, or both. The high percentage of nosocomial infections in Taiwan suggests that infection from hospitalized patients with suspected, but not yet classified, cases is a major factor in the spread of disease. Delays in reclassification also contributed to the problem. Because accurate diagnostic testing for SARS is currently lacking, intervention measures aimed at more efficient diagnosis, isolation of suspected SARS patients, and reclassification procedures could greatly reduce the number of infections in future outbreaks.

On April 22, 2003, the World Health Organization (WHO) reported 3,947 probable severe acute respiratory syndrome (SARS) cases with 229 deaths worldwide (

Many questions arose as to how SARS was able to spread so rapidly in Taiwan, a full 2 months after the global alert posted by WHO and >1 month after its passage through Hong Kong, Singapore, and other neighboring countries (

Riley et al. (

We proposed a dynamic model to reflect the actual sequence of events for a reported case-patient in Taiwan, from onset to admission at a hospital as a suspected case-patient to either reclassification as a probable case-patient or removal from the suspected SARS category, and finally reclassification from probable case to discharged case or fatality. Our goal was to evaluate the dynamics at work that resulted in rapid epidemic growth during the period observed. We chose to use a discrete difference equation model because the data used are the discrete daily numbers of reported suspected cases, probable cases, and accumulated deaths posted on the Taiwan Center for Disease Control Web site (

Starting from the Hoping Hospital cluster in Taipei on April 22, the large numbers of cases reported daily (

The number of new probable cases in Taiwan by reporting date, May 5–June 4, 2003.

To this end, we considered a model with susceptible patients (S_{n}), hospitalized suspected case-patients (H_{n}), reported probable SARS case-patients (I_{n}), and the accumulated SARS deaths (D_{n}). The exposed population was not considered since there had been no documented evidence of transmission before onset of symptoms (

Flow diagram for the model dynamics of the model proposed.

We used the daily cumulative numbers of reported suspected cases, probable cases, and deaths from May 5 to June 4 for the true data for the respective numbers for H_{n}, I_{n}, and D_{n} in our model. We chose the data period May 5–June 4 for expediency: it was the only period when all three numbers could be extracted from the Taiwan Center for Disease Control Web site data. We purposely used the number of probable cases by reporting date instead of by onset date to capture what truly happened clinically and in hospital at various stages of a patient’s clinical progression.

To simplify our estimation procedure, we discarded the time dependence (or subscript n) of each parameter, thus considering the parameters as mean estimates of the variable parameters over the period considered. The model equations were simplified to a linear system of simultaneous difference equations with which data can be easily implemented for the parameter estimation procedure. We used the three-stage least squares (3SLS) procedure commonly used in econometrics, which provides a useful parameter estimation procedure for simultaneous equations (

The parameters estimated, without the subscripts, are: λ and β (the respective admission rates due to contact with probable and suspected case-patients at time n-3); ξ (admission rate due to contact with probable case-patient at time n); α (rule out rate of uninfected hospitalized persons at time n); γ (reclassification rate of suspected SARS case-patients to probable at time n); σ (discharge rate of probable SARS patients at time n); ρ (death rate of probable SARS patients at time n). Note that, by their definitions, α, γ, σ, and ρ are proportions between 0 and 1.

From the estimation results, the contributions of contacts of probable case-patients to the suspected SARS population (λ and ξ) are not significantly different from zero. Hence, almost all SARS-infected persons had symptoms and were admitted before their infections were reclassified from suspected to probable SARS. This finding could indicate efficient admission, slow reclassification process, or a mixture of both. The high percentage of nosocomial infections in Taiwan (73% of all traceable cases) suggests that infection from hospitalized suspected case-patients while they waited to be reclassified (and were subsequently placed in negative-pressure rooms) is a major factor in the spread of disease. Most of the newly admitted suspected case-patients were found by onset of symptoms combined with record of contact with other suspected cases of _{n-3}). We also attempted to fit the data for possible contacts with I_{n-k} and H_{n-k} for k = 1 to 7 (given that the incubation time has been estimated at 2 to 7 days). Only H_{n-3} turned out to be a significant source of contact for the suspected case-patients. This finding gives a time from infection to onset of

The results of the parameter estimations are given in

Parameter | Estimated value | 90% CI | p value |
---|---|---|---|

SARS^{b} death rate | 0.0023 to 0.00101 | 0.0125 | |

Discharge rate of probable case-patients | 0.000^{c} to 0.1500 | <0.0001^{d} | |

Admission rate of suspected case-patients | 0.0814 to 0.5927 | 0.0336 | |

Reclassification rate from suspected to probable case | 0.0281 to 0.1311 | 0.0142^{e} | |

Rule-out rate of suspected cases | 0.3571 to 0.5927 | - | |

Proportion of probable cases in suspected class | - | - |

^{a}All rates are per day.
^{b}SARS, severe acute respiratory syndrome.
^{c}Max{0,-0.0046}.
^{d}p value for ^{e}p value for

A, number of hospitalized suspected case-patients (H_{n}) computed from the model compared with real data from May 5 to June 4, 2003. B, number of reported probable case-patients (I_{n}) computed from the model compared with real data from May 5 to June 4. C, cumulative number of deaths due to severe acute respiratory syndrome (D_{n}) computed from the model compared with real data from May 5 to June 4.

To make the results more transparent, we used the mean estimates of daily rates to calculate the mean interval for progression through various stages, given in

Interval for: | Mean estimate (days) |
---|---|

Admission to reclassification as probable case-patient | 12.56 |

Admission to removal from suspected case-patient category | 2.11 |

Probable case classification to death | 24.31 |

Probable case classification to discharge | 11.38 |

^{a}SARS, severe acute respiratory syndrome.

In our study, the gap between mean time from admission to reclassification as probable SARS case-patient was 12.56 days; and the mean time from admission to a case’s being ruled out as a SARS case was 2.11 days. When first admitted with symptoms, a patient is treated with an antimicrobial drug. When the symptoms subsequently subside, the patient status is usually downgraded and the patient is removed from the category of suspected SARS case-patients after a few days of observation. Moreover, anyone who is symptomatic, had contact with this person, but shows no lingering symptoms will also be subsequently quickly downgraded. Hence, a mean estimate of 2.11 days from admission to being ruled out as a case seems reasonable. On the other hand, if the antimicrobial treatment does not yield marked improvement, a person is kept under observation for

Days | ||
---|---|---|

Interval for: | Taiwan | Hong Kong |

Admission to designation as a probable case-patient to death | 36.87 | 35.9 |

Admission to designation as a probable case-patient to discharge | 23.94 | 23.5 |

^{a}By Donnelly et al. (

The total time from admission to discharge for a SARS patient was 23.94 days. To obtain a “mean effective reproductive number for the observed time period,” R*, we use the mean admission rate by suspected cases (β) and multiply it by the mean time the person spent as a suspected case-patient before reclassification (12.56 days) to get R* = 4.23. However, this figure might be an overestimate because of uncertainty regarding how infectious a SARS patient is, relative to the change in his or her viral load (_{t} (

The results for the mean effective reproductive number, R*, suggest that the easiest way to reduce infections is more efficient diagnosis of the probable SARS case-patients and their speedy isolation in negative-pressure rooms. In light of the present lack of accurate diagnostic testing for SARS, public health measures aimed at more efficient clinical diagnosis, isolation of suspected case-patients, and reclassification procedures could greatly reduce the number of infections in future outbreaks. Such steps could be accomplished by quickly identifying the true suspected SARS cases, speedy reporting, effective in-hospital isolation, and fast reclassification of the SARS patients.

The quarantine implemented in Taiwan resulted in only a small number of persons later diagnosed as suspected or probable case-patients. However, one can only speculate about the number of additional infections that the quarantine of these few patients prevented. Events in Canada, for example, demonstrated how one misreported case could lead to an entirely new wave of infections. While there is ample evidence that the quarantine implemented by several countries was instrumental in stopping the spread of SARS, the important public health policy decision of using quarantine as an intervention measure, weighed against its socioeconomic costs, requires further studies with better data and more detailed mathematical modeling.

We had attempted to obtain the estimates by splitting the observed time period into two distinct intervals to see if the three factors involved indeed show a decrease during the course of the observed period. Unfortunately, limited data size inhibits such an endeavor. With the help of Center for Disease Control of Taiwan, more extensive data are currently being collected and generated, including information on the chains of infections as well as clusters. Such data collection takes time, involving the difficult task of contact tracing, but it will form the basis of a more comprehensive modeling study in the future, one that can account for the complete sequence of events.

From the model, it is also clear that the estimated parameters should be time-dependent. However, given the limited data available, one must make simplifications to estimate the means of the parameters over the observed period. With more and better data, one could perhaps estimate the parameters over smaller periods of interest during the complete progression of the epidemic, if not the parameter values for each time n.

Another crucial factor in the outbreak is spatial heterogeneity (i.e., diversity in spatial dimension, brought on by the factor of distance). As Hoping Hospital was closed on April 24 in the aftermath of cluster infections, its patients were allowed to disperse freely to other hospitals; some transferred though the medical system, others on their own. This dispersal of infected persons was directly responsible for several hospital cluster infections in Taipei and even one in Kaohsiung, the southern port city, the effect of which cannot be examined without introducing spatial heterogeneity into the model. Dye and Gay (

The Model

Estimation Method

We thank Mei-Shang Ho for constructive comments; Chwan-Chuen King for providing the epidemiologic data; Center for Disease Control of Taiwan for use of its database; and the reviewers and the associate editor for their constructive comments.

The authors were supported by grants from National Science Council (NSC) and Center for Disease Control of Taiwan. This work is dedicated to the men and women of the medical profession everywhere in the world who lost their lives fighting on the frontline in the battle against SARS.

Dr. Hsieh is a professor of Applied Mathematics Department at National Chung Hsing University. His primary research interests are focused on mathematical and statistical modeling of infectious diseases epidemiology.