SARS epidemiology modeling.

To the Editor: To assess the effectiveness of intervention measures during the recent severe acute respiratory syndrome (SARS) pandemic, Zhou and Yan () used Richards model, a logistic-type model, to fit the cumulative number of SARS cases reported daily in Singapore, Hong Kong, and Beijing. The key to using mathematical models for SARS epidemiology is understanding the models (). In the Richards model (), the function F(S) in the model was described as measuring "the effectiveness of intervention measures." The parameters in F(S), namely the maximum cases load K and the exponent of deviation a, depict the actual progression of the epidemic as described by the reported data. In other words, the parameter estimates are used to quantify end results of the intervention measures implemented during the outbreaks. Simply put, the all-important question of "what if?" was not answered by their result. To gauge the effectiveness of intervention measures, one should consider a more complicated model with variable maximum case load and growth rate (r) that highlights the time-varying nature of an epidemic and its dependence on the intervention measures implemented during the epidemic.

Predicting the trend of an epidemic from limited data during early stages of the epidemic is often futile and sometimes misleading (). Nevertheless, early prediction of the magnitude of an epidemic outbreak is immeasurably more important than retrospective studies. But how early is too early? Intuitively, the cumulative case curve will always be S-shaped and well-described by a logistic-type model. The essential factor is the time when the inflection of the cumulative case curve occurs, i.e., the moment when a rapid increase in case numbers is replaced by a slower increase. Since the inflection point, approximated by t (), dictates the point in time when the rate of increase of cumulative case numbers reaches its maximum, the moment marks the key turning point when the spread of the disease starts to decline. As long as the data include this inflection point and a time interval shortly after, the curve fitting and predicting future case number will be reasonably accurate.

To illustrate this point more precisely, the cumulative SARS case data by onset date in Taiwan were obtained from the SARS databank of Taiwan Center for Disease Control. The data cover the time from February 25, 2003, the onset date of the first confirmed SARS case, to June 15, 2003, the onset date of the last confirmed case; a total of 346 SARS cases were confirmed during the 2003 outbreak in Taiwan (). The cumulative case data are fitted to the cumulative case function S(t) in Richards model with the initial time t = 0 being February 25 and the initial case number S = S(0) = 1. Description of the model, as well as the result of the parameter estimation, is shown in Tables A1-A6. The estimates for the parameters are r = 0.136 (95% confidence interval [CI] 0.121 to 0.150), K = 343.4 (95% CI 339.7 to 347.1), a = 1.07 (95% CI 0.80 to1.35), and the approximate inflection point at tm = 66.62 (95% CI 63.9 to 69.3) with adjusted r2 >0.998, p < 0.0001 for the goodness-of-fit of the model (Figure). The result indicates that the inflection point occurred on May 3, and the estimate for the maximum case number K = 343.3 is 0.8% off the actual total case numbers.

Table A1

Study 1, patient characteristics, methicillin-resistant Staphylococcus aureus (MRSA), controls not infected with S. aureus and controls with methicillin-susceptible S. aureus (MSSA) surgical site infections, bivariable analyses

Variable	Cases, MRSA (%) (n = 121)	Controls, uninfected patients (%) (n = 193)	p value, (MRSA vs. uninfected controls)	Controls, MSSA (%) (n = 165)	p value (MRSA vs. MSSA)
Age, mean ± SD, y	63.9 ± 15.4	57.3 ± 18.3	0.001	55.1 ± 17.4	<0.001
Male sex	55 (45.5)	92 (42.7)	0.73	90 (54.6)	0.15
Coexisting conditions
Diabetes mellitus	59 (48.8)	66 (34.2)	0.01	57 (34.6)	0.02
Hematologic disorder	1 (0.8)	1 (0.5)	1.00	2 (1.2)	1.00
HIV infection	0 (0.0)	1 (0.5)	1.00	0	1.00
Hypertension	64 (52.9)	75 (38.9)	0.02	80 (48.5)	0.48
Liver disease	4 (3.3)	1 (0.5)	0.07	2 (1.2)	0.25
Malignancy	15 (12.4)	14 (7.3)	0.16	13 (7.9)	0.23
Obesity	10 (8.3)	12 (6.2)	0.50	18 (10.9)	0.55
Peripheral vascular disease	12 (9.9)	3 (1.6)	0.002	9 (5.5)	0.17
Pulmonary disease	21 (17.4)	23 (11.9)	0.19	32 (19.4)	0.76
Renal disease	19 (15.7)	9 (4.7)	0.002	13 (7.9)	0.06
Transplant	1 (0.8)	0	0.39	0	0.42
Tobacco use	16 (13.2)	20 (10.4)	0.47	24 (14.6)	0.86
Alcohol abuse	4 (3.3)	2 (1.0)	0.21	6 (3.6)	1.00
Hospital-related risk factors
Treatment at the academic tertiary care hospital	94 (77.8)	125 (64.8)	0.02	109 (66.1)	0.04
LOS before surgery, median, IQR	1, 0–4	0, 0–3	0.02	0, 0–2	0.01
LOS before culture, median, IQR	8, 5–14	NA	NA	5, 3–10	<0.001
Proportion of patients with an ICU stay before surgery	11 (9.1)	13 (7.9)	0.83	18 (9.3)	1.0
ASA score, median, IQR	3, 3–4	3, 2–4	0.03	3, 2–4	0.15
Duration of surgery (min), median, IQR	240, 166–305	194, 113–276	0.004	202, 116–285	0.01
Wound class, median, IQR	1, 1–1	1, 1–1	0.82	1, 1–1	0.36
NNIS Risk Index, median, IQR	1, 1–2	1, 1–1	0.002	1, 1–2	0.06

aLOS, length of stay; IQR, interquartile range; ASA, American Society of Anesthesiologists-Physical Status score; NNIS, National Nosocomial Infections Surveillance System.

Table A6

Study 2, adjusted outcomes models for vancomycin-resistant enterococcus (VRE) wound infection compared to control patients with wound infection due to vancomycin-susceptible enterococcus (VSE)a

Variable	Deathsb		Length of Stayc		Costd
	Odds Ratio (OR) (95% Confidence Interval [CI])	Variable	ORe (95% CI)	Variable	ORe (95% CI)
VRE	2.5 (1.1, 6.1)	VRE	1.1 (0.9, 1.4)	VRE	1.4 (1.2, 1.6)
Intensive care unit stay (ICU)f	9.0 (3.0, 27.4)	ICU stayf	1.8 (1.3, 2.5)	Surgeryf	1.2 (1.1, 1.3)

aOR, odds ratio; CI, confidence interval; ICU, intensive care unit.
bModel includes the following confounding variables: gender and surgery before infection.
cModel includes the following confounding variable: malignancy and length of stay before infection.
dModel includes the following confounding variables: length of stay before cohort inclusion.
eFor length of hospital stay and cost, OR represents multiplicative effect.
fBefore infection for cases and before index date for controls.

Figure

SARS cases, Taiwan, 2003, using Richards model; t = real data. A, confirmed cases; B, estimated cases using the truncated data.

Moreover, the case number data are sorted by onset date. Given a mean SARS incubation of 5 days (4–6 days) (), the inflection point for SARS in Taiwan could be traced back to 5 days before May 3, namely April 28. On April 26, the first SARS patient in Taiwan died. Starting April 28, the government implemented a series of strict intervention measures, including household quarantine of all travelers from affected areas (). In retrospect, April 28 was indeed the turning point of the SARS outbreak in Taiwan.

To address making projections during an ongoing epidemic, we used the same dataset but used various time intervals (all starting February 25) but truncated at various dates around the inflection point of May 3. The resulting parameter estimates are given in Tables A1-A6. For the truncated data ending on April 28 before the inflection, an unreasonable estimate of K = 875.8 was obtained. However, if we use the data ending on May 5, May 10, May 15, and May 20, we obtain estimates of K = 204.9, 253.1, 334.2, and 342.1, respectively. These estimates improve as we move further past the inflection time of May 3 (Figure). Moreover, the last estimate, using data from February 25–May 20 only, produces a 1.1% error from the eventual cumulative case number of 346, with 95% CI of 321.5 to 362.6. This retrospective exercise demonstrates that if the cumulative case data used for predictive purpose during an outbreak contain information on the inflection point and approximately 2 weeks afterwards, the estimate for the total case number can be obtained with accuracy, well before the date of the last reported case. This procedure may be immensely useful for deciding future public health policies although correctly determining the true inflection point during a real ongoing epidemic calls for scrutiny and judicious use of the model, as with all mathematical epidemic models.

In Reply: Our analysis of the dynamics of reported severe acute respiratory syndrome (SARS) clinical cases was conducted in May 2003 during the height of the public panic (). Our primary goal in that study was to predict "when the epidemic might be brought under control if the current intervention measures were continued" (). We used the Richards model and successfully predicted the epidemic cessation dates in Beijing, Hong Kong, and Singapore. Our predicted total number of SARS cases was close to the actual number of cases. In addition, we estimated the basic reproductive rate (R0) of SARS infection, and our estimates based on the deterministic model were similar to those based on stochastic models (,). Therefore, our analysis provided useful information on the epidemiologic characteristic of SARS infections in three major Asian cities.

Hsieh et al. () commented that our article did not address the effect that specific intervention measures might have on the dynamics of SARS infection. Our study was not intended to measure this. As we stated in our article, "the transmission mechanism of the coronavirus that causes SARS and the epidemiologic determinants of spread of the virus are poorly understood." Any models built on these unknowns are not suitable for assessing the effects of specific intervention measures. A method suggested by Hsieh et al. () to merely "consider a more complicated model with variable maximum case load and growth rate" will not answer the question to any extent.

The retrospective analysis of SARS case dynamics in Taiwan by Hsieh et al. () found that "as long as the data include this inflection point and time interval shortly after, the curve fitting and predicting future case number will be reasonably accurate." This notion holds only if the true inflection point is known before an epidemic ends. The main difficulty is how the true inflection point is correctly determined, as noted by Hsieh et al. (). The time when inflection occurs varies tremendously if truncated data of cumulative SARS case numbers are used. To illustrate this point, we used the cumulative number of reported probable SARS cases in Hong Kong, starting March 17, 2003, but truncated at various dates, and calculated the date when inflection occurred (Table). For example, if the data period from the onset date (March 17, 2003) to the last case reported (June 12, 2003) was used, the date when inflection would occur was estimated as March 19, 2003. If the truncated data ending April 9, April 16, April 30, May 14, and May 28, 2003, were used, the dates when inflection would occur were estimated as April 2, February 7, March 3, March 23, and April 2, 2003, respectively (Table). Clearly, inflection point dates became a moving target as the epidemic progressed. When truncated data ending April 9, April 16, April 30, May 14, and May 28, 2003, were used, the corresponding estimated maximum numbers of cumulative cases (K) were 1,107, 1,907, 1,819, 1,749, and 1,733, respectively. Estimation of K improved when the data period used for prediction was at least one month past the March 19 inflection point obtained from the entire epidemic period. This analysis highlights the difficulty in identifying an optimal inflection point for prediction purposes during an ongoing epidemic when only a partial cumulative case number is available.

Table

Predicted inflection point and dates when inflection occurs based on truncated data of cumulative number of reported severe acute respiratory syndrome cases in Hong Kong

Data period (ending date)	tm a	Dateb	K c	r d	αe
April 9, 2003	16.62	April 2, 2003	1,107	0.20	0.74
April 16, 2003	–40.79	February 7, 2003	1,907	0.07	52.11
April 30, 2003	–13.52	March 3, 2003	1,819	0.07	10.21
May 14, 2003	6.80	March 23, 2003	1,749	0.09	2.84
May 28, 2003	17.31	April 2, 2003	1,733	0.10	1.38
June 12, 2003	2.63	March 19, 2003	1,751	0.09	3.77

at is the inflection point of the model.
bDate refers to the date when inflection occurs.
cK is the predicted maximum number of cumulative cases.
dr is the intrinsic growth rate.
eα measures the extent of deviation of S-shaped dynamics from the classic logistic growth curve.

We fully agree with Hsieh et al. () that the quantitative assessment of the effectiveness of public health intervention measures for SARS is a difficult task for modelers. To make models useful for assessing the effects of specific intervention measures and for predicting the future dynamics during an ongoing epidemic, we need improved knowledge on the transmission mechanisms, pathogenesis, and the epidemiologic determinants of the spread of the virus. Any retrospective analysis of the 2003 SARS epidemic that improves our knowledge of SARS epidemiology is welcome.

Table A2

Study 1: Adjusted outcomes models for methicillin-resistant Staphylococcus aureus (MRSA) surgical site infection (SSI) compared to uninfected control patientsa

Variable	Deaths	Length of stayb	Costc
	OR (95% CI)	ORd (95% CI)	OR (95% CI)
MRSA	11.4 (2.8 to 34.9)	3.2 (2.7 to 3.7)	2.2 (2.0 to 2.6)
ASA scoree,f		1.3 (1.2 to 1.5)	ASA score = 4 3.7 (1.5 to 8.9)
			ASA score = 2 2.0 (1.4 to 2.9)
			ASA score = 3 3.0 (2.1 to 4.3)
			ASA Score = 4 4.1 (2.8 to 6.0)
>73 y of age	4.8 (2.0 to 11.6)
Operative duration (min)g
211–400		(0.9 to 1.3)	1.4 (1.2 to 1.7)
401–590		1.7 (1.2 to 2.4)	2.2 (1.6 to 3.1)
>590		1.8 (1.1 to 2.9)	2.6 (1.6 to 4.0)
Length of stay before surgeryh
7–13 d		1.6 (1.1 to 2.1)	1.7 (1.3 to 2.3)
14–20 d		3.6 (1.4 to 9.6)	5.6 (2.3 to 13.4)
>20 d		0.7 (0.2 to 2.6)	1.2 (0.3 to 4.3)
Intensive care unit stay before surgery			1.5 (1.2 to 2.0)
Tertiary care hospital			1.5 (1.2 to 1.7)

aOR, odds ratio; CI, confidence interval; ASA, American Society of Anesthesiologists -Physical Status.
bModel includes the following confounding variables: admission to the tertiary care hospital, diabetes, and renal disease.
cModel includes the following confounding variable: renal disease.
dFor length of hospital stay and cost, OR represents multiplicative effect
eLength of stay increases by 1.3-fold for each point increase in ASA score.
fFor cost, reference category is ASA score = 1.
gReference category is operative duration < 211 min.
hReference category is length of stay before surgery < 7 d.

Table A3

Study 1, adjusted outcomes models for methicillin-resistant Staphylococcus. aureus (MRSA) surgical site infections (SSI) compared to patients with methicillin-resistant S. aureus (MSSA) SSIa

	Deathsb	Length of Stayc	Costd
Variable	OR (95% CI)	OR (95% CI)e	ORe (95% CI)
MRSA	3.4 (1.5 to 7.7)	1.2 (1.0 to 1.5)	1.2 (1.0 to 1.4)
ASA scoref	ASA score = 4 5.1 (2.1 to12.5)	ASA score = 2 0.9 (0.5 to 1.7)	ASA score = 2 1.0 (0.7 to 1.5)
		ASA score = 3 1.6 (0.9 to 2.9)	ASA score = 3 1.4 (1.0 to 2.1)
		Asa score = 4 1.8 (1.0 to 3.5)	ASA score = 4 2.1 (1.4 to 3.2)
Age > 61 years	3.0 (1.2 to 7.3)
Operative duration, ming
206–381		1.3 (1.0 to 1.6)	1.4 (1.1 to 1.6)
382–557		1.3 (0.8 to 2.1)	1.8 (1.3 to 2.5)
>557		1.1 (0.5 to 2.6)	1.6 (0.9 to 2.8)
Length (d) of stay before infectionh
11–20		1.4 (1.0 to 1.8)	1.6 (1.3 to 2.0)
21–30		1.6 (1.0 to 2.7)	1.7 (1.2 to 2.5)
>30		1.3 (0.5 to 3.1)	1.8 (0.9 to 3.8)
Renal disease		1.5 (1.0 to 2.2)
Length (d) of intensive care unit stay before infectioni
8–14			1.8 (1.1, 2.8)
15–21			2.1 (1.1, 8.8)
>21			1.9 (0.4, 8.0)
Tertiary care hospital			1.3 (1.1, 1.6)

aOR, odds ratio; CI, confidence interval; ASA, American Society of Anesthesiologists -Physical Status.
bModel includes the following confounding variable: operative duration >222 min.
cModel includes the following confounding variables: admission to tertiary care hospital and diabetes.
dModel includes the following confounding variables: diabetes and renal disease.
eFor length of hospital stay and cost, OR represents multiplicative effect.
fFor deaths, reference category is ASA score < 1; for length of stay and cost, reference category is ASA score = 1.
gReference category is operative duration < 206 min.
hReference category is length of stay prior to infection < 11 d.
iReference category is intensive care unit length of stay prior to infection < 8 d.

Table A4

Study 2, patient characteristics, vancomycin-resistant enterococci (VRE) wound infections, controls not infected with enterococci, and controls with vancomycin-susceptible enterococci (VSE) wound infections, bivariate analyses

Variable	Cases, VRE wound (%) (n = 99)	Controls, not infected (%) (n = 280)	P Value (VRE vs. controls not infected)	Controls, VSE (%) (n = 213)	p value (VRE vs. VSE)
Age, mean (y)	60.3	63.6	0.09	59.1	0.51
Sex (female)	46 (46)	124 (44.3)	0.7	127 (59.6)	0.03
Main diagnosis
Orthopedic condition	11 (11)	30 (10.7)		18 (8.4)
Cardiovascular condition	25 (25)	117 (41)		61 (28.6)
Endocrine disorder	3 (3)	6 (2.1)		4 (1.9)
Gastrointestinal disorder	25 (25)	60 (21.4)		62 (29.1)
Genitourinary disorder	6 (6)	12 (4.2)		9 (4.3)
Infectious disease	16 (16)	6 (2.1)		20 (9.4)
Hematologic disease	0 (0)	2 (.7)		0
Neurologic disease	11 (11)	32 (11.4)		34 (16)
Pulmonary disease	2 (2)	14 (5)		5 (2.4)
Coexisting conditions
Cardiovascular disease	73 (74)	204 (72.9)	0.86	150 (70.4)	0.55
Lung disease	11 (11)	33 (11.7)	0.9	26 (12.2)	0.78
Diabetes mellitus	67 (67.7)	139 (49.6)	0.002	127 (59.6)	0.17
Organ transplant recipient	14 (14)	21 (7.5)	0.08	18 (8.4)	0.12
Renal disease	18 (18.2)	39 (14)	0.7	28 (13.2)	0.24
Malignancy	7 (7.1)	27 (9.6)	0.5	32 (15)	0.05
AIDS	2 (2)	2 (0.7)	0.27	0	0.1
Hepatobiliary disease	16 (16.6)	40 (14.3)	0.8	31 (14.5)	0.71
Charlson comorbidity score, mean	3.17	2.66	0.07
Hospital-related risk factors
Transfer from another institution	34 (34.3)	102 (36.4)	0.5	34 (16)	<0.001
Surgery	29 (29.3)	94 (33.6)	0.08	90 (42.3)	0.03
Admission to ICU	26 (26.2)	58 (20.7)	0.9	53 (33.3)	0.8

Table A5

Study 2, adjusted outcomes models for vancomycin-resistant enterococcus (VRE) wound infection compared to uninfected control patientsa

Variable	Deathsb	Variable	Length of Stayc	Variable	Costd
	OR (95% CI)		ORe (95% CI)		ORe (95% CI)
VRE infection	2.0 (0.8 to 5.2)	VRE infection	1.8 (1.3 to 2.4)	VRE infection	1.5 (1.3, 1.8)
		Transfer from another hospital	1.5 (1.2 to 1.9)	Surgerye	1.4 (1.1, 1.8)
		Renal disease	2.0 (1.5 to 2.7)
		Malignancy	0.7 (0.5 to 0.9)
		Intensive care unit stayf	2.3 (1.6 to 3.3)

aOR, odds ratio; CI, confidence interval.
bModel includes the following confounding variables: intensive care unit (ICU) stay and number of coexisting conditions.
cModel includes the following confounding variable: propensity score (i.e., likelihood of being a VRE case).
dModel includes the following confounding variables: propensity score [i.e., likelihood of being a VRE case (Appendix)] and length of stay before infection (index date for controls).
eFor length of hospital stay and cost, OR represents multiplicative effect.
fBefore infection for cases and before index date for controls.