# Identifying index (source) patient location of SARS transmission in a hospital ward.

This article presents the application of a
probability-concept-based algorithm that can track the source origins of
viruses based on the number and location of infected individuals (cases)
observed in an enclosed indoor environment. This
probability-concept-based prediction uses the convective transfer
mechanism of virus-containing airborne aerosols to track inversely the
most probable locations of infection sources according to the
concentration value measured in certain location(s). A new approach is
proposed to convert the number of infection cases to the concentration
values of virus particles that can be used as the input for prediction,
making the concentration measurement unnecessary for source tracking in
a real outbreak of epidemic disease. The source-tracking method is
demonstrated and verified by using the SARS infection cases of medical
students in a hospital ward outbreak in Hong Kong in 2003. The model
successfully predicts the location of the virus origin, i.e., the
clinically identified SARS patient's bed in the outbreak. The
method shows a good potential for quickly identifying the index patient
location during an airborne disease outbreak, as well as confirming the
airborne transmission mechanism and dispersion of indoor virus- and
bacterium-containing aerosols.

Introduction

Cross-infection in hospital wards during an airborne-transmitted disease epidemic is a serious concern. When a hospital ward outbreak is suspected, identifying the location of the source patient(s) (index patient(s)) is generally an essential step in investigating the suspected disease outbreak. Identifying the source patient and his/her location is also essential for environmental investigations. Conventionally, the identification of the index patient is done using clinical data. This article proposes using an engineering calculation method for identifying source patient(s) for airborne disease outbreaks in a confined environment with statistical data of infections.

Through transportation of bacteria or viruses attached to airborne particles and/or droplet nuclei, a disease may be transmitted in the same room of the index patient or to other rooms in the same hospital through an airborne route. For example, during the 2003 SARS epidemics worldwide, a number of hospital outbreaks were reported. In at least one of the outbreaks, an airborne transmission hypothesis was proposed (Li et al. 2005). Hospital outbreaks have played an important role in the initial spread of the diseases in a number of locations, such as Hong Kong, Vancouver, and Beijing (WHO 2003). During a hospital ward outbreak of airborne diseases, if the in-patients were mostly confined to their own beds, the infection data may be used to understand the spatial spread of the disease.

A comprehensive literature review was carried out on various engineering contaminant tracking methods (kiu and Zhai 2007). The adjoint probability method (Neupauer and Wilson 1999, 2001) developed originally for groundwater pollutant source tracking was found to be appropriate for indoor contaminant source tracking. Stemming from the original concept of the adjoint probability, a set of theories and algorithms have been developed that can identify the locations of single contaminant sources in various indoor environments based on a limited number of contaminant sensor outputs (Liu and Zhai 2008, 2009; Zhai et al. 2012). The developed prediction algorithm for dynamically releasing sources (e.g., breathing SARS patient) has great potential to tracking infection origins in hospital disease outbreak using sensor reading in experiment as input concentration.

However, successfully locating the patient source during an outbreak is critical in controlling the spread, which requires providing the critical sensor (concentration) inputs for the simulation. Measurement is thus required during the outbreak to get the first-hand data on virus concentration. However, this is usually not feasible for two reasons. First, it is almost impossible to have a sensor that can detect any possible virus and record the concentration, and second, for some new types of disease, it would take a relatively long time for medical experts to figure out the characteristics of the virus and put it under monitor. To solve the problem, a new approach is developed here that can convert the first-hand statistic data of infection cases to the normalized quanta concentration values as the input concentration for the inverse simulation. The new application of the probability-based source-tracking algorithm using this quanta concentration shows that the index patient location during the disease outbreak can be identified quickly. The study further verifies the influence of error of the calculated quanta concentration to the source identification accuracy.

Methodology

Probability-based source-tracking method

The probability-based tracking method for continuously releasing sources (Zhai et al. 2012) is based on the tracking method for instant-releasing sources (Liu 2008). The first step of this source-tracking method requires an understanding of the flow field in the space, through either measurement or simulation (such as computational fluid dynamics [CFD] or multi-zone modeling). Previous study shows that the precise velocity field is desirable but not necessary for source tracking, as long as the general flow pattern is reasonably captured. Characteristics of contaminant sensors (e.g., number, location, accuracy, and reading) are critical for prompt and accurate prediction of source location. Sensor readings can also indicate the characteristics of the contaminants (e.g., instantaneously versus continuously releasing source), which affects the selection of proper prediction algorithms. In mathematics, a continuously releasing source can be temporarily discretized into many instantaneous sources. The contaminant mass accumulated at any location in a space is thus the result of dispersion of many such instantaneous sources. The algorithm first calculates the standard adjoint location probability (SALP) for each temporally discretized source based on each sensor location by solving the adjoint differential contaminant transport equation (Liu and Zhai 2008):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [psi]* is the SALP on the backward time [tau] ([tau] = t - T). [delta](x) is the impulse function, which equals 1 when x = 0 and 0 otherwise. [[??].sub.s] is an individual sensor location, and [V.sub.j] is velocity at the [x.sub.j] direction. The SALP describes the probability of contaminant detected at location [[??].sub.s] and time [t.sub.i] originated from source at location [??], which started the release at t = T.

Mathematically, the mass discharging rate of the source to the space follows this equation:

dM(t)/dt = [M.sub.0] x h (t), (2)

where [M.sub.0] is the initially released contaminant mass from the source, and h(t) is the source release strength profile (compared with the initial release mass). If h(t) is constant over a time period, the source has constant releasing strength, a special case of dynamic source.

For the contaminant mass released at any time t, the time it spreads in the space is T-t, so the contributed concentration from the dynamic source accumulated to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [f.sub.x] ([[??].sub.o]; T - t, [[??].sub.s]) is the forward location probability density function for instantaneous source strength at time t. It can be determined by the SALP value at location [[??].sub.s] as

[f.sub.x]([[??].sub.o]; T - t, [[??].sub.s]) = [[psi].sup.*.sub.x] ([[??].sub.0]; T - t, [[??].sub.x]). (4)

The SALP for a dynamic source (SALP-D) for each sensor location [[??].sub.s,i] can then be calculated through

SALP-D = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The SALP-D quantifies the probability distribution of a dynamic source in the space. Calculating the SALP and SALP-D does not require sensor concentration information. Adding the concentration readings and/or using multiple sensor outputs into the prediction can improve the accuracy and effectiveness of the algorithm in identifying the contaminant source. Such improved adjoint location probabilities, as defined in Equation 5, are named as conditioned adjoint location probabilities for dynamic source (CALP-D):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where P([C.sub.i]) is the probability for the measured concentration [C.sub.i] conditioned on source mass [M.sub.0] and source location x, which follows a normal distribution, as suggested by Neupauer (2000):

P([C.sub.i]|[M.sub.0], X; [[tau].sub.0], [X.sub.i], [[tau].sub.i]) ~ N([M.sub.0] x [f.sub.x] (X; [[tau].sub.0], [X.sub.i], [[tau].sub.i]), [[sigma].sup.2].sub.[epsilon]), (7)

where [[sigma].sup.2.sub.[epsilon]] is the variance for the N measurements. The general procedure of the algorithm is summarized in Figure 1.

[FIGURE 1 OMITTED]

Quanta concentration calculation

The source-tracking algorithm in this article requires concentration values at single or multiple sensing positions for accurate prediction of the location of the source causing the concentration. These values, however, are not always available, particularly for disease outbreak investigations. The statistical data available in the epidemic study can only provide binary numbers for either infected or non-infected or the ratio of infection. Conversion from statistical or binary data to useful relative concentration values can be obtained (Qian et al. 2009) using the Wells-Riley equation (Riley and Nardell 1989). The infection data can be used to estimate the relative quanta concentration by the following equation:

N = ln(1 - P),/-[p.sub.t] (8)

where p is the pulmonary ventilation rate of each susceptible group per minute ([m.sup.3]/min), N is the quanta concentration (quanta/[m.sup.3]), t is the duration of exposure (min), and P is the probability of infection (%). The probability of infection in a particular environment can thus be estimated based on the infection data.

Verification

This study used the data from the Ward 8A outbreak in the Prince of Wales Hospital in Hong Kong during the 2003 SARS epidemic to evaluate and demonstrate the performance of the developed algorithm for tracking indoor airborne virus. The study investigated the epidemiological features of the infected medical students who examined patients at the bedside on a relative fixed position during the infection period (Qian et al. 2009). Some of these students were infected by SARS after presenting in the ward while some were not. The probability of infection was calculated by dividing the number of infected students by the total student number presented at the same time. To investigate the transmitting principle of airborne disease in the hospital ward and provide suggestions to eliminate cross-infection during the SARS breakout, a field test was conducted by Li et al. (2005) using aerosols as contaminants. Both the post-measurement and the real infection case were simulated by this article for verification purposes.

The Ward 8 outbreak

Figure 2 shows the floor plan of the hospital's Ward 8A being studied. There were five beds along each side of the wall in each cubicle during the SARS outbreak. This number was reduced to four during the post-measurement by removing the beds numbered by "x," and other conditions remained the same. The supply and exhaust airflow rates in the ward were measured. The dark bed 11 indicates the actual location of the index patient. The mean supply air temperature was measured to be 57.7[degrees]F (14.3[degrees]C). A total heat gain of 39,744 Btu/h (11.64 kW) was identified in the ward, including 7621 Btu/h (2.232 kW) from lighting, 10,120 Btu/h (2.964 kW) from 39 patients (259.5 Btu/h [76 W] each), and 22,002 Btu/h (6.444 kW) from others. All of these parameters were assumed to be constant and the same during the breakout of SARS and during the post-experiment.

The hospital ward had four main cubicles, separated by a corridor, nurses' station, store, and storage/cleaning room. The ward had an overall dimension of 78.7 ft (24 m) length by 59.1 ft (18 m) width by 8.9 ft (2.7 m) height. The four cubicles were semi-enclosed, and each was 24.6 ft x 19.7 ft x 8.9 ft (7.5 m x 6m x 2.7 m).

There were normally eight beds in each cubicle, but at the time of the outbreak, ten beds were located in each cubicle, with extra beds labeled "x". The ward was centrally air conditioned. The supply air was delivered into the cubicles/nurses' station via a number of four-way rectangular air supply diffusers (1.97 ft x 1.97 ft [0.6 m x 0.6 m]) mounted in the suspended ceiling at the center of the cubicle and over the nurses' station. Return grilles (0.98 ft x 1.97 ft [0.3 m x 0.6 m]) were located in the suspended ceiling in the corridor outside each cubicle and the nurses' station. The airflow rate of each supply and return opening is also shown in Figure 2. The total supply airflow rate is greater than that exhausted from the return grille. Excess air escaped through two exhaust fans inside the patients' bathrooms and two fans in the store/cleaning room, as well as through the leakage of the ward entrance, which generated positive pressure in the ward.

Source tracking for the experiment case

The post-SARS experiment placed an aerosol generator on a bed next to the index patient's bed on the central line of the bed (3.6 ft [1.1 m] above the floor and 3.9 ft [1.2 m] away from the wall perpendicular to the bed) to mimic the virus exhalation of the original SARS patient. After a sufficiently long period of release time of aerosols, the concentration field was supposed to be steady, and the normalized particle concentration was measured at all bed locations, as indicated in Figure 3. The contour in Figure 3 is the predicted concentrations using CfD with the given source location and release intensity (Li et al. 2005). The simulation matches the measurement well, indicating that CFD is capable of simulating the dispersion of aerosols in these ventilated spaces if the source is known. However, in most real cases, contaminant sources are unknown prior. This study used the measured aerosol concentrations at several locations to predict the source origin.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

CFD simulation was first performed to capture the major flow field in the ward using the measured ventilation rate and room conditions as shown in Figure 2. Figure 4 shows the velocity vectors at the middle plane of the source cubicle. The prediction shows similar flow patterns, as in the study by Li et al. (2005). With the predicted airflow in the ward, the developed source-tracking program was applied to find the source origin using the three normalized concentrations measured at beds 12, 13, and 24. Figure 5 reveals the predicted source location at bed height z = 3.6 ft (1.1 m), which is very close to where the aerosol source was in the experiment. It should be noticed that the absolute CALP value indicates the probability distribution of potential source location in the space, where the largest value (0.24% for this case) implies the most possible source location. This case verifies that the algorithm is capable of predicting the contaminant source location using the measured concentration data.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Source tracking using the infection case data

The source-tracking algorithm was then applied to predict the SARS source during the outbreak. Equation 8 was used to obtain the relative quanta concentration that was needed for the source identification, where the infection probability P was calculated by dividing the number of infected people by the total number of people presented at the same time in the same space. The medical students presented in the investigation were divided into three groups, as shown in Figure 2.

* Group 1--the three students close to the wall. All of them were infected, and thus P = 100%. The location of the Group 1 students in the simulation was assigned to bed 12 (head location).

* Group 2--the three students including the one fight next to the index bed and the two immediately nearby. Among these three students, two were infected, and thus P = 2/3 = 67%. The location of the Group 2 students in the simulation was the middle point between beds 14 and 15 (head location).

* Group 3--the remaining four students with P = 50% who were located between beds 9, 16, and 16x.

The pulmonary ventilation rate used in Equation 8 was assumed to be 0.2118 cfm (6 1/min = 0.006 [m.sup.3]/min). The duration of the contact time was set to t = 45 min for the students. Table 1 presents the calculated quanta concentration value for each group (P = 99% was used for Group 1, rather than 100%, in order to apply the log function in Equation 8). The error values indicate the potential range of the concentration variations if a certain percentage of uncertainty in the statistical data is taken into account based on approximation.

The study simulated again the flow field using CFD for the SARS breakout case, which had similar operating conditions but with extra beds in each cubicle. The predicted flow field is almost identical as the experiment case so it is not presented again. The study then used this flow field to predict the source location. The calculated quanta concentrations in Table 1 were specified in the model at the height of 5.6 ft (1.7 m) (the height of a typical adult) at the bedside of the location of each group to simulate the virus concentration sensed by each student group. Figure 6 shows the source location prediction when assuming the statistical infection data has a 5% error. The 3D contour shows the iso-surfaces of CALP values, and the 2D contour shows the results at the plane of z = 3.6 ft (1.1 m), which is the height of beds. The results reveal that the simulation captures the correct index patient. At the height of beds where all the patients laid, the maximum source location probability was found to be at the bedside of the actual index patient identified in the clinic.

[FIGURE 6 OMITTED]

Figures 7 and 8 present the same type of results using the quanta data with different error percentages. The 3D plumes show approximately the same patterns as in Figure 6, and the 2D contours indicate a reasonably accurate source location, on bed 11 or 12 or between the two beds. The main influence of using larger uncertainties in the simulation is the reduction of the maximum value of calculated CALPs. The locations of the maximum CALPs, however, are within the same vicinity of the real source. It can thus be concluded that even with significant uncertainty in Equation 8, the algorithm is still be able to find the source with the calculated quanta concentration.

[FIGURE 7 OMITTED]

The study further tested the robustness and sensitivity of the algorithm by inputting wrong infection data, for instance, switching the infection probabilities for different infection locations. Figure 9 shows the source location prediction when the concentration data for Groups 1 and 2 were exchanged, which provides a wrong source location.

[FIGURE 8 OMITTED]

In terms of computing cost, the current algorithm requires only basic arithmetic calculation for each computational grid, and no iteration is involved in the algorithm, so the computational time for such source tracking is limited. For this hospital case with a grid resolution of 85 x 82 x 27, it only takes a minute to get the prediction result using an ordinary PC. The low computing cost gives great potential of real-time prediction.

[FIGURE 9 OMITTED]

The current algorithm applies only to steady-state airflow in an enclosed environment. For situations with unsteady airflow rate, the applicability of the algorithm is pending for further study. The current indoor airflow pattern result is from CFD simulation using those boundary conditions, which is not exactly the same as in reality. However, the source location prediction is quite close to the actual position of the index patient. The temperature difference caused two-way airflow in openings would influence the precision of airflow pattern by CFD, but the location prediction can be still reliable according to error analysis; details of such a study can be found in Liu (2008).

Conclusions

This study verifies and demonstrates the capability of a newly developed dynamic source-location identification algorithm for tracking indoor airborne viruses and bacteria using measured concentration as input as well as applying it to identifying the index patient location during the real outbreak of SARS in 2003 using statistic infection data. Both infection and experiment data from a hospital ward obtained during and after the SARS period in Hong Kong was used for the verification. With multiple measured aerosol concentration values, the algorithm is able to find the accurate source location. Furthermore, the algorithm is applied to the real SARS breakout situation. With the relative quanta concentration calculated from the number of infected students as input, it is able to identify the virus source location. The results show that the uncertainty in the estimation of the virus concentration in the space does not affect the identification of the source location. The successful application of the method in using statistical infection data to identify sources of infection diseases provides great potential for quickly pinpointing infection sources in an epidemiology investigation during an outbreak of an airborne infectious disease with first-hand data. It also confirms the airborne transmission mechanism and path of indoor viruses and bacteria.

Acknowledgment

Part of the work described in this article was supported by a William Mong Research Fellowship Award to the second author during his stay at the University of Hong Kong.

References

Li, Y., X. Huang, I.T.S. Yu, T.W. Wong, and H. Qian. 2005. Role of air distribution in SARS transmission during the largest nosocomial outbreak in Hong Kong. Indoor Air 15(2):83-95.

Liu, X. 2008. Identification of indoor airborne contaminant sources with probability-based inverse modeling methods. PhD dissertation. University of Colorado at Boulder, Boulder, CO.

Liu, X., and Z. Zhai. 2007. Inverse modeling methods for indoor airborne pollutant tracking: Literature review and fundamentals. Indoor Air 17(6):419-38.

Liu, X., and Z. Zhai. 2008. Location identification for indoor instantaneous point contaminant source by probability-based inverse computational fluid dynamics modeling. Indoor Air 18(1):2-11.

Liu, X., and J.Z. Zhai. 2009. Prompt tracking of indoor airborne contaminant source location with probability-based inverse multi-zone modeling. Building and Environment 44(6): 1135-43.

Neupauer, R.M. 2000. Receptor based modeling of groundwater contamination. PhD dissertation, New Mexico Institute of Mining and Technology, Socorro, NM.

Neupauer, R.M., and J.L. Wilson. 1999. Adjoint method for obtaining backward-in-time location and travel time probabilities of a conservative groundwater contaminant. Water Resource Research 35(11):3389-98.

Neupauer, R.M., and J.L. Wilson. 2001. Adjoint-derived location and travel time probabilities for a multidimensional groundwater system. Water Resource Research 37(6): 1657-68.

Qian, H., Y. Li, P.V. Nielsen., and X. Huang. 2009. Spatial distribution of infection risk of SARS transmission in a hospital ward. Building and Environment 44(8): 1651-8.

Riley, R.L., and E.A. Nardell. 1989. Clearing the air. The theory and application of ultraviolet air disinfection. The American Review of Respiratory Disease 139(5):1286-94.

World Health Organization (WHO). 2003. Summary of probable SARS cases with onset if illness from 1 November 2002 to 31 July 2003. http://www.who.int/csr/sars/country/table2003_09_23/en/.

Zhai, Z., X. Liu, H. Wang, Y. Li, and J. Liu. 2012. Experimental verification of tracking algorithm for dynamically-releasing single indoor contaminant. Building Simulation 5(1):5-14.

Haidong Wang, (1) Zhiqiang (John) Zhai, (1), * Yuguo Li, (2) and Xiang Liu (3)

(1) Department of Civil, Environmental and Architectural Engineering, University of Colorado at Boulder, 428UCB, Boulder, CO 80309-0428, USA

(2) Department of Mechanical Engineering, The University of Hong Kong, Hong Kong SAR, China

(3) University of Colorado at Boulder, Boulder, CO, USA

* Corresponding author e-mail: john.zhai@colorado.edu

Received February 21, 2011; accepted May 23, 2011

Haidong Wang, Student Member ASHRAE, is PhD student. Zhiqiang (John) Zhai, DrEng, PhD, Member ASHRAE, is Associate Professor. Yuguo Li, PhD, Fellow ASHRAE, is Professor. Xiang Liu, PhD, Student Member ASHRAE, is Energy Engineer at Nexant Inc.

DOI: 10.1080/10789669.2011.594699

Introduction

Cross-infection in hospital wards during an airborne-transmitted disease epidemic is a serious concern. When a hospital ward outbreak is suspected, identifying the location of the source patient(s) (index patient(s)) is generally an essential step in investigating the suspected disease outbreak. Identifying the source patient and his/her location is also essential for environmental investigations. Conventionally, the identification of the index patient is done using clinical data. This article proposes using an engineering calculation method for identifying source patient(s) for airborne disease outbreaks in a confined environment with statistical data of infections.

Through transportation of bacteria or viruses attached to airborne particles and/or droplet nuclei, a disease may be transmitted in the same room of the index patient or to other rooms in the same hospital through an airborne route. For example, during the 2003 SARS epidemics worldwide, a number of hospital outbreaks were reported. In at least one of the outbreaks, an airborne transmission hypothesis was proposed (Li et al. 2005). Hospital outbreaks have played an important role in the initial spread of the diseases in a number of locations, such as Hong Kong, Vancouver, and Beijing (WHO 2003). During a hospital ward outbreak of airborne diseases, if the in-patients were mostly confined to their own beds, the infection data may be used to understand the spatial spread of the disease.

A comprehensive literature review was carried out on various engineering contaminant tracking methods (kiu and Zhai 2007). The adjoint probability method (Neupauer and Wilson 1999, 2001) developed originally for groundwater pollutant source tracking was found to be appropriate for indoor contaminant source tracking. Stemming from the original concept of the adjoint probability, a set of theories and algorithms have been developed that can identify the locations of single contaminant sources in various indoor environments based on a limited number of contaminant sensor outputs (Liu and Zhai 2008, 2009; Zhai et al. 2012). The developed prediction algorithm for dynamically releasing sources (e.g., breathing SARS patient) has great potential to tracking infection origins in hospital disease outbreak using sensor reading in experiment as input concentration.

However, successfully locating the patient source during an outbreak is critical in controlling the spread, which requires providing the critical sensor (concentration) inputs for the simulation. Measurement is thus required during the outbreak to get the first-hand data on virus concentration. However, this is usually not feasible for two reasons. First, it is almost impossible to have a sensor that can detect any possible virus and record the concentration, and second, for some new types of disease, it would take a relatively long time for medical experts to figure out the characteristics of the virus and put it under monitor. To solve the problem, a new approach is developed here that can convert the first-hand statistic data of infection cases to the normalized quanta concentration values as the input concentration for the inverse simulation. The new application of the probability-based source-tracking algorithm using this quanta concentration shows that the index patient location during the disease outbreak can be identified quickly. The study further verifies the influence of error of the calculated quanta concentration to the source identification accuracy.

Methodology

Probability-based source-tracking method

The probability-based tracking method for continuously releasing sources (Zhai et al. 2012) is based on the tracking method for instant-releasing sources (Liu 2008). The first step of this source-tracking method requires an understanding of the flow field in the space, through either measurement or simulation (such as computational fluid dynamics [CFD] or multi-zone modeling). Previous study shows that the precise velocity field is desirable but not necessary for source tracking, as long as the general flow pattern is reasonably captured. Characteristics of contaminant sensors (e.g., number, location, accuracy, and reading) are critical for prompt and accurate prediction of source location. Sensor readings can also indicate the characteristics of the contaminants (e.g., instantaneously versus continuously releasing source), which affects the selection of proper prediction algorithms. In mathematics, a continuously releasing source can be temporarily discretized into many instantaneous sources. The contaminant mass accumulated at any location in a space is thus the result of dispersion of many such instantaneous sources. The algorithm first calculates the standard adjoint location probability (SALP) for each temporally discretized source based on each sensor location by solving the adjoint differential contaminant transport equation (Liu and Zhai 2008):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [psi]* is the SALP on the backward time [tau] ([tau] = t - T). [delta](x) is the impulse function, which equals 1 when x = 0 and 0 otherwise. [[??].sub.s] is an individual sensor location, and [V.sub.j] is velocity at the [x.sub.j] direction. The SALP describes the probability of contaminant detected at location [[??].sub.s] and time [t.sub.i] originated from source at location [??], which started the release at t = T.

Mathematically, the mass discharging rate of the source to the space follows this equation:

dM(t)/dt = [M.sub.0] x h (t), (2)

where [M.sub.0] is the initially released contaminant mass from the source, and h(t) is the source release strength profile (compared with the initial release mass). If h(t) is constant over a time period, the source has constant releasing strength, a special case of dynamic source.

For the contaminant mass released at any time t, the time it spreads in the space is T-t, so the contributed concentration from the dynamic source accumulated to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [f.sub.x] ([[??].sub.o]; T - t, [[??].sub.s]) is the forward location probability density function for instantaneous source strength at time t. It can be determined by the SALP value at location [[??].sub.s] as

[f.sub.x]([[??].sub.o]; T - t, [[??].sub.s]) = [[psi].sup.*.sub.x] ([[??].sub.0]; T - t, [[??].sub.x]). (4)

The SALP for a dynamic source (SALP-D) for each sensor location [[??].sub.s,i] can then be calculated through

SALP-D = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The SALP-D quantifies the probability distribution of a dynamic source in the space. Calculating the SALP and SALP-D does not require sensor concentration information. Adding the concentration readings and/or using multiple sensor outputs into the prediction can improve the accuracy and effectiveness of the algorithm in identifying the contaminant source. Such improved adjoint location probabilities, as defined in Equation 5, are named as conditioned adjoint location probabilities for dynamic source (CALP-D):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where P([C.sub.i]) is the probability for the measured concentration [C.sub.i] conditioned on source mass [M.sub.0] and source location x, which follows a normal distribution, as suggested by Neupauer (2000):

P([C.sub.i]|[M.sub.0], X; [[tau].sub.0], [X.sub.i], [[tau].sub.i]) ~ N([M.sub.0] x [f.sub.x] (X; [[tau].sub.0], [X.sub.i], [[tau].sub.i]), [[sigma].sup.2].sub.[epsilon]), (7)

where [[sigma].sup.2.sub.[epsilon]] is the variance for the N measurements. The general procedure of the algorithm is summarized in Figure 1.

[FIGURE 1 OMITTED]

Quanta concentration calculation

The source-tracking algorithm in this article requires concentration values at single or multiple sensing positions for accurate prediction of the location of the source causing the concentration. These values, however, are not always available, particularly for disease outbreak investigations. The statistical data available in the epidemic study can only provide binary numbers for either infected or non-infected or the ratio of infection. Conversion from statistical or binary data to useful relative concentration values can be obtained (Qian et al. 2009) using the Wells-Riley equation (Riley and Nardell 1989). The infection data can be used to estimate the relative quanta concentration by the following equation:

N = ln(1 - P),/-[p.sub.t] (8)

where p is the pulmonary ventilation rate of each susceptible group per minute ([m.sup.3]/min), N is the quanta concentration (quanta/[m.sup.3]), t is the duration of exposure (min), and P is the probability of infection (%). The probability of infection in a particular environment can thus be estimated based on the infection data.

Verification

This study used the data from the Ward 8A outbreak in the Prince of Wales Hospital in Hong Kong during the 2003 SARS epidemic to evaluate and demonstrate the performance of the developed algorithm for tracking indoor airborne virus. The study investigated the epidemiological features of the infected medical students who examined patients at the bedside on a relative fixed position during the infection period (Qian et al. 2009). Some of these students were infected by SARS after presenting in the ward while some were not. The probability of infection was calculated by dividing the number of infected students by the total student number presented at the same time. To investigate the transmitting principle of airborne disease in the hospital ward and provide suggestions to eliminate cross-infection during the SARS breakout, a field test was conducted by Li et al. (2005) using aerosols as contaminants. Both the post-measurement and the real infection case were simulated by this article for verification purposes.

The Ward 8 outbreak

Figure 2 shows the floor plan of the hospital's Ward 8A being studied. There were five beds along each side of the wall in each cubicle during the SARS outbreak. This number was reduced to four during the post-measurement by removing the beds numbered by "x," and other conditions remained the same. The supply and exhaust airflow rates in the ward were measured. The dark bed 11 indicates the actual location of the index patient. The mean supply air temperature was measured to be 57.7[degrees]F (14.3[degrees]C). A total heat gain of 39,744 Btu/h (11.64 kW) was identified in the ward, including 7621 Btu/h (2.232 kW) from lighting, 10,120 Btu/h (2.964 kW) from 39 patients (259.5 Btu/h [76 W] each), and 22,002 Btu/h (6.444 kW) from others. All of these parameters were assumed to be constant and the same during the breakout of SARS and during the post-experiment.

The hospital ward had four main cubicles, separated by a corridor, nurses' station, store, and storage/cleaning room. The ward had an overall dimension of 78.7 ft (24 m) length by 59.1 ft (18 m) width by 8.9 ft (2.7 m) height. The four cubicles were semi-enclosed, and each was 24.6 ft x 19.7 ft x 8.9 ft (7.5 m x 6m x 2.7 m).

There were normally eight beds in each cubicle, but at the time of the outbreak, ten beds were located in each cubicle, with extra beds labeled "x". The ward was centrally air conditioned. The supply air was delivered into the cubicles/nurses' station via a number of four-way rectangular air supply diffusers (1.97 ft x 1.97 ft [0.6 m x 0.6 m]) mounted in the suspended ceiling at the center of the cubicle and over the nurses' station. Return grilles (0.98 ft x 1.97 ft [0.3 m x 0.6 m]) were located in the suspended ceiling in the corridor outside each cubicle and the nurses' station. The airflow rate of each supply and return opening is also shown in Figure 2. The total supply airflow rate is greater than that exhausted from the return grille. Excess air escaped through two exhaust fans inside the patients' bathrooms and two fans in the store/cleaning room, as well as through the leakage of the ward entrance, which generated positive pressure in the ward.

Source tracking for the experiment case

The post-SARS experiment placed an aerosol generator on a bed next to the index patient's bed on the central line of the bed (3.6 ft [1.1 m] above the floor and 3.9 ft [1.2 m] away from the wall perpendicular to the bed) to mimic the virus exhalation of the original SARS patient. After a sufficiently long period of release time of aerosols, the concentration field was supposed to be steady, and the normalized particle concentration was measured at all bed locations, as indicated in Figure 3. The contour in Figure 3 is the predicted concentrations using CfD with the given source location and release intensity (Li et al. 2005). The simulation matches the measurement well, indicating that CFD is capable of simulating the dispersion of aerosols in these ventilated spaces if the source is known. However, in most real cases, contaminant sources are unknown prior. This study used the measured aerosol concentrations at several locations to predict the source origin.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

CFD simulation was first performed to capture the major flow field in the ward using the measured ventilation rate and room conditions as shown in Figure 2. Figure 4 shows the velocity vectors at the middle plane of the source cubicle. The prediction shows similar flow patterns, as in the study by Li et al. (2005). With the predicted airflow in the ward, the developed source-tracking program was applied to find the source origin using the three normalized concentrations measured at beds 12, 13, and 24. Figure 5 reveals the predicted source location at bed height z = 3.6 ft (1.1 m), which is very close to where the aerosol source was in the experiment. It should be noticed that the absolute CALP value indicates the probability distribution of potential source location in the space, where the largest value (0.24% for this case) implies the most possible source location. This case verifies that the algorithm is capable of predicting the contaminant source location using the measured concentration data.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Source tracking using the infection case data

The source-tracking algorithm was then applied to predict the SARS source during the outbreak. Equation 8 was used to obtain the relative quanta concentration that was needed for the source identification, where the infection probability P was calculated by dividing the number of infected people by the total number of people presented at the same time in the same space. The medical students presented in the investigation were divided into three groups, as shown in Figure 2.

* Group 1--the three students close to the wall. All of them were infected, and thus P = 100%. The location of the Group 1 students in the simulation was assigned to bed 12 (head location).

* Group 2--the three students including the one fight next to the index bed and the two immediately nearby. Among these three students, two were infected, and thus P = 2/3 = 67%. The location of the Group 2 students in the simulation was the middle point between beds 14 and 15 (head location).

* Group 3--the remaining four students with P = 50% who were located between beds 9, 16, and 16x.

The pulmonary ventilation rate used in Equation 8 was assumed to be 0.2118 cfm (6 1/min = 0.006 [m.sup.3]/min). The duration of the contact time was set to t = 45 min for the students. Table 1 presents the calculated quanta concentration value for each group (P = 99% was used for Group 1, rather than 100%, in order to apply the log function in Equation 8). The error values indicate the potential range of the concentration variations if a certain percentage of uncertainty in the statistical data is taken into account based on approximation.

The study simulated again the flow field using CFD for the SARS breakout case, which had similar operating conditions but with extra beds in each cubicle. The predicted flow field is almost identical as the experiment case so it is not presented again. The study then used this flow field to predict the source location. The calculated quanta concentrations in Table 1 were specified in the model at the height of 5.6 ft (1.7 m) (the height of a typical adult) at the bedside of the location of each group to simulate the virus concentration sensed by each student group. Figure 6 shows the source location prediction when assuming the statistical infection data has a 5% error. The 3D contour shows the iso-surfaces of CALP values, and the 2D contour shows the results at the plane of z = 3.6 ft (1.1 m), which is the height of beds. The results reveal that the simulation captures the correct index patient. At the height of beds where all the patients laid, the maximum source location probability was found to be at the bedside of the actual index patient identified in the clinic.

[FIGURE 6 OMITTED]

Figures 7 and 8 present the same type of results using the quanta data with different error percentages. The 3D plumes show approximately the same patterns as in Figure 6, and the 2D contours indicate a reasonably accurate source location, on bed 11 or 12 or between the two beds. The main influence of using larger uncertainties in the simulation is the reduction of the maximum value of calculated CALPs. The locations of the maximum CALPs, however, are within the same vicinity of the real source. It can thus be concluded that even with significant uncertainty in Equation 8, the algorithm is still be able to find the source with the calculated quanta concentration.

[FIGURE 7 OMITTED]

The study further tested the robustness and sensitivity of the algorithm by inputting wrong infection data, for instance, switching the infection probabilities for different infection locations. Figure 9 shows the source location prediction when the concentration data for Groups 1 and 2 were exchanged, which provides a wrong source location.

[FIGURE 8 OMITTED]

In terms of computing cost, the current algorithm requires only basic arithmetic calculation for each computational grid, and no iteration is involved in the algorithm, so the computational time for such source tracking is limited. For this hospital case with a grid resolution of 85 x 82 x 27, it only takes a minute to get the prediction result using an ordinary PC. The low computing cost gives great potential of real-time prediction.

[FIGURE 9 OMITTED]

The current algorithm applies only to steady-state airflow in an enclosed environment. For situations with unsteady airflow rate, the applicability of the algorithm is pending for further study. The current indoor airflow pattern result is from CFD simulation using those boundary conditions, which is not exactly the same as in reality. However, the source location prediction is quite close to the actual position of the index patient. The temperature difference caused two-way airflow in openings would influence the precision of airflow pattern by CFD, but the location prediction can be still reliable according to error analysis; details of such a study can be found in Liu (2008).

Conclusions

This study verifies and demonstrates the capability of a newly developed dynamic source-location identification algorithm for tracking indoor airborne viruses and bacteria using measured concentration as input as well as applying it to identifying the index patient location during the real outbreak of SARS in 2003 using statistic infection data. Both infection and experiment data from a hospital ward obtained during and after the SARS period in Hong Kong was used for the verification. With multiple measured aerosol concentration values, the algorithm is able to find the accurate source location. Furthermore, the algorithm is applied to the real SARS breakout situation. With the relative quanta concentration calculated from the number of infected students as input, it is able to identify the virus source location. The results show that the uncertainty in the estimation of the virus concentration in the space does not affect the identification of the source location. The successful application of the method in using statistical infection data to identify sources of infection diseases provides great potential for quickly pinpointing infection sources in an epidemiology investigation during an outbreak of an airborne infectious disease with first-hand data. It also confirms the airborne transmission mechanism and path of indoor viruses and bacteria.

Acknowledgment

Part of the work described in this article was supported by a William Mong Research Fellowship Award to the second author during his stay at the University of Hong Kong.

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Haidong Wang, (1) Zhiqiang (John) Zhai, (1), * Yuguo Li, (2) and Xiang Liu (3)

(1) Department of Civil, Environmental and Architectural Engineering, University of Colorado at Boulder, 428UCB, Boulder, CO 80309-0428, USA

(2) Department of Mechanical Engineering, The University of Hong Kong, Hong Kong SAR, China

(3) University of Colorado at Boulder, Boulder, CO, USA

* Corresponding author e-mail: john.zhai@colorado.edu

Received February 21, 2011; accepted May 23, 2011

Haidong Wang, Student Member ASHRAE, is PhD student. Zhiqiang (John) Zhai, DrEng, PhD, Member ASHRAE, is Associate Professor. Yuguo Li, PhD, Fellow ASHRAE, is Professor. Xiang Liu, PhD, Student Member ASHRAE, is Energy Engineer at Nexant Inc.

DOI: 10.1080/10789669.2011.594699

Table 1. Calculated quanta concentration value with the infection data by using the quanta method. 5% 10% 30% uncertainty uncertainty uncertainty Group P N ([+ or -]N) ([+ or -]N) ([+ or -]N) 1 0.99 17.06 5.96 8.43 12.60 2 0.67 4.11 0.60 1.30 3.13 3 0.50 2.57 0.39 0.82 1.89