Effective containment explains subexponential growth in recent confirmed COVID-19 cases in China
Identifieur interne : 000821 ( Pmc/Corpus ); précédent : 000820; suivant : 000822Effective containment explains subexponential growth in recent confirmed COVID-19 cases in China
Auteurs : Benjamin F. Maier ; Dirk BrockmannSource :
- Science (New York, N.y.) [ 0036-8075 ] ; 2020.
Abstract
The recent outbreak of COVID-19 in Mainland China was characterized by a distinctive subexponential increase of confirmed cases during the early phase of the epidemic, contrasting an initial exponential growth expected for an unconstrained outbreak. We show that this effect can be explained as a direct consequence of containment policies that effectively deplete the susceptible population. To this end, we introduce a parsimonious model that captures both, quarantine of symptomatic infected individuals as well as population-wide isolation practices in response to containment policies or behavioral changes and show that the model captures the observed growth behavior accurately. The insights provided here may aid the careful implementation of containment strategies for ongoing secondary outbreaks of COVID-19 or similar future outbreaks of other emergent infectious diseases.
Url:
DOI: 10.1126/science.abb4557
PubMed: 32269067
PubMed Central: 7164388
Links to Exploration step
PMC:7164388Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Effective containment explains subexponential growth in recent confirmed
COVID-19 cases in China</title><author><name sortKey="Maier, Benjamin F" sort="Maier, Benjamin F" uniqKey="Maier B" first="Benjamin F." last="Maier">Benjamin F. Maier</name><affiliation><nlm:aff id="aff1">Robert Koch Institute, Nordufer 20, D-13353 Berlin, Germany.</nlm:aff></affiliation></author><author><name sortKey="Brockmann, Dirk" sort="Brockmann, Dirk" uniqKey="Brockmann D" first="Dirk" last="Brockmann">Dirk Brockmann</name><affiliation><nlm:aff id="aff1">Robert Koch Institute, Nordufer 20, D-13353 Berlin, Germany.</nlm:aff></affiliation><affiliation><nlm:aff id="aff2">Institute for Theoretical Biology, Humboldt-University of Berlin, Philippstr. 13, D-10115 Berlin, Germany.</nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">32269067</idno><idno type="pmc">7164388</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7164388</idno><idno type="RBID">PMC:7164388</idno><idno type="doi">10.1126/science.abb4557</idno><date when="2020">2020</date><idno type="wicri:Area/Pmc/Corpus">000821</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000821</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Effective containment explains subexponential growth in recent confirmed
COVID-19 cases in China</title><author><name sortKey="Maier, Benjamin F" sort="Maier, Benjamin F" uniqKey="Maier B" first="Benjamin F." last="Maier">Benjamin F. Maier</name><affiliation><nlm:aff id="aff1">Robert Koch Institute, Nordufer 20, D-13353 Berlin, Germany.</nlm:aff></affiliation></author><author><name sortKey="Brockmann, Dirk" sort="Brockmann, Dirk" uniqKey="Brockmann D" first="Dirk" last="Brockmann">Dirk Brockmann</name><affiliation><nlm:aff id="aff1">Robert Koch Institute, Nordufer 20, D-13353 Berlin, Germany.</nlm:aff></affiliation><affiliation><nlm:aff id="aff2">Institute for Theoretical Biology, Humboldt-University of Berlin, Philippstr. 13, D-10115 Berlin, Germany.</nlm:aff></affiliation></author></analytic><series><title level="j">Science (New York, N.y.)</title><idno type="ISSN">0036-8075</idno><idno type="eISSN">1095-9203</idno><imprint><date when="2020">2020</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>The recent outbreak of COVID-19 in Mainland China was characterized by a distinctive
subexponential increase of confirmed cases during the early phase of the epidemic,
contrasting an initial exponential growth expected for an unconstrained outbreak. We show
that this effect can be explained as a direct consequence of containment policies that
effectively deplete the susceptible population. To this end, we introduce a parsimonious
model that captures both, quarantine of symptomatic infected individuals as well as
population-wide isolation practices in response to containment policies or behavioral
changes and show that the model captures the observed growth behavior accurately. The
insights provided here may aid the careful implementation of containment strategies for
ongoing secondary outbreaks of COVID-19 or similar future outbreaks of other emergent
infectious diseases.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Cohen, J" uniqKey="Cohen J">J. Cohen</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Hsu, J" uniqKey="Hsu J">J. Hsu</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Lewis, T" uniqKey="Lewis T">T. Lewis</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Chen, N" uniqKey="Chen N">N. Chen</name></author><author><name sortKey="Zhou, M" uniqKey="Zhou M">M. Zhou</name></author><author><name sortKey="Dong, X" uniqKey="Dong X">X. Dong</name></author><author><name sortKey="Qu, J" uniqKey="Qu J">J. Qu</name></author><author><name sortKey="Gong, F" uniqKey="Gong F">F. Gong</name></author><author><name sortKey="Han, Y" uniqKey="Han Y">Y. Han</name></author><author><name sortKey="Qiu, Y" uniqKey="Qiu Y">Y. Qiu</name></author><author><name sortKey="Wang, J" uniqKey="Wang J">J. Wang</name></author><author><name sortKey="Liu, Y" uniqKey="Liu Y">Y. Liu</name></author><author><name sortKey="Wei, Y" uniqKey="Wei Y">Y. Wei</name></author><author><name sortKey="Xia, J" uniqKey="Xia J">J. Xia</name></author><author><name sortKey="Yu, T" uniqKey="Yu T">T. Yu</name></author><author><name sortKey="Zhang, X" uniqKey="Zhang X">X. Zhang</name></author><author><name sortKey="Zhang, L" uniqKey="Zhang L">L. Zhang</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Zhao, S" uniqKey="Zhao S">S. Zhao</name></author><author><name sortKey="Lin, Q" uniqKey="Lin Q">Q. Lin</name></author><author><name sortKey="Ran, J" uniqKey="Ran J">J. Ran</name></author><author><name sortKey="Musa, S S" uniqKey="Musa S">S. S. Musa</name></author><author><name sortKey="Yang, G" uniqKey="Yang G">G. Yang</name></author><author><name sortKey="Wang, W" uniqKey="Wang W">W. Wang</name></author><author><name sortKey="Lou, Y" uniqKey="Lou Y">Y. Lou</name></author><author><name sortKey="Gao, D" uniqKey="Gao D">D. Gao</name></author><author><name sortKey="Yang, L" uniqKey="Yang L">L. Yang</name></author><author><name sortKey="He, D" uniqKey="He D">D. He</name></author><author><name sortKey="Wang, M H" uniqKey="Wang M">M. H. Wang</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Dong, E" uniqKey="Dong E">E. Dong</name></author><author><name sortKey="Du, H" uniqKey="Du H">H. Du</name></author><author><name sortKey="Gardner, L" uniqKey="Gardner L">L. Gardner</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="De Picoli Junior, S" uniqKey="De Picoli Junior S">S. de Picoli Junior</name></author><author><name sortKey="Teixeira, J X0a J" uniqKey="Teixeira J">J.
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<front><journal-meta><journal-id journal-id-type="nlm-ta">Science</journal-id><journal-id journal-id-type="iso-abbrev">Science</journal-id><journal-id journal-id-type="publisher-id">SCIENCE</journal-id><journal-title-group><journal-title>Science (New York, N.y.)</journal-title></journal-title-group><issn pub-type="ppub">0036-8075</issn><issn pub-type="epub">1095-9203</issn><publisher><publisher-name>American Association for the Advancement of Science</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">32269067</article-id><article-id pub-id-type="pmc">7164388</article-id><article-id pub-id-type="publisher-id">abb4557</article-id><article-id pub-id-type="doi">10.1126/science.abb4557</article-id><article-categories><subj-group subj-group-type="article-type"><subject>Research Article</subject></subj-group><subj-group subj-group-type="heading"><subject>Research Articles</subject></subj-group><subj-group subj-group-type="legacy-article-type"><subject>R-Articles</subject></subj-group><subj-group subj-group-type="field"><subject>Epidemiology</subject><subject>Medicine</subject></subj-group></article-categories><title-group><article-title>Effective containment explains subexponential growth in recent confirmed
COVID-19 cases in China</article-title></title-group><contrib-group><contrib contrib-type="author"><contrib-id contrib-id-type="orcid" authenticated="true">https://orcid.org/0000-0001-7414-8823</contrib-id><name><surname>Maier</surname><given-names>Benjamin F.</given-names></name><xref ref-type="award" rid="award591535"></xref><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="corresp" rid="cor1">*</xref></contrib><contrib contrib-type="author"><contrib-id contrib-id-type="orcid" authenticated="true">https://orcid.org/0000-0001-5708-2922</contrib-id><name><surname>Brockmann</surname><given-names>Dirk</given-names></name><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="aff" rid="aff2">2</xref></contrib><aff id="aff1"><label>1</label>Robert Koch Institute, Nordufer 20, D-13353 Berlin, Germany.</aff><aff id="aff2"><label>2</label>Institute for Theoretical Biology, Humboldt-University of Berlin, Philippstr. 13, D-10115 Berlin, Germany.</aff></contrib-group><author-notes><corresp id="cor1"><label>*</label>Corresponding author. Email: <email xlink:href="bfmaier@physik.hu-berlin.de">bfmaier@physik.hu-berlin.de</email></corresp></author-notes><pub-date pub-type="epub"><day>08</day><month>4</month><year>2020</year></pub-date><elocation-id>eabb4557</elocation-id><history><date date-type="received"><day>25</day><month>2</month><year>2020</year></date><date date-type="accepted"><day>04</day><month>4</month><year>2020</year></date></history><permissions><copyright-statement> Copyright © 2020 The Authors, some rights reserved; exclusive
licensee American Association for the Advancement of Science. No claim to original U.S.
Government Works</copyright-statement><copyright-year>2020</copyright-year><copyright-holder>AAAS</copyright-holder><license license-type="open-access" xlink:href="http://creativecommons.org/licenses/by/4.0/"><ali:license_ref specific-use="vor" start_date="2020-03-04">https://creativecommons.org/licenses/by/4.0/</ali:license_ref><license-p>This open access article is distributed under <ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution License 4.0 (CC BY)</ext-link>.</license-p></license></permissions><abstract><p>The recent outbreak of COVID-19 in Mainland China was characterized by a distinctive
subexponential increase of confirmed cases during the early phase of the epidemic,
contrasting an initial exponential growth expected for an unconstrained outbreak. We show
that this effect can be explained as a direct consequence of containment policies that
effectively deplete the susceptible population. To this end, we introduce a parsimonious
model that captures both, quarantine of symptomatic infected individuals as well as
population-wide isolation practices in response to containment policies or behavioral
changes and show that the model captures the observed growth behavior accurately. The
insights provided here may aid the careful implementation of containment strategies for
ongoing secondary outbreaks of COVID-19 or similar future outbreaks of other emergent
infectious diseases.</p></abstract><funding-group><award-group id="award591535"><funding-source><institution-wrap><institution-id institution-id-type="doi">http://dx.doi.org/10.13039/100008662</institution-id><institution>Joachim Herz Stiftung</institution></institution-wrap></funding-source></award-group></funding-group><custom-meta-group><custom-meta><meta-name>Number-color-figs</meta-name><meta-value>2</meta-value></custom-meta><custom-meta><meta-name>Number-figs</meta-name><meta-value>2</meta-value></custom-meta></custom-meta-group></article-meta></front><body><sec sec-type="intro" disp-level="1"><title>Introduction</title><p>The outbreak of COVID-19 caused by the coronavirus SARS-CoV-2 in Mainland China was closely
monitored by governments, researchers, and the public alike (<xref rid="R1" ref-type="bibr"><italic>1</italic></xref>–<xref rid="R8" ref-type="bibr"><italic>8</italic></xref>). The rapid increase of positively diagnosed cases and
subsequent rise of secondary outbreaks in many countries worldwide raised concern on an
international scale. The World Health Organization (WHO) therefore announced the COVID-19
outbreak a <italic>Public Health Emergency of International Concern</italic> on Jan. 31st
and eventually classified it as a pandemic on Mar. 11th (<xref rid="R2" ref-type="bibr"><italic>2</italic></xref>, <xref rid="R3" ref-type="bibr"><italic>3</italic></xref>).</p><p>In Mainland China, confirmed cases increased from approx. 330 on Jan. 21st, 2020 to more
than 17,000 on Feb. 2nd, 2020 in a matter of two weeks (<xref rid="R9" ref-type="bibr"><italic>9</italic></xref>). In Hubei Province, the epicenter of the COVID-2019
outbreak, confirmed cases rose from 270 to 11,000 in this period, in all other Chinese
provinces the cumulated case count increased from 60 to 6,000 in the same period. Yet, as of
Mar. 28th, the total case count has saturated at 67,800 cases in Hubei with no new cases per
day and reached 13,600 in the remaining Chinese provinces with about 50 new cases per
day.</p><p>An initial exponential growth of confirmed cases is generically expected for an
uncontrolled outbreak, as observed e.g., during the 2009 Influenza A (H1N1) pandemic (<xref rid="R10" ref-type="bibr"><italic>10</italic></xref>) or the 2014 Ebola outbreak in West
Africa (<xref rid="R11" ref-type="bibr"><italic>11</italic></xref>). This initial outbreak
is in most cases mitigated with a time delay by effective containment strategies and
policies that reduce transmission and effective reproduction of the virus, commonly yielding
a saturation in the cumulative case count and an exponential decay in the number of new
infections (<xref rid="R12" ref-type="bibr"><italic>12</italic></xref>). Although in Hubei
the number of laboratory-confirmed cases <inline-formula><mml:math id="m1"><mml:mrow><mml:mi>C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> was observed to grow exponentially in early January (<xref rid="R13" ref-type="bibr"><italic>13</italic></xref>), the subsequent rise followed a
subexponential, super-linear, algebraic scaling law <inline-formula><mml:math id="m2"><mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mi>μ</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula> with an exponent <inline-formula><mml:math id="m3"><mml:mrow><mml:mi>μ</mml:mi><mml:mo>=</mml:mo><mml:mn>2.3</mml:mn></mml:mrow></mml:math></inline-formula> (between Jan. 24th and Feb. 9th), cf. <xref ref-type="fig" rid="F1">Fig. 1A</xref>. For the majority of the affected Chinese provinces of Mainland
China, however, this type of algebraic rise occurred from the beginning of case reporting on
Jan. 21st. The exponents <inline-formula><mml:math id="m4"><mml:mi>μ</mml:mi></mml:math></inline-formula> fluctuate around a typical value of <inline-formula><mml:math id="m5"><mml:mrow><mml:mi>μ</mml:mi><mml:mo>=</mml:mo><mml:mn>2.1</mml:mn><mml:mo>±</mml:mo><mml:mn>0.3</mml:mn></mml:mrow></mml:math></inline-formula> for the confirmed case curves in other substantially affected
provinces (confirmed case counts larger than 500 on Feb. 12th), displaying algebraic growth
despite geographical, socio-economical differences, possible differences in containment
strategies, and heterogeneities that may have variable impacts on how the local epidemic
unfolds, cf. <xref ref-type="fig" rid="F1">Fig. 1B-C</xref>. Eventually, case counts began
to deviate from the observed scaling laws around Feb. 9th for Hubei and in early February
for the remaining provinces, approaching a saturation behavior.</p><fig id="F1" fig-type="figure" orientation="portrait" position="float"><label>Fig. 1</label><caption><p><bold>Confirmed cases of COVID-19 infections <inline-formula><mml:math id="m6"><mml:mrow><mml:mi>C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula></bold> in Mainland China in late January/early
February. (<bold>A</bold>) Confirmed case numbers in Hubei. The increase in cases
follows a scaling law <inline-formula><mml:math id="m7"><mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mi>μ</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula> with an exponent <inline-formula><mml:math id="m8"><mml:mrow><mml:mi>μ</mml:mi><mml:mo>=</mml:mo><mml:mn>2.32</mml:mn></mml:mrow></mml:math></inline-formula> after a short initial exponential growth phase. On Feb
9th the case count starts deviating toward lower values. (<bold>B</bold>) Aggregated
confirmed cases in all other affected provinces except Hubei. <inline-formula><mml:math id="m9"><mml:mrow><mml:mi>C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> follows a scaling law with exponent
<inline-formula><mml:math id="m10"><mml:mrow><mml:mi>μ</mml:mi><mml:mo>=</mml:mo><mml:mn>1.92</mml:mn></mml:mrow></mml:math></inline-formula> until Feb. 2nd when case counts deviate to lower values.
The insets in <bold>A</bold> and <bold>B</bold> depict <inline-formula><mml:math id="m11"><mml:mrow><mml:mi>C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> on a log-log scale and show example exponential growth
curves for comparison. (<bold>C</bold>) Confirmed cases as a function of time for the 8
remaining most affected provinces in China. The curves roughly follow a scaling law with
exponents <inline-formula><mml:math id="m12"><mml:mrow><mml:mi>μ</mml:mi><mml:mo>≈</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></inline-formula> with the exception of Chongqing Province
(<inline-formula><mml:math id="m13"><mml:mrow><mml:mi>μ</mml:mi><mml:mo>=</mml:mo><mml:mn>1.45</mml:mn></mml:mrow></mml:math></inline-formula>) and Jiangxi Province (<inline-formula><mml:math id="m14"><mml:mrow><mml:mi>μ</mml:mi><mml:mo>=</mml:mo><mml:mn>2.75</mml:mn></mml:mrow></mml:math></inline-formula>).</p></caption><graphic xlink:href="abb4557-F1"></graphic></fig><p>The fact that the observed growth behavior appears for all provinces during the transient
phase between onset and saturation suggests that this aspect of the dynamics is determined
by fundamental principles that are at work and robust with respect to variation of other
parameters that typically shape the temporal evolution of epidemic processes. Three
questions immediately arise, (i) what may be the reason for this functional dependency, (ii)
are provinces other than Hubei mostly driven by export cases from Hubei and therefore follow
a similar functional form in case counts as suggested by preliminary studies discussing the
influence of human travel (<xref rid="R14" ref-type="bibr"><italic>14</italic></xref>–<xref rid="R16" ref-type="bibr"><italic>16</italic></xref>), or, alternatively, (iii) is the scaling law a consequence of
endogenous and basic epidemiological processes, caused by a balance between transmission
events and containment efforts.</p><p>In the following, we will provide evidence that the implementation of effective containment
strategies that target both susceptibles and infecteds can account for the observed growth
behavior.</p><p>The Chinese government put several mitigation policies in place to suppress the spread of
the epidemic (<xref rid="R17" ref-type="bibr"><italic>17</italic></xref>). In particular,
positively diagnosed cases were either quarantined in specialized hospital wards or put
under a form of monitored self-quarantine at home. Similarly, suspicious cases were confined
in monitored house arrest, e.g., individuals who arrived from Hubei before all traffic from
its capital Wuhan was effectively restricted. These measures aimed at the removal of
infectious individuals from the transmission process.</p><p>Additionally, introduced social distancing measures aimed at the protection of the
susceptible population, induced by behavioral changes as well as the partial shutdown of
public life (<xref rid="R17" ref-type="bibr"><italic>17</italic></xref>). For instance, many
people wore face masks in public spaces and followed stricter hygiene procedures concerning
hand washing, universities remained closed, many businesses closed down, and people were
asked to remain in their homes for as much time as possible, in several places enforced by
mandatory curfews. Another standard strategy that Chinese authorities applied was contact
tracing (<xref rid="R17" ref-type="bibr"><italic>17</italic></xref>), where possible
transmission chains between known infecteds and their contacts were identified and suspected
cases were isolated at home before symptom onset. While very effective in interrupting the
transmission process and thereby shielding large numbers of susceptibles from acquiring the
infection, contact tracing becomes infeasible when the number of infected grows rapidly in a
short amount of time or when an undetected outbreak leads to a large number of
unidentifiable infecteds, as was the case in Hubei.</p><p>The latter containment efforts that affect both, susceptibles and asymptomatic infectious
individuals, not only protect susceptibles from acquiring the infection but also remove a
substantial fraction of the entire pool of susceptibles from the transmission process,
indirectly mitigating the proliferation of the virus in the population in much the same way
that herd immunity is effective in the context of vaccine-preventable diseases.</p></sec><sec sec-type="other1" disp-level="1"><title>Modeling epidemic spread under containment efforts</title><p>On a very basic level, an outbreak as the one in Hubei is captured by SIR dynamics where
the population is divided into three compartments that differentiate the state of
individuals with respect to the contagion process: (I)nfected, (S)usceptible to infection,
and (R)emoved (i.e., not taking part in the transmission process) (<xref rid="R18" ref-type="bibr"><italic>18</italic></xref>, <xref rid="R19" ref-type="bibr"><italic>19</italic></xref>). The corresponding variables <inline-formula><mml:math id="m15"><mml:mi>S</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="m16"><mml:mi>I</mml:mi></mml:math></inline-formula>, and <inline-formula><mml:math id="m17"><mml:mi>R</mml:mi></mml:math></inline-formula> quantify the respective compartments’ fraction of the
total population such that <inline-formula><mml:math id="m18"><mml:mrow><mml:mi>S</mml:mi><mml:mo>+</mml:mo><mml:mi>I</mml:mi><mml:mo>+</mml:mo><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>. The temporal evolution of the number of cases is governed by
two processes: The infection that describes the transmission from an infectious to a
susceptible individual with basic reproduction number <inline-formula><mml:math id="m19"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> and the recovery of an infected after an infectious period of
average length <inline-formula><mml:math id="m20"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>I</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>. The basic reproduction number <inline-formula><mml:math id="m21"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> captures the average number of secondary infections an
infected will cause before he or she recovers or is effectively removed from the
population.</p><p>Initially, a small fraction of infecteds yields an exponential growth if the basic
reproduction number is larger than unity. A simple reduction of contacts caused by
quarantine policies without additional shielding of susceptibles could be associated with a
reduction in the effective reproduction number, which would, however, still yield an
exponential growth in <inline-formula><mml:math id="m22"><mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> if <inline-formula><mml:math id="m23"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>, inconsistent with the observed transient scaling law
<inline-formula><mml:math id="m24"><mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mi>μ</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula> discussed above. To test the hypothesis that the observed
growth behavior can be caused by mitigation policies that apply to both, infected and
susceptible individuals, we extend the SIR model by two additional mechanisms one of which
can be interpreted as a process of removing susceptibles from the transmission process:
First, we assume that general public containment efforts or individual behavioral changes in
response to the epidemic effectively remove individuals from the interaction dynamics or
significantly reduce their participation in the transmission dynamics. We will refer to this
mechanism as ‘containment‘ in the following. Secondly, we account for the
removal of symptomatic infected individuals, which we will refer to as
‘quarantine‘ procedures. The dynamics are governed by the system of ordinary
differential equations:<disp-formula id="E1"><mml:math id="m25"><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi>α</mml:mi><mml:mi>S</mml:mi><mml:mi>I</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>S</mml:mi></mml:mrow></mml:math><label>(1)</label></disp-formula><disp-formula id="E2"><mml:math id="m26"><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mi>I</mml:mi><mml:mo>=</mml:mo><mml:mi>α</mml:mi><mml:mi>S</mml:mi><mml:mi>I</mml:mi><mml:mo>−</mml:mo><mml:mi>β</mml:mi><mml:mi>I</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>I</mml:mi><mml:mo>−</mml:mo><mml:mi>κ</mml:mi><mml:mi>I</mml:mi></mml:mrow></mml:math><label>(2)</label></disp-formula><disp-formula id="E3"><mml:math id="m27"><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mi>β</mml:mi><mml:mi>I</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>S</mml:mi></mml:mrow></mml:math><label>(3)</label></disp-formula><disp-formula id="E4"><mml:math id="m28"><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mi>X</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>κ</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math><label>(4)</label></disp-formula>a generalization of the standard SIR model,
henceforth referred to as the SIR-X model. The rate parameters <inline-formula><mml:math id="m29"><mml:mi>α</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="m30"><mml:mi>β</mml:mi></mml:math></inline-formula> quantify the transmission rate and the recovery rate of the
standard SIR model, respectively. Additionally, the impact of containment efforts is
captured by the terms proportional to the containment rate <inline-formula><mml:math id="m31"><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> that is effective in both <inline-formula><mml:math id="m32"><mml:mi>I</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="m33"><mml:mi>S</mml:mi></mml:math></inline-formula> populations, since measures like social distancing and
curfews affect the whole population alike. Infected individuals are removed at rate
<inline-formula><mml:math id="m34"><mml:mi>κ</mml:mi></mml:math></inline-formula> corresponding to quarantine measures that only affect
symptomatic infecteds. The new compartment <inline-formula><mml:math id="m35"><mml:mi>X</mml:mi></mml:math></inline-formula> quantifies symptomatic, quarantined infecteds. Here we assume
that the fraction <inline-formula><mml:math id="m36"><mml:mrow><mml:mi>X</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> is proportional to the empirically confirmed and reported
cases <inline-formula><mml:math id="m37"><mml:mrow><mml:mi>C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> and that the time period between sampling and test results
can be neglected. The case <inline-formula><mml:math id="m38"><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula> corresponds to a scenario in which the general population is
unaffected by policies or does not commit behavioral changes in response to an epidemic. The
case <inline-formula><mml:math id="m39"><mml:mrow><mml:mi>κ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula> corresponds to a scenario in which symptomatic infecteds are
not isolated specifically.</p><p>In the basic SIR model that captures unconstrained, free spread of the disease, the basic
reproduction number <inline-formula><mml:math id="m40"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> is related to transmission and recovery rate by
<inline-formula><mml:math id="m41"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext>free</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>α</mml:mi><mml:mo>/</mml:mo><mml:mi>β</mml:mi></mml:mrow></mml:math></inline-formula> because <inline-formula><mml:math id="m42"><mml:mrow><mml:msup><mml:mi>β</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mi>I</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is the average time an infected individual remains infectious
before recovery or removal. Here, the time period that an infected individual remains
infectious is <inline-formula><mml:math id="m43"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>I</mml:mi><mml:mo>,</mml:mo><mml:mtext>eff</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>β</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>κ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> such that the effective, or “observed”
reproduction number <inline-formula><mml:math id="m44"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext>eff</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>α</mml:mi><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>I</mml:mi><mml:mo>,</mml:mo><mml:mtext>eff</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is smaller than <inline-formula><mml:math id="m45"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext>free</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> since both <inline-formula><mml:math id="m46"><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="m47"><mml:mrow><mml:mi>κ</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula>.</p><p>The key mechanism at work in the model defined by <xref ref-type="disp-formula" rid="E1">Eqs. (1)</xref>-<xref ref-type="disp-formula" rid="E4">(4)</xref> is the exponentially
fast depletion of susceptibles in addition to isolation of infecteds. This effect is
sufficient to account for the observed scaling law in the number of confirmed cases for a
plausible range of model parameters as discussed below.</p></sec><sec sec-type="other2" disp-level="1"><title>Effective protection of susceptibles leads to subexponential growth</title><p>We assume that a small number of infected individuals traveled from Hubei to each of the
other affected provinces before traffic restrictions were effective but at a time when
containment measures were just being implemented. <xref ref-type="fig" rid="F2">Figure
2</xref> illustrates the degree to which the case count for Hubei Province and the
aggregated case count for all other provinces is captured by the SIR-X model as defined by
<xref ref-type="disp-formula" rid="E1">Eqs. (1)</xref>-<xref ref-type="disp-formula" rid="E4">(4)</xref>.</p><fig id="F2" fig-type="figure" orientation="portrait" position="float"><label>Fig. 2</label><caption><title>Case numbers in Hubei compared to model predictions.</title><p>The quarantined compartment <inline-formula><mml:math id="m48"><mml:mrow><mml:mi>X</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> and the unidentified infectious compartment
<inline-formula><mml:math id="m49"><mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> are obtained from fits to the model defined by <xref ref-type="disp-formula" rid="E1">Eqs. (1)</xref>-<xref ref-type="disp-formula" rid="E4">(4)</xref> as described in the Materials and Methods. All fits were performed
for case numbers predating Feb. 12th at which the case number definition was temporarily
changed for Hubei, adding approximately <inline-formula><mml:math id="m50"><mml:mrow><mml:mn>15</mml:mn><mml:mo>,</mml:mo><mml:mn>000</mml:mn></mml:mrow></mml:math></inline-formula> cases at once. Consequently, confirmed case numbers after
Feb. 12th are well captured by the model for all provinces but Hubei. Fit parameters are
given in table S1. (<bold>A</bold>) In Hubei, the model captures both, the initial rise
of confirmed cases as well as the subsequent algebraic growth. The confirmed cases were
predicted to saturate at <inline-formula><mml:math id="m51"><mml:mrow><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mn>51</mml:mn><mml:mo>,</mml:mo><mml:mn>000</mml:mn></mml:mrow></mml:math></inline-formula>. The model also predicts the time-course of the number of
unidentified infectious individuals <inline-formula><mml:math id="m52"><mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> which peaks on Feb. 7th and declines exponentially
afterwards. While the magnitude of <inline-formula><mml:math id="m53"><mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> is associated with rather large fluctuations due to
uncertainties in the fitting parameters, the predicted peak time is robust, consistently
around Feb. 7th. (<bold>B</bold>) Model prediction for case numbers aggregated over all
affected provinces other than Hubei. The case numbers’ algebraic growth is well
reflected and predicted to saturate at <inline-formula><mml:math id="m54"><mml:mrow><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mn>12</mml:mn><mml:mo>,</mml:mo><mml:mn>600</mml:mn></mml:mrow></mml:math></inline-formula>. In contrast to Hubei, the fraction of unidentified
infecteds peaks around Feb. 1st, approximately one week earlier. The insets in
<bold>A</bold> and <bold>B</bold> depict both data and fits on a log-log scale.
(<bold>C</bold>) Fits for confirmed cases as a function of time for the remaining 8
most affected provinces in China. All curves are well captured by the model fits that
predict similar values for the peak time of unidentified infecteds.</p></caption><graphic xlink:href="abb4557-F2"></graphic></fig><p>For a wide range of model parameters, the empirical case count is well reproduced,
displaying the observed scaling law <inline-formula><mml:math id="m55"><mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mi>μ</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula> for a significant period of time before saturating to a
constant level. Remarkably, the model is able to reproduce both growth behaviors observed in
the data: It predicts the expected initial growth of case numbers in Hubei Province followed
by an algebraic growth episode for <inline-formula><mml:math id="m56"><mml:mrow><mml:mo>≈</mml:mo><mml:mn>11</mml:mn></mml:mrow></mml:math></inline-formula> days until the saturation sets in, a consequence of the decay
of unidentified infected individuals after a peak time around Feb. 7th (see <xref ref-type="fig" rid="F2">Fig. 2A</xref>). Furthermore, the model also captures the
immediate subexponential growth observed in the remaining most affected provinces (<xref ref-type="fig" rid="F2">Fig. 2B-C</xref>). Again, saturation is induced by a decay of
unidentified infecteds after peaks that occur several days before peak time in Hubei,
ranging from Jan. 31st to Feb. 4th. For all provinces, following their respective peaks, the
number of unidentified infecteds <inline-formula><mml:math id="m57"><mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> decays over a time period that is longer than the reported
estimation of maximum incubation period of <inline-formula><mml:math id="m58"><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:math></inline-formula> days (<xref rid="R4" ref-type="bibr"><italic>4</italic></xref>, <xref rid="R20" ref-type="bibr"><italic>20</italic></xref>).
It is important to note that the numerical value of unidentified infecteds is sensitive to
parameter variations—the general shape of <inline-formula><mml:math id="m59"><mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>, however, is robust for a wide choice of parameters, as
discussed in the Materials and Methods.</p><p>Parameter choices for best fits were a fixed basic reproduction number of
<inline-formula><mml:math id="m60"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext>free</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>6.2</mml:mn></mml:mrow></mml:math></inline-formula> (note that this reproduction number corresponds to an
unconstrained epidemic) and a fixed mean infection duration of <inline-formula><mml:math id="m61"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>I</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>8</mml:mn><mml:mo> </mml:mo><mml:mtext>d</mml:mtext></mml:mrow></mml:math></inline-formula> consistent with previous reports concerning the incubation
period of COVID-19 (<xref rid="R4" ref-type="bibr"><italic>4</italic></xref>, <xref rid="R20" ref-type="bibr"><italic>20</italic></xref>). The remaining fit parameters are
shown in table S1. For these values, the effective basic reproduction number is found to
range between <inline-formula><mml:math id="m62"><mml:mrow><mml:mn>1.4</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext>eff</mml:mtext></mml:mrow></mml:msub><mml:mo>≤</mml:mo><mml:mn>3.3</mml:mn></mml:mrow></mml:math></inline-formula> for the discussed provinces, consistent with estimates found
in previous early assessment studies (<xref rid="R4" ref-type="bibr"><italic>4</italic></xref>, <xref rid="R8" ref-type="bibr"><italic>8</italic></xref>,
<xref rid="R21" ref-type="bibr"><italic>21</italic></xref>, <italic>22</italic>). We
discuss these parameters and their possible range in the Materials and Methods.</p><p>On Feb. 12th, case counting procedures were altered regarding the outbreak in Hubei to
include cases that were clinically diagnosed and not laboratory-confirmed. Consequently,
about 15,000 new cases were added on a single day, compared to about 1,500 new cases on the
day before. We therefore only use data predating Feb. 12 to estimate model parameters for
the most affected provinces and compare the obtained predictions with subsequent cases
numbers. We find that for all provinces other than Hubei, our predictions accurately reflect
the empirical observations (<xref ref-type="fig" rid="F2">Fig. 2B-C</xref>). Omitting the
aforementioned discontinuity that arises in the empirical data, the saturating behavior of
cases in Hubei is consistent with the prediction, as well. Subtracting the clinically
diagnosed cases of Feb. 12 from the number of cases in Hubei, our model underestimates the
final count of laboratory-confirmed cases in Hubei by 4%. In the remaining part of Mainland
China we underestimate the final case count by 7% (as of Mar. 28th).</p><p>A detailed analysis of the model parameters indicates that a wide range of values can
account for similar shapes of the respective case counts (see Materials and Methods).
Consequently, the model is structurally stable with respect to these parameters and the
numerical value is of less importance than the quality of the mechanism they control. In
particular, we find that an exponential decay of available susceptibles is responsible for
the observed subexponential growth behavior, i.e., a nonzero containment rate
<inline-formula><mml:math id="m63"><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula>. We present model analyses for all affected Chinese provinces
in the SM (c.f. figs. S1-S2, table S2). We further provide analytical arguments for the
emergence of subexponential growth of confirmed cases and derive an approximate expression
that relates the scaling-law exponent <inline-formula><mml:math id="m64"><mml:mi>μ</mml:mi></mml:math></inline-formula> to model parameters, finding reasonable agreement with the
empirical values shown in <xref ref-type="fig" rid="F1">Fig. 1</xref> (c.f. table S5).</p><p>Additionally, we analyzed two model variants where (i) containment strategies affect the
whole population equally (<inline-formula><mml:math id="m65"><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="m66"><mml:mrow><mml:mi>κ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula>) and (ii) only infecteds are quarantined
(<inline-formula><mml:math id="m67"><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="m68"><mml:mrow><mml:mi>κ</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula>). The first model captures the case number growth slightly
less accurately compared to the complete model (c.f. figs. S3-S4, table S3). The second
model requires the assumption that shielding susceptibles from transmission was sufficiently
effective that only a very small number was at risk of infection, i.e., the effective
population size <inline-formula><mml:math id="m69"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mtext>eff</mml:mtext></mml:mrow></mml:msub><mml:mo>≪</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:math></inline-formula> becomes an additional model parameter (c.f. figs. S5-S6,
table S4). In this case, the immediate exponential depletion of susceptibles due to the
transmission process itself causes the empirically observed growth in case numbers. Both of
these limiting cases strengthen the point that the fast removal of susceptibles from the
population is responsible for the observed subexponential growth.</p><p>The described saturation behavior of confirmed cases requires that eventually all
susceptibles will effectively be removed from the transmission process. In reality, not
every susceptible person can be shielded or shield themselves for such an extended period of
time as the model suggests. One might expect instead that the number of unidentified
infecteds will decay more slowly and saturate to a small, yet nonzero level, which is why we
expect systematic but small underestimations regarding the final empirical case count of
epidemic outbreaks.</p></sec><sec sec-type="discussion|conclusions" disp-level="1"><title>Discussion and conclusion</title><p>In summary, we find that one of the key features of the dynamics of the COVID-19 epidemic
in Hubei Province but also in all other provinces is the robust subexponential rise in the
number of confirmed cases according to a scaling law <inline-formula><mml:math id="m70"><mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mi>μ</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula> during the transient episode of the epidemic before assuming
saturating behavior. This general shape of growth suggests that fundamental principles are
at work associated with this particular outbreak that are dominated by the interplay of the
contagion process with endogenous behavioral changes in the susceptible population and
external containment policies. While the explicit shape of the growth curves discussed here
can be influenced by factors such as seasonal effects, systematic delay in reporting, or
heterogeneities in demographic structure and population mixing, the fact that total case
numbers eventually reached a stable value suggests that containment strategies that shielded
the susceptible population from the transmission process were rather
effective—compared to potential case numbers of an unmitigated outbreak, only a small
fraction of the Chinese population that was at risk has been infected up to date (Mar.
29th). Nevertheless, we cannot rule out that other factors contributed to the growth
behavior displayed in the data that was collected in a short amount of time during a tense
situation.</p><p>The model defined by <xref ref-type="disp-formula" rid="E1">Eqs. (1)</xref>-<xref ref-type="disp-formula" rid="E4">(4)</xref> and discussed here indicates that the type of
observed growth behavior can generally be expected if the supply of susceptible individuals
is systematically decreased by means of implemented containment strategies or behavioral
changes in response to information about the ongoing epidemic. Unlike contagion processes
that develop without external interference at all or processes that merely lead to
parametric changes in the dynamics, our analysis suggests that non-exponential growth is
expected when the supply of susceptibles is depleted on a timescale comparable to the
infectious period of a disease.</p><p>The model reproduces the empirical case counts in all provinces well for plausible
parameter values. The quality of the reproduction of the case counts in all 29 affected
provinces can be used to estimate the peak time of the number of asymptomatic or
oligo-symptomatic infected individuals in the population, which is the key quantity for
estimating the time when an outbreak will wane. The current analysis indicates that this
peak time was reached around Feb. 7th for Hubei and within the first days of February in the
remaining affected provinces.</p><p>The model further suggests that the public response to the epidemic and the containment
measures put in place were effective despite the increase in confirmed cases. That this
behavior was observed in all provinces also indicates that containment strategies were
universally effective. Based on our analysis, such strategies would have to stay in effect
for a longer time than the maximum incubation period after the saturation in confirmed cases
sets in.</p><p>Our analysis shows that mitigation strategies that target the susceptible population and
induce behavioral changes at this “end” of the transmission process can be
very effective to contain an epidemic—especially in situations when asymptomatic or
mildly symptomatic infectious periods are long or their duration unknown. While standard
containment strategies such as contact tracing may become infeasible during large-scale
outbreaks of such diseases, the implementation of stricter measures can aid in the fast
reduction of the number of new infections, thereby quickly increasing the feasibility of
interventions that do not affect the general public as drastically. This may be of
importance for developing containment strategies for currently developing large-scale
secondary outbreaks of COVID-19 in several regions of the world or future outbreaks of other
infectious diseases.</p><p>We do want to stress that our model describes the general effects of containment
mechanisms, effectively averaged over many applied strategies or individual changes of
behavior. Our analysis can therefore not identify the efficacy of specific actions. As the
implementation of drastic measures such as mandatory curfews can have severe consequences
for both, individuals as well as a country’s society and economy, decisions about
their application should not be made lightly.</p></sec></body><back><ack><title>Acknowledgments</title><p>We want to express our gratitude to L . H. Wieler and L. Schaade for helpful comments
regarding the manuscript. BFM thanks L. Drescher and M. Borinsky for helpful remarks
regarding the analysis. <bold>Funding:</bold> BFM is financially supported as an
<italic>Add-on Fellow for Interdisciplinary Life Science</italic> by the Joachim Herz
Stiftung. <bold>Author contributions:</bold> BFM developed the initial idea. Both authors
contributed to the research process and the manuscript equally. <bold>Competing
interests:</bold> The authors declare no competing interests. <bold>Data and materials
availability</bold>: We make both data and analysis material available online (<xref rid="R23" ref-type="bibr"><italic>23</italic></xref>). This work is licensed under a
Creative Commons Attribution 4.0 International (CC BY 4.0) license, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original work
is properly cited. To view a copy of this license, visit <ext-link ext-link-type="uri" xlink:href="https://creativecommons.org/licenses/by/4.0/">https://creativecommons.org/licenses/by/4.0/</ext-link>. This license does not apply to
figures/photos/artwork or other content included in the article that is credited to a third
party; obtain authorization from the rights holder before using such material.</p></ack><app-group><app><title>Supplementary Material</title><supplementary-material content-type="local-data"><p><ext-link ext-link-type="uri" xlink:href="https://science.sciencemag.org/cgi/content/full/science.abb4557/DC1">science.sciencemag.org/cgi/content/full/science.abb4557/DC1</ext-link></p><p>Materials and Methods</p><p>Supplementary Text</p><p>Fig. S1 to S6</p><p>Tables S1 to S5</p><p>Reference (<xref rid="R24" ref-type="bibr"><italic>24</italic></xref>)</p><p>MDAR Reproducibility Checklist</p></supplementary-material><supplementary-material content-type="local-data" id="S1"><media xlink:href="abb4557_Reproducibility_Checklist.pdf"><caption><p>Click here for additional data file.</p></caption></media></supplementary-material><supplementary-material content-type="local-data" id="S2"><media xlink:href="abb4557_Maier_SM.pdf"><caption><p>Click here for additional data file.</p></caption></media></supplementary-material></app></app-group><ref-list><title>References and Notes</title><ref id="R1"><label>1</label><mixed-citation publication-type="journal"><person-group person-group-type="author"><name name-style="western"><surname>Cohen</surname><given-names>J.</given-names></name></person-group>, <article-title>Scientists are
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HfdIndexSelect -h $EXPLOR_AREA/Data/Pmc/Corpus/RBID.i -Sk "pubmed:32269067" \
| HfdSelect -Kh $EXPLOR_AREA/Data/Pmc/Corpus/biblio.hfd \
| NlmPubMed2Wicri -a StressCovidV1
| This area was generated with Dilib version V0.6.33. |  |