Does homologous reinfection drive multiple-wave influenza outbreaks? Accounting for immunodynamics in epidemiological models☆
Identifieur interne : 000613 ( Pmc/Checkpoint ); précédent : 000612; suivant : 000614Does homologous reinfection drive multiple-wave influenza outbreaks? Accounting for immunodynamics in epidemiological models☆
Auteurs : A. Camacho [France, Royaume-Uni] ; B. Cazelles [France]Source :
- Epidemics [ 1755-4365 ] ; 2013.
Abstract
We model the primary immune responses to influenza infection in humans. We examine the interplay between immunological and epidemiological dynamics. The model explains cases of homologous reinfection reported during past pandemics. Three epidemic profiles can arise depending on the degree of population mixing. A substantial proportion of infected host would remain unprotected after the 2009 influenza pandemic.
Url:
DOI: 10.1016/j.epidem.2013.09.003
PubMed: 24267875
PubMed Central: 3863957
Affiliations:
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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Does homologous reinfection drive multiple-wave influenza outbreaks? Accounting for immunodynamics in epidemiological models<sup><xref ref-type="fn" rid="d32e809">☆</xref></sup></title><author><name sortKey="Camacho, A" sort="Camacho, A" uniqKey="Camacho A" first="A." last="Camacho">A. Camacho</name><affiliation wicri:level="3"><nlm:aff id="aff0005">Eco-Evolution Mathématique, UMR 7625, CNRS-UPMC-ENS, 75230 Paris Cedex 05, France</nlm:aff><country xml:lang="fr">France</country><wicri:regionArea>Eco-Evolution Mathématique, UMR 7625, CNRS-UPMC-ENS, 75230 Paris Cedex 05</wicri:regionArea><placeName><region type="region" nuts="2">Île-de-France</region><settlement type="city">Paris</settlement></placeName></affiliation><affiliation wicri:level="3"><nlm:aff id="aff0010">Department of Infectious Disease Epidemiology, London School of Hygiene and Tropical Medicine, London, United Kingdom</nlm:aff><country xml:lang="fr">Royaume-Uni</country><wicri:regionArea>Department of Infectious Disease Epidemiology, London School of Hygiene and Tropical Medicine, London</wicri:regionArea><placeName><settlement type="city">Londres</settlement><region type="country">Angleterre</region><region type="région" nuts="1">Grand Londres</region></placeName></affiliation></author><author><name sortKey="Cazelles, B" sort="Cazelles, B" uniqKey="Cazelles B" first="B." last="Cazelles">B. Cazelles</name><affiliation wicri:level="3"><nlm:aff id="aff0005">Eco-Evolution Mathématique, UMR 7625, CNRS-UPMC-ENS, 75230 Paris Cedex 05, France</nlm:aff><country xml:lang="fr">France</country><wicri:regionArea>Eco-Evolution Mathématique, UMR 7625, CNRS-UPMC-ENS, 75230 Paris Cedex 05</wicri:regionArea><placeName><region type="region" nuts="2">Île-de-France</region><settlement type="city">Paris</settlement></placeName></affiliation><affiliation wicri:level="3"><nlm:aff id="aff0015">UMMISCO UMI 209 IRD-UPMC, F-93142 Bondy, France</nlm:aff><country xml:lang="fr">France</country><wicri:regionArea>UMMISCO UMI 209 IRD-UPMC, F-93142 Bondy</wicri:regionArea><placeName><region type="region" nuts="2">Île-de-France</region><settlement type="city">Bondy</settlement></placeName></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">24267875</idno><idno type="pmc">3863957</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3863957</idno><idno type="RBID">PMC:3863957</idno><idno type="doi">10.1016/j.epidem.2013.09.003</idno><date when="2013">2013</date><idno type="wicri:Area/Pmc/Corpus">000A27</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000A27</idno><idno type="wicri:Area/Pmc/Curation">000A27</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Curation">000A27</idno><idno type="wicri:Area/Pmc/Checkpoint">000613</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Checkpoint">000613</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Does homologous reinfection drive multiple-wave influenza outbreaks? Accounting for immunodynamics in epidemiological models<sup><xref ref-type="fn" rid="d32e809">☆</xref></sup></title><author><name sortKey="Camacho, A" sort="Camacho, A" uniqKey="Camacho A" first="A." last="Camacho">A. Camacho</name><affiliation wicri:level="3"><nlm:aff id="aff0005">Eco-Evolution Mathématique, UMR 7625, CNRS-UPMC-ENS, 75230 Paris Cedex 05, France</nlm:aff><country xml:lang="fr">France</country><wicri:regionArea>Eco-Evolution Mathématique, UMR 7625, CNRS-UPMC-ENS, 75230 Paris Cedex 05</wicri:regionArea><placeName><region type="region" nuts="2">Île-de-France</region><settlement type="city">Paris</settlement></placeName></affiliation><affiliation wicri:level="3"><nlm:aff id="aff0010">Department of Infectious Disease Epidemiology, London School of Hygiene and Tropical Medicine, London, United Kingdom</nlm:aff><country xml:lang="fr">Royaume-Uni</country><wicri:regionArea>Department of Infectious Disease Epidemiology, London School of Hygiene and Tropical Medicine, London</wicri:regionArea><placeName><settlement type="city">Londres</settlement><region type="country">Angleterre</region><region type="région" nuts="1">Grand Londres</region></placeName></affiliation></author><author><name sortKey="Cazelles, B" sort="Cazelles, B" uniqKey="Cazelles B" first="B." last="Cazelles">B. Cazelles</name><affiliation wicri:level="3"><nlm:aff id="aff0005">Eco-Evolution Mathématique, UMR 7625, CNRS-UPMC-ENS, 75230 Paris Cedex 05, France</nlm:aff><country xml:lang="fr">France</country><wicri:regionArea>Eco-Evolution Mathématique, UMR 7625, CNRS-UPMC-ENS, 75230 Paris Cedex 05</wicri:regionArea><placeName><region type="region" nuts="2">Île-de-France</region><settlement type="city">Paris</settlement></placeName></affiliation><affiliation wicri:level="3"><nlm:aff id="aff0015">UMMISCO UMI 209 IRD-UPMC, F-93142 Bondy, France</nlm:aff><country xml:lang="fr">France</country><wicri:regionArea>UMMISCO UMI 209 IRD-UPMC, F-93142 Bondy</wicri:regionArea><placeName><region type="region" nuts="2">Île-de-France</region><settlement type="city">Bondy</settlement></placeName></affiliation></author></analytic><series><title level="j">Epidemics</title><idno type="ISSN">1755-4365</idno><idno type="eISSN">1878-0067</idno><imprint><date when="2013">2013</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><title>Highlights</title><p><list list-type="simple"><list-item id="listitem0005"><label>•</label><p>We model the primary immune responses to influenza infection in humans.</p></list-item><list-item id="listitem0010"><label>•</label><p>We examine the interplay between immunological and epidemiological dynamics.</p></list-item><list-item id="listitem0015"><label>•</label><p>The model explains cases of homologous reinfection reported during past pandemics.</p></list-item><list-item id="listitem0020"><label>•</label><p>Three epidemic profiles can arise depending on the degree of population mixing.</p></list-item><list-item id="listitem0025"><label>•</label><p>A substantial proportion of infected host would remain unprotected after the 2009 influenza pandemic.</p></list-item></list></p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Anonymous" uniqKey="Anonymous">Anonymous</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Baguelin, M" uniqKey="Baguelin M">M. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">Epidemics</journal-id><journal-id journal-id-type="iso-abbrev">Epidemics</journal-id><journal-title-group><journal-title>Epidemics</journal-title></journal-title-group><issn pub-type="ppub">1755-4365</issn><issn pub-type="epub">1878-0067</issn><publisher><publisher-name>Elsevier</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">24267875</article-id><article-id pub-id-type="pmc">3863957</article-id><article-id pub-id-type="publisher-id">S1755-4365(13)00042-X</article-id><article-id pub-id-type="doi">10.1016/j.epidem.2013.09.003</article-id><article-categories><subj-group subj-group-type="heading"><subject>Article</subject></subj-group></article-categories><title-group><article-title>Does homologous reinfection drive multiple-wave influenza outbreaks? Accounting for immunodynamics in epidemiological models<sup><xref ref-type="fn" rid="d32e809">☆</xref></sup></article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Camacho</surname><given-names>A.</given-names></name><email>anton.camacho@lshtm.ac.uk</email><xref rid="aff0005" ref-type="aff">a</xref><xref rid="aff0010" ref-type="aff">b</xref><xref rid="cor0005" ref-type="corresp">⁎</xref></contrib><contrib contrib-type="author"><name><surname>Cazelles</surname><given-names>B.</given-names></name><xref rid="aff0005" ref-type="aff">a</xref><xref rid="aff0015" ref-type="aff">c</xref></contrib></contrib-group><aff id="aff0005"><label>a</label>Eco-Evolution Mathématique, UMR 7625, CNRS-UPMC-ENS, 75230 Paris Cedex 05, France</aff><aff id="aff0010"><label>b</label>Department of Infectious Disease Epidemiology, London School of Hygiene and Tropical Medicine, London, United Kingdom</aff><aff id="aff0015"><label>c</label>UMMISCO UMI 209 IRD-UPMC, F-93142 Bondy, France</aff><author-notes><corresp id="cor0005"><label>⁎</label>Corresponding author at: Department of Infectious Disease Epidemiology, London School of Hygiene and Tropical Medicine, London, United Kingdom. Tel.: +44 2079272407. <email>anton.camacho@lshtm.ac.uk</email></corresp></author-notes><pub-date pub-type="pmc-release"><day>1</day><month>12</month><year>2013</year></pub-date><pmc-comment> PMC Release delay is 0 months and 0 days and was based on .</pmc-comment>
<pub-date pub-type="ppub"><month>12</month><year>2013</year></pub-date><volume>5</volume><issue>4</issue><fpage>187</fpage><lpage>196</lpage><history><date date-type="received"><day>30</day><month>6</month><year>2012</year></date><date date-type="rev-recd"><day>6</day><month>9</month><year>2013</year></date><date date-type="accepted"><day>23</day><month>9</month><year>2013</year></date></history><permissions><copyright-statement>© 2013 The Authors</copyright-statement><copyright-year>2013</copyright-year><license xlink:href="https://creativecommons.org/licenses/by/3.0/"><license-p>Open Access under <ext-link ext-link-type="uri" xlink:href="https://creativecommons.org/licenses/by/3.0/">CC BY 3.0</ext-link> license</license-p></license></permissions><abstract abstract-type="author-highlights"><title>Highlights</title><p><list list-type="simple"><list-item id="listitem0005"><label>•</label><p>We model the primary immune responses to influenza infection in humans.</p></list-item><list-item id="listitem0010"><label>•</label><p>We examine the interplay between immunological and epidemiological dynamics.</p></list-item><list-item id="listitem0015"><label>•</label><p>The model explains cases of homologous reinfection reported during past pandemics.</p></list-item><list-item id="listitem0020"><label>•</label><p>Three epidemic profiles can arise depending on the degree of population mixing.</p></list-item><list-item id="listitem0025"><label>•</label><p>A substantial proportion of infected host would remain unprotected after the 2009 influenza pandemic.</p></list-item></list></p></abstract><abstract><p>Epidemiological models of influenza transmission usually assume that recovered individuals instantly develop a fully protective immunity against the infecting strain. However, recent studies have highlighted host heterogeneity in the development of this immune response, characterized by delay and even absence of protection, that could lead to homologous reinfection (HR). Here, we investigate how these immunological mechanisms at the individual level shape the epidemiological dynamics at the population level. In particular, because HR was observed during the successive waves of past pandemics, we assess its role in driving multiple-wave influenza outbreaks. We develop a novel mechanistic model accounting for host heterogeneity in the immune response. Immunological parameters are inferred by fitting our dynamical model to a two-wave influenza epidemic that occurred on the remote island of Tristan da Cunha (TdC) in 1971, and during which HR occurred in 92 of 284 islanders. We then explore the dynamics predicted by our model for various population settings. We find that our model can explain HR over both short (e.g. week) and long (e.g. month) time-scales, as reported during past pandemics. In particular, our results reveal that the HR wave on TdC was a natural consequence of the exceptional contact configuration and high susceptibility of this small and isolated community. By contrast, in larger, less mixed and partially protected populations, HR alone cannot generate multiple-wave outbreaks. However, in the latter case, we find that a significant proportion of infected hosts would remain unprotected at the end of the pandemic season and should therefore benefit from vaccination. Crucially, we show that failing to account for these unprotected individuals can lead to large underestimation of the magnitude of the first post-pandemic season. These results are relevant in the context of the 2009 A/H1N1 influenza post-pandemic era.</p></abstract><kwd-group><title>Keywords</title><kwd>Influenza</kwd><kwd>Mechanistic modelling</kwd><kwd>Multiple-wave outbreak</kwd><kwd>Pandemic</kwd><kwd>Primary immune response</kwd><kwd>Reinfection</kwd></kwd-group></article-meta></front><floats-group><fig id="fig0005"><label>Fig. 1</label><caption><p>Mechanistic modelling of the primary immune response to influenza. (A) Schematized dynamics of the viral load as well as the innate and adaptive immune responses, as described in section “The primary immune response to influenza infection in humans”. (B) The SEICWH model. The six immunological stages are <italic>S</italic>: susceptible, <italic>E</italic>: exposed, <italic>I</italic>: clinically ill and infectious, <italic>C</italic>: temporarily protected by the cellular response, <italic>W</italic>: temporarily susceptible, <italic>H</italic>: protected on the long-term by the humoral response. The number of sub-compartments in each immunological stages corresponds to the shape of the Erlang distribution for the residence time in this stage (see section “Mechanistic modelling”). The infection force is <italic>λ</italic> = <italic>β</italic>(<italic>I</italic><sub>1</sub> + <italic>I</italic><sub>2</sub>)/<italic>Ω</italic>. A description of the parameters can be found in <xref rid="tbl0010" ref-type="table">Table 2</xref>. The transition rates used to stochastically simulate the model are provided in <xref rid="tbl0005" ref-type="table">Table 1</xref>. The set of ordinary differential equations used for deterministic simulations can be found in Text S3.</p></caption><graphic xlink:href="gr1"></graphic></fig><fig id="fig0010"><label>Fig. 2</label><caption><p>Detailed analysis of the 10<sup>5</sup> realizations of the stochastic SEICWH model for the 1971 TdC epidemic using Gillespie's algorithm (<xref rid="bib0100" ref-type="bibr">Gillespie, 1977</xref>). Upper panel: original incidence data (black dots) and expected incidence (red line) conditioned on non-extinction under the best fit model together with 50 and 95 percentile intervals (red envelopes) due to demographic stochasticity. This figure demonstrates that the best fit of the SEICWH model captures the shape and the dynamics of the data. Lower panel: time course of the extinction probability <italic>p</italic>(<italic>t</italic>) defined as the probability that the epidemic has faded out by time <italic>t</italic> and estimated by the proportion of fade-out realizations at time <italic>t</italic>. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)</p></caption><graphic xlink:href="gr2"></graphic></fig><fig id="fig0015"><label>Fig. 3</label><caption><p>Dynamics of the immune response, under the SEICWH model, inferred from the 1971 TdC epidemic. At the population level, our framework allows us to reconstruct the proportion of individuals that are infectious (dotted black line), short-term protected thanks to the innate and cellular immunity (dot-dashed black line), protected on the long-term thanks to antibodies (dashed black line) and unprotected (solid black line) by interval since symptom onset. These proportions correspond, respectively, to the probabilities <italic>P</italic><sub>1–4</sub>(<italic>τ</italic>) described in section “Quantities of epidemiological interest”. The dashed red line corresponds to the proportion of individuals seroconverted by interval since symptom onset obtained by <xref rid="bib0015" ref-type="bibr">Baguelin et al. (2011)</xref>. This study involved 115 individuals infected during the 2009 A/H1N1 pandemic and the seroconversion interval of each individual was defined as the time taken to reach an HI titre of ≥32. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)</p></caption><graphic xlink:href="gr3"></graphic></fig><fig id="fig0020"><label>Fig. 4</label><caption><p>Example of the three typical epidemic profiles generated by the SEICWH model, depending on the value of <italic>R</italic><sub>0</sub>. A: <italic>R</italic><sub>0</sub> = 1.4, B: <italic>R</italic><sub>0</sub> = 4 and C: <italic>R</italic><sub>0</sub> = 10. These values reflect the tendency of the contact rate among individuals to increase as the community becomes smaller. Upper panels: contribution of infection and HR to the time series of the prevalence. The dashed line represents the average daily inflow of unprotected individuals (<inline-formula><mml:math id="M20" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi mathvariant="script">U</mml:mi><mml:mi>d</mml:mi></mml:msub></mml:math></inline-formula>). Lower panels: time course of the effective reproduction number <italic>R</italic><sub><italic>e</italic></sub>(<italic>t</italic>). The solid line represents the threshold <italic>R</italic><sub><italic>e</italic></sub> = 1, above which the epidemic can grow. (For interpretation of the references to color in text, the reader is referred to the web version of the article.)</p></caption><graphic xlink:href="gr4"></graphic></fig><fig id="fig0025"><label>Fig. 5</label><caption><p>Change in the fractions of infected (<italic>F</italic><sub><italic>I</italic>.</sub>, upper panel), reinfected (<italic>F</italic><sub><italic>II</italic>.</sub>, middle panel) and unprotected (<italic>F</italic><sub><italic>IS</italic></sub>, lower panel) individuals at the end of the epidemic as a function of <italic>R</italic><sub>0</sub>. Each colour refers to an epidemic profile of <xref rid="fig0020" ref-type="fig">Fig. 4</xref> (bell: green, tail end: orange, two-peaks: violet). The impact of the reinfection dynamics on <italic>F</italic><sub><italic>I</italic>.</sub> can be obtained by subtracting the same fraction expected under the SEIR model (solid line, upper panel). The expected fraction of individuals reinfected following a lack of humoral response corresponds to <italic>αF</italic><sub><italic>I</italic>.</sub> − <italic>F</italic><sub><italic>IS</italic></sub> (dashed line, middle panel). The HR threshold <inline-formula><mml:math id="M21" altimg="si2.gif" overflow="scroll"><mml:msubsup><mml:mi>R</mml:mi><mml:mn>0</mml:mn><mml:mo>*</mml:mo></mml:msubsup></mml:math></inline-formula> is plotted as a dotted line and estimates of <italic>R</italic><sub>0</sub> for the 2009 A/H1N1 pandemic (≈1.4, black triangle) and for the 1971 TdC epidemic (≈12, black dot) are also mapped. Note the log-scale on the <italic>x</italic>-axis. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)</p></caption><graphic xlink:href="gr5"></graphic></fig><fig id="fig0030"><label>Fig. 6</label><caption><p>Implication of immunodynamics for the first post-pandemic season. Upper panels: expected fraction of individuals infected at least once by a mutant strain during the post-pandemic season (<italic>F</italic><sub><italic>I</italic>.</sub> post-pdm, colour coded), as a function of the relative increase in transmissibility (Δ<italic>R</italic><sub>0</sub>/<italic>R</italic><sub>0</sub> ∈ [0, 1]) and the level of immune escape (<italic>σ</italic> ∈ [0, 0.5]) to the pandemic strain (the choice of these parameter ranges is justified in Text S5 based on published studies). Results are given for five reasonable values of <italic>R</italic><sub>0</sub> for pandemic scenarios (between 1.2 and 2). For each value of <italic>R</italic><sub>0</sub>, the expected fraction of protected (<inline-formula><mml:math id="M22" altimg="si3.gif" overflow="scroll"><mml:mover accent="true"><mml:mi>H</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math></inline-formula>) and unprotected (<inline-formula><mml:math id="M23" altimg="si4.gif" overflow="scroll"><mml:mi mathvariant="normal">Δ</mml:mi><mml:mover accent="true"><mml:mi>H</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math></inline-formula>) individuals at the end of the pandemic are also given. The fraction <inline-formula><mml:math id="M24" altimg="si5.gif" overflow="scroll"><mml:mover accent="true"><mml:mi>H</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math></inline-formula> is therefore partially protected against the mutant strain by a factor of reduction of susceptibility 1 − <italic>σ</italic> (see also Text S5). Finally, the fraction of infected individuals during the pandemic season (<italic>F</italic><sub><italic>I</italic>.</sub> pdm) is also mapped as a black dotted isocline for comparison with <italic>F</italic><sub><italic>I</italic>.</sub> post-pdm (colour-coded). Lower panels: effect of assuming that all infected individuals develop an efficient humoral response during the pandemic and thus a partial-protection against the mutant strain. We compared the SEICWH model with initially <inline-formula><mml:math id="M25" altimg="si6.gif" overflow="scroll"><mml:mover accent="true"><mml:mi>H</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math></inline-formula> partially protected individuals with a SEIR model with <inline-formula><mml:math id="M26" altimg="si7.gif" overflow="scroll"><mml:mover accent="true"><mml:mi>H</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mover accent="true"><mml:mi>H</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math></inline-formula> as initial condition and plotted the difference (Δ<italic>F</italic><sub><italic>I</italic>.</sub> post-pdm, colour-coded). Since the SEIR model overestimates the population herd immunity, it predicts a greater invasion threshold for the post-pandemic mutant (isocline <italic>R</italic><sub><italic>e</italic></sub> = 1, black solid line) than the SEICWH model. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)</p></caption><graphic xlink:href="gr6"></graphic></fig><table-wrap id="tbl0005" position="float"><label>Table 1</label><caption><p>Transitions between classes in the stochastic SEICWH model. The notation <italic>A</italic> → <italic>B</italic> means that when the event occurs one individual is transferred from compartment <italic>A</italic> to compartment <italic>B</italic>.</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left">Event</th><th align="left">Transition</th><th align="left">Rate at which event occurs</th></tr></thead><tbody><tr><td align="left">(re)Infection</td><td align="left"><italic>S</italic> → <italic>E</italic><sub>1</sub></td><td align="left"><italic>βs</italic>(<italic>i</italic><sub>1</sub> + <italic>i</italic><sub>2</sub>)/<italic>Ω</italic></td></tr><tr><td align="left"></td><td align="left"><italic>W</italic> → <italic>E</italic><sub>1</sub></td><td align="left"><inline-formula><mml:math id="M27" altimg="si8.gif" overflow="scroll"><mml:mi>β</mml:mi><mml:mi>w</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>i</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>Ω</mml:mi></mml:math></inline-formula></td></tr><tr><td align="left">Progression of incubation</td><td align="left"><italic>E</italic><sub>1</sub> → <italic>E</italic><sub>2</sub></td><td align="left">2<italic>ϵe</italic><sub>1</sub></td></tr><tr><td align="left">Start of infectiosity</td><td align="left"><italic>E</italic><sub>2</sub> → <italic>I</italic><sub>1</sub></td><td align="left">2<italic>ϵe</italic><sub>2</sub></td></tr><tr><td align="left">Progression of infectiosity</td><td align="left"><italic>I</italic><sub>1</sub> → <italic>I</italic><sub>2</sub></td><td align="left">2<italic>νi</italic><sub>1</sub></td></tr><tr><td align="left">Recovery</td><td align="left"><italic>I</italic><sub>2</sub> → <italic>C</italic><sub>1</sub></td><td align="left">2<italic>νi</italic><sub>2</sub></td></tr><tr><td align="left">Progressive loss of cellular protection</td><td align="left"><italic>C</italic><sub><italic>k</italic></sub> → <italic>C</italic><sub><italic>k</italic>+1</sub></td><td align="left">5<italic>γc</italic><sub><italic>k</italic></sub></td></tr><tr><td align="left">Deficiency of humoral response</td><td align="left"><italic>C</italic><sub>5</sub> → <italic>S</italic></td><td align="left">(1 − <italic>α</italic>)5<italic>γc</italic><sub>5</sub></td></tr><tr><td align="left">Start of the window of susceptibility</td><td align="left"><italic>C</italic><sub>5</sub> → <italic>W</italic></td><td align="left"><italic>α</italic>5<italic>γc</italic><sub>5</sub></td></tr><tr><td align="left">Start of the humoral protection</td><td align="left"><italic>W</italic> → <italic>H</italic></td><td align="left"><inline-formula><mml:math id="M28" altimg="si9.gif" overflow="scroll"><mml:mi>ω</mml:mi><mml:mi>w</mml:mi></mml:math></inline-formula></td></tr></tbody></table></table-wrap><table-wrap id="tbl0010" position="float"><label>Table 2</label><caption><p>Results of the maximum likelihood statistical inference for the 1971 TdC epidemic. ML estimates and 95% confidence intervals for the SEICWH model parameters.</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left">Symbol</th><th align="left">Description</th><th align="left">Estimate</th><th align="left">CI<sub>95%</sub></th></tr></thead><tbody><tr><td align="left"><italic>R</italic><sub>0</sub> = <italic>β</italic>/<italic>ν</italic></td><td align="left">Basic reproduction number</td><td align="char">11.78</td><td align="char">7.70–25.50</td></tr><tr><td align="left">1/<italic>ϵ</italic></td><td align="left">Mean latent period (days)</td><td align="char">2.18</td><td align="char">1.53–2.96</td></tr><tr><td align="left">1/<italic>ν</italic></td><td align="left">mean infectious period (days)</td><td align="char">2.32</td><td align="char">0.70–5.03</td></tr><tr><td align="left">1/<italic>γ</italic></td><td align="left">Mean temporary removed period (days)</td><td align="char">13.37</td><td align="char">10.37–16.31</td></tr><tr><td align="left">1/<italic>ω</italic></td><td align="left">Mean duration of the reinfection window (days)</td><td align="char">2.75</td><td align="char">0–6.03</td></tr><tr><td align="left"><italic>α</italic></td><td align="left">Probability to develop long-term immunity</td><td align="char">0.83</td><td align="char">0.49–1</td></tr><tr><td align="left"><italic>ρ</italic></td><td align="left">Reporting rate for observation</td><td align="char">0.71</td><td align="char">0.62–0.82</td></tr><tr><td align="left"><italic>I</italic><sub>0</sub></td><td align="left">Number of initially infectious individuals</td><td align="char">1</td><td align="char">1–3</td></tr><tr><td align="left"><italic>S</italic><sub>0</sub></td><td align="left">Number of initially susceptible individuals</td><td align="char">277</td><td align="char">275–280</td></tr><tr><td colspan="4" align="left">
</td></tr><tr><td align="left"><italic>l</italic>(<italic>θ</italic><sub>ML</sub>)</td><td align="left">Maximized log-likelihood</td><td align="char">−112.19</td><td align="char">–</td></tr></tbody></table></table-wrap></floats-group></pmc><affiliations><list><country><li>France</li><li>Royaume-Uni</li></country><region><li>Angleterre</li><li>Grand Londres</li><li>Île-de-France</li></region><settlement><li>Bondy</li><li>Londres</li><li>Paris</li></settlement></list><tree><country name="France"><region name="Île-de-France"><name sortKey="Camacho, A" sort="Camacho, A" uniqKey="Camacho A" first="A." last="Camacho">A. Camacho</name></region><name sortKey="Cazelles, B" sort="Cazelles, B" uniqKey="Cazelles B" first="B." last="Cazelles">B. Cazelles</name><name sortKey="Cazelles, B" sort="Cazelles, B" uniqKey="Cazelles B" first="B." last="Cazelles">B. Cazelles</name></country><country name="Royaume-Uni"><region name="Angleterre"><name sortKey="Camacho, A" sort="Camacho, A" uniqKey="Camacho A" first="A." last="Camacho">A. Camacho</name></region></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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