Corpus
Pmc
Racine
Corpus
Checkpoint
Pmc
Racine
Pmc
Exploration
Accueil
Racine
Racine

Serveur d'exploration Covid (26 mars)

Attention, ce site est en cours de développement !
Attention, site généré par des moyens informatiques à partir de corpus bruts.
Les informations ne sont donc pas validées.

How Inhomogeneous Site Percolation Works on Bethe Lattices: Theory and Application

Identifieur interne : 000326 ( Pmc/Corpus ); précédent : 000325; suivant : 000327

How Inhomogeneous Site Percolation Works on Bethe Lattices: Theory and Application

Auteurs : Jingli Ren ; Liying Zhang ; Stefan Siegmund

Source :

RBID : PMC:4772486

Abstract

Inhomogeneous percolation, for its closer relationship with real-life, can be more useful and reasonable than homogeneous percolation to illustrate the critical phenomena and dynamical behaviour of complex networks. However, due to its intricacy, the theoretical framework of inhomogeneous percolation is far from being complete and many challenging problems are still open. In this paper, we first investigate inhomogeneous site percolation on Bethe Lattices with two occupation probabilities, and then extend the result to percolation with m occupation probabilities. The critical behaviour of this inhomogeneous percolation is shown clearly by formulating the percolation probability with given occupation probability p, the critical occupation probability , and the average cluster size where p is subject to . Moreover, using the above theory, we discuss in detail the diffusion behaviour of an infectious disease (SARS) and present specific disease-control strategies in consideration of groups with different infection probabilities.


Url:
DOI: 10.1038/srep22420
PubMed: 26926785
PubMed Central: 4772486

Links to Exploration step

PMC:4772486

Le document en format XML

<record>
<TEI>
<teiHeader>
<fileDesc>
<titleStmt>
<title xml:lang="en">How Inhomogeneous Site Percolation Works on Bethe Lattices: Theory and Application</title>
<author>
<name sortKey="Ren, Jingli" sort="Ren, Jingli" uniqKey="Ren J" first="Jingli" last="Ren">Jingli Ren</name>
<affiliation>
<nlm:aff id="Aff1">
<institution-wrap>
<institution-id institution-id-type="GRID">grid.207374.5</institution-id>
<institution-id institution-id-type="ISNI">0000 0001 2189 3846</institution-id>
<institution>School of Mathematics and Statistics, Zhengzhou University,</institution>
</institution-wrap>
Zhengzhou, 450001 P.R. China</nlm:aff>
</affiliation>
</author>
<author>
<name sortKey="Zhang, Liying" sort="Zhang, Liying" uniqKey="Zhang L" first="Liying" last="Zhang">Liying Zhang</name>
<affiliation>
<nlm:aff id="Aff1">
<institution-wrap>
<institution-id institution-id-type="GRID">grid.207374.5</institution-id>
<institution-id institution-id-type="ISNI">0000 0001 2189 3846</institution-id>
<institution>School of Mathematics and Statistics, Zhengzhou University,</institution>
</institution-wrap>
Zhengzhou, 450001 P.R. China</nlm:aff>
</affiliation>
</author>
<author>
<name sortKey="Siegmund, Stefan" sort="Siegmund, Stefan" uniqKey="Siegmund S" first="Stefan" last="Siegmund">Stefan Siegmund</name>
<affiliation>
<nlm:aff id="Aff2">Department of Mathematics, Center for Dynamics & Institute for Analysis, TU Dresden, Dresden, 01062 Germany</nlm:aff>
</affiliation>
</author>
</titleStmt>
<publicationStmt>
<idno type="wicri:source">PMC</idno>
<idno type="pmid">26926785</idno>
<idno type="pmc">4772486</idno>
<idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4772486</idno>
<idno type="RBID">PMC:4772486</idno>
<idno type="doi">10.1038/srep22420</idno>
<date when="2016">2016</date>
<idno type="wicri:Area/Pmc/Corpus">000326</idno>
<idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000326</idno>
</publicationStmt>
<sourceDesc>
<biblStruct>
<analytic>
<title xml:lang="en" level="a" type="main">How Inhomogeneous Site Percolation Works on Bethe Lattices: Theory and Application</title>
<author>
<name sortKey="Ren, Jingli" sort="Ren, Jingli" uniqKey="Ren J" first="Jingli" last="Ren">Jingli Ren</name>
<affiliation>
<nlm:aff id="Aff1">
<institution-wrap>
<institution-id institution-id-type="GRID">grid.207374.5</institution-id>
<institution-id institution-id-type="ISNI">0000 0001 2189 3846</institution-id>
<institution>School of Mathematics and Statistics, Zhengzhou University,</institution>
</institution-wrap>
Zhengzhou, 450001 P.R. China</nlm:aff>
</affiliation>
</author>
<author>
<name sortKey="Zhang, Liying" sort="Zhang, Liying" uniqKey="Zhang L" first="Liying" last="Zhang">Liying Zhang</name>
<affiliation>
<nlm:aff id="Aff1">
<institution-wrap>
<institution-id institution-id-type="GRID">grid.207374.5</institution-id>
<institution-id institution-id-type="ISNI">0000 0001 2189 3846</institution-id>
<institution>School of Mathematics and Statistics, Zhengzhou University,</institution>
</institution-wrap>
Zhengzhou, 450001 P.R. China</nlm:aff>
</affiliation>
</author>
<author>
<name sortKey="Siegmund, Stefan" sort="Siegmund, Stefan" uniqKey="Siegmund S" first="Stefan" last="Siegmund">Stefan Siegmund</name>
<affiliation>
<nlm:aff id="Aff2">Department of Mathematics, Center for Dynamics & Institute for Analysis, TU Dresden, Dresden, 01062 Germany</nlm:aff>
</affiliation>
</author>
</analytic>
<series>
<title level="j">Scientific Reports</title>
<idno type="eISSN">2045-2322</idno>
<imprint>
<date when="2016">2016</date>
</imprint>
</series>
</biblStruct>
</sourceDesc>
</fileDesc>
<profileDesc>
<textClass></textClass>
</profileDesc>
</teiHeader>
<front>
<div type="abstract" xml:lang="en">
<p id="Par1">Inhomogeneous percolation, for its closer relationship with real-life, can be more useful and reasonable than homogeneous percolation to illustrate the critical phenomena and dynamical behaviour of complex networks. However, due to its intricacy, the theoretical framework of inhomogeneous percolation is far from being complete and many challenging problems are still open. In this paper, we first investigate inhomogeneous site percolation on Bethe Lattices with two occupation probabilities, and then extend the result to percolation with
<italic>m</italic>
occupation probabilities. The critical behaviour of this inhomogeneous percolation is shown clearly by formulating the percolation probability
<inline-formula id="IEq1">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq1_HTML.gif"></inline-graphic>
</inline-formula>
with given occupation probability
<italic>p</italic>
, the critical occupation probability
<inline-formula id="IEq2">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq2_HTML.gif"></inline-graphic>
</inline-formula>
, and the average cluster size
<inline-formula id="IEq3">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq3_HTML.gif"></inline-graphic>
</inline-formula>
where
<italic>p</italic>
is subject to
<inline-formula id="IEq4">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq4_HTML.gif"></inline-graphic>
</inline-formula>
. Moreover, using the above theory, we discuss in detail the diffusion behaviour of an infectious disease (SARS) and present specific disease-control strategies in consideration of groups with different infection probabilities.</p>
</div>
</front>
<back>
<div1 type="bibliography">
<listBibl>
<biblStruct>
<analytic>
<author>
<name sortKey="Stauffer, D" uniqKey="Stauffer D">D Stauffer</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Essam, Jw" uniqKey="Essam J">JW Essam</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Ghanbarzadeh, S" uniqKey="Ghanbarzadeh S">S Ghanbarzadeh</name>
</author>
<author>
<name sortKey="Prodanovi, M" uniqKey="Prodanovi M">M Prodanović</name>
</author>
<author>
<name sortKey="Hesse, Ma" uniqKey="Hesse M">MA Hesse</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Moon, K" uniqKey="Moon K">K Moon</name>
</author>
<author>
<name sortKey="Girvin, Sm" uniqKey="Girvin S">SM Girvin</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Moore, C" uniqKey="Moore C">C Moore</name>
</author>
<author>
<name sortKey="Newman, Mej" uniqKey="Newman M">MEJ Newman</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Janssen, Hk" uniqKey="Janssen H">HK Janssen</name>
</author>
<author>
<name sortKey="Muller, M" uniqKey="Muller M">M Müller</name>
</author>
<author>
<name sortKey="Stenull, O" uniqKey="Stenull O">O Stenull</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Newman, Mej" uniqKey="Newman M">MEJ Newman</name>
</author>
<author>
<name sortKey="Jensen, I" uniqKey="Jensen I">I Jensen</name>
</author>
<author>
<name sortKey="Ziff, Mr" uniqKey="Ziff M">MR Ziff</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Odagaki, T" uniqKey="Odagaki T">T Odagaki</name>
</author>
<author>
<name sortKey="Toyofuku, S" uniqKey="Toyofuku S">S Toyofuku</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Tobochnik, J" uniqKey="Tobochnik J">J Tobochnik</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Bondt, Sd" uniqKey="Bondt S">SD Bondt</name>
</author>
<author>
<name sortKey="Froyen, L" uniqKey="Froyen L">L Froyen</name>
</author>
<author>
<name sortKey="Deruyttere, A" uniqKey="Deruyttere A">A Deruyttere</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Bentz, Dp" uniqKey="Bentz D">DP Bentz</name>
</author>
<author>
<name sortKey="Garboczi, Ej" uniqKey="Garboczi E">EJ Garboczi</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Bramwell, St" uniqKey="Bramwell S">ST Bramwell</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Squires, S" uniqKey="Squires S">S Squires</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Guisoni, N" uniqKey="Guisoni N">N Guisoni</name>
</author>
<author>
<name sortKey="Loscar, Es" uniqKey="Loscar E">ES Loscar</name>
</author>
<author>
<name sortKey="Albano, Ev" uniqKey="Albano E">EV Albano</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Sahimi, M" uniqKey="Sahimi M">M Sahimi</name>
</author>
<author>
<name sortKey="Robertson, Mc" uniqKey="Robertson M">MC Robertson</name>
</author>
<author>
<name sortKey="Sammis, Cg" uniqKey="Sammis C">CG Sammis</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Fisher, Me" uniqKey="Fisher M">ME Fisher</name>
</author>
<author>
<name sortKey="Essam, Jw" uniqKey="Essam J">JW Essam</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Sykes, Mf" uniqKey="Sykes M">MF Sykes</name>
</author>
<author>
<name sortKey="Essam, Jw" uniqKey="Essam J">JW Essam</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Gerald, P" uniqKey="Gerald P">P Gerald</name>
</author>
<author>
<name sortKey="Ziff, Rm" uniqKey="Ziff R">RM Ziff</name>
</author>
<author>
<name sortKey="Stanley, He" uniqKey="Stanley H">HE Stanley</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Grimmett, Gr" uniqKey="Grimmett G">GR Grimmett</name>
</author>
<author>
<name sortKey="Manolescu, I" uniqKey="Manolescu I">I Manolescu</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Iliev, Gk" uniqKey="Iliev G">GK Iliev</name>
</author>
<author>
<name sortKey="Rensburg, Ejjv" uniqKey="Rensburg E">EJJV Rensburg</name>
</author>
<author>
<name sortKey="Madras, N" uniqKey="Madras N">N Madras</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Gennadiy, B" uniqKey="Gennadiy B">B Gennadiy</name>
</author>
<author>
<name sortKey="Yessica, Cs" uniqKey="Yessica C">CS Yessica</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Ziff, Rm" uniqKey="Ziff R">RM Ziff</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Scullard, Cr" uniqKey="Scullard C">CR Scullard</name>
</author>
<author>
<name sortKey="Ziff, Rm" uniqKey="Ziff R">RM Ziff</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Bender, A" uniqKey="Bender A">A Bender</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Radicchi, F" uniqKey="Radicchi F">F Radicchi</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Karrer, B" uniqKey="Karrer B">B Karrer</name>
</author>
<author>
<name sortKey="Newman, Mej" uniqKey="Newman M">MEJ Newman</name>
</author>
<author>
<name sortKey="Zdeborova, L" uniqKey="Zdeborova L">L Zdeborová</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Hamilton, Ke" uniqKey="Hamilton K">KE Hamilton</name>
</author>
<author>
<name sortKey="Pryadko, Lp" uniqKey="Pryadko L">LP Pryadko</name>
</author>
</analytic>
</biblStruct>
<biblStruct></biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Gong, Jh" uniqKey="Gong J">JH Gong</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Hamby, Dm" uniqKey="Hamby D">DM Hamby</name>
</author>
</analytic>
</biblStruct>
<biblStruct>
<analytic>
<author>
<name sortKey="Saltelli, A" uniqKey="Saltelli A">A Saltelli</name>
</author>
<author>
<name sortKey="Tarantola, S" uniqKey="Tarantola S">S Tarantola</name>
</author>
<author>
<name sortKey="Campolongo, F" uniqKey="Campolongo F">F Campolongo</name>
</author>
</analytic>
</biblStruct>
</listBibl>
</div1>
</back>
</TEI>
<pmc article-type="research-article">
<pmc-dir>properties open_access</pmc-dir>
<front>
<journal-meta>
<journal-id journal-id-type="nlm-ta">Sci Rep</journal-id>
<journal-id journal-id-type="iso-abbrev">Sci Rep</journal-id>
<journal-title-group>
<journal-title>Scientific Reports</journal-title>
</journal-title-group>
<issn pub-type="epub">2045-2322</issn>
<publisher>
<publisher-name>Nature Publishing Group UK</publisher-name>
<publisher-loc>London</publisher-loc>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="pmid">26926785</article-id>
<article-id pub-id-type="pmc">4772486</article-id>
<article-id pub-id-type="publisher-id">BFsrep22420</article-id>
<article-id pub-id-type="doi">10.1038/srep22420</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Article</subject>
</subj-group>
</article-categories>
<title-group>
<article-title>How Inhomogeneous Site Percolation Works on Bethe Lattices: Theory and Application</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author" corresp="yes">
<name>
<surname>Ren</surname>
<given-names>Jingli</given-names>
</name>
<address>
<email>renjl@zzu.edu.cn</email>
</address>
<xref ref-type="aff" rid="Aff1">1</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Zhang</surname>
<given-names>Liying</given-names>
</name>
<xref ref-type="aff" rid="Aff1">1</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Siegmund</surname>
<given-names>Stefan</given-names>
</name>
<xref ref-type="aff" rid="Aff2">2</xref>
</contrib>
<aff id="Aff1">
<label>1</label>
<institution-wrap>
<institution-id institution-id-type="GRID">grid.207374.5</institution-id>
<institution-id institution-id-type="ISNI">0000 0001 2189 3846</institution-id>
<institution>School of Mathematics and Statistics, Zhengzhou University,</institution>
</institution-wrap>
Zhengzhou, 450001 P.R. China</aff>
<aff id="Aff2">
<label>2</label>
Department of Mathematics, Center for Dynamics & Institute for Analysis, TU Dresden, Dresden, 01062 Germany</aff>
</contrib-group>
<pub-date pub-type="epub">
<day>1</day>
<month>3</month>
<year>2016</year>
</pub-date>
<pub-date pub-type="pmc-release">
<day>1</day>
<month>3</month>
<year>2016</year>
</pub-date>
<pub-date pub-type="collection">
<year>2016</year>
</pub-date>
<volume>6</volume>
<elocation-id>22420</elocation-id>
<history>
<date date-type="received">
<day>29</day>
<month>10</month>
<year>2015</year>
</date>
<date date-type="accepted">
<day>15</day>
<month>2</month>
<year>2016</year>
</date>
</history>
<permissions>
<copyright-statement>© The Author(s) 2016</copyright-statement>
<license license-type="OpenAccess">
<license-p>This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit
<ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/4.0/">http://creativecommons.org/licenses/by/4.0/</ext-link>
</license-p>
</license>
</permissions>
<abstract id="Abs1">
<p id="Par1">Inhomogeneous percolation, for its closer relationship with real-life, can be more useful and reasonable than homogeneous percolation to illustrate the critical phenomena and dynamical behaviour of complex networks. However, due to its intricacy, the theoretical framework of inhomogeneous percolation is far from being complete and many challenging problems are still open. In this paper, we first investigate inhomogeneous site percolation on Bethe Lattices with two occupation probabilities, and then extend the result to percolation with
<italic>m</italic>
occupation probabilities. The critical behaviour of this inhomogeneous percolation is shown clearly by formulating the percolation probability
<inline-formula id="IEq1">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq1_HTML.gif"></inline-graphic>
</inline-formula>
with given occupation probability
<italic>p</italic>
, the critical occupation probability
<inline-formula id="IEq2">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq2_HTML.gif"></inline-graphic>
</inline-formula>
, and the average cluster size
<inline-formula id="IEq3">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq3_HTML.gif"></inline-graphic>
</inline-formula>
where
<italic>p</italic>
is subject to
<inline-formula id="IEq4">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq4_HTML.gif"></inline-graphic>
</inline-formula>
. Moreover, using the above theory, we discuss in detail the diffusion behaviour of an infectious disease (SARS) and present specific disease-control strategies in consideration of groups with different infection probabilities.</p>
</abstract>
<kwd-group kwd-group-type="npg-subject">
<title>Subject terms</title>
<kwd>Viral infection</kwd>
<kwd>Applied mathematics</kwd>
</kwd-group>
<custom-meta-group>
<custom-meta>
<meta-name>issue-copyright-statement</meta-name>
<meta-value>© The Author(s) 2016</meta-value>
</custom-meta>
</custom-meta-group>
</article-meta>
</front>
<body>
<sec id="Sec1">
<title>Introduction</title>
<p id="Par2">Percolation (for reviews see Stauffer 1979
<sup>
<xref ref-type="bibr" rid="CR1">1</xref>
</sup>
, Essam 1980
<sup>
<xref ref-type="bibr" rid="CR2">2</xref>
</sup>
) is the random occupation of sites or bonds on lattices or networks, named as site percolation or bond percolation, respectively. Site percolation is more general than bond percolation because every bond model may be reformulated as a site model (on a different graph) and the converse is in general not true
<sup>
<xref ref-type="bibr" rid="CR3">3</xref>
</sup>
. The appeal of percolation is the occurrence of a critical phenomenon, which has attracted attention for a wide range of applications: liquid flows in porous media
<sup>
<xref ref-type="bibr" rid="CR4">4</xref>
,
<xref ref-type="bibr" rid="CR5">5</xref>
</sup>
, epidemic spread
<sup>
<xref ref-type="bibr" rid="CR6">6</xref>
,
<xref ref-type="bibr" rid="CR7">7</xref>
,
<xref ref-type="bibr" rid="CR8">8</xref>
</sup>
, granular and composite materials
<sup>
<xref ref-type="bibr" rid="CR9">9</xref>
,
<xref ref-type="bibr" rid="CR10">10</xref>
,
<xref ref-type="bibr" rid="CR11">11</xref>
,
<xref ref-type="bibr" rid="CR12">12</xref>
</sup>
, forest fires
<sup>
<xref ref-type="bibr" rid="CR13">13</xref>
,
<xref ref-type="bibr" rid="CR14">14</xref>
,
<xref ref-type="bibr" rid="CR15">15</xref>
</sup>
and fracture patterns and earthquakes in rocks
<sup>
<xref ref-type="bibr" rid="CR16">16</xref>
</sup>
.</p>
<p id="Par3">The research originated from homogeneous percolation, i.e., percolation with a single occupation probability. For example, Fisher and Essam (1961) solved homogeneous percolation problems on Bethe lattices
<sup>
<xref ref-type="bibr" rid="CR17">17</xref>
</sup>
; Sykes and Essam (1964) studied the exact critical occupation probabilities in two dimensions
<sup>
<xref ref-type="bibr" rid="CR18">18</xref>
</sup>
; Gerald (
<italic>et al.</italic>
, 2001) derived the value of critical occupation probabilities for a four-dimensional percolation problem on hyper-cubic lattices
<sup>
<xref ref-type="bibr" rid="CR19">19</xref>
</sup>
.</p>
<p id="Par4">Building on those results, researchers began to study inhomogeneous percolation
<sup>
<xref ref-type="bibr" rid="CR20">20</xref>
,
<xref ref-type="bibr" rid="CR21">21</xref>
,
<xref ref-type="bibr" rid="CR22">22</xref>
,
<xref ref-type="bibr" rid="CR23">23</xref>
,
<xref ref-type="bibr" rid="CR24">24</xref>
,
<xref ref-type="bibr" rid="CR25">25</xref>
,
<xref ref-type="bibr" rid="CR26">26</xref>
</sup>
, in which sites (or bonds) may have different occupation probabilities. In 1982, Kesten
<sup>
<xref ref-type="bibr" rid="CR20">20</xref>
</sup>
obtained a critical surface of occupation probabilities for inhomogeneous percolation on square lattices. In 2013, Grimmett
<sup>
<xref ref-type="bibr" rid="CR21">21</xref>
</sup>
extended the result of inhomogeneous bond percolation to triangular and hexagonal lattices utilizing Russo-Seymour-Welsh (RSW) theory of box-crossings. Recently, Radicchi
<sup>
<xref ref-type="bibr" rid="CR27">27</xref>
</sup>
studied percolation on not necessarily infinite graphs and used graph decomposition methods to identify abrupt and continuous changes in percolation.</p>
<p id="Par5">In the current paper, we focus on inhomogeneous percolation on Bethe lattices. The interest in a good understanding of this inhomogeneous percolation process is twofold. Firstly, Bethe lattices might hold fractal structures, which allow more complex behaviour than square or triangular lattices. The results of this inhomogeneous percolation on Bethe lattices have more extensive applications, e.g., the spreading problem of social networks or infectious diseases (see section III). Secondly, as is well known, most results on percolation are obtained using the heuristic approximation or numerical approaches
<sup>
<xref ref-type="bibr" rid="CR27">27</xref>
,
<xref ref-type="bibr" rid="CR28">28</xref>
,
<xref ref-type="bibr" rid="CR29">29</xref>
</sup>
and it is difficult to get the exact solutions for average cluster sizes and the percolation probability even for homogeneous percolation. In this paper, we investigate inhomogeneous site percolation on Bethe lattices from two occupation probabilities to
<italic>m</italic>
occupation probabilities: for the case of two occupation probabilities, we present the explicit formula of the critical occupation probability and the exact solution of average cluster size
<inline-formula id="IEq5">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq5_HTML.gif"></inline-graphic>
</inline-formula>
; and for the case of
<italic>m</italic>
occupation probabilities, because of computational complexity, we formulate and obtain the numerical solutions of average cluster size and percolation probability, which might shed light on deriving the results of inhomogeneous percolation. Besides, we analyse in detail the spread of SARS (an infectious disease) using this inhomogeneous percolation on Bethe lattices with dynamically changing parameters. We present specific control strategies for SARS by comparing the critical infection probabilities, the average numbers of infected individuals for each day, and the probability of the large-scale outbreak of SARS.</p>
</sec>
<sec id="Sec2">
<title>Basic theory</title>
<p id="Par6">In ref.
<xref ref-type="bibr" rid="CR30">30</xref>
a Bethe lattices is defined to be a tree where each site has
<italic>Z</italic>
neighbours,
<italic>Z</italic>
is also named coordination number. For the sake of convenience, we denote it as
<inline-formula id="IEq6">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq6_HTML.gif"></inline-graphic>
</inline-formula>
.</p>
<p id="Par7">On a Bethe lattices, if all sites are occupied randomly with the same probability
<italic>p</italic>
, independent of its neighbours, we call the percolation process as homogeneous site percolation and write it as
<inline-formula id="IEq7">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq7_HTML.gif"></inline-graphic>
</inline-formula>
. If the sites of a Bethe lattices
<inline-formula id="IEq8">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq8_HTML.gif"></inline-graphic>
</inline-formula>
are occupied with different probabilities, then the percolation is inhomogeneous. To be concrete, without loss of generality, assume that
<italic>Z</italic>
neighbours of each site are occupied randomly with
<italic>m</italic>
<inline-formula id="IEq9">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq9_HTML.gif"></inline-graphic>
</inline-formula>
occupation probabilities
<inline-formula id="IEq10">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq10_HTML.gif"></inline-graphic>
</inline-formula>
, and then, we divide
<italic>Z</italic>
neighbours of each site into
<italic>m</italic>
different groups (
<italic>m</italic>
is the number of types of neighbours according to occupation probabilities), where,
<inline-formula id="IEq11">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq11_HTML.gif"></inline-graphic>
</inline-formula>
of
<italic>Z</italic>
neighbours are sites with occupation probability
<inline-formula id="IEq12">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq12_HTML.gif"></inline-graphic>
</inline-formula>
,
<inline-formula id="IEq13">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq13_HTML.gif"></inline-graphic>
</inline-formula>
are sites with occupation probability
<inline-formula id="IEq14">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq14_HTML.gif"></inline-graphic>
</inline-formula>
and
<inline-formula id="IEq15">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq15_HTML.gif"></inline-graphic>
</inline-formula>
are sites with occupation probability
<inline-formula id="IEq16">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq16_HTML.gif"></inline-graphic>
</inline-formula>
. The inhomogeneous percolation is indicated with
<inline-formula id="IEq17">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq17_HTML.gif"></inline-graphic>
</inline-formula>
. See
<xref rid="Fig1" ref-type="fig">Fig. 1</xref>
for an illustration of
<inline-formula id="IEq18">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq18_HTML.gif"></inline-graphic>
</inline-formula>
<fig id="Fig1">
<label>Figure 1</label>
<caption>
<p>
<italic>PBL</italic>
(3; (1,
<italic>p</italic>
<sub>1</sub>
), (2,
<italic>p</italic>
<sub>2</sub>
)) with 3 sub-generations.</p>
</caption>
<graphic xlink:href="41598_2016_Article_BFsrep22420_Fig1_HTML" id="d29e489"></graphic>
</fig>
</p>
<p id="Par8">In this paper, we first consider inhomogeneous percolation on a Bethe lattices with
<inline-formula id="IEq19">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq19_HTML.gif"></inline-graphic>
</inline-formula>
, that is
<inline-formula id="IEq20">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq20_HTML.gif"></inline-graphic>
</inline-formula>
. Clearly, in this case if
<inline-formula id="IEq21">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq21_HTML.gif"></inline-graphic>
</inline-formula>
or
<inline-formula id="IEq22">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq22_HTML.gif"></inline-graphic>
</inline-formula>
or
<inline-formula id="IEq23">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq23_HTML.gif"></inline-graphic>
</inline-formula>
, inhomogeneous percolation will specialize to homogeneous percolation on
<italic>BL(Z)</italic>
<sup>
<xref ref-type="bibr" rid="CR30">30</xref>
</sup>
. In a second step, we generalize the results of
<inline-formula id="IEq24">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq24_HTML.gif"></inline-graphic>
</inline-formula>
to the case of
<inline-formula id="IEq25">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq25_HTML.gif"></inline-graphic>
</inline-formula>
.</p>
<sec id="Sec3">
<title>Critical surface of occupation probability</title>
<p id="Par9">Occupation probability is the probability with which the sites of a network are occupied. For
<inline-formula id="IEq26">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq26_HTML.gif"></inline-graphic>
</inline-formula>
, assume that a grandparent site is a first type-site on an infinite lattice, then for the parent site, there are
<inline-formula id="IEq27">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq27_HTML.gif"></inline-graphic>
</inline-formula>
sub-branches that begin with the first type-sites and
<inline-formula id="IEq28">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq28_HTML.gif"></inline-graphic>
</inline-formula>
sub-branches beginning with the second type-sites (
<xref rid="Fig1" ref-type="fig">Fig. 1</xref>
.). According to the binomial distribution, only
<inline-formula id="IEq29">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq29_HTML.gif"></inline-graphic>
</inline-formula>
branches are accessible on average. On the other hand, if the grandparent site is a second type-site, then for the parent site, there are
<inline-formula id="IEq30">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq30_HTML.gif"></inline-graphic>
</inline-formula>
sub-branches that begin with first type-sites and
<inline-formula id="IEq31">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq31_HTML.gif"></inline-graphic>
</inline-formula>
sub-branches which begin with second type-sites. In this case, on average, only
<inline-formula id="IEq32">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq32_HTML.gif"></inline-graphic>
</inline-formula>
branches are accessible. Recalling that the ratio of these two types of sites is
<inline-formula id="IEq33">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq33_HTML.gif"></inline-graphic>
</inline-formula>
, and according to expectation theory, on average, only
<disp-formula id="Equ1">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ1_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par10">branches are accessible at each step. In order to get an infinite cluster, it is necessary that the quantity (1) is equal or greater than one. Therefore, the critical condition such that an infinite cluster (percolating cluster) first occurs is
<disp-formula id="Equ2">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ2_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par11">from equation (2), we derive the critical surface of occupation probability
<disp-formula id="Equ3">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ3_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par12">this critical surface of
<inline-formula id="IEq34">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq34_HTML.gif"></inline-graphic>
</inline-formula>
is a line with slope
<inline-formula id="IEq35">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq35_HTML.gif"></inline-graphic>
</inline-formula>
in the occupation probabilities set
<inline-formula id="IEq36">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq36_HTML.gif"></inline-graphic>
</inline-formula>
. As an example, see
<xref rid="Fig2" ref-type="fig">Fig. 2(a)</xref>
for an illustration of the critical surface of
<inline-formula id="IEq37">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq37_HTML.gif"></inline-graphic>
</inline-formula>
.
<fig id="Fig2">
<label>Figure 2</label>
<caption>
<title>Critical surface of occupation probability.</title>
<p>(
<bold>a</bold>
) Critical surface of occupation probability of
<inline-formula id="IEq170">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq170_HTML.gif"></inline-graphic>
</inline-formula>
. (
<bold>b</bold>
) Critical surface of occupation probability of
<inline-formula id="IEq171">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq171_HTML.gif"></inline-graphic>
</inline-formula>
.</p>
</caption>
<graphic xlink:href="41598_2016_Article_BFsrep22420_Fig2_HTML" id="d29e627"></graphic>
</fig>
</p>
<p id="Par13">In a similar way, we can derive the following result.</p>
<p id="Par14">
<bold>Theorem 1</bold>
. The critical surface of
<inline-formula id="IEq38">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq38_HTML.gif"></inline-graphic>
</inline-formula>
is given by
<disp-formula id="Equ4">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ4_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
</sec>
<sec id="Sec4">
<title>Proof of Theorem 1</title>
<p id="Par15">For
<inline-formula id="IEq39">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq39_HTML.gif"></inline-graphic>
</inline-formula>
, if the grandparent site is an
<italic>i</italic>
th
<inline-formula id="IEq40">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq40_HTML.gif"></inline-graphic>
</inline-formula>
type-site, then for the parent site, there are
<inline-formula id="IEq41">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq41_HTML.gif"></inline-graphic>
</inline-formula>
sub-branches that begin with an
<italic>i</italic>
th type-site and
<italic>n</italic>
<sub>
<italic>k</italic>
</sub>
<inline-formula id="IEq42">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq42_HTML.gif"></inline-graphic>
</inline-formula>
sub-branches beginning with
<italic>k</italic>
th type-sites (
<xref rid="Fig1" ref-type="fig">Fig. 1</xref>
). According to the multi-binomial distribution, in this case only
<inline-formula id="IEq43">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq43_HTML.gif"></inline-graphic>
</inline-formula>
branches are accessible on average.</p>
<p id="Par16">Recalling the occupation probabilities for all types of sites and according to expectation theory, overall, only
<disp-formula id="Equ5">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ5_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par17">branches are accessible on average. In view that
<inline-formula id="IEq44">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq44_HTML.gif"></inline-graphic>
</inline-formula>
is equal to one on the critical surfaces, we have equation (4), the critical surface of
<inline-formula id="IEq45">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq45_HTML.gif"></inline-graphic>
</inline-formula>
.</p>
<p id="Par18">It is clear that
<inline-formula id="IEq46">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq46_HTML.gif"></inline-graphic>
</inline-formula>
if
<inline-formula id="IEq47">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq47_HTML.gif"></inline-graphic>
</inline-formula>
;
<inline-formula id="IEq48">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq48_HTML.gif"></inline-graphic>
</inline-formula>
if
<inline-formula id="IEq49">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq49_HTML.gif"></inline-graphic>
</inline-formula>
, where,
<inline-formula id="IEq50">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq50_HTML.gif"></inline-graphic>
</inline-formula>
;
<inline-formula id="IEq51">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq51_HTML.gif"></inline-graphic>
</inline-formula>
is the percolation probability (existing infinite clusters). The theory generalizes the concept of exact critical value
<inline-formula id="IEq52">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq52_HTML.gif"></inline-graphic>
</inline-formula>
in homogeneous percolation. For consistency, we also indicated the critical surface of
<inline-formula id="IEq53">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq53_HTML.gif"></inline-graphic>
</inline-formula>
by
<inline-formula id="IEq54">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq54_HTML.gif"></inline-graphic>
</inline-formula>
. The critical surface
<inline-formula id="IEq55">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq55_HTML.gif"></inline-graphic>
</inline-formula>
is the subset of the occupation-probability set
<inline-formula id="IEq56">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq56_HTML.gif"></inline-graphic>
</inline-formula>
with all
<inline-formula id="IEq57">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq57_HTML.gif"></inline-graphic>
</inline-formula>
satisfying equation (4). Then, in the supercritical phase,
<inline-formula id="IEq58">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq58_HTML.gif"></inline-graphic>
</inline-formula>
stands for
<inline-formula id="IEq59">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq59_HTML.gif"></inline-graphic>
</inline-formula>
, and there exists almost surely at least one infinite cluster of occupied sites. Contrarily, in the subcritical phase,
<inline-formula id="IEq60">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq60_HTML.gif"></inline-graphic>
</inline-formula>
stands for
<inline-formula id="IEq61">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq61_HTML.gif"></inline-graphic>
</inline-formula>
, and all clusters of occupied sites are almost surely finite. For an illustration,
<xref rid="Fig3" ref-type="fig">Fig. 3(b)</xref>
is the critical surface of
<inline-formula id="IEq62">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq62_HTML.gif"></inline-graphic>
</inline-formula>
.
<fig id="Fig3">
<label>Figure 3</label>
<caption>
<title>Average cluster size
<italic>χ</italic>
(
<italic>p</italic>
) of
<italic>PBL</italic>
(3; (1,
<italic>p</italic>
<sub>1</sub>
), (2,
<italic>p</italic>
<sub>2</sub>
)).</title>
<p>The left-hand curve is a sketch of the average cluster size
<inline-formula id="IEq172">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq172_HTML.gif"></inline-graphic>
</inline-formula>
and the right-hand curve is a sketch of the mean size of finite clusters of occupied sites when
<inline-formula id="IEq173">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq173_HTML.gif"></inline-graphic>
</inline-formula>
.</p>
</caption>
<graphic xlink:href="41598_2016_Article_BFsrep22420_Fig3_HTML" id="d29e814"></graphic>
</fig>
</p>
<p id="Par19">For an intuitive understanding, take
<inline-formula id="IEq63">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq63_HTML.gif"></inline-graphic>
</inline-formula>
as a reference, and
<inline-formula id="IEq64">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq64_HTML.gif"></inline-graphic>
</inline-formula>
, we have from equation (4) that
<inline-formula id="IEq65">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq65_HTML.gif"></inline-graphic>
</inline-formula>
. In this case
<inline-formula id="IEq66">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq66_HTML.gif"></inline-graphic>
</inline-formula>
is degenerated to a critical point, which is consistent with homogeneous percolation (in the supercritical phase,
<inline-formula id="IEq67">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq67_HTML.gif"></inline-graphic>
</inline-formula>
and in the subcritical phase,
<inline-formula id="IEq68">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq68_HTML.gif"></inline-graphic>
</inline-formula>
.</p>
<p id="Par20">The critical surface of occupation probabilities consists of those probabilities at which the percolation behaviour of the system changes essentially. We can control the percolation behaviour if we know the critical surface of occupation probabilities.</p>
<sec id="Sec5">
<title>Average cluster size of occupied sites</title>
<p id="Par21">The average cluster size
<inline-formula id="IEq69">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq69_HTML.gif"></inline-graphic>
</inline-formula>
of occupied sites is the mean size of the finite (non-percolating) clusters of occupied sites. It is closely related to the critical surface of occupation probabilities. We consider
<inline-formula id="IEq70">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq70_HTML.gif"></inline-graphic>
</inline-formula>
and the relationship between
<inline-formula id="IEq71">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq71_HTML.gif"></inline-graphic>
</inline-formula>
and critical surface.</p>
</sec>
</sec>
<sec id="Sec6">
<title>Theorem 2</title>
<p id="Par22">Assume that the Bethe lattice
<inline-formula id="IEq72">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq72_HTML.gif"></inline-graphic>
</inline-formula>
is infinite with occupation probabilities
<inline-formula id="IEq73">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq73_HTML.gif"></inline-graphic>
</inline-formula>
such that
<inline-formula id="IEq74">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq74_HTML.gif"></inline-graphic>
</inline-formula>
, then the average cluster size of occupied sites of
<inline-formula id="IEq75">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq75_HTML.gif"></inline-graphic>
</inline-formula>
satisfies
<disp-formula id="Equ6">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ6_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par23">where
<inline-formula id="IEq76">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq76_HTML.gif"></inline-graphic>
</inline-formula>
,
<inline-formula id="IEq77">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq77_HTML.gif"></inline-graphic>
</inline-formula>
.</p>
<p id="Par24">
<bold>Proof of Theorem 2</bold>
. If the Bethe lattice is infinite, all sites are equivalent for evaluating the average cluster size of occupied sites. Let
<inline-formula id="IEq78">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq78_HTML.gif"></inline-graphic>
</inline-formula>
be the average cluster size of the centre sites which are of type
<italic>i</italic>
<inline-formula id="IEq79">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq79_HTML.gif"></inline-graphic>
</inline-formula>
, and
<inline-formula id="IEq80">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq80_HTML.gif"></inline-graphic>
</inline-formula>
be the contribution (to the average cluster size) from a sub-branch which begins with a
<italic>j</italic>
th type-site and whose parent site is an
<italic>i</italic>
th type-site (
<xref rid="Fig1" ref-type="fig">Fig. 1</xref>
). Then
<disp-formula id="Equ7">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ7_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par25">where, the first term is the contribution from the centre site itself, the second term is the contribution from the first type branches, and the third term is the contribution of the second type branches.</p>
<p id="Par26">According to expectation theory, on average, the average cluster size is
<disp-formula id="Equ8">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ8_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par27">Based on definition of inhomogeneous percolation on a Bethe lattice, the following recurrence relations can be concluded
<disp-formula id="Equ9">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ9_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par28">Solving
<inline-formula id="IEq81">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq81_HTML.gif"></inline-graphic>
</inline-formula>
from equation (9), we have from (7) and (8) that
<disp-formula id="Equ10">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ10_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par29">and
<disp-formula id="Equ11">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ11_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par30">where,
<inline-formula id="IEq82">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq82_HTML.gif"></inline-graphic>
</inline-formula>
and
<inline-formula id="IEq83">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq83_HTML.gif"></inline-graphic>
</inline-formula>
.</p>
<p id="Par31">For
<inline-formula id="IEq84">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq84_HTML.gif"></inline-graphic>
</inline-formula>
or
<inline-formula id="IEq85">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq85_HTML.gif"></inline-graphic>
</inline-formula>
or
<inline-formula id="IEq86">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq86_HTML.gif"></inline-graphic>
</inline-formula>
, equation (6) reduces to
<inline-formula id="IEq87">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq87_HTML.gif"></inline-graphic>
</inline-formula>
.</p>
<p id="Par32">The average cluster size (of occupied sites)
<inline-formula id="IEq88">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq88_HTML.gif"></inline-graphic>
</inline-formula>
, which is a function of the occupation probability
<inline-formula id="IEq89">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq89_HTML.gif"></inline-graphic>
</inline-formula>
, can reveal the intensity of percolation. For
<inline-formula id="IEq90">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq90_HTML.gif"></inline-graphic>
</inline-formula>
,
<inline-formula id="IEq91">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq91_HTML.gif"></inline-graphic>
</inline-formula>
increases rapidly with
<italic>p</italic>
(
<xref rid="Fig3" ref-type="fig">Fig. 3</xref>
), and diverges in a power law of the distance between
<italic>p</italic>
and
<italic>p</italic>
<sub>
<italic>c</italic>
</sub>
, as
<italic>p</italic>
approaches
<italic>p</italic>
<sub>
<italic>c</italic>
</sub>
from below. For
<inline-formula id="IEq92">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq92_HTML.gif"></inline-graphic>
</inline-formula>
, there exist infinite clusters of occupied sites and their number increases as
<inline-formula id="IEq93">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq93_HTML.gif"></inline-graphic>
</inline-formula>
. On the other hand, the numbers of finite clusters (of occupied sites) and their sizes are reducing. Therefore, for
<inline-formula id="IEq94">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq94_HTML.gif"></inline-graphic>
</inline-formula>
, the average sizes of finite clusters
<inline-formula id="IEq95">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq95_HTML.gif"></inline-graphic>
</inline-formula>
decrease with
<italic>p</italic>
increasing (
<xref rid="Fig3" ref-type="fig">Fig. 3</xref>
).</p>
<p id="Par33">Similarly, generalizing the result to
<inline-formula id="IEq96">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq96_HTML.gif"></inline-graphic>
</inline-formula>
, we first solve
<inline-formula id="IEq97">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq97_HTML.gif"></inline-graphic>
</inline-formula>
from equation (10), where the
<inline-formula id="IEq98">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq98_HTML.gif"></inline-graphic>
</inline-formula>
have the same meaning as above
<disp-formula id="Equ12">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ12_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par34">Then, substitute the solution of equation (10) into equation (11) to get the values of
<inline-formula id="IEq99">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq99_HTML.gif"></inline-graphic>
</inline-formula>
, here
<inline-formula id="IEq100">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq100_HTML.gif"></inline-graphic>
</inline-formula>
.
<disp-formula id="Equ13">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ13_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par35">According to expectation theory, the average cluster size is
<disp-formula id="Equ14">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ14_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par36">Because the explicit expression of
<inline-formula id="IEq101">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq101_HTML.gif"></inline-graphic>
</inline-formula>
is very complex, we provide only the derivation process.</p>
</sec>
<sec id="Sec7">
<title>Percolation probability</title>
<p id="Par37">In this part, we mainly discuss the percolation probability, i.e., the probability that the origin site belongs to a percolating infinite cluster. Percolation probability is indicated as
<inline-formula id="IEq102">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq102_HTML.gif"></inline-graphic>
</inline-formula>
that can reveal the intensity of percolating, for
<inline-formula id="IEq103">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq103_HTML.gif"></inline-graphic>
</inline-formula>
(
<xref rid="Fig4" ref-type="fig">Fig. 4</xref>
).
<fig id="Fig4">
<label>Figure 4</label>
<caption>
<p>Percolation probability
<italic>P</italic>
<sub>∞</sub>
(
<italic>p</italic>
) of
<italic>PBL</italic>
(3; (1,
<italic>p</italic>
<sub>1</sub>
), (2,
<italic>p</italic>
<sub>2</sub>
)).</p>
</caption>
<graphic xlink:href="41598_2016_Article_BFsrep22420_Fig4_HTML" id="d29e1135"></graphic>
</fig>
</p>
<p id="Par38">For
<inline-formula id="IEq104">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq104_HTML.gif"></inline-graphic>
</inline-formula>
, in order to determine
<inline-formula id="IEq105">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq105_HTML.gif"></inline-graphic>
</inline-formula>
, let
<inline-formula id="IEq106">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq106_HTML.gif"></inline-graphic>
</inline-formula>
denote the probability that an
<italic>i</italic>
th type origin site belongs to a percolating infinite cluster, where
<inline-formula id="IEq107">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq107_HTML.gif"></inline-graphic>
</inline-formula>
,
<inline-formula id="IEq108">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq108_HTML.gif"></inline-graphic>
</inline-formula>
.</p>
<p id="Par39">A site belongs to a percolating infinite cluster, which means, not only the site itself is occupied, but also at least one of the
<inline-formula id="IEq109">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq109_HTML.gif"></inline-graphic>
</inline-formula>
branches (originating from the site) connects to the percolating cluster. Both of these are independent of each other, so,
<inline-formula id="IEq110">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq110_HTML.gif"></inline-graphic>
</inline-formula>
. According to mean theory, it can be concluded that
<disp-formula id="Equ15">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ15_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par40">where,
<inline-formula id="IEq111">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq111_HTML.gif"></inline-graphic>
</inline-formula>
, is the probability that a sub-branch does not connect to the percolating cluster and the sub-branch begins with a
<italic>j</italic>
th type-site, whose parent site is an
<italic>i</italic>
th type-site. Then, we have
<disp-formula id="Equ16">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ16_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par41">for each equation of (14), the first term is the probability that the root site of a sub-branch is not occupied and the second term is the probability that the root site of the sub-branch is occupied but no child sub-branch connects to the percolating cluster.</p>
<p id="Par42">If
<inline-formula id="IEq112">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq112_HTML.gif"></inline-graphic>
</inline-formula>
, equation (14) has only the trivial solution
<inline-formula id="IEq113">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq113_HTML.gif"></inline-graphic>
</inline-formula>
, then from (13) we have
<inline-formula id="IEq114">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq114_HTML.gif"></inline-graphic>
</inline-formula>
(
<xref rid="Fig4" ref-type="fig">Fig. 4</xref>
). If
<inline-formula id="IEq115">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq115_HTML.gif"></inline-graphic>
</inline-formula>
, there is a nontrivial solution for equation (14) and then equation (13) has a nonzero solution. In this case, it is not easy to get the exact nontrivial solution of equation (14) for it is a set of multivariable high-order equations. By employing fixed-point iteration, we get a numerical solution of equation (14) instead and then obtain the percolation probability from (13). From the numerical solution (
<xref rid="Fig4" ref-type="fig">Fig. 4</xref>
), it can be seen that
<inline-formula id="IEq116">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq116_HTML.gif"></inline-graphic>
</inline-formula>
picks up abruptly at
<italic>p</italic>
<sub>
<italic>c</italic>
</sub>
then increases rapidly with
<italic>p</italic>
increasing.</p>
<p id="Par43">We extend the result to
<inline-formula id="IEq117">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq117_HTML.gif"></inline-graphic>
</inline-formula>
by a similar analysis. In this general case,
<inline-formula id="IEq118">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq118_HTML.gif"></inline-graphic>
</inline-formula>
satisfy
<inline-formula id="IEq119">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq119_HTML.gif"></inline-graphic>
</inline-formula>
meaning the same as above)
<disp-formula id="Equ17">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ17_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par44">Firstly, derive the solution of (15) by fixed-point iteration, and then substitute
<inline-formula id="IEq120">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq120_HTML.gif"></inline-graphic>
</inline-formula>
into (16),
<disp-formula id="Equ18">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ18_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par45">This way, we get
<inline-formula id="IEq121">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq121_HTML.gif"></inline-graphic>
</inline-formula>
, where
<inline-formula id="IEq122">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq122_HTML.gif"></inline-graphic>
</inline-formula>
.</p>
</sec>
<sec id="Sec8">
<title>Percolation model of the disease spreading (SARS in Beijing, 2003)</title>
<p id="Par46">Severe acute respiratory syndrome (SARS) is a viral respiratory illness caused by a corona-virus. In Beijing (China), about 2523 cases have been infected with SARS in 2003. At the beginning of emergence, because of the lack of understanding of SARS and the high mobility of the modern-social activities, SARS spread rapidly. Afterwards, when people found the high infectivity and death rate of SARS, they begun to limit social activity of the public and take strict isolated measure to prevent the spreading of disease then the disease was contained. Is this the proper infection control measure of SARS? Which kind of infectious diseases are suitable for this approach?</p>
<p id="Par47">In fact, the spreading of SARS is a percolation process. Considering the differences of physical resistibility or intimate contact with the infected individual, we divided people into two groups: the people with higher infection probability (e.g., infants and elderly or healthcare workers), named as susceptible persons; and the people with lower infection probability, named as common persons. Then this disease is modelled as inhomogeneous percolation on Bethe lattices with two occupied probabilities, i.e.,
<inline-formula id="IEq123">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq123_HTML.gif"></inline-graphic>
</inline-formula>
. Here,
<italic>Z</italic>
is the average contact number of each person,
<italic>S</italic>
(of
<italic>Z</italic>
) denotes susceptible persons with infection probability
<italic>p</italic>
and
<italic>Z-S</italic>
(of
<italic>Z</italic>
) is common persons with infection probability
<italic>kp</italic>
<inline-formula id="IEq124">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq124_HTML.gif"></inline-graphic>
</inline-formula>
.</p>
<p id="Par48">The first case of SARS was confirmed on March 5 (in Beijing, 2003) and the government started reporting the cases from April 20. According to case-reporting data from April 20 to June 23 (in Beijing, 2003) (from the government bulletin) and the control activities of government, we find three other critical time points of SARS: May 1 (it is legal holidays from May 1 to May 7), May 14 (people generally panicked over SARS and limited their social activities), and May 30 (new cases of SARS considerably decreased). Correspondingly, the spreading process of SARS was divided into five stages. Then, by random simulation, we found that the spreading of SARS has a fifteen-day time delay. Therefore, we changed the five stages of the spreading process of SARS into: stage 1—from March 20 to May 4, stage 2—from May 5 to May 15, stage 3—from May 16 to May 28, stage 4—from May 29 to June 13, and stage 5—from June 14 to June 23.</p>
<p id="Par49">In the initial stage, the average contact number of one infected patient was around fifteen
<inline-formula id="IEq125">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq125_HTML.gif"></inline-graphic>
</inline-formula>
, in which two or three person had infected SARS (Gong
<italic>et al.</italic>
<sup>
<xref ref-type="bibr" rid="CR31">31</xref>
</sup>
), so the infection probability was around
<inline-formula id="IEq126">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq126_HTML.gif"></inline-graphic>
</inline-formula>
. We take
<inline-formula id="IEq127">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq127_HTML.gif"></inline-graphic>
</inline-formula>
according to the percentage of susceptible people in the population and we set
<inline-formula id="IEq128">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq128_HTML.gif"></inline-graphic>
</inline-formula>
by statistical investigation. It is worth noting that the infection probability
<italic>p</italic>
is the manifestation of the spreading intensity of the disease, which can only be slightly affected by the protective approach. Therefore, the infection probability was adjusted to 0.1425 in the second stage and third stage, and was adjusted to 0.141 in the fourth stage and fifth stage. The other parameters, i.e.,
<italic>Z</italic>
and
<italic>S</italic>
would change with prevention (isolation of infected patient and restriction of travel) and
<italic>k</italic>
remains unchanged in different stages. By statistical investigation, we set
<italic>Z</italic>
 = 13 and
<italic>S</italic>
 = 4 in second stage,
<italic>Z</italic>
 = 12 and
<italic>S</italic>
 = 3 in third stage,
<italic>Z</italic>
 = 11 and
<italic>S</italic>
 = 3 in fourth stage,
<italic>Z</italic>
 = 8 and
<italic>S</italic>
 = 2 in fifth stage.</p>
<p id="Par50">Obviously, the spreading model of SARS (in Beijing) is inhomogeneous percolation on a Bethe lattice with dynamically changing parameters. See
<xref rid="Tab1" ref-type="table">Table 1</xref>
for the model division and the parameters.
<table-wrap id="Tab1">
<label>Table 1</label>
<caption>
<p>The results of inhomogeneous percolation of SARS and some parameters by simulation and statistical analysis.</p>
</caption>
<table frame="hsides" rules="groups">
<thead>
<tr>
<th align="center"></th>
<th align="center">Mar. 20-May 4</th>
<th align="center">May 5-May 15</th>
<th align="center">May 16-May 28</th>
<th align="center">May 29-June 13</th>
<th align="center">June 14-June 23</th>
</tr>
</thead>
<tbody>
<tr>
<td>Infection probability—
<italic>p</italic>
</td>
<td>0.149</td>
<td>0.1425</td>
<td>0.1425</td>
<td>0.141</td>
<td>0.141</td>
</tr>
<tr>
<td>Average contact number—
<italic>Z</italic>
(
<italic>S, Z-S</italic>
)</td>
<td>15(4, 11)</td>
<td>13(4, 9)</td>
<td>12(3, 9)</td>
<td>11(3, 8)</td>
<td>8(2, 6)</td>
</tr>
<tr>
<td>Critical infection probability—
<italic>p</italic>
<sub>
<italic>c</italic>
</sub>
</td>
<td>0.1276</td>
<td>0.1425</td>
<td>0.1653</td>
<td>0.1774</td>
<td>0.2597</td>
</tr>
<tr>
<td>Average infected persons by one patient</td>
<td>infinite</td>
<td>infinite</td>
<td>74.77</td>
<td>17.4</td>
<td>3.0277</td>
</tr>
<tr>
<td>Probability of large-scale outbreak—
<italic>p</italic>
<sub>∞</sub>
</td>
<td>0.0233</td>
<td>0.0045</td>
<td>0</td>
<td>0</td>
<td>0</td>
</tr>
</tbody>
</table>
</table-wrap>
</p>
</sec>
<sec id="Sec9">
<title>Results of percolation model and control measures</title>
<p id="Par51">From equation (3), equation (6), and equation (13), we acquire the critical infection probabilities, average number of infected individuals, and the probability of large-scale outbreak for the SARS percolation model
<inline-formula id="IEq129">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq129_HTML.gif"></inline-graphic>
</inline-formula>
in different spreading stages (
<xref rid="Tab1" ref-type="table">Table 1</xref>
).</p>
<p id="Par52">It can be concluded from
<xref rid="Tab1" ref-type="table">Table 1</xref>
, that, if one is infected with SARS and would live as a normal person, then the disease would infect a massive crowd of healthy people except for
<inline-formula id="IEq130">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq130_HTML.gif"></inline-graphic>
</inline-formula>
and
<inline-formula id="IEq131">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq131_HTML.gif"></inline-graphic>
</inline-formula>
. That warns us, at the initial stage of SARS, humankind should try their best to find SARS patients as early as possible and isolate them from healthy people. Nevertheless, during the incubation period, it is inevitable that some infected persons, who cannot be found and live as the normal person around us that is quite dangerous. At this time, the most effective way is to reduce outdoor activities of public then cut down the average contact number.</p>
<p id="Par53">For more accurate and meticulous disease-control strategies, we scrutinized the dynamic-dependent relationship between
<inline-formula id="IEq132">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq132_HTML.gif"></inline-graphic>
</inline-formula>
and the average number of infected cases with subtle parameters by a Monte-Carlo simulation. In the initial stage of the SARS process, from March 20 to April 8, according to the above analysis,
<inline-formula id="IEq133">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq133_HTML.gif"></inline-graphic>
</inline-formula>
, and
<inline-formula id="IEq134">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq134_HTML.gif"></inline-graphic>
</inline-formula>
, based on these, the average number of infected cases of each day was simulated. See
<xref rid="Fig5" ref-type="fig">Fig. 5(a)</xref>
for an illustration. It is clear that the random variation of the infected number has an incremental trend and that implies that the disease would infect a large amount of persons. Then from April 9 to April 19, during the second stage of SARS, some protective measures were taken with the understanding of the disease, so parameters changed to
<inline-formula id="IEq135">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq135_HTML.gif"></inline-graphic>
</inline-formula>
, and
<inline-formula id="IEq136">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq136_HTML.gif"></inline-graphic>
</inline-formula>
. By simulation, we found that the infected number changes chaotically as in
<xref rid="Fig5" ref-type="fig">Fig. 5(b)</xref>
. With the time going by, from April 20 to May 4, the severity of the disease gradually being known, more protective measures were taken and parameters reduced further to
<inline-formula id="IEq137">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq137_HTML.gif"></inline-graphic>
</inline-formula>
, and
<inline-formula id="IEq138">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq138_HTML.gif"></inline-graphic>
</inline-formula>
. In this stage, the simulation revealed that the average infected number of each day fluctuates with a trend of decline and it would be zero after a period, which predicates the infectious disease can be controlled without any additional measures (
<xref rid="Fig5" ref-type="fig">Fig. 5(c)</xref>
). In fact, by simulating the percolation, we acquire
<inline-formula id="IEq139">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq139_HTML.gif"></inline-graphic>
</inline-formula>
in the initial stage,
<inline-formula id="IEq140">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq140_HTML.gif"></inline-graphic>
</inline-formula>
in the second stage, and
<inline-formula id="IEq141">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq141_HTML.gif"></inline-graphic>
</inline-formula>
in the third stage, i.e., the initial stage is a supercritical phase of the spread of SARS, the second stage is a critical phase, and the third stage is a subcritical phase, which agrees with the simulation. We could conclude that, near the critical point, slightly adjusting of the system parameters would cause a fundamental change of the trend of infectious diseases. Therefore, in order control the large-scale outbreak of the disease, we must try our best to make
<inline-formula id="IEq142">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq142_HTML.gif"></inline-graphic>
</inline-formula>
, even if the infection probability is only a little smaller than the critical infection probability. Actually, if
<inline-formula id="IEq143">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq143_HTML.gif"></inline-graphic>
</inline-formula>
, the probability of a large outbreak of SARS is zero; and if
<italic>p</italic>
reaches and crosses
<inline-formula id="IEq144">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq144_HTML.gif"></inline-graphic>
</inline-formula>
, the probability picks up as a power law with exponent one in terms of
<italic>p-p</italic>
<sub>
<italic>c</italic>
</sub>
, and the disease will outbreak rapidly.
<fig id="Fig5">
<label>Figure 5</label>
<caption>
<p>The average number of infected individuals versus every day with infection probabilities that are larger, equal, and smaller than critical infection probability.</p>
</caption>
<graphic xlink:href="41598_2016_Article_BFsrep22420_Fig5_HTML" id="d29e1625"></graphic>
</fig>
</p>
<p id="Par54">As we know, reducing outdoor activities of the public is a powerful strategy for infectious disease with latent period but it severely obstructs the people’s daily life and social economy. The methods in this paper will supply a quantitative measure for the risk of disease outbreaks and to guide our practice more appropriately.</p>
</sec>
</sec>
<sec id="Sec10">
<title>Predictions</title>
<p id="Par55">Based on the model of the spreading of SARS, we could supplement the data of cases from March 5 to April 20, during which the recording data was missing. See
<xref rid="Fig6" ref-type="fig">Fig. 6</xref>
.
<fig id="Fig6">
<label>Figure 6</label>
<caption>
<p>Simulation results (
<bold>a</bold>
) Cumulative cases of SARS with simulation data and actual data. (
<bold>b</bold>
) Daily new cases of simulation and report. (
<bold>c</bold>
) The cumulative cases and daily new cases by simulation.</p>
</caption>
<graphic xlink:href="41598_2016_Article_BFsrep22420_Fig6_HTML" id="d29e1653"></graphic>
</fig>
</p>
<p id="Par56">
<xref rid="Fig6" ref-type="fig">Figure 6(a)</xref>
is the actual cumulative case-reporting data and simulative data. New cases of each day are displayed in
<xref rid="Fig6" ref-type="fig">Fig. 6(b)</xref>
. We can find that the inhomogeneous percolation is in good fitting with the dynamic process of SARS spreading and time delay is the considerable feature of SARS. The relationship between cumulative confirmed cases and cases out of effective control displays in
<xref rid="Fig6" ref-type="fig">Fig. 6(c)</xref>
. It can be seen that there will be many persons infected with SARS even if only a few cases of SARS are out of effective control.</p>
<sec id="Sec11">
<title>Sensitivity analysis</title>
<p id="Par57">The sensitivity index is the ratio of the change in output to the change in input of parameters or variables
<sup>
<xref ref-type="bibr" rid="CR32">32</xref>
</sup>
. Taking into account the characteristics of the model, we employ a one factor at a time (OAT) approach, which is more agile and easy to interpret. The popular sensitivity index of OAT approach is
<inline-formula id="IEq145">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq145_HTML.gif"></inline-graphic>
</inline-formula>
, where
<italic>Y</italic>
is the output,
<italic>θ</italic>
is the input,
<inline-formula id="IEq146">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq146_HTML.gif"></inline-graphic>
</inline-formula>
is the sensitivity index of
<italic>Y</italic>
to
<italic>θ</italic>
, and
<inline-formula id="IEq147">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq147_HTML.gif"></inline-graphic>
</inline-formula>
is the partial derivative of
<italic>Y</italic>
with respect to
<italic>θ</italic>
. The quotient
<inline-formula id="IEq148">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq148_HTML.gif"></inline-graphic>
</inline-formula>
is introduced to normalize the index by removing the affects of units
<sup>
<xref ref-type="bibr" rid="CR33">33</xref>
</sup>
.</p>
<p id="Par58">First, we get
<inline-formula id="IEq149">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq149_HTML.gif"></inline-graphic>
</inline-formula>
from equation (3), and then derive that:
<disp-formula id="Equ19">
<graphic xlink:href="41598_2016_Article_BFsrep22420_Equ19_HTML.gif" position="anchor"></graphic>
</disp-formula>
</p>
<p id="Par59">Where,
<inline-formula id="IEq150">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq150_HTML.gif"></inline-graphic>
</inline-formula>
is the critical infection probability, one of the output of the SARS-percolation model;
<inline-formula id="IEq151">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq151_HTML.gif"></inline-graphic>
</inline-formula>
denote the average contact number of each patient, the number of susceptible persons of
<italic>Z</italic>
, the ratio of infection probabilities of susceptible person to common person (see page 12), respectively. They are all input parameters.</p>
<p id="Par60">Based on the
<xref rid="Tab1" ref-type="table">Table 1</xref>
, the sensitivity indexes of
<inline-formula id="IEq152">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq152_HTML.gif"></inline-graphic>
</inline-formula>
to three input parameters at five critical time points are obtained, which are all negative scalars. These suggest that the decreasing of each input parameters correspond to the increasing of
<inline-formula id="IEq153">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq153_HTML.gif"></inline-graphic>
</inline-formula>
. Among them, the absolute value of
<inline-formula id="IEq154">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq154_HTML.gif"></inline-graphic>
</inline-formula>
is about 0.8, which is maximum,
<inline-formula id="IEq155">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq155_HTML.gif"></inline-graphic>
</inline-formula>
is about 0.5, and
<inline-formula id="IEq156">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq156_HTML.gif"></inline-graphic>
</inline-formula>
is about 0.3. It is clear that
<inline-formula id="IEq157">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq157_HTML.gif"></inline-graphic>
</inline-formula>
has greater sensitivity to
<italic>Z</italic>
. See
<xref rid="Fig7" ref-type="fig">Fig. 7(a)</xref>
for an illustration.
<fig id="Fig7">
<label>Figure 7</label>
<caption>
<p>Sensitivity index (
<bold>a</bold>
) Sensitivity index of critical infection probability to input parameters-
<italic>Z</italic>
,
<italic>S</italic>
and
<italic>k</italic>
on five critical days of SARS in Beijing (2003) (
<bold>b</bold>
) Sensitivity index of the average infected cases by one patient to input variable
<italic>p</italic>
(infection probability). (
<bold>c</bold>
) Sensitivity index of the probability of SARS large-scale outbreak to input variable
<italic>p</italic>
.</p>
</caption>
<graphic xlink:href="41598_2016_Article_BFsrep22420_Fig7_HTML" id="d29e1804"></graphic>
</fig>
</p>
<p id="Par61">By a similar analysis with a numerical approach, we obtain
<inline-formula id="IEq158">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq158_HTML.gif"></inline-graphic>
</inline-formula>
and
<inline-formula id="IEq159">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq159_HTML.gif"></inline-graphic>
</inline-formula>
. The sensitivity index
<inline-formula id="IEq160">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq160_HTML.gif"></inline-graphic>
</inline-formula>
is shown in
<xref rid="Fig7" ref-type="fig">Fig. 7(b)</xref>
. It indicates that
<inline-formula id="IEq161">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq161_HTML.gif"></inline-graphic>
</inline-formula>
is more sensitive near the
<inline-formula id="IEq162">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq162_HTML.gif"></inline-graphic>
</inline-formula>
. Since
<inline-formula id="IEq163">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq163_HTML.gif"></inline-graphic>
</inline-formula>
is the average size of all-finite clusters of infected cases, the
<inline-formula id="IEq164">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq164_HTML.gif"></inline-graphic>
</inline-formula>
exhibits negative value when
<italic>p</italic>
is greater than
<inline-formula id="IEq165">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq165_HTML.gif"></inline-graphic>
</inline-formula>
. By a numerical approach, the sensitivity of the probability of large-scalar outbreak of SARS
<inline-formula id="IEq166">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq166_HTML.gif"></inline-graphic>
</inline-formula>
is displayed in
<xref rid="Fig7" ref-type="fig">Fig. 7(c)</xref>
.
<inline-formula id="IEq167">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq167_HTML.gif"></inline-graphic>
</inline-formula>
has similar characteristics as
<inline-formula id="IEq168">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq168_HTML.gif"></inline-graphic>
</inline-formula>
. They are both more sensitive near the
<inline-formula id="IEq169">
<inline-graphic xlink:href="41598_2016_Article_BFsrep22420_IEq169_HTML.gif"></inline-graphic>
</inline-formula>
.</p>
</sec>
</sec>
<sec id="Sec12">
<title>Conclusion</title>
<p id="Par62">In this paper, we present a theoretical framework for inhomogeneous site percolation on Bethe Lattices, and apply it to investigate a diffusion problem of an infectious disease. It is found that the inhomogeneous percolation on Bethe lattices serves as an appropriate model to describe the dynamic spreading behaviour of the infectious disease (SARS). The percolation model of SARS is not only in good agreement with the actual recorded data, but also can be used to predict the future trend of the disease and supply the missing data of the past. Moreover, it can provide quantitative results for government to make more proper disease-control strategies.</p>
</sec>
<sec id="Sec13">
<title>Additional Information</title>
<p id="Par63">
<bold>How to cite this article</bold>
: Ren, J.
<italic>et al.</italic>
How Inhomogeneous Site Percolation Works on Bethe Lattices: Theory and Application.
<italic>Sci. Rep.</italic>
<bold>6</bold>
, 22420; doi: 10.1038/srep22420 (2016).</p>
</sec>
</body>
<back>
<ack>
<title>Acknowledgements</title>
<p>This work is supported by the NSFC (No. 11271339) project, the Plan for Scientific Innovation Talent of Henan Province, and the ZDGD 13001 program.</p>
</ack>
<notes notes-type="author-contribution">
<title>Author Contributions</title>
<p>R.J.L. conceived the project and derived the theory. Z.L.Y. established and analysed the model. S.S. partly contributed to the theory and gave some advices on the model. All authors contributed to writing the manuscript.</p>
</notes>
<notes notes-type="COI-statement">
<title>Competing interests</title>
<p id="Par64">The authors declare no competing financial interests.</p>
</notes>
<ref-list id="Bib1">
<title>References</title>
<ref id="CR1">
<label>1</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Stauffer</surname>
<given-names>D</given-names>
</name>
</person-group>
<article-title>Scaling theory of percolation clusters</article-title>
<source>Phys. Rep.</source>
<year>1979</year>
<volume>54</volume>
<fpage>1</fpage>
<lpage>74</lpage>
<pub-id pub-id-type="doi">10.1016/0370-1573(79)90060-7</pub-id>
</element-citation>
</ref>
<ref id="CR2">
<label>2</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Essam</surname>
<given-names>JW</given-names>
</name>
</person-group>
<article-title>Percolation theory</article-title>
<source>Rep. Prog. Phys.</source>
<year>1980</year>
<volume>43</volume>
<fpage>833</fpage>
<lpage>912</lpage>
<pub-id pub-id-type="doi">10.1088/0034-4885/43/7/001</pub-id>
</element-citation>
</ref>
<ref id="CR3">
<label>3</label>
<mixed-citation publication-type="other">Grimmett, G. R.
<italic>Percolation</italic>
<italic>2nd edn</italic>
, 23–24 (Springer, Verlag Berlin Heidelberg, 1999).</mixed-citation>
</ref>
<ref id="CR4">
<label>4</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Ghanbarzadeh</surname>
<given-names>S</given-names>
</name>
<name>
<surname>Prodanović</surname>
<given-names>M</given-names>
</name>
<name>
<surname>Hesse</surname>
<given-names>MA</given-names>
</name>
</person-group>
<article-title>Percolation and grain boundary wetting in anisotropic texturally equilibrated pore networks</article-title>
<source>Phys. Rev. Lett.</source>
<year>2014</year>
<volume>113</volume>
<fpage>048001</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevLett.113.048001</pub-id>
<pub-id pub-id-type="pmid">25105655</pub-id>
</element-citation>
</ref>
<ref id="CR5">
<label>5</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Moon</surname>
<given-names>K</given-names>
</name>
<name>
<surname>Girvin</surname>
<given-names>SM</given-names>
</name>
</person-group>
<article-title>Critical behaviour of superfluid
<sup>4</sup>
He in aerogel</article-title>
<source>Phys. Rev. Lett.</source>
<year>1995</year>
<volume>75</volume>
<fpage>1328</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevLett.75.1328</pub-id>
<pub-id pub-id-type="pmid">10060265</pub-id>
</element-citation>
</ref>
<ref id="CR6">
<label>6</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Moore</surname>
<given-names>C</given-names>
</name>
<name>
<surname>Newman</surname>
<given-names>MEJ</given-names>
</name>
</person-group>
<article-title>Epidemics and percolation in small-world networks</article-title>
<source>Phys. Rev. E.</source>
<year>2000</year>
<volume>61</volume>
<fpage>5678</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevE.61.5678</pub-id>
</element-citation>
</ref>
<ref id="CR7">
<label>7</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Janssen</surname>
<given-names>HK</given-names>
</name>
<name>
<surname>Müller</surname>
<given-names>M</given-names>
</name>
<name>
<surname>Stenull</surname>
<given-names>O</given-names>
</name>
</person-group>
<article-title>Generalized epidemic process and tricritical dynamic percolation</article-title>
<source>Phys. Rev. E.</source>
<year>2004</year>
<volume>70</volume>
<fpage>026114</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevE.70.026114</pub-id>
</element-citation>
</ref>
<ref id="CR8">
<label>8</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Newman</surname>
<given-names>MEJ</given-names>
</name>
<name>
<surname>Jensen</surname>
<given-names>I</given-names>
</name>
<name>
<surname>Ziff</surname>
<given-names>MR</given-names>
</name>
</person-group>
<article-title>Percolation and epidemics in a two-dimensional small world</article-title>
<source>Phys. Rev. E.</source>
<year>2002</year>
<volume>65</volume>
<fpage>021904</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevE.65.021904</pub-id>
</element-citation>
</ref>
<ref id="CR9">
<label>9</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Odagaki</surname>
<given-names>T</given-names>
</name>
<name>
<surname>Toyofuku</surname>
<given-names>S</given-names>
</name>
</person-group>
<article-title>Properties of percolation clusters in a model granular system in two dimensions</article-title>
<source>J. Phys-Condens Mat.</source>
<year>1998</year>
<volume>10</volume>
<fpage>6447</fpage>
<lpage>6452</lpage>
<pub-id pub-id-type="doi">10.1088/0953-8984/10/28/022</pub-id>
</element-citation>
</ref>
<ref id="CR10">
<label>10</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Tobochnik</surname>
<given-names>J</given-names>
</name>
</person-group>
<article-title>Granular collapse as a percolation transition</article-title>
<source>Phys. Rev. E.</source>
<year>1999</year>
<volume>60</volume>
<fpage>7137</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevE.60.7137</pub-id>
</element-citation>
</ref>
<ref id="CR11">
<label>11</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Bondt</surname>
<given-names>SD</given-names>
</name>
<name>
<surname>Froyen</surname>
<given-names>L</given-names>
</name>
<name>
<surname>Deruyttere</surname>
<given-names>A</given-names>
</name>
</person-group>
<article-title>Electrical conductivity of composites: a percolation approach</article-title>
<source>J. Mater. Sci.</source>
<year>1992</year>
<volume>27</volume>
<fpage>1983</fpage>
<lpage>1988</lpage>
<pub-id pub-id-type="doi">10.1007/BF01107228</pub-id>
</element-citation>
</ref>
<ref id="CR12">
<label>12</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Bentz</surname>
<given-names>DP</given-names>
</name>
<name>
<surname>Garboczi</surname>
<given-names>EJ</given-names>
</name>
</person-group>
<article-title>Modelling the leaching of calcium hydroxide from cement paste: effects on pore space percolation and diffusivity</article-title>
<source>Mater. Struct.</source>
<year>1992</year>
<volume>25</volume>
<fpage>523</fpage>
<lpage>533</lpage>
<pub-id pub-id-type="doi">10.1007/BF02472448</pub-id>
</element-citation>
</ref>
<ref id="CR13">
<label>13</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Bramwell</surname>
<given-names>ST</given-names>
</name>
<etal></etal>
</person-group>
<article-title>Universal fluctuations in correlated systems</article-title>
<source>Phys. Rev. Lett.</source>
<year>2000</year>
<volume>84</volume>
<fpage>3744</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevLett.84.3744</pub-id>
<pub-id pub-id-type="pmid">11019195</pub-id>
</element-citation>
</ref>
<ref id="CR14">
<label>14</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Squires</surname>
<given-names>S</given-names>
</name>
<etal></etal>
</person-group>
<article-title>Weakly explosive percolation in directed networks</article-title>
<source>Phys. Rev. E.</source>
<year>2013</year>
<volume>87</volume>
<fpage>052127</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevE.87.052127</pub-id>
</element-citation>
</ref>
<ref id="CR15">
<label>15</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Guisoni</surname>
<given-names>N</given-names>
</name>
<name>
<surname>Loscar</surname>
<given-names>ES</given-names>
</name>
<name>
<surname>Albano</surname>
<given-names>EV</given-names>
</name>
</person-group>
<article-title>Phase diagram and critical behavior of a forest-fire model in a gradient of immunity</article-title>
<source>Phys. Rev. E.</source>
<year>2011</year>
<volume>83</volume>
<fpage>011125</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevE.83.011125</pub-id>
</element-citation>
</ref>
<ref id="CR16">
<label>16</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Sahimi</surname>
<given-names>M</given-names>
</name>
<name>
<surname>Robertson</surname>
<given-names>MC</given-names>
</name>
<name>
<surname>Sammis</surname>
<given-names>CG</given-names>
</name>
</person-group>
<article-title>Fractal distribution of earthquake hypocenters and its relation to fault patterns and percolation</article-title>
<source>Phys. Rev. Lett.</source>
<year>1993</year>
<volume>70</volume>
<fpage>2186</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevLett.70.2186</pub-id>
<pub-id pub-id-type="pmid">10053492</pub-id>
</element-citation>
</ref>
<ref id="CR17">
<label>17</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Fisher</surname>
<given-names>ME</given-names>
</name>
<name>
<surname>Essam</surname>
<given-names>JW</given-names>
</name>
</person-group>
<article-title>Some cluster size and percolation problems</article-title>
<source>J. Math. Phys.</source>
<year>1961</year>
<volume>2</volume>
<fpage>609</fpage>
<lpage>619</lpage>
<pub-id pub-id-type="doi">10.1063/1.1703745</pub-id>
</element-citation>
</ref>
<ref id="CR18">
<label>18</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Sykes</surname>
<given-names>MF</given-names>
</name>
<name>
<surname>Essam</surname>
<given-names>JW</given-names>
</name>
</person-group>
<article-title>Exact critical percolation probabilities for site and bond problems in two dimensions</article-title>
<source>J. Math. Phys.</source>
<year>1964</year>
<volume>5</volume>
<fpage>1117</fpage>
<lpage>1127</lpage>
<pub-id pub-id-type="doi">10.1063/1.1704215</pub-id>
</element-citation>
</ref>
<ref id="CR19">
<label>19</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Gerald</surname>
<given-names>P</given-names>
</name>
<name>
<surname>Ziff</surname>
<given-names>RM</given-names>
</name>
<name>
<surname>Stanley</surname>
<given-names>HE</given-names>
</name>
</person-group>
<article-title>Percolation threshold, fisher exponent, and shortest path exponent for four and five dimensions</article-title>
<source>Phys. Rev. E.</source>
<year>2001</year>
<volume>64</volume>
<fpage>026115</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevE.64.026115</pub-id>
</element-citation>
</ref>
<ref id="CR20">
<label>20</label>
<mixed-citation publication-type="other">Harry, K.
<italic>Percolation Theory for Mathematicians.</italic>
53–55 (Birkh01user: Boston,, 1982).</mixed-citation>
</ref>
<ref id="CR21">
<label>21</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Grimmett</surname>
<given-names>GR</given-names>
</name>
<name>
<surname>Manolescu</surname>
<given-names>I</given-names>
</name>
</person-group>
<article-title>Inhomogeneous bond percolation on square, triangular and hexagonal lattices</article-title>
<source>Ann. Probab.</source>
<year>2013</year>
<volume>41</volume>
<fpage>2990</fpage>
<lpage>3025</lpage>
<pub-id pub-id-type="doi">10.1214/11-AOP729</pub-id>
</element-citation>
</ref>
<ref id="CR22">
<label>22</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Iliev</surname>
<given-names>GK</given-names>
</name>
<name>
<surname>Rensburg</surname>
<given-names>EJJV</given-names>
</name>
<name>
<surname>Madras</surname>
<given-names>N</given-names>
</name>
</person-group>
<article-title>Phase diagram of inhomogeneous percolation with a defect plane</article-title>
<source>J. Stat. Phys.</source>
<year>2014</year>
<volume>158</volume>
<fpage>255</fpage>
<lpage>299</lpage>
<pub-id pub-id-type="doi">10.1007/s10955-014-1125-5</pub-id>
</element-citation>
</ref>
<ref id="CR23">
<label>23</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Gennadiy</surname>
<given-names>B</given-names>
</name>
<name>
<surname>Yessica</surname>
<given-names>CS</given-names>
</name>
</person-group>
<article-title>Percolation and lasing in real 3D crystals with inhomogeneous distributed random pores</article-title>
<source>Phys. B: Condens. Mat.</source>
<year>2014</year>
<volume>453</volume>
<fpage>12</fpage>
</element-citation>
</ref>
<ref id="CR24">
<label>24</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Ziff</surname>
<given-names>RM</given-names>
</name>
<etal></etal>
</person-group>
<article-title>The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices</article-title>
<source>J. Phys. A: Math. Theor.</source>
<year>2012</year>
<volume>45</volume>
<fpage>494005</fpage>
<pub-id pub-id-type="doi">10.1088/1751-8113/45/49/494005</pub-id>
</element-citation>
</ref>
<ref id="CR25">
<label>25</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Scullard</surname>
<given-names>CR</given-names>
</name>
<name>
<surname>Ziff</surname>
<given-names>RM</given-names>
</name>
</person-group>
<article-title>Critical surfaces for general inhomogeneous bond percolation problems</article-title>
<source>J. Stat. Mech.</source>
<year>2010</year>
<volume>03</volume>
<fpage>p03021</fpage>
</element-citation>
</ref>
<ref id="CR26">
<label>26</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Bender</surname>
<given-names>A</given-names>
</name>
<etal></etal>
</person-group>
<article-title>Inhomogeneous markov approach to percolation theory based propagation in random media</article-title>
<source>IEEE. T. Antenn. Propag.</source>
<year>2008</year>
<volume>56</volume>
<fpage>3271</fpage>
<lpage>3284</lpage>
<pub-id pub-id-type="doi">10.1109/TAP.2008.929439</pub-id>
</element-citation>
</ref>
<ref id="CR27">
<label>27</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Radicchi</surname>
<given-names>F</given-names>
</name>
</person-group>
<article-title>Percolation in real interdependent networks</article-title>
<source>Nature Phys.</source>
<year>2015</year>
<volume>11</volume>
<fpage>597</fpage>
<lpage>602</lpage>
<pub-id pub-id-type="doi">10.1038/nphys3374</pub-id>
</element-citation>
</ref>
<ref id="CR28">
<label>28</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Karrer</surname>
<given-names>B</given-names>
</name>
<name>
<surname>Newman</surname>
<given-names>MEJ</given-names>
</name>
<name>
<surname>Zdeborová</surname>
<given-names>L</given-names>
</name>
</person-group>
<article-title>Percolation on sparse networks</article-title>
<source>Phys. Rev. Lett.</source>
<year>2014</year>
<volume>113</volume>
<fpage>208702</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevLett.113.208702</pub-id>
<pub-id pub-id-type="pmid">25432059</pub-id>
</element-citation>
</ref>
<ref id="CR29">
<label>29</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Hamilton</surname>
<given-names>KE</given-names>
</name>
<name>
<surname>Pryadko</surname>
<given-names>LP</given-names>
</name>
</person-group>
<article-title>Tight lower bound for percolation threshold on an infinite graph</article-title>
<source>Phys. Rev. Lett.</source>
<year>2014</year>
<volume>113</volume>
<fpage>208701</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevLett.113.208701</pub-id>
<pub-id pub-id-type="pmid">25432058</pub-id>
</element-citation>
</ref>
<ref id="CR30">
<label>30</label>
<mixed-citation publication-type="other">Christensen, K. & Moloney, N. R.
<italic>Complexity and Criticality</italic>
15–22 (Imperial College Press, London, 2005).</mixed-citation>
</ref>
<ref id="CR31">
<label>31</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Gong</surname>
<given-names>JH</given-names>
</name>
<etal></etal>
</person-group>
<article-title>Simulation and analysis of control of severe acute respiratory syndrome</article-title>
<source>J. Remote Sens.</source>
<year>2003</year>
<volume>7</volume>
<fpage>260</fpage>
<lpage>265</lpage>
</element-citation>
</ref>
<ref id="CR32">
<label>32</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Hamby</surname>
<given-names>DM</given-names>
</name>
</person-group>
<article-title>A review of techniques for parameter sensitivity analysis of environmental models</article-title>
<source>Environ. Monit. Assess.</source>
<year>1994</year>
<volume>32</volume>
<fpage>135</fpage>
<lpage>154</lpage>
<pub-id pub-id-type="doi">10.1007/BF00547132</pub-id>
<pub-id pub-id-type="pmid">24214086</pub-id>
</element-citation>
</ref>
<ref id="CR33">
<label>33</label>
<element-citation publication-type="journal">
<person-group person-group-type="author">
<name>
<surname>Saltelli</surname>
<given-names>A</given-names>
</name>
<name>
<surname>Tarantola</surname>
<given-names>S</given-names>
</name>
<name>
<surname>Campolongo</surname>
<given-names>F</given-names>
</name>
</person-group>
<article-title>Sensitivity analysis as an ingredient of modelling</article-title>
<source>Stat. Sci.</source>
<year>2000</year>
<volume>15</volume>
<fpage>377</fpage>
<lpage>395</lpage>
<pub-id pub-id-type="doi">10.1214/ss/1009213004</pub-id>
</element-citation>
</ref>
</ref-list>
</back>
</pmc>
</record>

Pour manipuler ce document sous Unix (Dilib)

EXPLOR_STEP=$WICRI_ROOT/Wicri/Sante/explor/CovidV2/Data/Pmc/Corpus

HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 000326 | SxmlIndent | more

Ou

HfdSelect -h $EXPLOR_AREA/Data/Pmc/Corpus/biblio.hfd -nk 000326 | SxmlIndent | more

Pour mettre un lien sur cette page dans le réseau Wicri

{{Explor lien
   |wiki=    Wicri/Sante
   |area=    CovidV2
   |flux=    Pmc
   |étape=   Corpus
   |type=    RBID
   |clé=     PMC:4772486
   |texte=   How Inhomogeneous Site Percolation Works on Bethe Lattices: Theory and Application
}}

Pour générer des pages wiki

HfdIndexSelect -h $EXPLOR_AREA/Data/Pmc/Corpus/RBID.i   -Sk "pubmed:26926785" \
       | HfdSelect -Kh $EXPLOR_AREA/Data/Pmc/Corpus/biblio.hfd   \
       | NlmPubMed2Wicri -a CovidV2
Wicri

This area was generated with Dilib version V0.6.33.
Data generation: Sat Mar 28 17:51:24 2020. Site generation: Sun Jan 31 15:35:48 2021