[Source: ProMedMail.org, full page: (LINK). Extract, edited.]
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Novel Coronavirus, Middle East: Transmissibility and cluster sizes
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[4] Transmissibility and cluster sizes
Date: Sun 26 May 2013  From: David N Fisman <david.fisman@utoronto.ca> [edited],
Thoughts on transmissibility and cluster sizes seen with MERSCoV
I have watched the emergence of the MERS coronavirus with a concern that is shared by many colleagues in the communicable disease control world. However, given recent concerns expressed regarding the communicability of this virus, I wished to share some thoughts on the relationship between disease clusters, sustained chains of transmission, and reproductive numbers of pathogens.
The basic reproductive number (R0), the number of secondary cases of infection produced by a primary case in a completely susceptible population, is an important index of epidemic potential. When R0 is greater than 1 (even slightly) selfsustaining exponential growth of case numbers becomes likely (that is, an epidemic is expected to occur). However, it is often forgotten that transmission can (and is expected to) occur when R0 less than 1, with the result being (generally short) chains of transmission (observed as disease clusters) rather than exponential increases in case numbers. For R0 less than 1, expected cluster size (including the index case) can be expressed as the sum of a geometric series such that:
N = 1/(1R0)
For instance, if R0 = 0.5, each primary case infects, on average, 0.5 secondary cases before recovering and an average cluster size would be expected to be 2 cases (that is, one primary case, and on average, one secondary case). If R0 is 0, the average cluster size is zero (the index case only, with no onward transmission) and if R0 is 1, the expected cluster size is infinite as the disease is endemic (each infectious case "replaces itself" before recovering). I present these relationships graphically in MERS figures 1A and 1B (available at http://davidfisman.tumblr.com/)
We can manipulate the expression above in order to estimate R0 based on average cluster size using the relation:
R0 = [(1/N)  1]
Thus we can empirically estimate the current R0 for MERSCoV based on average cluster sizes, as presented in the table below. These have been largely derived from the European Centres for Disease Control Rapid Risk Assessment on MERSCoV (1), with the addition of 3 additional hospitalassociated cases to the large Saudi cluster as of [21 May 2013] (2), and the addition of a 3person cluster recently identified in Tunisia described on ProMED on [20 May 2013 (3). Note that I assume that 5 cases in Riyadh, Saudi Arabia are unclustered, based on the information available to me. Furthermore, although only 2 confirmed cases have been identified retrospectively from Jordan, these cases were part of an 11person cluster of respiratory illness in a hospital in that country and I have assumed that the total cluster size was thus 11.
Cluster size / Linked countries / Notes
It is possible to model an average chain of transmission as a "Markov process", with sequential individuals classed as "transmitters" (who create an additional case with probability equal to R0) or "nontransmitters" who result in termination of the chain of transmission (MERS figure 2, available at http://davidfisman.tumblr.com/ ). The probability of nontransmission for a given case is (1R0). If we run such a model the average cumulative cluster size is identical to that calculated using the geometric series approach. What is perhaps more interesting is to run this model as a 1storder Monte Carlo simulation, where at each generation transmission may occur (with probability = R0) or not (with probability 1R0). The resulting model outputs can be interpreted as the probability of seeing clusters of a given size, for a given R0.
What we see is that, for a disease transmission process with an R0 of 0.76, the occurrence of chains of transmission resulting in 22 cases is incredibly unlikely (approximately 0.08 per cent) (MERS figure 3, available at http://davidfisman.tumblr.com/ ). Indeed, as far as I am aware, no cluster of 5 cases has been observed, but we would expect to have seen approximately 140 such clusters before seeing a single cluster of 22 cases. If the Jordan cluster was indeed 11 cases in size, we would have expected to have seen 20 such clusters before seeing a single 22 case cluster (MERS figure 4, available at http://davidfisman.tumblr.com/ ).
While it is possible that a 22 case cluster could occur by chance (parenthetically, somewhere between the probability of getting 10 and 11 heads in a row when tossing a fair coin), the low likelihood of such an event (and indeed of the 11 case cluster) with a disease with R0 of 0.76 leads me to suggest that the ability of this pathogen to cause large clusters or sustained chains of transmission in the context of the Eastern Saudi healthcare outbreak is quite different from that seen elsewhere, and especially outside the healthcare environment.
I can think of 3 possible mechanisms for this difference:
References
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[4] Transmissibility and cluster sizes
Date: Sun 26 May 2013  From: David N Fisman <david.fisman@utoronto.ca> [edited],
Thoughts on transmissibility and cluster sizes seen with MERSCoV
I have watched the emergence of the MERS coronavirus with a concern that is shared by many colleagues in the communicable disease control world. However, given recent concerns expressed regarding the communicability of this virus, I wished to share some thoughts on the relationship between disease clusters, sustained chains of transmission, and reproductive numbers of pathogens.
The basic reproductive number (R0), the number of secondary cases of infection produced by a primary case in a completely susceptible population, is an important index of epidemic potential. When R0 is greater than 1 (even slightly) selfsustaining exponential growth of case numbers becomes likely (that is, an epidemic is expected to occur). However, it is often forgotten that transmission can (and is expected to) occur when R0 less than 1, with the result being (generally short) chains of transmission (observed as disease clusters) rather than exponential increases in case numbers. For R0 less than 1, expected cluster size (including the index case) can be expressed as the sum of a geometric series such that:
N = 1/(1R0)
For instance, if R0 = 0.5, each primary case infects, on average, 0.5 secondary cases before recovering and an average cluster size would be expected to be 2 cases (that is, one primary case, and on average, one secondary case). If R0 is 0, the average cluster size is zero (the index case only, with no onward transmission) and if R0 is 1, the expected cluster size is infinite as the disease is endemic (each infectious case "replaces itself" before recovering). I present these relationships graphically in MERS figures 1A and 1B (available at http://davidfisman.tumblr.com/)
We can manipulate the expression above in order to estimate R0 based on average cluster size using the relation:
R0 = [(1/N)  1]
Thus we can empirically estimate the current R0 for MERSCoV based on average cluster sizes, as presented in the table below. These have been largely derived from the European Centres for Disease Control Rapid Risk Assessment on MERSCoV (1), with the addition of 3 additional hospitalassociated cases to the large Saudi cluster as of [21 May 2013] (2), and the addition of a 3person cluster recently identified in Tunisia described on ProMED on [20 May 2013 (3). Note that I assume that 5 cases in Riyadh, Saudi Arabia are unclustered, based on the information available to me. Furthermore, although only 2 confirmed cases have been identified retrospectively from Jordan, these cases were part of an 11person cluster of respiratory illness in a hospital in that country and I have assumed that the total cluster size was thus 11.
Cluster size / Linked countries / Notes
 11 / Jordan / only 2 confirmed retrospectively
 22 / Saudi Arabia / true cluster size only 20 (2 unlinked cases?)
 3 / Saudi Arabia, Tunisia ex Qatar / 
 1 / Germany ex UAE / 
 2 / France ex UAE / secondary transmission in French hospital
 3 / Pakistan, UK ex Saudi Arabia / two 2ndary transmissions in the UK
 1 / UK ex Qatar / 
 3 / Saudi Arabia / family cluster
 1 / Saudi Arabia / isolated case in Jeddah
 1 / Saudi Arabia / Riyadh
 1 / Saudi Arabia / Riyadh
 1 / Saudi Arabia / Riyadh
 1 / Saudi Arabia / Riyadh
 1 / Saudi Arabia / Riyadh
 1 / Germany ex Qatar / 
It is possible to model an average chain of transmission as a "Markov process", with sequential individuals classed as "transmitters" (who create an additional case with probability equal to R0) or "nontransmitters" who result in termination of the chain of transmission (MERS figure 2, available at http://davidfisman.tumblr.com/ ). The probability of nontransmission for a given case is (1R0). If we run such a model the average cumulative cluster size is identical to that calculated using the geometric series approach. What is perhaps more interesting is to run this model as a 1storder Monte Carlo simulation, where at each generation transmission may occur (with probability = R0) or not (with probability 1R0). The resulting model outputs can be interpreted as the probability of seeing clusters of a given size, for a given R0.
What we see is that, for a disease transmission process with an R0 of 0.76, the occurrence of chains of transmission resulting in 22 cases is incredibly unlikely (approximately 0.08 per cent) (MERS figure 3, available at http://davidfisman.tumblr.com/ ). Indeed, as far as I am aware, no cluster of 5 cases has been observed, but we would expect to have seen approximately 140 such clusters before seeing a single cluster of 22 cases. If the Jordan cluster was indeed 11 cases in size, we would have expected to have seen 20 such clusters before seeing a single 22 case cluster (MERS figure 4, available at http://davidfisman.tumblr.com/ ).
While it is possible that a 22 case cluster could occur by chance (parenthetically, somewhere between the probability of getting 10 and 11 heads in a row when tossing a fair coin), the low likelihood of such an event (and indeed of the 11 case cluster) with a disease with R0 of 0.76 leads me to suggest that the ability of this pathogen to cause large clusters or sustained chains of transmission in the context of the Eastern Saudi healthcare outbreak is quite different from that seen elsewhere, and especially outside the healthcare environment.
I can think of 3 possible mechanisms for this difference:
 1. The pathogen has mutated and is now more transmissible (I know of no evidence to suggest that this is the case).
 2. The healthcare environment provides a rich environment for recognized transmission of this pathogen, due to the frequency of contacts and vulnerability of individuals to the development of symptomatic recognized infection.
 3. As with SARS, there may have been point source exposures of large numbers of individuals, which resulted acutely in infection of a large cluster of individuals with subsequent decay in the risk of infection of secondary cases (as R0 less than 1). With SARS, such events were certainly seen in Toronto (in the context of emergency endotracheal intubation of an individual with SARS and respiratory failure, with infection of 11 secondary cases, 6 of whom were present at the emergency procedure (4)), and in Hong Kong (such as in the Amoy Gardens episode (5)).
 1. Persontoperson transmission may occur with R0 less than 1, and indeed clusters are expected. This does not imply that epidemic transmission (that is, selfsustaining, exponentially increasing case counts) will occur, as such chains, while concerning (and potentially fairly large with R0 greater than 0.75), are expected to be selflimited.
 2. The processes observed in the current Saudi healthcare outbreak (and one presumes, in the context of the retrospectively identified Jordanian healthcare cluster) are epidemiologically distinct, with respect to communicability, from transmission processes occurring outside the healthcare environment. Indeed, there may be a distinct "withinhealthcare R0" for this pathogen that is close to 1, or even exceeds 1.
 a) This may have been the case with SARS: here in Toronto, we did see spillover cases in the community, but these did not result in communitybased outbreaks of SARS. The occurrence of community cases did, however, result in a WHO travel advisory (6), which caused substantial economic losses to the city. I would suggest that the relevant health authorities recognize the distinct epidemiology of disease transmission inside and outside healthcare, and exercise caution prior to the issuance of any travel advisories, which can cause tremendous economic damage to communities.
 b) If indeed MERSCoV echo those of SARSCoV, it is important to recognize that a syndromic approach to excellent infection control, not targeted at those with suspected MERSCoV, but applied to all individuals with asyetundiagnosed febrile respiratory illness, is likely to be effective at preventing transmission of this pathogen in the healthcare environment.
References
 European Centre for Disease Prevention and Control. Rapid risk assessment: Severe respiratory disease associated with Middle East respiratory syndrome coronavirus (MERSCoV). 17 May 2013. Available at http://www.ecdc.europa.eu/en/publications/Publications/Forms/ECDC_DispForm.aspx?ID=1121. Last accessed 25 May 2013.
 ProMEDmail. MERSCoV  Eastern Mediterranean (06): Saudi Arabia, new fatality, meeting; archive no 20130521.1726656. 21 May 2013. Available at http://www.promedmail.org/direct.php?id=20130521.1726656. Last accessed 25 May 2013.
 ProMEDmail. MERSCoV  Eastern Mediterranean (05): Tunisia ex Saudi Arabia/Qatar, fatal, RFI. archive no 20130520.1725864. 20 May 2013. Available at http://www.promedmail.org/direct.php?id=20130520.1725864. Last accessed 20 May 2013.
 US Centers for Disease Control and Prevention. Cluster of severe acute respiratory syndrome cases among protected healthcare workers  Toronto, Canada, April 2003. MMWR Morb Mortal Wkly Rep 2003; 52(19): 4336. Available at http://www.cdc.gov/mmwr/preview/mmwrhtml/mm5219a1.htm.
 McKinney KR, Gong YY, Lewis TG. Environmental transmission of SARS at Amoy Gardens. J Environ Health 2006; 68(9): 2630; quiz 512. Available at http://www.thefreelibrary.com/Environmental+transmission+of+SARS+at+Amoy+Gardens .a0145934763.
 Rodier GR. Why was Toronto included in the World Health Organization's SARSrelated travel advisory? CMAJ. 2003; 168(11): 14345. Available at http://www.cmaj.ca/content/168/11/1434.full.pdf+html.

David N Fisman, MD MPH FRCPC, Associate Professor of Epidemiology, Medicine, and Health Policy, Dalla Lana School of Public Health and Institute of Health Policy, Management and Evaluation, University of Toronto, Canada, david.fisman@utoronto.ca
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[ProMEDmail would like to thank Dr Fisman for sharing his analyses with the ProMEDmail readership. This moderator is in agreement with Dr Fisman's conclusion of the need to implement strict infection control procedures when confronted with severe acute respiratory infections (SARI) that are as yet undiagnosed, in order to prevent potential additional transmission of respiratory agents in the healthcare setting (see moderator comment in ProMEDmail MERSCoV  Eastern Mediterranean (09): Saudi Arabia, WHO, Jordan 20130523.1733317).  Mod.MPP]David N Fisman, MD MPH FRCPC, Associate Professor of Epidemiology, Medicine, and Health Policy, Dalla Lana School of Public Health and Institute of Health Policy, Management and Evaluation, University of Toronto, Canada, david.fisman@utoronto.ca
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