CRITICAL ROLE OF NOSOCOMIAL TRANSMISSION IN THE TORONTO SARS OUTBREAK. Glenn F. Webb. Martin J. Blaser. Huaiping Zhu. Sten Ardal.

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1 MATHEMATICAL BIOSCIENCES mbe/ AND ENGINEERING Volume 1, Number 1, June 2004 pp CRITICAL ROLE OF NOSOCOMIAL TRANSMISSION IN THE TORONTO SARS OUTBREAK Glenn F. Webb Department of Mathematics, Vanerbilt University 1326 Stevenson Center, Nashville, TN Martin J. Blaser Department of Meicine, New York University School of Meicine OBV A606, 550 First Avenue, New York, NY Huaiping Zhu Laboratory for Inustrial an Applie Mathematics, Department of Mathematics an Statistics York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canaa Sten Aral Central East Health Information Partnership Box 159, 4950 Yonge Street, Suite 610, Toronto, ON M2N 6K1, Canaa Jianhong Wu Laboratory for Inustrial an Applie Mathematics, Department of Mathematics an Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canaa Communicate by Denise Kirschner Abstract. We evelop a compartmental mathematical moel to aress the role of hospitals in severe acute respiratory synrome SARS transmission ynamics, which partially explains the heterogeneity of the epiemic. Comparison of the effects of two major policies, strict hospital infection control proceures an community-wie quarantine measures, implemente in Toronto two weeks into the initial outbreak, shows that their combination is the key to short-term containment an that quarantine is the key to long-term containment. 1. Introuction. One of the salient features of the outbreak of severe acute respiratory synrome SARS in the Greater Toronto Area GTA is the role of the hospital in transmission. Of 144 early patients, % were expose to SARS in the hospital setting; of these, 73 patients 51% were health-care workers, incluing nurses, respiratory therapists, physicians, raiology an electrocariogram technicians, housekeepers, clerical staff, security personnel, parameics, an research assistants [1]. The high risk of transmission within the health-care setting has a significant impact on the conuct of public-health interventions in the continuing SARS epiemic [1, 2] an potentially for other emerging respiratory iseases Mathematics Subject Classification. 92D30. Key wors an phrases. severe acute respiratory synrome SARS, coronavirus, epiemiology, mathematical moel, nonlinear ynamics, containment, quarantine. 1

2 2 G. F. WEBB, M. J. BLASER, H. ZHU, S. ARDAL AND J. WU To examine the SARS outbreak in GTA, we evelop a compartmental moel iviing the entire population into classes of susceptibles, expose, infectives, hospitalize, an remove an subclasses representing the general public an iniviuals in the hospital setting incluing health-care workers an patients HCWP. The moel reflects the extremely intense exposure of HCWP to infecte iniviuals prior to awareness of SARS by the meical community; their heightene risk continue until aequate precautions were fully operant in hospitals. In two recent analyses of SARS epiemiology [3, 4], all members of the public were consiere as one class, espite evient heterogeneity in transmission, an the rapi initial sprea of SARS in Vietnam, Hong Kong, an Canaa in hospital wars [5]. In our analysis, the seconary infection inuce by a hospitalize patient for the HCWP R is much larger than the seconary infection inuce by an average infective for the general public R uring the first two weeks of the SARS outbreak in GTA. These seconary infection rates ecrease when hospital infection control proceures an community-wie quarantine measures were introuce. 2. Mathematical moels an analysis of ynamics. Moels were evelope to correspon to the two stages of the SARS outbreak in GTA: pre-moel I an intra-moel II quarantine. Moel I consists of the following compartments: Susceptibles S iniviuals not yet infecte; Expose E susceptibles who have become infecte an are not yet infectious; Infectives I expose iniviuals who have become infecte an can sprea the SARS coronavirus; Remove R iniviuals who have been expose or infective an who are no longer consiere to be susceptible; an Hospitalize U infectives who are in the immeiate environment of HCWP; these iniviuals are not consiere to pose any risk to the general public, but may infect HCWP. For each class, subinices g an h represent general public an HCWP, respectively. Moel I consists of 8 couple nonlinear ifferential equations escribing the transfer of iniviuals from one compartment to another Fig. 1: t S gt = a g S g t I g t + I h t t S ht = a h S h t I g t + I h t a u S h t U h t + U g t t E gt = a g S g t I g t + I h t b g E g t t E ht = a h S h t I g t + I h t + a u S h t U h t + U g t b h E h t t I gt = b g E g t c g I g t r g I g t t I ht = b h E h t c h I h t r h I h t t U gt = r g I g t ɛ g U g t t U ht = r h I h t ɛ h U h t where a g, a h, an a u are the transmission coefficients for the general public an HCWP infectives, an for hospitalize infectives for HCWP, respectively; b g an b h are the transmission coefficients for expose iniviuals to the infective class; c g an 1

3 NOSOCOMIAL TRANSMISSION IN THE SARS OUTBREAK 3 Sg S h Eg E h I g Q I g I h Ih Q R U g R U h R Figure 1. Schematic of SARS transmission when the total population is ivie into the categories of the general public g an HCWP h who have irect contact with hospitalize infecte patients. Each category is ivie into classes of susceptibles S, expose E, infectives I, an hospitalize U. In Moel I, no quarantine or special hospital infection control proceures is implemente an the subclasses of infectives Ig Q, I Q h uner quarantine o not exist. Circles R inicate all iniviuals remove from the infective an hospitalize classes. c h are the transmission coefficients for infective iniviuals to the remove class, an r g an r h are the transition coefficients for infectives to hospitalization. The transition coefficients for the remove class are ɛ g an ɛ h, reflecting the effectiveness of treatments. The secon equation in 1 escribes the aitional risk of HCWP resulting from their irect contact with SARS patients in the health-care setting. The time scale consiere in Moel I 1 is short enough that all emographic etails can be ignore. As a result, each of the total populations in the subclasses I g t, I h t, E g t, E h t, U g t, an U h t eventually approaches zero, an explicit formulae can be obtaine to calculate the total numbers of infecte an hospitalize infectives uring the entire course of the infection see Appenix for more etails. Although simulations base on Moel I correlate with the actual ata in GTA [6] for the first two weeks of the outbreak Fig. 2, the moel preicte a much larger number of infectives subsequently. On March 31 two weeks after the first cases, Ontario eclare a provincial emergency to contain the sprea of SARS an introuce public-health measures, incluing extensive contact tracing, isolation of suspect an probable cases, an voluntary home quarantine for asymptomatic contacts. Hospital infection control proceures also were enforce. Clinicians, initially unaware of the communicability of the SARS coronavirus, ha use positive-pressure ventilation methos to alleviate respiratory symptoms, inavertently augmenting ispersion of contagious roplets. Such treatments were stoppe, negative-pressure

4 4 G. F. WEBB, M. J. BLASER, H. ZHU, S. ARDAL AND J. WU rooms were use, an face shiels were introuce after masking alone prove insufficient. These more stringent hospital measures may have been important in controlling the sprea of SARS in GTA [7] as well as elsewhere [3, 8], an are reflecte in Moel II by significant reuction of the coefficients a h an a u. In Moel II 2, S, E, I, R, an U represent the same groups as in Moel I Fig. 1. We introuce a secon category of infectives I Q representing infecte iniviuals quarantine before they are amitte into or never amitte to the hospital. They pose no risk to HCWP an a low risk to the general public. Moel II reas t S gt = a g S g t I g t + I h t t S ht = a h S h t I g t + I h t a u S h t U h t + U g t t E gt = a g S g t I g t + I h t q g b g E g t 1 q g b g E g t t E ht = a h S h t I g t + I h t + a u S h t U h t + U g t q h b h E h t 1 q h b h E h t t IQ g t = q g b g E g t c g I Q g t r g I Q g t 2 t IQ h t = q hb h E h t c h I Q h t r hi Q h t t I gt = 1 q g b g E g t c g I g t r g I g t t I ht = 1 q h b h E h t c h I h t r h I h t t U gt = r g I g t + I Q g t ɛ g U g t t U ht = r h I h t + I Q h t ɛ hu h t where parameters q g an q h are the fractions of expose general public an HCWP that have been quarantine, respectively. Note that in the case where q g = q h = 0, Ig Q = I Q h = 0 if their initial values are zero; thus Moel II reuces to Moel I. The rates of transfer from the E class to I class are escribe by new equations on Ig Q t an I Q h t to reflect the effect of quarantine measures see Appenix. This moel allows analysis of the epenence of the total number of infecte an hospitalize iniviuals Xt at time t on the parameters a g, a h, a u, q g, an q h. This moel also provies an explicit formula for the lowest possible ultimate number of infecte an hospitalize X when hospital infection control measures an quarantine measures are strictly enforce, an for the most conservative estimation of the quarantine fraction for X to fall below a specifie level see Appenix for etails. 3. Results. The parameters an initial conitions for the simulations are base on the 1996 census ajuste by 1999 intercensus estimates for the year The total population in GTA by PHU is the following: PEEL: 1,107,504; CITY OF

5 NOSOCOMIAL TRANSMISSION IN THE SARS OUTBREAK 5 TORONTO: 2,620,228; DURHAM: 544,069; HALTON: 398,592; an YORK RE- GION: 778,295. Also, accoring to the statistics from Health Canaa [6], we have the initial population ata in Table 1, where the initial t = 0 correspons to March 18. S g 0 S h 0 E g 0 E h 0 I g 0 I h 0 U g 0 U h 0 5, 443, 104 3, Table 1. Initial population ata for Moel I, t = 0 correspons to March 18. Although it is ifficult to estimate a h, a g an a u, we know that a g a h << a u. In a crue approximation in which the total population is ivie into three classes susceptible, infective, an remove an where the transition coefficients from S class to I class an from I class to R class are a an c, respectively, the reprouctive number R 0 is given by an/c. The reprouctive number is efine as the average number of seconary infections that occur when one infective is introuce into a completely susceptible host population [11, 10, 13, 12]. If we take R 0 = 1.2 a value very similar to that calculate for some strains of influenza virus [9] espite the heterogeneity in transmission of the SARS coronavirus, an if we use N = 5, 446, 104 an an estimate 3 ays as the meian time for symptomatic iniviuals to be amitte to hospitals [1], then c = 1/3 an a = 1.2c/N = We use this value for a g in the simulations for both moels. In the stuy carrie out in [3, 14], R 0 is estimate as 3, 2.7, an 1.2. These estimate values of R 0 o not aress ifferent transmission rates in the general community an in the health care setting. However, the ratio a u /a g reflects the effectiveness of hospital infection control measures, an is use as a parameter in the simulation of the secon moel. Since stringent hospital control measures were absent pre-march 30 in the first moel, hence we take a h = 100a g, an a u = 1000a g. Hence the average infective inuce seconary infection is R for the general community, an the average hospitalize patient inuce seconary infection is R for the health care setting. Because the average expose perio prior to becoming infectious is 1/b, an the meian time from self-reporte earliest known exposure to symptoms onset is 6 ays [1], we use b g = b h = 1/6. The average perio before an infective iniviual ies without being amitte to a hospital is 1/c. This number is small, reflecting a relatively low eath rate. The 21-ay survival in a Toronto cohort of 144 hospitalize cases was 93.5%, with negative outcomes most often associate with iabetes or other co-morbi conitions [1]. Estimates for case fatality rates are also associate with age, with a fatality rate of 13.2% for those uner 60, rising to 43.3% for those 60 reporte in Hong Kong [8]. We use c g = c h = Because 1/r is the average perio for an infective iniviual before amission to a hospital, r g = r h = 1/3 [1]. Assuming that this rate improve over the course of the epiemic, we use r g = r h = 1/4 for the first 12 ays. For the remaining course simulation for Moel II, r g = r h = 1/3. Finally, ɛ g = ɛ h = 0.1. The meian hospital stay is 10 ays [1], but assuming that the hospital stay for the infectives in the early stage is longer, r 1 = 20 ays. Therefore, for Moel I, ɛ g = ɛ h = All the parameters involve in Moel I an II are summarize in Table 2. Base on these parameters given in Table 2, the numerical simulations were carrie out using Maple an Mathematica. The simulate result from Moel I an

6 6 G. F. WEBB, M. J. BLASER, H. ZHU, S. ARDAL AND J. WU a g a h a u b g b h r g r h ɛ g ɛ g q g q h Moel I 7.5E-8 100a g 1000a g Moel II 7.5E-8 10a g 10a g Table 2. The parameters value for both of the Moels I an II use in the numerical simulations Cumulative Number Probable case Probable + Suspect Cases Preiction Date:March 18 - April 2 Figure 2. SARS in GTA: Comparison of the reporte ata base on Health Canaa statistics [6] an Moel I preictions of hospitalize patients an remove iniviuals from March 18 t = 0 until April 1 t = 14. On March 30, the number of probable cases from the Health Canaa statistics is 42; the moel preiction number, X12 = 46. the actual ata Fig. 2 compare well for the first two weeks of the outbreak. After March 30 t = 12, however, the simulation yiels a much higher number of infectives than actually occurre. This inconsistency clearly inicates the effectiveness of the combination of hospital control proceures an quarantine measures, proviing the basis for Moel II. The ratio of infectives between HCWP an the general public is close to constant 50% which inicates the proportional risk of HCWP uring the initial outbreak, but it changes after an effective quarantine policy is establishe for the public an infection controls are use in health-care facilities. After March 30, both hospital control measures an quarantine were implemente in GTA; thus, for Moel II we use the same parameters as in Moel I, except a h = 10a g, a u = 10a g, q g = 0.8, an q h = 0.9, to reflect the effectiveness of quarantine measures. Initial conitions for Moel II correspon to the values from Moel I reache on March 30 t = 12 an are in agreement with the actual ata. From the preictions using Moel I, we get the initial ata for Moel II Table 3, where t = 0 correspons to March 30. However, the actual ata available o not enumerate the infectives/hospitalize/remove an HCWP/public istinctions.

7 NOSOCOMIAL TRANSMISSION IN THE SARS OUTBREAK 7 Cumulative Number Probable case Probable + Suspect Cases Preiction Date:March 30 - May 8 Figure 3. SARS in GTA: Comparison of actual ata an simulations from March 30 t = 0, t = 39 correspons to May 8, when Health Canaa statistics [6] inicate the cumulative number of probable an suspect cases in GTA were 140 an 122, respectively. The moel preiction inicates on May 08, X39 = 198. On May 23 when our work was nearly complete, 33 new cases were reporte. S g 0 S h 0 E g 0 E h 0 I Q g 0 I Q h 0 I g0 I h 0 U g 0 U h 0 5, 443, 041 2, Table 3. Initial population ata for Moel II, t = 0 correspons to March 30. The actual ata an simulation results for Moel II until t = 40, t = 39 correspons to May 8, when the Worl Health Organization [WHO] officially lifte its travel warning after two incubation perios ha elapse since the last new reporte probable case are closely aligne Fig. 3 an illustrate the effects of reucing transmission rates for HCWP from a h = 100a g an a u = 1000a g to a h = a u = 10a g ue to strict hospital infection control measures an raising quarantine parameters from q g = q h = 0.0 to q g = 0.8 an q h = 0.9 corresponing to the implementation of aggressive quarantine measures. A key issue in the GTA SARS epiemic, an for future outbreaks, is ientifying which of these two policies was most effective in containing the outbreak. Further simulations of Moel II show that hospital control measures must be strict to contain the virus. The results of the simulations in Fig. 3 pre-march 30 values a h = 100a g an a u = 1000a g change to a u = a h = 10a g an Fig. 4a no change post-march 30 are sharply ifferent. In the latter case, even when the quarantine fraction is high q g = 0.8 an q h = 0.9 but nosocomial transmission rates remain as for Moel I, the total hospitalize an remove iniviuals on May 8th woul be 1324, the outbreak woul last > 100 ays, an about 3, 000 iniviuals woul

8 8 G. F. WEBB, M. J. BLASER, H. ZHU, S. ARDAL AND J. WU Xt 1500 Igt t t a t, Xt b t, I gt Iht 8 Ugt t t c t, I h t t, U gt Figure 4. Moel II simulations starting March 30 t = 0 with a h = 100a g an a u = 1000a g. a shows that if strict hospital control measures are not taken, there woul be 3000 infecte, an the epiemic woul last > 100 ays; b- epict I g, I h, an U g as functions of the time, an illustrate multiple peaks of infection. be either hospitalize or remove Fig. 4a. The multiple local maxima inicating possible multiple peaks shoul hospital control proceures not be taken Fig. 4b-, mirror the actual patterns in other regions: an initial peak reflective of early exposures mostly to HCWP followe by a secon peak that inclues both those expose through hospital an general population contacts [3]. Following this secon peak, with improve control measures, the number of new cases eclines as the outbreak comes uner control R 0 < 1. The simulations of Moel II inicate that both hospital infection control an quarantine measures are important in containing the epiemic. In Fig. 5, we set q g = q h = q, an all other parameters an initial values are as in Moel I, except that a h = a g. The total number Xt of infecte an hospitalize cases are functions of a u an q. By 40 ays, hospital control an quarantine measures, if sufficiently strong, substantially reuce Xt Fig. 5a, but quarantine measures are essential for long-term 150 ays containment of the epiemic Fig. 5b. If enhance measures for hospital infection control an quarantine are relaxe prematurely, the number of hospitalize iniviuals continue to iminish for several ays, an then rise again Fig. 6. The reporte SARS cases in GTA on May 23 illustrate the consequences of premature relaxation of infection control measures.

9 NOSOCOMIAL TRANSMISSION IN THE SARS OUTBREAK qg=qh ag 80 ag 60 ag au 40 ag 20 ag a 40 ays qg=qh ag 80 ag 60 ag au 40 ag 20 ag b 150 ays Figure 5. The number Xt of infecte an hospitalize at a: 40 ays, b: 150 ays, after the first two weeks t = 0 = March 30 as a function of the hospital transmission parameter a u vs. the fraction quarantine q = q g = q h. Substantially reucing either a u or increasing q significantly is sufficient to reuce the number of cases to low levels at 40 ays. At 140 ays reucing hospital transmission to 0 without quarantine results in 25, 000 cases, whereas increasing the hospital transmission parameter to a u = 100a g an the quarantine fraction to.5 results in only 3, 000 cases. To contain the epiemic over time, quarantine of expose iniviuals is necessary, because R 0 > 1 for the general public. 4. Discussion. In total, our simulations show that the combination of moerate quarantine but strict hospital infection control proceures was the key to the containment of SARS in GTA; increasing the effectiveness of quarantine > 85% i

10 10 G. F. WEBB, M. J. BLASER, H. ZHU, S. ARDAL AND J. WU Ugt 12 Uht t t a t, U gt b t, U h t Figure 6. This simulation shows the consequences of relaxation of hospital infection control measures an quarantine measures. The simulation starts on May 9 t = 0, using the initial ata obtaine from Moel II an uner the assumption that parameters a h an a u are increase to a h = 100a g an a u = 500a g an q g an q h are ecrease to q g = 20% an q h = 40%, respectively. The total number of the remove an hospitalize iniviuals will increase by 32 after 20 ays on May 29. not significantly reuce the total number of hospitalize an remove iniviuals [15]. Conversely, without strict hospital control proceures, the outbreak uration will be significantly longer > 100 ays, with multiple peaks of infection uring the entire course an affecting greater numbers of iniviuals Fig. 4. Our consierations here also may be relevant to the sprea of other respiratory infections, incluing pneumonic plague or other emerging respiratory infections. Hospitals have been amplifiers of iseases that involve the general public [such as influenza respiratory, an enteropathogenic E. coli enteric]; our moels shoul be useful to aress issues relate to their control. The epiemiology of SARS has been complex with rapi sprea in some areas e.g., Beijing but not others, e.g. Shanghai espite introuction of the causative agent, as well as the control of the epiemic in many localities without continuous community sprea [16]. These patterns are more consistent with seconary infection in the health-care setting being much greater than in the general community, with an overall R 0 > 1 [3]. The ability of relatively moest quarantine measures to ecrease R 0 < 1 in the general community suggests a pathogen that is not well-aapte to human hosts, an implies strong selection for transmission. Thus, stringent control measures in hospital settings-the major amplifiers of transmission-are neee to minimize the risk for panemic sprea of SARS. 5. Appenix: Analysis of ynamics for Moels I an II. In this section, we give the analysis of ynamics for Moels I an II. In particularly, we give the optional quarantine fractions to control the total number of hospitalize an remove iniviuals below a specifie level.

11 NOSOCOMIAL TRANSMISSION IN THE SARS OUTBREAK Total numbers of infecte an hospitalize infectives, without quarantine. If U g 0 = U h 0 = 0, then lim I gt, I h t, E g t, E h t, U g t, U h t = 0, 0, 0, 0, 0, 0, t lim S gt = S g 0e agi g +I h, t lim S ht = S h 0e a hi g +I h auu g +U h t where Ig = I 0 g ss, Ug = U 0 g ss, Ih = I 0 h ss, Uh = U 0 h ss escribe the total numbers of infecte an hospitalize infectives uring the entire course of infection, an these values are etermine as follows: Ug = rg ɛ g Ig, Uh = r h ɛh Ih, S g 0 + E g 0 + I g 0 = S g 0e agi g +I h + c g + ɛ g Ig, S h 0 + E h 0 + I h 0 = S h 0e a hi g +I h auu g +U h + c h + ɛ h Ig Total numbers of infecte an hospitalize infectives, with quarantine. For Moel II, we have where lim I gt, I h t, Ig Q t, I Q t h t, E gt, E h t, U g t, U h t = 0, 0, 0, 0, 0, 0, 0, 0, S g := lim t S g t = S g 0e agi g +I h, S h := lim t S h t = S h 0e a hi g +I h auu g +U h I g = I 0 g ss, Ig Q = I Q 0 g tt, Eg = E 0 g tt, Ug = U 0 g ss, Ih = I 0 h ss, I Q h = I Q 0 h tt, E h = E 0 h tt, Uh = U 0 h ss an these parameters are etermine by U g = rg ɛ g I g + I Q g + 1 ɛ g U g 0, E g = 1 b g E g b g S g 0[1 e agi g +I h ], c g + r g I g I g 0 1 q g b g E g = 0, c h + r h I h I h0 1 q h b h E h = 0, c g + r g I Q g q g gb g ge g I Q g 0 = 0, c h + r h I Q h q h b h Eh IQ h 0 = Optional quarantine fractions. The epenence of X on the parameters a h, a u, q g, q h can be calculate. Since Ẋ = U g + U h + c g Ig Q + I Q h + c gi g + c h I h + ɛ g U g + ɛ h U h

12 12 G. F. WEBB, M. J. BLASER, H. ZHU, S. ARDAL AND J. WU then X = X0 + c g + r g Ig + Ig Q + c h + r h I h + I Q h. For the sake of simplicity, we let q = q g = q h, b = b g = b h, ɛ = ɛ g = ɛ h, r = r g = r h, α = c g + r g = c h + r h. Denote by I0 = I g 0 + I h 0, U0 = U g 0 + U h 0, E0 = E g 0 + E h 0, an I = Ig + Ih, IQ = Ig Q + I Q h, U = Ug + Uh, E = Eg + Eh an recall that IQ g 0 = I Q h 0 = 0, we get αi = I0 + 1 qbe, αi Q = qbe, U = r ɛ I + I Q + 1 ɛ U0 = r ɛα I0 + be + 1 ɛ U0. Therefore, be = E0 + S g 01 e ag α I0 e ag α 1 qbe +S h 01 e a h I0 α e au ɛα ri0 au ɛ U0 e a h α 1 qbe e au ɛα rbe. It is straightforwar to show that the above equation always has a unique positive solution Y = be, an hence when all others are fixe, be = fa h, a u, q is a function of a h, a u, an q. This is an increasing function of either a u or a h, when all others are fixe, an a ecreasing function of q, when all others are fixe. The same is true for X, since X = X0 + I0 + be. The above equation also allows calculations of the minimal quarantine fraction to control the total number of hospitalize an remove iniviuals below a specifie level. As an illustration, we note that there clearly is a limit to what can be achieve by both quarantine an hospital control proceures. In the best possible case, in which a u = 0 an q = 1, be = E0+S g 01 e ag α I0 +S h 01 e ag α I0 = E0+S01 e a α I0 an X = X0 + E0 + I0 + S01 e a α I0 := Xbest. Assuming the hospital control proceures are so strictly enforce that a u = 0, an we want to control the outbreak so that X X, a given number. Note that necessarily, X Xbest. Let Y = X X0 I0. Then, we can fin q [0, 1] so that Y = E0 + S g 01 e a α I0 e a α 1 qy + S h 01 e a α I0 e a α 1 qy. Thus, X X if an only if q q. Acknowlegments. We thank MITACS an all members of the MITACS project Transmission Dynamics an Spatial Sprea of Infectious Diseases: Moelling, Preiction an Control team for their support an iscussions. This work was partially supporte by NSERC H. Zhu an J. Wu, CRC J. Wu, Ontario Ministry of Health an Long Term Care S. Aral, NSF G. Webb, an R01GM63270 from NIH M. Blaser an G. Webb. REFERENCES [1] C. M. Booth, L. M. Matukas, G. A. Tomlinson, A. R. Rachlis, D. B. Rose, H. A. Dwosh, et al., Clinical features an short-term outcomes of 144 patients with SARS in the Greater Toronto area. JAMA [2] R. Schabas, Pruence: not panic. Can. Me. Assoc. J [3] S. Riley, C. Fraser, C. A. Donnelly, A. C. Ghani, L. J. Abu-Raa, A. J. Heley, et al., Transmission Dynamics of the Etiological Agent of SARS in Hong Kong: Impact of Public Health Interventions. Science

13 NOSOCOMIAL TRANSMISSION IN THE SARS OUTBREAK 13 [4] M. Lipstich, T. Cohen, B. Cooper, J. M. Robbins, S. Ma, J. Lyn, et al., Transmission ynamics an control of severe acute respiratory synrome. Science [5] C. Dye an N. Gay, Moeling the SARS epiemic. Science [6] Health Canaa. [Accesse May 1, 2003] Available from: [7] CDC. Cluster of Severe Acute Respiratory Synrome Cases Among Protecte Health-Care Workers, Toronto, Canaa. MMWR Morb Mortal Wkly Rep [8] N. Lee, D. Hui, A. Wu, P. Chan, P. Cameron, G. M. Joynt, et al., A Major Outbreak of Severe Acute Respiratory Synrome in Hong Kong. N Eng J Me [9] C. Castillo-Chavez, H. W. Hethcote, V. Anreasen, S. A. Levin, an W. N. Liu, Epiemiological moels with age structure, proportionate mixing, an cross-immunity. J Math Biol [10] S. Busenberg an K. Cooke, Vertically Transmitte Diseases. Springer-Verlag, New York, [11] F. Brauer an C. Castillo-Chavez, Mathematical Moels in Population Biology an Epiemics. Springer-Verlag, New York, [12] H. W. Hethcote, The mathematics of infectious iseases. SIAM Review [13] O. Diekmann an J. A. P. Heesterbeek, Mathematical Epiemiolgy of Infectious Diseases: Moel Builing, Analysis an Interpretation. Wiley, New York, [14] G. Chowell, P. W. Fenimore, M. A. Castillo-Garsow, an C. Castillo-Chavez, SARS outbreaks in Ontario, Hong Kong an Singapore: the role of iagnosis an isolation as a control mechanism. J Theoret Biol [15] H. A. Dwosh, H. H. Hong, D. Austgaren, S. Herman, an R. Schabas, Ientification an containment of an outbreak of SARS in a community hospital. Can Me Assoc J [16] WHO. Epiemic curves: Severe Acute Respiratory Synrome SARS [Accesse May 1, 2003] Available from: URL: Receive on Dec. 7, Revise on Jan. 6, aress: glenn.f.webb@vanerbilt.eu aress: Martin.Blaser@msnyuhealth.org aress: huaiping@mathstat.yorku.ca aress: Sten@Cehip.org aress: wujh@mathstat.yorku.ca