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Temporal Variability and Social Heterogeneity in Disease Transmission: The Case of SARS in Hong Kong

Identifieur interne : 001325 ( Pmc/Corpus ); précédent : 001324; suivant : 001326

Temporal Variability and Social Heterogeneity in Disease Transmission: The Case of SARS in Hong Kong

Auteurs : Anne Cori ; Pierre-Yves Boëlle ; Guy Thomas ; Gabriel M. Leung ; Alain-Jacques Valleron

Source :

RBID : PMC:2717369

Abstract

The extent to which self-adopted or intervention-related changes in behaviors affect the course of epidemics remains a key issue for outbreak control. This study attempted to quantify the effect of such changes on the risk of infection in different settings, i.e., the community and hospitals. The 2002–2003 severe acute respiratory syndrome (SARS) outbreak in Hong Kong, where 27% of cases were healthcare workers, was used as an example. A stochastic compartmental SEIR (susceptible-exposed-infectious-removed) model was used: the population was split into healthcare workers, hospitalized people and general population. Super spreading events (SSEs) were taken into account in the model. The temporal evolutions of the daily effective contact rates in the community and hospitals were modeled with smooth functions. Data augmentation techniques and Markov chain Monte Carlo (MCMC) methods were applied to estimate SARS epidemiological parameters. In particular, estimates of daily reproduction numbers were provided for each subpopulation. The average duration of the SARS infectious period was estimated to be 9.3 days (±0.3 days). The model was able to disentangle the impact of the two SSEs from background transmission rates. The effective contact rates, which were estimated on a daily basis, decreased with time, reaching zero inside hospitals. This observation suggests that public health measures and possible changes in individual behaviors effectively reduced transmission, especially in hospitals. The temporal patterns of reproduction numbers were similar for healthcare workers and the general population, indicating that on average, an infectious healthcare worker did not infect more people than any other infectious person. We provide a general method to estimate time dependence of parameters in structured epidemic models, which enables investigation of the impact of control measures and behavioral changes in different settings.


Url:
DOI: 10.1371/journal.pcbi.1000471
PubMed: 19696879
PubMed Central: 2717369

Links to Exploration step

PMC:2717369

Le document en format XML

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<name>
<surname>Cori</surname>
<given-names>Anne</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
<xref ref-type="corresp" rid="cor1">
<sup>*</sup>
</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Boëlle</surname>
<given-names>Pierre-Yves</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Thomas</surname>
<given-names>Guy</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Leung</surname>
<given-names>Gabriel M.</given-names>
</name>
<xref ref-type="aff" rid="aff3">
<sup>3</sup>
</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Valleron</surname>
<given-names>Alain-Jacques</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
</contrib>
</contrib-group>
<aff id="aff1">
<label>1</label>
<addr-line>INSERM, Paris, France</addr-line>
</aff>
<aff id="aff2">
<label>2</label>
<addr-line>Université Pierre et Marie Curie-Paris6, Paris, France</addr-line>
</aff>
<aff id="aff3">
<label>3</label>
<addr-line>School of Public Health, Li Ka Shing Faculty of Medicine, The University of Hong Kong, Hong Kong, Special Administrative Region, People's Republic of China</addr-line>
</aff>
<contrib-group>
<contrib contrib-type="editor">
<name>
<surname>Ferguson</surname>
<given-names>Neil M.</given-names>
</name>
<role>Editor</role>
<xref ref-type="aff" rid="edit1"></xref>
</contrib>
</contrib-group>
<aff id="edit1">Imperial College London, United Kingdom</aff>
<author-notes>
<corresp id="cor1">* E-mail:
<email>cori@u707.jussieu.fr</email>
</corresp>
<fn fn-type="con">
<p>Conceived and designed the experiments: AC P-YB GT A-JV. Performed the experiments: AC P-YB. Analyzed the data: AC P-YB GT. Contributed reagents/materials/analysis tools: P-YB GML. Wrote the paper: AC. Contributed to drafting the manuscript: P-YB GT GML A-JF.</p>
</fn>
</author-notes>
<pub-date pub-type="collection">
<month>8</month>
<year>2009</year>
</pub-date>
<pub-date pub-type="epub">
<day>21</day>
<month>8</month>
<year>2009</year>
</pub-date>
<volume>5</volume>
<issue>8</issue>
<elocation-id>e1000471</elocation-id>
<history>
<date date-type="received">
<day>13</day>
<month>10</month>
<year>2008</year>
</date>
<date date-type="accepted">
<day>15</day>
<month>7</month>
<year>2009</year>
</date>
</history>
<permissions>
<copyright-statement>Cori et al.</copyright-statement>
<copyright-year>2009</copyright-year>
<license xlink:href="http://creativecommons.org/licenses/by/4.0/">
<license-p>This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited.</license-p>
</license>
</permissions>
<abstract>
<p>The extent to which self-adopted or intervention-related changes in behaviors affect the course of epidemics remains a key issue for outbreak control. This study attempted to quantify the effect of such changes on the risk of infection in different settings, i.e., the community and hospitals. The 2002–2003 severe acute respiratory syndrome (SARS) outbreak in Hong Kong, where 27% of cases were healthcare workers, was used as an example. A stochastic compartmental SEIR (susceptible-exposed-infectious-removed) model was used: the population was split into healthcare workers, hospitalized people and general population. Super spreading events (SSEs) were taken into account in the model. The temporal evolutions of the daily effective contact rates in the community and hospitals were modeled with smooth functions. Data augmentation techniques and Markov chain Monte Carlo (MCMC) methods were applied to estimate SARS epidemiological parameters. In particular, estimates of daily reproduction numbers were provided for each subpopulation. The average duration of the SARS infectious period was estimated to be 9.3 days (±0.3 days). The model was able to disentangle the impact of the two SSEs from background transmission rates. The effective contact rates, which were estimated on a daily basis, decreased with time, reaching zero inside hospitals. This observation suggests that public health measures and possible changes in individual behaviors effectively reduced transmission, especially in hospitals. The temporal patterns of reproduction numbers were similar for healthcare workers and the general population, indicating that on average, an infectious healthcare worker did not infect more people than any other infectious person. We provide a general method to estimate time dependence of parameters in structured epidemic models, which enables investigation of the impact of control measures and behavioral changes in different settings.</p>
</abstract>
<abstract abstract-type="summary">
<title>Author Summary</title>
<p>Recent epidemics have shown that healthcare workers may be overrepresented among cases and how critical it is to protect them. For example, during the 2002–2003 severe acute respiratory syndrome (SARS) epidemics in Hong Kong, 27%of cases were healthcare workers when they were <1% of the population. Better means of protection require understanding how healthcare workers were infected and assessing their role in disease transmission. Here, we describe a method for estimating the temporal profile of the risk of infection and probability of transmission in the community and hospitals. The 2002–2003 SARS outbreak in Hong Kong is used as an example. For the SARS epidemic, we show that the risk of infection in the community and hospitals decreased with time down to zero in hospitals but remained larger in the community. This observation suggests that public health measures and behavioural changes most effectively reduced transmission in hospitals. Besides, we find that the large number of cases observed among healthcare workers is more likely a result of large and sustained exposure to hospitalized cases than to transmission among healthcare workers. These results are of interest to design control measures in the event of an influenza pandemic.</p>
</abstract>
<counts>
<page-count count="8"></page-count>
</counts>
</article-meta>
</front>
<body>
<sec id="s1">
<title>Introduction</title>
<p>Emerging infectious diseases have been defined as, “infections that have newly appeared in a population or have existed previously but are rapidly increasing in incidence or geographic range.
<xref rid="pcbi.1000471-Morse1" ref-type="bibr">[1]</xref>
” Several features may make them particularly threatening. First, recognizing the disease can be difficult when the first cases appear, especially when the symptoms are non-specific. Second, no vaccine or specific treatment may be known initially. Moreover, heterogeneities in disease transmission may create high-risk groups, such as healthcare workers
<xref rid="pcbi.1000471-ChanYeung1" ref-type="bibr">[2]</xref>
<xref rid="pcbi.1000471-Varia1" ref-type="bibr">[5]</xref>
and high-risk geographical areas, thereby dramatically enhancing the impact of the outbreak
<xref rid="pcbi.1000471-Anderson1" ref-type="bibr">[6]</xref>
.</p>
<p>The 2003 severe acute respiratory syndrome (SARS) outbreak in Hong Kong is remarkably illustrative of the above issues: symptoms were similar to pneumonia
<xref rid="pcbi.1000471-Chowell1" ref-type="bibr">[7]</xref>
; the incubation period was long enough for local and international transmission to occur
<xref rid="pcbi.1000471-World1" ref-type="bibr">[8]</xref>
; no vaccine or treatment was available; as much as 21% of cases worldwide were healthcare workers
<xref rid="pcbi.1000471-World2" ref-type="bibr">[9]</xref>
. The outbreak also demonstrated the possible existence of super-spreading events (SSEs)
<xref rid="pcbi.1000471-LloydSmith1" ref-type="bibr">[10]</xref>
, during which a few infectious individuals contaminated a high number of secondary cases. Hong Kong had two SSEs: the first occurred in Hospital X around March 3 and led to about 125 cases
<xref rid="pcbi.1000471-Lee1" ref-type="bibr">[11]</xref>
; the second occurred in Housing Estate Y on March 19, and led to over 300 cases
<xref rid="pcbi.1000471-Riley1" ref-type="bibr">[12]</xref>
,
<xref rid="pcbi.1000471-Leung1" ref-type="bibr">[13]</xref>
. Despite its particularly threatening features, the outbreak was brought under control.</p>
<p>In this context, once the epidemic is detected, spontaneous changes in behavior will occur, and non-pharmacological measures are usually initiated to control the outbreak. The resulting effects of these two phenomena on disease transmission is not easily quantified.</p>
<p>The effective contact rate, which reflects the combined influences of social proximity (the number of contacts per time unit) and the probability of infection through each contact, is an essential determinant of disease spread. Our aim was to estimate the temporal variation of this parameter in the community and hospitals, over the course of the outbreak.</p>
<p>Previously published mathematical models of parameter estimation addressed the issues of temporal variability
<xref rid="pcbi.1000471-Riley1" ref-type="bibr">[12]</xref>
,
<xref rid="pcbi.1000471-Cauchemez1" ref-type="bibr">[14]</xref>
or social heterogeneity
<xref rid="pcbi.1000471-ChanYeung1" ref-type="bibr">[2]</xref>
,
<xref rid="pcbi.1000471-LloydSmith2" ref-type="bibr">[15]</xref>
. Here we present an approach that deals with both issues, together with the occurrence of SSEs. Then the method is applied to the 2003 SARS epidemic in Hong Kong (SARSID database
<xref rid="pcbi.1000471-Leung1" ref-type="bibr">[13]</xref>
).</p>
</sec>
<sec id="s2">
<title>Materials and Methods</title>
<sec id="s2a">
<title>Data</title>
<p>Among the 1755 patients admitted to Hong Kong hospitals in 2003 for suspected SARS, 1467 serologically confirmed SARS cases were retained for analysis. For each case, occupation, date of symptom onset, date of hospital admission, duration of hospital stay and discharge status (dead or alive) were recorded. Durations of hospital stay were missing for 12 cases and imputed to 100 days.</p>
</sec>
<sec id="s2b">
<title>Transmission Model</title>
<p>The epidemic process was cast into a discrete time stochastic susceptible-exposed-infectious-removed (SEIR) compartmental model, designed to reflect a two-way classification of individuals according to disease status and ‘social’ category (
<xref ref-type="fig" rid="pcbi-1000471-g001">Figure 1</xref>
).</p>
<fig id="pcbi-1000471-g001" position="float">
<object-id pub-id-type="doi">10.1371/journal.pcbi.1000471.g001</object-id>
<label>Figure 1</label>
<caption>
<title>Compartmental Model for SARS Transmission in Hong Kong.</title>
<p>Superscript letters denote social categories:
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e001.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, general population;
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e002.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, healthcare workers;
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e003.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, hospitalized patients. Disease states are:
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e004.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, susceptible;
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e005.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, exposed (infected but not yet infectious);
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e006.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, infectious not hospitalized;
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e007.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, infectious hospitalized, and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e008.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, removed (recovered or dead).</p>
</caption>
<graphic xlink:href="pcbi.1000471.g001"></graphic>
</fig>
<p>The latter was defined in three categories: hospitalized patients (hp), healthcare workers (hw), and the general population (gp).</p>
<p>According to these three social categories, SARS cases were qualified: nosocomial when the patient had been hospitalized for ≥5 days before symptom onset
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e009.jpg" mimetype="image"></inline-graphic>
</inline-formula>
; healthcare workers when the subjects were indeed healthcare workers and not nosocomial
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e010.jpg" mimetype="image"></inline-graphic>
</inline-formula>
; or general population, all others
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e011.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Their corresponding epidemic curves are shown in
<xref ref-type="fig" rid="pcbi-1000471-g002">Figure 2</xref>
.</p>
<fig id="pcbi-1000471-g002" position="float">
<object-id pub-id-type="doi">10.1371/journal.pcbi.1000471.g002</object-id>
<label>Figure 2</label>
<caption>
<title>Daily Incidence of SARS Symptom Onset (Observed and 5×10
<sup>4</sup>
Simulated Epidemics), Hong Kong, 2003.</title>
<p>Cases were defined as: nosocomial when patients had been hospitalized for ≥5 days before symptom onset
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e012.jpg" mimetype="image"></inline-graphic>
</inline-formula>
; healthcare workers when they were indeed healthcare workers and not nosocomial
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e013.jpg" mimetype="image"></inline-graphic>
</inline-formula>
; and general population, otherwise
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e014.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. The grey cloud surrounding the observed epidemic curve corresponds to simulated epidemic curves.</p>
</caption>
<graphic xlink:href="pcbi.1000471.g002"></graphic>
</fig>
<p>Disease status was described in five compartments: susceptible (S), exposed (E), infectious non-hospitalized (I), infectious hospitalized (H), and removed (R). Individuals are initially susceptible to the disease and infected through contact with infectious subjects. Once infected, individuals are first exposed (infected, non-infectious) and then become infectious. The infectious stage is defined as the period of time during which infectious individuals can transmit the disease through contact with susceptibles. Finally, the infectious individuals are removed, either through recovery or death. Quarantine or isolation was not documented in the database, and was not specifically described: possibly isolated infectious individuals remain in stage
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e015.jpg" mimetype="image"></inline-graphic>
</inline-formula>
or
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e016.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and quarantined contacts remain in stage
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e017.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<p>Thus, depending on social category, susceptible individuals may be in compartments
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e018.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(general population),
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e019.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(healthcare workers), or
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e020.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(hospitalized patients); similarly, exposed and recovered individuals may be in compartments
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e021.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e022.jpg" mimetype="image"></inline-graphic>
</inline-formula>
or
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e023.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e024.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e025.jpg" mimetype="image"></inline-graphic>
</inline-formula>
or
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e026.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, respectively; while infectious subjects are in compartments
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e027.jpg" mimetype="image"></inline-graphic>
</inline-formula>
or
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e028.jpg" mimetype="image"></inline-graphic>
</inline-formula>
before hospitalization, and in compartments
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e029.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e030.jpg" mimetype="image"></inline-graphic>
</inline-formula>
or
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e031.jpg" mimetype="image"></inline-graphic>
</inline-formula>
once hospitalized.</p>
<p>The size of the Hong Kong population (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e032.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) was obtained from local census data (
<ext-link ext-link-type="uri" xlink:href="http://www.info.gov.hk/info/hkbrief/eng/living2.htm">http://www.info.gov.hk/info/hkbrief/eng/living2.htm</ext-link>
). The number of hospitalized patients (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e033.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) equaled the number of hospital beds in Hong Kong (
<ext-link ext-link-type="uri" xlink:href="http://www.info.gov.hk/info/hkbrief/eng/living2.htm">http://www.info.gov.hk/info/hkbrief/eng/living2.htm</ext-link>
). The number of healthcare workers (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e034.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) was derived from the healthcare worker-to-bed ratio in the Hospital X
<xref rid="pcbi.1000471-Leung1" ref-type="bibr">[13]</xref>
.
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e035.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e036.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e037.jpg" mimetype="image"></inline-graphic>
</inline-formula>
were assumed to be constant throughout the epidemic. Under this steady-state assumption, transitions between compartments
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e038.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e039.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e040.jpg" mimetype="image"></inline-graphic>
</inline-formula>
did not have to be included explicitly in the model.</p>
<p>The model assumes that there is no direct contact between hospitalized individuals and non hospitalized individuals from the general population. In particular, susceptible individuals in the general population (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e041.jpg" mimetype="image"></inline-graphic>
</inline-formula>
compartment) cannot be infected by infectious hospitalized SARS cases (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e042.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e043.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e044.jpg" mimetype="image"></inline-graphic>
</inline-formula>
compartments), and susceptible hospitalized patients (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e045.jpg" mimetype="image"></inline-graphic>
</inline-formula>
compartment) cannot be infected by infectious not-yet-hospitalized cases from the general population (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e046.jpg" mimetype="image"></inline-graphic>
</inline-formula>
compartment).</p>
</sec>
<sec id="s2c">
<title>Statistical Model</title>
<p>In the following,
<bold>1</bold>
<sub>{.}</sub>
denotes the indicator function, defined by
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e047.jpg" mimetype="image"></inline-graphic>
</inline-formula>
if
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e048.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is true, and 0 otherwise.</p>
<p>For each Hong Kong inhabitant
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e049.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e050.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, let
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e051.jpg" mimetype="image"></inline-graphic>
</inline-formula>
be the time of symptom onset,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e052.jpg" mimetype="image"></inline-graphic>
</inline-formula>
the day of hospital admission,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e053.jpg" mimetype="image"></inline-graphic>
</inline-formula>
the day of hospital discharge,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e054.jpg" mimetype="image"></inline-graphic>
</inline-formula>
the day of death (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e055.jpg" mimetype="image"></inline-graphic>
</inline-formula>
if the case did not die from SARS), and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e056.jpg" mimetype="image"></inline-graphic>
</inline-formula>
the social category (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e057.jpg" mimetype="image"></inline-graphic>
</inline-formula>
if
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e058.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e059.jpg" mimetype="image"></inline-graphic>
</inline-formula>
if
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e060.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and inhabitant
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e061.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is a healthcare worker, and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e062.jpg" mimetype="image"></inline-graphic>
</inline-formula>
otherwise). For all inhabitants who were not infected by SARS, we let
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e063.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<p>For each individual
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e064.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e065.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, let
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e066.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. The observed data
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e067.jpg" mimetype="image"></inline-graphic>
</inline-formula>
were augmented with
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e068.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, where
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e069.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e070.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e071.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e072.jpg" mimetype="image"></inline-graphic>
</inline-formula>
correspond to the dates of transition into the
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e073.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e074.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e075.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e076.jpg" mimetype="image"></inline-graphic>
</inline-formula>
states respectively ;
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e077.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the date of death, and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e078.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the social category for case
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e079.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e080.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e081.jpg" mimetype="image"></inline-graphic>
</inline-formula>
or
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e082.jpg" mimetype="image"></inline-graphic>
</inline-formula>
).</p>
<p>Letting
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e083.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e084.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the joint density
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e085.jpg" mimetype="image"></inline-graphic>
</inline-formula>
of
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e086.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e087.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and of the vector
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e088.jpg" mimetype="image"></inline-graphic>
</inline-formula>
of unknown parameters is written as the following product:
<disp-formula>
<graphic xlink:href="pcbi.1000471.e089.jpg" mimetype="image" position="float"></graphic>
</disp-formula>
where
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e090.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e091.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is a prior distribution for
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e092.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<p>As defined by Auranen et al.
<xref rid="pcbi.1000471-Auranen1" ref-type="bibr">[17]</xref>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e093.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e094.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e095.jpg" mimetype="image"></inline-graphic>
</inline-formula>
refer to the observation level, the transmission level and the prior level respectively.</p>
<p>The observation level ensures that the observed data are consistent with the augmented data.</p>
<p>During the SARS outbreak, few cases were reportedly infected by asymptomatic persons, but cases rapidly became infectious after symptom onset
<xref rid="pcbi.1000471-Riley1" ref-type="bibr">[12]</xref>
,
<xref rid="pcbi.1000471-Anderson2" ref-type="bibr">[18]</xref>
,
<xref rid="pcbi.1000471-Lipsitch1" ref-type="bibr">[19]</xref>
. Therefore, for each case
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e096.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the onset of symptom was considered acceptable if
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e097.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<p>The day
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e098.jpg" mimetype="image"></inline-graphic>
</inline-formula>
of hospital admission was consistent with the augmented data if
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e099.jpg" mimetype="image"></inline-graphic>
</inline-formula>
when the case was infectious prior to hospitalization (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e100.jpg" mimetype="image"></inline-graphic>
</inline-formula>
)
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e101.jpg" mimetype="image"></inline-graphic>
</inline-formula>
when the case was infectious only after hospitalization (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e102.jpg" mimetype="image"></inline-graphic>
</inline-formula>
)
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e103.jpg" mimetype="image"></inline-graphic>
</inline-formula>
when the case was not infectious anymore at the time of hospitalization (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e104.jpg" mimetype="image"></inline-graphic>
</inline-formula>
).</p>
<p>It was also assumed that the infectious period did not outlast hospital discharge, that is
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e105.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</p>
<p>The date of death was
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e106.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<p>Finally, the professional category
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e107.jpg" mimetype="image"></inline-graphic>
</inline-formula>
was acceptable if
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e108.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<p>Hence:
<disp-formula>
<graphic xlink:href="pcbi.1000471.e109.jpg" mimetype="image" position="float"></graphic>
</disp-formula>
</p>
<p>The transmission level describes SARS transmission, assuming
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e110.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e111.jpg" mimetype="image"></inline-graphic>
</inline-formula>
are known, conditional on the day
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e112.jpg" mimetype="image"></inline-graphic>
</inline-formula>
of infection of the first case.</p>
<p>A deterministic latent period of 5 days was assumed for all cases (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e113.jpg" mimetype="image"></inline-graphic>
</inline-formula>
for
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e114.jpg" mimetype="image"></inline-graphic>
</inline-formula>
such that
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e115.jpg" mimetype="image"></inline-graphic>
</inline-formula>
)
<xref rid="pcbi.1000471-Leung1" ref-type="bibr">[13]</xref>
.</p>
<p>The duration of the infectious period (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e116.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) for SARS cases was gamma-distributed, with mean
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e117.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and variance
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e118.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. We let
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e119.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e120.jpg" mimetype="image"></inline-graphic>
</inline-formula>
denote its density and cumulative distribution function respectively. For SARS patients dead on discharge, the infectious period was considered censored by death. Since the infectious period was defined as the period during which infectious cases can transmit the disease through contact with susceptibles, its distribution was assumed to remain the same over the course of the epidemic.</p>
<p>The specific stochastic infection rates on day
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e121.jpg" mimetype="image"></inline-graphic>
</inline-formula>
for susceptible individuals in compartments
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e122.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e123.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e124.jpg" mimetype="image"></inline-graphic>
</inline-formula>
are:
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e125.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e126.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e127.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, where
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e128.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e129.jpg" mimetype="image"></inline-graphic>
</inline-formula>
denote the numbers of individuals in compartments
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e130.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e131.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, respectively;
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e132.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e133.jpg" mimetype="image"></inline-graphic>
</inline-formula>
are the daily effective contact rates in the community and hospitals, respectively;
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e134.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e135.jpg" mimetype="image"></inline-graphic>
</inline-formula>
are temporary level shift interventions
<xref rid="pcbi.1000471-Nishiura1" ref-type="bibr">[4]</xref>
reflecting the increment of infectiousness during the Hospital X and Housing Estate Y SSEs, i.e. from days
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e136.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e137.jpg" mimetype="image"></inline-graphic>
</inline-formula>
to days
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e138.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e139.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<p>This leads to:
<disp-formula>
<graphic xlink:href="pcbi.1000471.e140.jpg" mimetype="image" position="float"></graphic>
</disp-formula>
where
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e141.jpg" mimetype="image"></inline-graphic>
</inline-formula>
;
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e142.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e143.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e144.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e145.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) is the incidence in
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e146.jpg" mimetype="image"></inline-graphic>
</inline-formula>
on day
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e147.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<p>The vector
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e148.jpg" mimetype="image"></inline-graphic>
</inline-formula>
comprised about 228 unknown parameters, the epidemic lasting about
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e149.jpg" mimetype="image"></inline-graphic>
</inline-formula>
days.</p>
<p>For all model parameters except the effective contact rates, independent prior distributions were chosen. For the time of start of SSEs, the prior distributions were informative (see
<xref ref-type="table" rid="pcbi-1000471-t001">Table 1</xref>
). The effective contact rates
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e150.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e151.jpg" mimetype="image"></inline-graphic>
</inline-formula>
were modeled as second-order Gaussian random walks, on the log scale, with flat exponential priors on the first two states of the random walk. In this approach, the respective variances
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e152.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e153.jpg" mimetype="image"></inline-graphic>
</inline-formula>
of innovations correspond to the smoothing parameters of cubic smoothing splines
<xref rid="pcbi.1000471-KnorrHeld1" ref-type="bibr">[21]</xref>
; smaller values of
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e154.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e155.jpg" mimetype="image"></inline-graphic>
</inline-formula>
are associated with smoother trajectories. For the two precision parameters
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e156.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e157.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, exponential hyperpriors with mean
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e158.jpg" mimetype="image"></inline-graphic>
</inline-formula>
were selected. A sensitivity analysis of the hyperparameter value was performed (see
<xref ref-type="supplementary-material" rid="pcbi.1000471.s001">Text S1</xref>
).</p>
<table-wrap id="pcbi-1000471-t001" position="float">
<object-id pub-id-type="doi">10.1371/journal.pcbi.1000471.t001</object-id>
<label>Table 1</label>
<caption>
<title>Prior Distributions for Model Parameters.</title>
</caption>
<alternatives>
<graphic id="pcbi-1000471-t001-1" xlink:href="pcbi.1000471.t001"></graphic>
<table frame="hsides" rules="groups">
<colgroup span="1">
<col align="left" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
</colgroup>
<thead>
<tr>
<td align="left" rowspan="1" colspan="1">Parameters</td>
<td align="left" rowspan="1" colspan="1">Description</td>
<td align="left" rowspan="1" colspan="1">Prior Distributions</td>
</tr>
</thead>
<tbody>
<tr>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e159.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e160.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">Inverse variance of innovations of random walks</td>
<td align="left" rowspan="1" colspan="1">Exponential distribution (mean
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e161.jpg" mimetype="image"></inline-graphic>
</inline-formula>
)</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e162.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e163.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">Mean and standard deviation of the infectious period</td>
<td align="left" rowspan="1" colspan="1">Gamma distribution (mean 200, variance 40000)</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e164.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e165.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">Effective contact rates in the community during the first two days</td>
<td align="left" rowspan="1" colspan="1">Exponential distribution (mean 1000)</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e166.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e167.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">Effective contact rates in hospitals during the first two days</td>
<td align="left" rowspan="1" colspan="1">Exponential distribution (mean 1000)</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e168.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e169.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">Area under SSE curves</td>
<td align="left" rowspan="1" colspan="1">Exponential distribution (mean 1000)</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e170.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">Day Hospital X SSE starts</td>
<td align="left" rowspan="1" colspan="1">Uniform distribution on 02/18–03/10</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e171.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">Day Housing Estate Y SSE starts</td>
<td align="left" rowspan="1" colspan="1">Uniform distribution on 03/14–03/24</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e172.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e173.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">SSE durations</td>
<td align="left" rowspan="1" colspan="1">Poisson distribution (mean 3)</td>
</tr>
</tbody>
</table>
</alternatives>
</table-wrap>
</sec>
<sec id="s2d">
<title>Parameter Estimation</title>
<p>A Markov chain Monte Carlo (MCMC) method was used to sample the joint posterior distribution
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e174.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref rid="pcbi.1000471-Gilks1" ref-type="bibr">[22]</xref>
,
<xref rid="pcbi.1000471-Robert1" ref-type="bibr">[23]</xref>
. More details on the sampler are provided in
<xref ref-type="supplementary-material" rid="pcbi.1000471.s002">Text S2</xref>
.</p>
<p>From the joint posterior distribution of the parameters, a number of meaningful epidemiological quantities, such as daily case-reproduction numbers
<xref rid="pcbi.1000471-Fraser1" ref-type="bibr">[24]</xref>
in each category (see
<xref ref-type="supplementary-material" rid="pcbi.1000471.s003">Text S3</xref>
), could be derived. In particular, the number of cases generated by each SSE could be estimated.</p>
</sec>
</sec>
<sec id="s3">
<title>Results</title>
<p>Estimates of the days of SSE starts and ends, increments (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e175.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e176.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), and the number of SSE cases in Hospital X and Housing Estate Y are shown in
<xref ref-type="table" rid="pcbi-1000471-t002">Table 2</xref>
. Despite the somewhat shorter SSE duration for Housing Estate Y than for Hospital X, 2.5 times more cases occurred in Housing Estate Y than Hospital X.</p>
<table-wrap id="pcbi-1000471-t002" position="float">
<object-id pub-id-type="doi">10.1371/journal.pcbi.1000471.t002</object-id>
<label>Table 2</label>
<caption>
<title>Estimated Parameters for Super Spreading Events in Hospital X and Housing Estate Y.</title>
</caption>
<alternatives>
<graphic id="pcbi-1000471-t002-2" xlink:href="pcbi.1000471.t002"></graphic>
<table frame="hsides" rules="groups">
<colgroup span="1">
<col align="left" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
</colgroup>
<thead>
<tr>
<td align="left" rowspan="1" colspan="1">Site</td>
<td align="left" rowspan="1" colspan="1">Parameter</td>
<td align="left" rowspan="1" colspan="1">Unit</td>
<td align="left" rowspan="1" colspan="1">Mean</td>
<td align="left" rowspan="1" colspan="1">95% Credible Intervals</td>
</tr>
</thead>
<tbody>
<tr>
<td align="left" rowspan="1" colspan="1">Hospital X</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e177.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">March 1, 2003</td>
<td align="left" rowspan="1" colspan="1">February 28–March 2, 2003</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e178.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">March 11, 2003</td>
<td align="left" rowspan="1" colspan="1">March 10–March 13, 2003</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e179.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">Days
<sup>−1</sup>
</td>
<td align="left" rowspan="1" colspan="1">1.4×10
<sup>−4</sup>
</td>
<td align="left" rowspan="1" colspan="1">(1.1×10
<sup>−4</sup>
–1.8×10
<sup>−4</sup>
)</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">Size</td>
<td align="left" rowspan="1" colspan="1">Cases</td>
<td align="left" rowspan="1" colspan="1">94</td>
<td align="left" rowspan="1" colspan="1">(72–118)</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">Housing Estate Y</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e180.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">March 18, 2003</td>
<td align="left" rowspan="1" colspan="1">March 17–March 18, 2003</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e181.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">March 23, 2003</td>
<td align="left" rowspan="1" colspan="1">March 23–March 24, 2003</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e182.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">Days
<sup>−1</sup>
</td>
<td align="left" rowspan="1" colspan="1">5.2×10
<sup>−6</sup>
</td>
<td align="left" rowspan="1" colspan="1">(4.1×10
<sup>−6</sup>
–6.4×10
<sup>−6</sup>
)</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">Size</td>
<td align="left" rowspan="1" colspan="1">Cases</td>
<td align="left" rowspan="1" colspan="1">235</td>
<td align="left" rowspan="1" colspan="1">(185–288)</td>
</tr>
</tbody>
</table>
</alternatives>
</table-wrap>
<p>The estimated mean of the infectious period was 9.3 days (95% credible interval: (8.6–9.9)), with an estimated standard deviation of 2.3 days (95% credible interval: (1.8–2.9)). The proportion of the infectious period spent in the community decreased continuously with time (>60% at the beginning, <20% as early as early April). Toward the end of the epidemic, >95% of the infectious period was spent inside hospitals (see
<xref ref-type="fig" rid="pcbi-1000471-g003">Figure 3</xref>
).</p>
<fig id="pcbi-1000471-g003" position="float">
<object-id pub-id-type="doi">10.1371/journal.pcbi.1000471.g003</object-id>
<label>Figure 3</label>
<caption>
<title>Proportion of Infectious Period Spent in the Community before Hospitalization as a Function of Week of Symptom Onset.</title>
</caption>
<graphic xlink:href="pcbi.1000471.g003"></graphic>
</fig>
<p>The daily effective contact rates in the community (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e183.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) and hospitals (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e184.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) exhibited progressive a decrease in time, as shown in
<xref ref-type="fig" rid="pcbi-1000471-g004">Figure 4</xref>
. However, while the contact rate was almost 0 by late March inside hospitals, it remained >0.17 in the community.</p>
<fig id="pcbi-1000471-g004" position="float">
<object-id pub-id-type="doi">10.1371/journal.pcbi.1000471.g004</object-id>
<label>Figure 4</label>
<caption>
<title>Mean Effective SARS Contact Rates (solid line) and 95% Credible Intervals (dashed line) in the Community and Hospitals as a Function of Time and Dates of Important Events.</title>
</caption>
<graphic xlink:href="pcbi.1000471.g004"></graphic>
</fig>
<p>The case-reproduction number first increased to 5.1 in the general population in late February and to 3.0 for healthcare workers in early March (see
<xref ref-type="supplementary-material" rid="pcbi.1000471.s003">Text S3</xref>
). It then decreased until the end of the epidemic. The case-reproduction numbers was <1 on March 13 for healthcare workers and on March 20 for the general population. Among nosocomial cases, the case-reproduction number was always <1, with a maximum value of 0.2 on March 14.</p>
<p>The model's ability to reproduce the main features of the epidemic was checked by simulating 5000 epidemics with parameters sampled from the estimated joint posterior distribution, as described in
<xref ref-type="supplementary-material" rid="pcbi.1000471.s004">Text S4</xref>
. The size and duration of simulated epidemics, as well as cases breakdown in categories (
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e185.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e186.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pcbi.1000471.e187.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) mirrored the Hong Kong epidemic (see
<xref ref-type="fig" rid="pcbi-1000471-g002">Figure 2</xref>
).</p>
<p>We also simulated 100 epidemics with a single set of parameters, sampled from the posterior distribution. Then, the estimation procedure was applied to each simulated epidemic in order to reestimate the parameters. The original parameters were in the estimated corresponding 95% credible interval in 87% of cases.</p>
</sec>
<sec id="s4">
<title>Discussion</title>
<p>To rapidly and economically design and assess control measures for epidemics in modern societies, added insight into the dynamics of disease transmission is needed. These dynamics are conveniently summarized by critical, albeit non-observable, characteristics, such as the duration of the infectious period and effective contact rates. Estimation of these parameters from the observed data requires the development of mathematical models. Herein, we presented a model for epidemics that provides for social heterogeneity and time variability of transmission parameters. As a working example, the model was applied to the 2003 SARS epidemic in Hong Kong.</p>
<p>The effect of interventions and/or changes in behavior during the 2003 SARS outbreak may be modelled as time varying contact rates
<xref rid="pcbi.1000471-Riley1" ref-type="bibr">[12]</xref>
,
<xref rid="pcbi.1000471-LloydSmith2" ref-type="bibr">[15]</xref>
,
<xref rid="pcbi.1000471-Gumel1" ref-type="bibr">[25]</xref>
or involve shortening of the infectious period
<xref rid="pcbi.1000471-Lipsitch1" ref-type="bibr">[19]</xref>
. Here, we adopted the first view. To assess if the data supported this choice, a model was fit where, in addition to time varying contact rates, we allowed the mean infectious period to change over three consecutive periods. The three posterior means were 9.5 days (before March 20), 9.2 (March 21 to April 9) and 10 days (after April 10), indicating that the time varying contact rates alone model the data adequately.</p>
<p>While the duration of the infectious period is an obvious determinant of disease transmission, no estimate has been available for SARS. The distribution of the viral load was found to peak 8–10 days after symptom onset
<xref rid="pcbi.1000471-Leung1" ref-type="bibr">[13]</xref>
,
<xref rid="pcbi.1000471-Anderson2" ref-type="bibr">[18]</xref>
,
<xref rid="pcbi.1000471-Peiris1" ref-type="bibr">[26]</xref>
,
<xref rid="pcbi.1000471-Pitzer1" ref-type="bibr">[27]</xref>
. Here, assuming that the infectious period started between 1 day before and 4 days after symptom onset, it was estimated to extend over an average period of 9.3 days. We also found that the proportion of time infectious people spent outside hospitals decreased during the outbreak and was <5% at the very end, in agreement with Anderson et al.
<xref rid="pcbi.1000471-Anderson2" ref-type="bibr">[18]</xref>
and Leung et al.
<xref rid="pcbi.1000471-Leung1" ref-type="bibr">[13]</xref>
who showed that the time from symptom onset to admission was shorter at the end of the epidemic.</p>
<p>One of the most striking features of the Hong Kong SARS epidemic was the occurrence of two SSEs. By definition, SSEs correspond to exceptional circumstances that are usually limited to well-circumscribed areas, such as Hospital X and Housing Estate Y, and last for only a few days
<xref rid="pcbi.1000471-LloydSmith1" ref-type="bibr">[10]</xref>
. In this respect, the very high contact rates generated by the SSEs were modeled as ‘innovation outliers’
<xref rid="pcbi.1000471-Fox1" ref-type="bibr">[28]</xref>
, to avoid spurious overestimation of contact rates among the Hong Kong population.</p>
<p>Whether SSEs are a result of a few particularly highly infectious cases (excreting much virus and/or highly connected socially), or of particular environmental circumstances, or maybe both, remains unclear
<xref rid="pcbi.1000471-Kwok1" ref-type="bibr">[16]</xref>
,
<xref rid="pcbi.1000471-Li1" ref-type="bibr">[29]</xref>
,
<xref rid="pcbi.1000471-Wong1" ref-type="bibr">[30]</xref>
. In our model, the force of infection associated with each SSE was independent on the number of currently infectious cases. The duration of SSEs was estimated independently for each SSE, and was independent on the duration of the infectious period. Therefore, our model was consistent with all possible causes of SSEs: one or several super-spreaders, or particular environmental circumstances, etc.</p>
<p>The level shift interventions
<xref rid="pcbi.1000471-Box1" ref-type="bibr">[20]</xref>
that were superimposed on the process describing the time evolution of the infection rates differed significantly from zero. Taking into account only serologically confirmed cases, we estimated that the Hospital X SSE began on March 1
<sup>st</sup>
, lasted 11 days and was responsible for 94 cases; and that the Housing Estate Y SSE began on March 18, lasted 6 days and caused 235 cases. Previous studies investigating SSEs in Hong Kong used all cases. By contact tracing, Lee et al.
<xref rid="pcbi.1000471-Lee1" ref-type="bibr">[11]</xref>
found that the Hospital X SSE started on March 4 and involved 125 cases; the Housing Estate Y SSE had been estimated to start on March 19
<xref rid="pcbi.1000471-Leung1" ref-type="bibr">[13]</xref>
and to involve 312–330
<xref rid="pcbi.1000471-Leung1" ref-type="bibr">[13]</xref>
or 331
<xref rid="pcbi.1000471-Riley1" ref-type="bibr">[12]</xref>
cases.</p>
<p>Effective contact rates were estimated on a daily basis, in the community and hospitals. Both rates tended to decline, probably reflecting the effect of control measures (listed in
<xref ref-type="fig" rid="pcbi-1000471-g004">Figure 4</xref>
<xref rid="pcbi.1000471-World3" ref-type="bibr">[31]</xref>
,
<xref rid="pcbi.1000471-WSARS1" ref-type="bibr">[32]</xref>
) or self-adopted behavioral changes. The measures seem to have been particularly effective in hospitals, where the effective contact rate was 0 by late March, whereas the risk in the community did not decrease as sharply. In both settings, the effective contact rate was almost constant after late March, when no more control measures were introduced.</p>
<p>Others who studied the dependence of disease transmission on time reported reproduction numbers rather than effective contact rates
<xref rid="pcbi.1000471-Riley1" ref-type="bibr">[12]</xref>
,
<xref rid="pcbi.1000471-Cauchemez1" ref-type="bibr">[14]</xref>
. While the daily effective contact rates are sensitive to short-term day-to-day variations in transmission, the reproduction numbers reflect the integrated influences of the temporal evolution of effective contact rates, the infectious period duration and other factors, such as time spent in the community before hospitalization. Here, estimates of daily reproduction numbers were obtained for each social category. Notably, unlike Cauchemez et al.
<xref rid="pcbi.1000471-Cauchemez1" ref-type="bibr">[14]</xref>
, it was not necessary to assume prior knowledge or constancy of the generation interval. The reproduction numbers showed a trend similar to the effective contact rates, with a clearly decreasing trend over time, suggesting that the epidemic was under control as early as mid-March (see Figure in
<xref ref-type="supplementary-material" rid="pcbi.1000471.s003">Text S3</xref>
). Moreover, the temporal patterns for the general population and healthcare workers were similar, with the reproduction numbers being higher for the general population, thereby indicating that on average, an infectious healthcare worker did not infect more people than any other infectious person. The reproduction numbers for nosocomial cases were much lower, either because they had fewer contacts or because the people they were in contact with were protected (typically healthcare workers wearing masks).</p>
<p>Our estimation procedure, applied to a set of 100 simulated epidemics, showed that in 87% of cases, the parameters used for simulation were inside the corresponding posterior 95% credible intervals. While most parameters were well estimated, the procedure tended to overestimate the duration of each SSE, while simultaneously underestimating its strength. The number of people affected by each SSE (i.e. population×duration×strength) was therefore correct, but its extent in time less robust. Ignoring the 17 days corresponding to both SSEs, 98% of the remaining parameters used for simulation were inside the posterior corresponding 95% credible intervals, indicating very little bias in our estimation procedure.</p>
<p>Herein, we described an approach to estimate the role of time variability and social heterogeneity in epidemic dynamics. Our model's simplifying assumptions such as the fixed duration of the latency period or the constant probability of transmission throughout the infectious period of cases, can be relaxed at the price of increasing complexity. Similarly, a more detailed model taking into account household transmission, and transmission inside and between hospitals, rather than assuming homogeneous mixing in the community and in hospitals, could be implemented, at the cost of a dramatic increase in the number of model parameters. More generally, the model can be easily accommodated to fit the specificities of any transmissible disease.</p>
</sec>
<sec sec-type="supplementary-material" id="s5">
<title>Supporting Information</title>
<supplementary-material content-type="local-data" id="pcbi.1000471.s001">
<label>Text S1</label>
<caption>
<p>Choice of Hyperparameter
<italic>η</italic>
</p>
<p>(0.09 MB PDF)</p>
</caption>
<media xlink:href="pcbi.1000471.s001.pdf">
<caption>
<p>Click here for additional data file.</p>
</caption>
</media>
</supplementary-material>
<supplementary-material content-type="local-data" id="pcbi.1000471.s002">
<label>Text S2</label>
<caption>
<p>Sampler Used for Parameter Estimation</p>
<p>(0.07 MB PDF)</p>
</caption>
<media xlink:href="pcbi.1000471.s002.pdf">
<caption>
<p>Click here for additional data file.</p>
</caption>
</media>
</supplementary-material>
<supplementary-material content-type="local-data" id="pcbi.1000471.s003">
<label>Text S3</label>
<caption>
<p>Estimation of the Case-Reproduction Number
<italic>R
<sup>A→B</sup>
<sub>t</sub>
</italic>
</p>
<p>(0.39 MB PDF)</p>
</caption>
<media xlink:href="pcbi.1000471.s003.pdf">
<caption>
<p>Click here for additional data file.</p>
</caption>
</media>
</supplementary-material>
<supplementary-material content-type="local-data" id="pcbi.1000471.s004">
<label>Text S4</label>
<caption>
<p>Epidemic Simulation</p>
<p>(0.04 MB PDF)</p>
</caption>
<media xlink:href="pcbi.1000471.s004.pdf">
<caption>
<p>Click here for additional data file.</p>
</caption>
</media>
</supplementary-material>
</sec>
</body>
<back>
<ack>
<p>The authors thank the Hong Kong Department of Health and Hong Kong Hospital Authority which were involved with the public health control of the SARS epidemic, and data collection and processing.</p>
</ack>
<fn-group>
<fn fn-type="COI-statement">
<p>The authors have declared that no competing interests exist.</p>
</fn>
<fn fn-type="financial-disclosure">
<p>The authors thank the following for research funding: The Research Fund for the Control of Infectious Diseases of the Food and Health Bureau of the Hong Kong SAR Government (GML); The University of Hong Kong SARS Research Fund (GML); The EU Sixth Framework Programme for Research for Policy Support (contracts SP22-CT-2004-511066 and FP6-2003-SSP2-513715) (AC, P-YB, GT, GML, A-JV). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.</p>
</fn>
</fn-group>
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