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    <notes>
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    <div class="dc:title">Restif2006 - Whooping cough</div>
<div class="dc:bibliographicCitation">
  <p>This model is described in the article:</p>
  <div class="bibo:title">
    <a href="http://identifiers.org/pubmed/16615206" title="Access to this publication">Integrating life history and
    cross-immunity into the evolutionary dynamics of pathogens.</a>
  </div>
  <div class="bibo:authorList">Restif O, Grenfell BT.</div>
  <div class="bibo:Journal">Proc. Biol. Sci. 2006 Feb; 273(1585):
  409-416</div>
  <p>Abstract:</p>
  <div class="bibo:abstract">
    <p>Models for the diversity and evolution of pathogens have
    branched into two main directions: the adaptive dynamics of
    quantitative life-history traits (notably virulence) and the
    maintenance and invasion of multiple, antigenically diverse
    strains that interact with the host's immune memory. In a first
    attempt to reconcile these two approaches, we developed a
    simple modelling framework where two strains of pathogens,
    defined by a pair of life-history traits (infectious period and
    infectivity), interfere through a given level of
    cross-immunity. We used whooping cough as a potential example,
    but the framework proposed here could be applied to other acute
    infectious diseases. Specifically, we analysed the effects of
    these parameters on the invasion dynamics of one strain into a
    population, where the second strain is endemic. Whereas the
    deterministic version of the model converges towards stable
    coexistence of the two strains in most cases, stochastic
    simulations showed that transient epidemic dynamics can cause
    the extinction of either strain. Thus ecological dynamics,
    modulated by the immune parameters, eventually determine the
    adaptive value of different pathogen genotypes. We advocate an
    integrative view of pathogen dynamics at the crossroads of
    immunology, epidemiology and evolution, as a way towards
    efficient control of infectious diseases.</p>
  </div>
</div>
<div class="bm:curation">
  <p style="font-size: 12px; color: rgb(0, 0, 0); font-family: helvetica, 'trebuchet MS', arial, sans-serif;">This
  version of the model can be used for both the stochastic and the
  deterministic simulations described in the article. For
  deterministic interpretations with infinite population sizes, set
  the population size 
  <em>N</em> = 1. The model reproduces the deterministic time
  courses. Stochastic interpretation with Copasi UI gave results
  similar to the article, but was not extensively tested. The
  initial conditions for competition simulations can be derived by
  equilibrating the system for one pathogen and then adding a
  starting concentration for the other.</p>
  <p style="font-size: 12px; color: rgb(0, 0, 0); font-family: helvetica, 'trebuchet MS', arial, sans-serif;">Originally
  created by libAntimony v1.3 (using libSBML 4.1.0-b1)</p>
</div>
<div class="dc:publisher">
  <p>This model is hosted on 
  <a href="http://www.ebi.ac.uk/biomodels/">BioModels Database</a>
  and identified by: 
  <a href="http://identifiers.org/biomodels.db/BIOMD0000000249">BIOMD0000000249</a>.</p>
  <p>To cite BioModels Database, please use: 
  <a href="http://identifiers.org/pubmed/20587024" title="Latest BioModels Database publication">BioModels Database:
  An enhanced, curated and annotated resource for published
  quantitative kinetic models</a>.</p>
</div>
<div class="dc:license">
  <p>To the extent possible under law, all copyright and related or
  neighbouring rights to this encoded model have been dedicated to
  the public domain worldwide. Please refer to 
  <a href="http://creativecommons.org/publicdomain/zero/1.0/" title="Access to: CC0 1.0 Universal (CC0 1.0), Public Domain Dedication">CC0
  Public Domain Dedication</a> for more information.</p>
</div>
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            <apply>
              <times/>
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            </apply>
          </math>
                </kineticLaw>
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              <times/>
              <ci> mu </ci>
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            <apply>
              <times/>
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            <apply>
              <times/>
              <ci> mu </ci>
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                </kineticLaw>
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            <apply>
              <times/>
              <ci> mu </ci>
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                </kineticLaw>
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        <annotation>
	<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:bqbiol="http://biomodels.net/biology-qualifiers/">
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            <apply>
              <times/>
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              <times/>
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            <apply>
              <times/>
              <ci> mu </ci>
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                </kineticLaw>
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                <divide/>
                <apply>
                  <plus/>
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                </apply>
                <ci> N </ci>
              </apply>
              <ci> S </ci>
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          </math>
                </kineticLaw>
      </reaction>
      <reaction id="r11" metaid="metaid_0000061" name="Primary Infection with strain 2" reversible="false" sboTerm="SBO:0000375">
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                <divide/>
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                  <plus/>
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                <ci> N </ci>
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          </math>
                </kineticLaw>
      </reaction>
      <reaction id="r12" metaid="metaid_0000062" name="Secondary Infection with strain 1" reversible="false" sboTerm="SBO:0000375">
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            <apply>
              <times/>
              <apply>
                <divide/>
                <apply>
                  <times/>
                  <apply>
                    <minus/>
                    <cn type="integer"> 1 </cn>
                    <ci> psi </ci>
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                  <ci> beta_1 </ci>
                  <apply>
                    <plus/>
                    <ci> I_1 </ci>
                    <ci> I_1p </ci>
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                </apply>
                <ci> N </ci>
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              <ci> R_2 </ci>
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          </math>
                </kineticLaw>
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      <reaction id="r13" metaid="metaid_0000063" name="Secondary Infection with strain 2" reversible="false" sboTerm="SBO:0000375">
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              <apply>
                <divide/>
                <apply>
                  <times/>
                  <apply>
                    <minus/>
                    <cn type="integer"> 1 </cn>
                    <ci> psi </ci>
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                  <ci> beta_2 </ci>
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                    <plus/>
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                    <ci> I_2p </ci>
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                <ci> N </ci>
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              <ci> R_1 </ci>
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          </math>
                </kineticLaw>
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      <reaction id="r14" metaid="metaid_0000064" name="Recovery (I_1)" reversible="false" sboTerm="SBO:0000375">
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          <math xmlns="http://www.w3.org/1998/Math/MathML">          
            <apply>
              <times/>
              <ci> gamma_1 </ci>
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                </kineticLaw>
      </reaction>
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            <apply>
              <times/>
              <ci> gamma_2 </ci>
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                </kineticLaw>
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          <math xmlns="http://www.w3.org/1998/Math/MathML">          
            <apply>
              <times/>
              <ci> gamma_1 </ci>
              <ci> I_1p </ci>
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                </kineticLaw>
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          <math xmlns="http://www.w3.org/1998/Math/MathML">          
            <apply>
              <times/>
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              <ci> I_2p </ci>
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                </kineticLaw>
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        <kineticLaw metaid="_0ab5e956-6514-4707-b2a0-ed1402ddc952">
          <math xmlns="http://www.w3.org/1998/Math/MathML">          
            <apply>
              <times/>
              <ci> sigma </ci>
              <ci> R_1 </ci>
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          </math>
                </kineticLaw>
      </reaction>
      <reaction id="r19" metaid="metaid_0000069" name="Loss of Immunity (R_2)" reversible="false" sboTerm="SBO:0000375">
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        <kineticLaw metaid="_2b94f20b-dec9-4f3e-852b-7338b81c126b">
          <math xmlns="http://www.w3.org/1998/Math/MathML">          
            <apply>
              <times/>
              <ci> sigma </ci>
              <ci> R_2 </ci>
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                </kineticLaw>
      </reaction>
      <reaction id="r20" metaid="metaid_0000070" name="Loss of Immunity (R_p)" reversible="false" sboTerm="SBO:0000375">
        <listOfReactants>
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        <kineticLaw metaid="_3687c9ac-7318-4cde-bfca-457b323c3088">
          <math xmlns="http://www.w3.org/1998/Math/MathML">          
            <apply>
              <times/>
              <ci> sigma </ci>
              <ci> R_p </ci>
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                </kineticLaw>
      </reaction>
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</sbml>