5.1 Comparing Simulation Results with Actual Cases
After initializing our model and setting up system and epi-demic disease parameters (Table 3) according to informa-tion distributed by WHO and the U.S. Centers for Disease Control and Prevention (CDC) [7, 10, 41-48], we simulated the transmission dynamics of SARS in different areas and compared the effectiveness of various public health poli-cies and disease prevention strategies. We used the simula-tion definisimula-tions and parameters identified in secsimula-tion 4 and assumed that one time step = 1 day in the real world.
Since SARS originated in China’s Guangdong province, we viewed the SARS viruses in all other coun-tries as being imported and used the number of imported cases announced by local health authorities to determine transmission source information—for example, number of infectious people entering a country, the time step during which they entered, and whether they entered as incubated or infected individuals (Tables 6-9). We incorporated pub-lic health popub-licies at certain time steps according to actual announcements made by local health authorities and ad-justed our simulation environment, epidemic, and public health policy parameters according to data from the CDC [42, 44, 45, 47] and Sebastian and Hoffmann [10].
5.1.1 Statistical Analyses for Epidemic Simulation We used five statistical tests to examine the reliability and validity of time-series data generated by the simulation system (Table 10): a chi-square test for homogeneity of proportions, a correlation coefficient (CC, equation (6)), coefficient of efficiency (CE, equation (7)), mean square error (MSE, equation (8)), and mean absolute error (MAE, equation (9)). {Xt| t = 1 . . . n ∧ t ∈ ℵ} represents time-series data for the number of individuals who were actually infected each day. {Yt|t = 1 . . . n∧t ∈ ℵ} represents time-series data for the numbers of infected individuals each day generated by our simulation system. In both data sets, t represents the time step (ranging from 1 to a maximum value of n), Xt represents the number of actual infected individuals at time step t , Yt represents the number of in-fected individuals generated by the simulation system at time step t, X represents the mean number of actual in-fected individuals, and Y represents the mean number of infected individuals in the simulation.
Table 6. Input data for simulating SARS epidemic curves in Taiwan, Singapore, and Toronto
Data Default
Category Attribute Type Description Value
Imported Cases
Time Point Date Date when imported case occurred.
Amount Integer Number of patients.
Phase Symbol Imported during incubation or illness period. Infected Super-spreader Boolean Determine whether the imported patient is a
super-spreader.
False Public Health Policy Related
Attributes
See Table 4
Run Day Integer Number of execution days.
Table 7. Singapore simulation input data
Time Public Health Special Description
Step Action Persons State Policy on the Simulator
2003/3/1 Trigger 1 Infectious Super-spreader
2 Trigger 2 Infectious
11 Set Reduced public contact Efficacy = 0.9, Participation = 0.5
15 Trigger 1 Incubation Mask-wearing policy Efficacy = 0.9, Participation = 0.9 for health care workers
22 Trigger 2 Incubation
23 Set
Home quarantine 10 days, Participation = 0.9 Controlling hospital access Efficacy = 0.9, Participation = 0.9 Mask-wearing policy for Efficacy = 0.9, Participation = 0.5 general public
25 Trigger 2 Infectious
52 Set Taking body temperature Efficacy = 0.9, Participation = 0.5
Table 8. Taipei simulation input data
Time Public Health Special Description
Step Action Persons State Policy on the Simulator
2003/3/20 Trigger 1 Infectious
2 Trigger 4 Incubation
9 Trigger 1 Incubation
11 Trigger 2 Infectious
12 Trigger 2 Infectious Home quarantine 10 days, Participation = 0.9
14 Trigger 1 Infectious
27 Trigger 1 Infectious Mask-wearing policy for health Efficacy = 0.9, Participation = 0.9 care workers
47 Set Controlling hospital access Efficacy = 0.9, Participation = 0.9
53 Set Home quarantine 14 days, Participation = 0.9
Mask-wearing policy for general Efficacy = 0.9, Participation = 0.5 public
74 Set Home quarantine 10 days, Participation = 0.9
88 Set Taking body temperature Efficacy = 0.9, Participation = 0.5
CC=
With the exception of the chi-square test, none of the sta-tistical tests requires a table lookup to evaluate simulation
Table 9. Toronto simulation input data
Time Public Health Special Description
Step Action Persons State Policy on the Simulator
2003/2/23 Trigger 1 Infectious
6 Trigger 1 Infectious
19 Trigger 1 Infectious Mask-wearing policy for health
care workers
Efficacy = 0.9, Participation = 0.9 Reduced public contact Efficacy = 0.9, Participation = 0.5
30 Trigger 1 Infectious
37 Set Controlling hospital access Efficacy = 0.9, Participation = 0.9
Home quarantine 10 days, Participation = 0.9
38 Trigger 1 Infectious
68 Close All public health policies
previously opened
91 Set Mask-wearing policy for health Efficacy = 0.9, Participation = 0.9
care workers
112 Set All public health policies
previously closed
Table 10. Reliability and validity tests for epidemic simulation using CASMIM
Reliability Test Validity Test
Chi-Square Test for Homogeneity of Proportions Simulated Degree of
Area Freedom χχχ222 χχχ20.05, degree of freedom20.05, degree of freedom20.05, degree of freedom P CC CE MSE MAE
Singapore 70 55.54 90.53 0.896 0.6943 0.9926 6.31 1.75
Taipei 87 100.48 109.77 0.153 0.7698 0.9948 15.00 2.36
Toronto 111 107.39 136.59 0.500 0.4201 0.9923 4.96 1.69
Note: CC = correlation coefficient; CE = coefficient of efficiency; MSE = mean square error; MAE = mean absolute error.
reliability or validity; in other words, the statistical esti-mates can be directly applied for evaluation. The closer the CC approaches 1, the higher the positive correlation between the actual and simulation data; the closer to –1, the more likely a negative correlation will result; and the closer to 0, the lower the chances of any correlation be-tween the two. The estimated value of the CE is a real number between 0 and 1. The closer it approaches 1, the higher the accuracy of the simulation. Both MSE and MAE use real numbers between 0 and infinity to represent de-gree of inaccuracy. The closer it approaches 0, the more accurate the simulation.
We adopted a chi-square test for homogeneity of pro-portions and a correlation coefficient (CC) to examine the actual number of daily SARS-infected individuals in each city and the numbers that were generated by the simulation system in order to determine whether the distribution pro-portions for the two sets of time-series data were consistent and reflected a positive correlation. According to the results shown in Table 10, the chi-square test valuesχ2for each city were smaller thanχ2(0.05, degree of freedom). We therefore ac-cepted the null hypothesis that the distribution proportions for the time-series data for the number of actual and
sim-ulated affected individuals in each city were consistent at aα = 0.05 level of significance. After examining the simu-lation time-series data for the three cities, we found three positive correlations with the actual time-series data.
In terms of simulation validity, if we only examined simulation accuracy according to the MSE and MAE, the respective accuracy data for Toronto (4.96 and 1.69) were higher than for Singapore (6.31 and 1.75) and Taipei (15.00 and 2.36). However, when the CE was applied, the results were the opposite: the value for Taipei (0.9948) was higher than for Singapore (0.9926) or Toronto (0.9923). The rea-son for this is that the efficiency coefficient primarily con-siders variables, while MSE and MAE focus on average total error values.
5.1.2 Singapore SARS Outbreak
A comparison of actual and simulated SARS cases in Sin-gapore (Fig. 16) shows that our simulated curve had a very close fit with data published by the city-state’s health authority for the two outbreaks that occurred between February 25 and May 5, 2003 (Table 7) [7, 10, 42, 46, 48]. Emergency public health policies were not activated
Figure 16. A comparison of actual and simulated epidemic results for the SARS outbreak in Singapore. The bars represent actual reported cases; the line represents an average of results from 20 simulation runs.
following the first outbreak, which was attributed to im-ported cases. The second outbreak was attributed to the compound effects of secondary infections. Several emer-gency policies were put into effect on March 24, including a ban on visits to patients in hospitals or under home quar-antine. The number of new cases dropped dramatically at the beginning of June; soon afterwards, WHO announced that the disease was under control.
5.1.3 Taipei SARS Outbreak
Our Taipei simulation included several public health poli-cies enforced by that city’s government, including several grades of home quarantine and a mask-wearing require-ment for all bus and train passengers (Table 8) [7, 10, 44, 46-48]. As shown in Figure 17, our simulated results had a close fit with the probable cases curve published by the Taiwanese health authority on September 28, 2003—a ma-jor spike followed by several smaller outbreaks. The higher concentration in the Taipei curve compared to Singapore’s is likely due to late case discoveries, delays in seeking treat-ment, illness cover-ups, public interactions, and the large number of cases imported by travelers returning from Hong Kong. In Singapore, all imported cases were reported prior to the first outbreak, and the second wave resulted from compound infections. The S-curve for the Taiwan situa-tion is more representative of a typical infecsitua-tion pattern.
5.1.4 Toronto SARS Outbreak
The SARS scenario in Toronto consisted of two major waves with almost no new cases in between (Fig. 18) [7, 10,
43, 45-46, 48]. However, after a reexamination of the data in August 2003, the Canadian authorities acknowledged several additional cases during the lull period. According to our simulation, the second wave would not have been as severe if strict public health policies had been enforced for a longer period following the first wave. In our simulation (Table 9), we relaxed epidemic control measures (espe-cially restricted hospital access and reduced public contact with infected persons) after the first wave subsided. As a result, a second spike occurred in our simulation within a few days of the actual spike reported by the Toronto health authorities. Our results support Sebastian and Hoffmann’s [10] conclusion that the Toronto government lifted its con-trol measures too quickly. Because of increased contact between patients and visitors and relaxed rules on wearing masks and/or respirators by health care workers, Toronto experienced a second nosocomial transmission period.
5.1.5 Home Quarantines
After releasing data on the global SARS outbreak on March 12, 2003, WHO officials recommended that home quarantine periods be at least twice as long as the then-average 4- to 6-day incubation period [7, 10, 42, 47]. The governments of Singapore, Taiwan, and Canada accepted this recommendation and enforced 10-day quarantine poli-cies for the duration of the epidemic; for a short period, the Taiwanese government enforced a 14-day policy. We used the home quarantine policy to test our model and observed that a minimum 10-day quarantine period was required to suppress the number of new cases—the same time period
Figure 17. A comparison of actual and simulated epidemic results for the SARS outbreak in Taipei
Figure 18. A comparison of actual and simulated epidemic results for the SARS outbreak in Toronto. We assumed that the second outbreak occurred because preventive policies were relaxed too soon following the first outbreak.
recommended by WHO (Fig. 19). Our simulation showed that the disease became endemic when the 10-day quaran-tine policy was enforced.
5.2 Analyzing Public Health Policies 5.2.1 Taking Body Temperature
The Singaporean and Taiwanese governments both imple-mented temperature measurement policies during the epi-demic, going so far as to launch national campaigns that included installing temperature-monitoring equipment and setting up manual temperature measurement stations at var-ious government buildings, clinics, and public transporta-tion facilities [7, 10]. According to our simulatransporta-tion results, when such policies were both comprehensive and com-pulsory, they reduced the number of feverish individuals entering public places. However, in the real world, this pol-icy is difficult to set up and enforce since implementation methods tend to vary, oversights are common, and an un-known number of individuals manage to evade having their temperatures taken.
Our simulation results suggest that a participation rate of between 80% and 90% is required for this public health policy to have a positive effect in controlling a SARS epi-demic (Fig. 20). At a rate of 65% or lower, it had little effect. This policy incurs significant social costs—for in-stance, distributing inexpensive thermometers, setting up temperature screening stations, and employing workers to take manual temperature measurements at various public facilities and medical clinics.
5.2.2 Wearing Masks with Different Protection Levels—General Public vs. Health Care Workers The efforts of the governments of Taiwan and Hong Kong to promote general mask-wearing policies led to hoarding and panic buying [7, 10]. Masks are categorized according to grade—ordinary, surgical, N95 respirator masks, and so on. In Taiwan, a serious shortage of professional masks for medical staff occurred following a mad rush by the general population to purchase masks regardless of grade;
this triggered a debate on the necessity of wearing N95 respirator masks outside of hospitals and clinics.
According to the results of a simulation that we ran to analyze this policy, ordinary and surgical masks assisted in controlling the epidemic outbreak as long as wearing them was a strong habit for the desired time period (Fig. 21). At a prevention efficiency of 65% or more (i.e., the mask cov-ered the mouth and nose), the epidemic could be controlled but not eliminated. When wearing ordinary masks, medi-cal staff members still had relatively high infection rates (Figs. 21 and 22); these personnel clearly benefited from wearing N95 and other high-resistance masks in hospitals and other medical centers. From our simulation, we sug-gest that the general public should not be required to wear high-resistance masks and that higher grade masks should be reserved for medical staff and health care workers.
5.3 Assessing Public Health Suites
Different public health policies entail different social costs.
Home quarantining is very effective but requires consider-able amounts of labor and material resources compared to temperature measurement and mask-wearing policies. We ran simulations of various prevention strategies to identify an optimal combination of public health policies in terms of efficacy and cost. According to our results, a combination of mask wearing by the general public and reduced contact in public places was the best combination for suppressing the spread of SARS (Fig. 23). Some costs are involved in purchasing masks, but few costs are associated with lim-ited public contact. In addition, mask wearing addresses an epidemic at its source—disease transmission.
The combination of temperature measurement, re-stricted hospital visitations, and mask wearing by health care workers should be considered a remedial reaction to a SARS outbreak since it is ineffective in terms of pre-venting patients in the incubation stage or patients suffer-ing from minor symptoms from spreadsuffer-ing the disease to others. In addition, this suite requires substantial amounts of labor and material resources. Furthermore, the combi-nation of home quarantine and reduced contact in public places also has high social costs, with results dependent on how well isolation guidelines are followed. Numerous instances of intrafamily infections were reported during the actual 2002-2003 SARS outbreak—evidence that cer-tain prevention strategies were ineffective in controlling the epidemic.
6. Conclusion
In this article, we proposed a novel small-world model consisting of cellular automata with social mirror identities representing daily-contact social networks for running epi-demiological simulations. We established the social mirror identity concept to integrate long-distance movement and geographic mobility into the model, which can be used to simulate the transmission dynamics of infectious dis-eases among social networks and to investigate the effica-cies of various public health polieffica-cies and epidemic preven-tion strategies—alone and in combinapreven-tion. The model suc-cessfully exhibits epidemiological behaviors in the form of daily interactions among heterogeneous individuals and expresses such present-day small-world properties as high degrees of clustering, low degrees of separation, and long-distance movement.
According to the results of simulations that we ran based on data collected during the 2002-2003 SARS outbreaks in Singapore, Taipei, and Toronto, we suggest that this model can be applied to different infection scenarios and used to simulate the development of epidemics with considerable accuracy. A comparison of simulation and real-world data indicates that our model can be used to test epidemic report systems and to identify the best public health policy suites for specific scenarios. The simulation results also indicate
Figure 19. Results from a simulation based on various home quarantine policies. Simulation period was 250 days, with a 5-day default incubation period. According to the results, (a) different home quarantine restriction levels exerted different impacts on the SARS epidemic, and (b) a home quarantine policy by itself was insufficient for suppressing the epidemic.
Figure 20. Results from a simulation focused on temperature measurement policy at different participation levels. We used the eight imported cases reported in Singapore to trigger the simulation. In each 66-day simulation run, the policy was activated on day 24.
Figure 21. Results from a simulation focused on the impact of mask wearing by the general public, comparing different mask protection levels
Figure 22. Results from a simulation focused on the impact of mask wearing by health care workers in health care facilities, comparing different mask protection levels
Figure 23. A comparison of various public health policy suites. We used the eight imported cases reported in Singapore to trigger the simulation. Policy suites went into effect on day 24 of our 66-day simulations. Suite 1 (cyan): A-class home quarantine for 10 days and reduced public contact; suite 2 (red): wide-scale taking of body temperatures and a restriction on hospital visitations; suite 3 (green): wide-scale taking of body temperatures, a restriction on hospital visitations, and mask wearing by health care workers;
suite 4 (pink): public mask wearing and reduced public contact.
considerable flexibility in the model—that is, we believe it can be applied to a wide range of contagious diseases (e.g., influenza, enteroviruses, and HIV/AIDS) that have well-defined epidemic parameters.
7. Acknowledgments
This work was supported in part by the National Science Council, Taiwan, Republic of China under grant NSC 92-2524-S-009-004 and NSC 93-2520-S-009-003.
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