Severe Acute Respiratory Syndrome (SARS)
Typical pneumonia is usually caused by bacteria such as Streptococcus pneumoniae. Atypical pneumonia is usually caused by influenza virus, mycoplasma, chlamydia, adenovirus and other unknown micro-organisms. Common symptoms include fever, chills, cough, headache and general malaise.
Atypical pneumonia can be transmitted by respiratory droplets over a short distance of one metre, or by contact with a patient's respiratory secretions.
For more information, Hong Kong Government's Atypical pneumonia Web site
The outbreak of SARS in Block E, Amoy Gardens, Ngau Tau Kok was first reported on 26 March. I have done a simple mathematical study of this incident based on the following assumptions.
Cumulative Number of SARS Admission
Hong Kong 18 Districts Intra-spreading Dynamics
Calculated Probability
| Symbol | Assumption | Value |
| Number of flat, N | Number of floor = 26, Number of flat per floor = 10 | 260 |
| Incubation period, d days | 2 - 10 | 2 |
| Number of lift, L | 1 - 4 | 1 |
| Fraction of coincident time, T | Average time taken to and from the lift and flat = 5 minutes. Consider one takes maximum 90 minutes from home to the workplace, the overlapping time at peak = 90 mminutes. Each day the morning and evening is the peak period of lift usage. |
10 / 180 = 1/18 |
| Probability of infection by droplets, P | As calculated by the formula | 0.094 |
| Probability of inflection by touching droplets in lift, t | It is (1/No of floor) of touching the virus pad. After touching it, there are four cases that may cause infection: touching nose, eye, mouth, and his family member. | 1 /26 * 1/4 = 1/104 |
| Number of flat infected, I | According to the newspaper, there were four infected families. Not counting the one who had member visited Wales Hospital, there were three infected. | 3 |
Formula
For incubation 2, 3, 4, 5 days and no. of flat per floor 8-12, below is the calculated probability of infection by droplets.
- The chance to be infected by a patient when in the closed small area is quite high, 9.4% for incubation of 2 days (6% for 3 days, 4.2% for 4 days, and 3.2% for 5 days). Consider if this value is inversely proportional to the volume of the enclosure. Then a lift that is double in area and the same height will three-quarter less than the original. As I had been to Amoy Gardens before. The lift was very small only allow for 4-5 person inside.
- Consider if the number of lift in operate is more than one, L>1. Then the Probability of infection by droplets P, will be larger. For L=2, P is 19.8% for 2 days, 12.9% for 3 days, 9.4% for 4 days and 7.3% for 5 days. In this case, the calculated value shown that the chance of infected SARS in closed area by droplets is very high.
- After the first outbreak, the rate of spreading is proportional to the number of infected. However, the family members of the infected has a very great chance to get the disease too. So the number of infected will increase in a multiple magnitude in a short time.
- The duration of the incubation period, the probability of infection, and the increasing number of infected determine the rate of spreading of the disease.
- Statistical modeling (incubation period, population density, incident reports, etc) and simulation (traffic pattern, wind velocity, infection rate, etc) can be used to locate the source(s) of the SARS virus. The results can also be used in the planning and control processes. Do not know whether we have this kind of study carrying in Hong Kong now.
- As we know virus is a fragment of DNA with no cell wall. So they cannot live without the living host. Virus does no harm to the host so the latter is the source of the human disease. The Hong Kong Government announced that they had found the origin of the first atypical pneumonia case. But it is not the source. The source is in mainland China. Or there is a new virus variant in Hong Kong.
- I think the HK Government is correct that only those has the SARS symptons to put on the mask and wear the glove. The best way to slow down the spread of the virus is to isolate them from the environment. A health people if not stay with the patient wearing a mask is a passive way to prevent the disease. By the way, do you know how safe is your mask? How long can a mask be used? What is the correct way of putting on the mask? How is your habit could affect the infection?
- From the Information Systems Security we know that the strength of your systems against attacks is determined by the weakest link of the systems.
- It would be interesting to calculate how many people are required to put on the mask (in ideal case) in order to reduce the number of infected per day by half.
Cumulative Number of SARS Admission
Logarithmic plot of the cumulative number of SARS admission from March 14 to May 22
May 23, 2003
On May 23, WHO removed Hong Kong and Guangdong from SARS travel advisory list.
- The average number of new added patient from May 2 to May 8 is 9.
- The average number of new added patient from May 9 to May 15 is 6.
- The average number of new added patient from May 16 to May 22 is 3.
- During the period from May 10 to May 22 the average number of suspected patient is 12.
- Where is the knowledge we have lost in information?
Although statistics indicated that the average number of SARS patients is below 5, it does not means that we have defeated the virus. The fact is there is no effective way to control the virus in the body. The death rate is high as about 20% (>50% for the elder). We are not sure what is the host(s) of the virus. Therefore, we are still like fighting in the dark. I'm also doubt the figures released by China. Hong Kong and China has different classification criteria. So before any conclusion could be made there must be some ways to gauge the data. - Where is the wisdom we have lost in Knowledge?
I think the government is too early (only just the travel advisory was lowered) and too high profile to jump into the boat of the business people to say that we have won the battle. The fact is our doing is no difference of running away from the virus (this is the best way we can do at this moment). This can buy us more time in fighting the virus. If we blamed Mr. Tung slowness in response to the outbreak. Then we should praise him this time if he could react "careful and well planned". Everyone hiking knows getting down from the top is more dangerous than climbing up (venture capitalists also say that getting out from the investment is more critical than getting in). Now Hong Kong is too weak to withstand the impact of the second outbreak.
May 1, 2003
Ministry of Health of mainland China announces the number of confirmed and number of suspected of SARS daily. Department of Health of Hong Kong has followed this way of doing on April 25.
- On April 25, Department of Health began to separate the number of the new added and suspected. The plot shows that the suspected number is increasing during the period. In this case, the situation is tending to stabilize rather than dropping sharply.
- The average number of new added patient from April 25 to May 1 is 16. The number is decreasing steady.
- It is possible that the average number of new added patient will less than 5 at the end of May.
April 24, 2003
- The average number of new added patient from April 19 to April 24 is 27 (round to integer). It is the same value before the outbreak of Amoy Gardens.
- It is a good sign that the growth rate fails below the lower bound line. If this trend was followed. Then at the end of April, the total number of inflected SARS will be smaller than 1,800.
- The turning point was on April 14
April 18, 2003
According to the government released data the total number of deaths up to April 18 is 69 and the total number of discharged patients is 322. The cumulative number of SARS patients is 1327. Department of Health calculates the death rate of the disease by 69 over 1327 equal to 0.052. In terms of percentage it is about 5%.
- Since a disease is either curable or causing death. Therefore the death rate should be calculated by using (69 + 322) as base. The value of 69 over 391 is 0.176. The 5% death rate underestimate the seriousness of the disease.
- The probability of death of someone who have close contact with the SARS patients is equal to the probability of infection by droplets P times the average death rate.
- For P = 0.198 (as calculated from above for number of lifts L equal to 2) and percentage of death = 0.176, the expected number of deaths = 1327 x 0.2 x 0.176 = 47.
- For the expected number of deaths = 69 and the percentage of death = 0.176, the calculated P = 0.294. This value is corresponding to (L,d) = (3,2)
- It is a good sign to see that the trend is falling below the green line and get closer to the lower bound red line.
April 11, 2003
On April 9, the Hong Kong Hospital Authority (HA) estimated that there would be about 1,800 to 3,000 SARS patients in Hong Kong at the end of April.
- From the graph above we can estimate that at the end of April that will be about 2,000 to 6,000 SARS patients (red lines) in Hong Kong.
- Consider if they are correspond to two infection tracks, then the mean number of SARS patients is 3,500 (the green line).
- To reach the number of 1,800 at the end of April, the new added number each day is 40. For 3,000, the new added number is 100 each day. It is possible to happen if two (2) Amoy Gardens incidents are present.
- The average number of new added patient from April 2 to April 11 is 38 (round to integer). It is 35% higher than the value before the outbreak found in the Amoy Gardens
- Does HA has enough Intensive Care Unit (ICU) rooms for the needed patients?
April 3, 2003
Today local newspapers report government saying that in Wales Hospital during the SARS outbreak statistically every 25 persons contacted the patient had one infected atypical pneumonia, the probability is 4%.
For P = 0.04, (number of lift L, incubation period d)
(L,d) = (1,4), (2,8), (3,12), (4,16)
The corresponding incubation periods matches with the Health Department estimation, i.e. 2-12 days.
- After March 18, the number of SARS patient increased linearly in logarithmic scale.
- The average number of new added patient from March 17 to March 26 is 28 (round to integer). The double is 56.
- Outbreak find in Block E of Amoy Gardens. During the period, from March 27 to April 1, the average number of new added patient is 62 (round to integer). All the people in the building were removed to recreation camps on April 2.
- The average number of new added patient from April 2 to April 3 is 25(round to integer). The number is the same as before the Amoy Gardens incident. So there is no reason to say that, at this moment, the virus spreading was slowing down.
- The linearity of the plot suggests that the spreading mechanism of the virus in the Amoy Gardens is similar to the Wales Hospital.
- However, Amoy Gardens incident was an exceptional case indicating the virus may has rooted in the building. So the Health Department need to remove all people from the building to immediate stop the infection. It is very important that the researchers could find out the origin of the virus.
Epidemic Model
The linearity of the logarithmic plot of above suggests that
where k, C are constants, t is time, and I is the number of infected.
Consider, the rate of spreading is proportional to the number of infected and also the number of uninfected. Each infected person will contact fraction of people c in the incubation period. The differential equation, therefore, can be written as
where a is a constant, N is the number of people who are susceptible to infection, c is the fraction of the people that the patient contact in the incubation period. In the Amoy Gardens case, N is equal to the total number of residents of Block E.
The solution of this autonomous first order differential equation is
therefore, k = acN, is the slope of the logarithmic plot and C is the intercept of the t-axis.
- This formulation does not consider removal of infected people from the public (e.g. treating in hospitals, death, quarantine , etc).
Hong Kong 18 Districts Intra-spreading Dynamics
Department of Health produces a list of the names of the buildings with SARS patients who have been hospitalized for treatment everyday. The list has been compiled from information provided by patients with confirmed diagnosis of SARS. If there are no further cases in 10 days, the names of the buildings will be taken off the list.
I have done a brief study of the spread of SARS in HK by using this list. The study is based on the idea of data clustering and dispersion of incidents. At the beginning, the data set was arranged in a normal distribution form, with the center at Kwun Tong and New Territories on the right and Kowloon and HK on the left. Date-sequential plotting of these data sets reveals both geographic and eventual patterns. The results may be used in the tracking and strategic planning of the control of spreading of the disease.
Initial sorting and arrangement of the 18 Districts data (April 15)
| Southern | Central & Western | Wan Chai | Yau Tsim Mong | Wong Tai Sin | Sai Kung | Kowloon City | Sham Shui Po | Eastern | Kwun Tong | Tai Po | Shatin | Kwai Tsing | Tuen Mun | Tsuen Wan | North | Yuen Long | Islands |
| 0 | 0 | 1 | 9 | 9 | 10 | 11 | 12 | 13 | 29 | 23 | 22 | 16 | 9 | 5 | 4 | 2 | 1 |
The value was peak at Kwun Tong. On the left are districts from Hong Kong & Kowloon except Sai Kung. On the right are districts from New Territories. Sai Kung is put on the HK & Kowloon side because there is a main road from Kowloon City to Wong Tai Sin to Sai Kung. Later we shall see this arrangement gives a meaningful result.
- LEFT TAIL - Southern, Central & Western, Wan Chai, Yau Tsim Mong. Except Yau Tsim Mong it compose of the whole Hong Kong districts.
- CENTRAL (left) - Wong Tai Sin, Sai Kung, Kowloon City, Sham Shui Po, Eastern, Kwun Tong. Except Sai Kung and Eastern it compose of the Kowloon districts.
- CENTRAL (right) - Tai Po, Shatin, Kwai Tsing. Central right and right tail forming the New Territories districts (except Sai Kung is in the central left).
- RIGHT TAIL - Tuen Mun, Tsuen Wan, North, Yuen Long, Islands
Note: I did not expect to see the distribution of Hong Kong, Kowloon, NT districts at the beginning of the data preparation. I just thinking of Amoy Gardens in Kwun Tong be the main cause of spreading should be placed at the central.
- The single central (Kwun Tong) peaked distribution transformed to three isolated clusters centered at Sai Kung, Shatin, and North on May 8, 2003
- The peak at Sai Kung flated on May 17. Although the numbers of the other two were decreasing, they persist to exist.
- Different district has different contact rate between infectious individuals and the rest of the subpopulation due to spatial variation in transmission.
- The number of confirmed building is decreasing steady in all districts except Yuen Long. We can also see the numbers decreasing sharply in Kwun Tung and Shatin.
- As the confirmed building's name will be removed from the list if there are no new cases reported in ten days, so we should take this time factor into consideration when studying the spread of the virus.
- A time correlation function is defined as sqrt [f(t)*f(t-10)] *number of new added(t). Multiply by the number of new added patient is because there could be multiple infected in one building. This function is integrated and plotted as below
- These values does not take into account of the suspected buildings. The curve remain flat from April 27 to April 30, while the number of suspected patient increased during the period.
- I am gathering the data of SARS in China. The patterns and the results of calculation of cases in Hong Kong can be applied to the study of the mainland China's.
- On May 8, the single central peak distribution transformed to three isolated clusters.
- The central peak remain strong during the period. But the most noticeable change of the mono distribution is the development of the peaks at the two tails. At the right is the [North, Yuen Long] region appeared on April 17 two days after April 15. At the left is the [Wong Tai Sin, Sai Kung] region appeared on April 21 six days after April 15. Both 2 and 6 days lay within the SARS incubation period.
- We must put special attention on these two tail-peak regions before it was too late.
- There are minimize zones find between the peaks. They are currently in Kowloon City, Tuen Mun, Southern, and Islands (consider the plot as in circular). Shall we do more to protect these districts to use them as firewalls to isolate different affected zones.
- For the central peak, on the slope of the right is [Tai Po, Shatin, Kwai Tsing] and on the left is [Kwun Tong, Eastern, Sham Shui Po]. The former group is where the SARS specialized hospitals are located. And the latter group consist of some residential densely populated areas.
- First on April 23, all 18 districts has reported SARS buildings.
- The western and southern part of Hong kong (the whole) have smaller number of SARS affected buildings.
- GIS could be used to map out the environment, habitat, medical, and hygiene factors of the above identified regions. We do not need a supercomputer to do this calculation and analysis.
HongKong - China Inter-spreading Dynamics
Consider SARS carriers or patients arrived from China is a Poisson process. ...
Census-Taker Problem
A census-taker knocks on a door, and asks the woman inside how many children she has and how old they are.
Mother said, "I have three daughers, their ages are whole numbers, and the product of the ages is 36."
"That's not enough information," responded the census-taker.
"I'd tell you the sum of their ages, but you's still be stumped."
"I wish you'd tell me something more."
"Okay, my oldest daughter Annie likes dogs."
What are the ages of the three daughters?
Managing Risk
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