Location: Wodarz, 2003 @ 737f4cea6ab1 / wodarz_2003.cellml

Author:

Tessa Paris <tpar054@aucklanduni.ac.nz>

Date:

2010-01-12 14:54:13+13:00

Desc:

Added initial conditions. The model can replicate part A of Figure 2 in the paper. To replicate the long memory g = 0.01, to replicate the short memory g = 40 (note this is larger than what is specified in the figure legend. Using the log of P_2 the model shows that the abundance of pathogens goes below zero which is not shown in the paper.)

Permanent Source URI:

https://models.cellml.org/workspace/wodarz_2003/rawfile/737f4cea6ab170b07fe86ff803f9ee8d4109b57a/wodarz_2003.cellml

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COR (0.9.31.1333)
Copyright 2002-2010 Dr Alan Garny
http://cor.physiol.ox.ac.uk/ - cor@physiol.ox.ac.uk

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http://www.cellml.org/
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				<title>Modelling the Evolution of Immunological Memory</title>
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					<firstname>Catherine</firstname>
					<surname>Lloyd</surname>
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						<shortaffil>Bioengineering Institute, University of Auckland</shortaffil>
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            This is the original unchecked version of the model imported from the previous
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Memory is an important characteristic of immune responses.  Immunological memory is defined as an increased number of specific immune cells that remain in the body after an infection has been resolved, and they function to protect the host against reinfection.  The exact process by which immunological memory evolves is not fully known.  To date, suggestions for the advantages which might drive the evolution of memory include: protection from reinfection, control of long-term infections, and the transfer of immune function to the next generation.    
</para>
				<para>
In this study by Wodarz, two mathematical models are developed to investigate new potential advantages that could drive the evolution of immunological memory (see figure 1 and figure 2 below).  Although mathematical analysis clearly shows that a longer duration of memory is advantageous for the host in that it protects against reinfection, it may also allow another, inferior pathogen species to persist.  This can be a significant disadvantage to the host if the inferior pathogen is more virulent.  In the absence of pathogen diversity, memory is expected to evolve to a long duration, but with increased pathogen diversity and under certain specific conditions, memory can evolve towards shorter durations. 
</para>
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The complete original paper reference is cited below:
</para>
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Evolution of Immunological Memory and the Regulation of Competition between Pathogens, Dominik Wodarz, 2003, <emphasis>Current Biology</emphasis>, 13, 1648-1652.  <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;list_uids=13678598&amp;dopt=Abstract">PubMed ID: 13678598</ulink>
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					<caption>Interactions between two pathogen populations and a host population.</caption>
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      <variable name="d" units="first_order_rate_constant" public_interface="in"/>
      <variable name="g" units="first_order_rate_constant" public_interface="in"/>
      <variable name="I_1" units="dimensionless" public_interface="in"/>
      <variable name="P_2" units="dimensionless" public_interface="in"/>
      <variable name="time" units="day" public_interface="in"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>R_1</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <minus/>
                  <apply>
                     <minus/>
                     <apply>
                        <times/>
                        <ci>alpha_1</ci>
                        <ci>I_1</ci>
                     </apply>
                     <apply>
                        <times/>
                        <ci>d</ci>
                        <ci>R_1</ci>
                     </apply>
                  </apply>
                  <apply>
                     <times/>
                     <ci>g</ci>
                     <ci>R_1</ci>
                  </apply>
               </apply>
               <apply>
                  <times/>
                  <ci>beta_2</ci>
                  <ci>R_1</ci>
                  <ci>P_2</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component xmlns="http://www.cellml.org/cellml/1.0#" name="R2">
      <variable name="R_2" units="dimensionless" initial_value="0" public_interface="out"/>
      <variable name="alpha_2" units="first_order_rate_constant" public_interface="in"/>
      <variable name="beta_1" units="first_order_rate_constant" public_interface="in"/>
      <variable name="d" units="first_order_rate_constant" public_interface="in"/>
      <variable name="g" units="first_order_rate_constant" public_interface="in"/>
      <variable name="I_2" units="dimensionless" public_interface="in"/>
      <variable name="P_1" units="dimensionless" public_interface="in"/>
      <variable name="time" units="day" public_interface="in"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>R_2</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <minus/>
                  <apply>
                     <minus/>
                     <apply>
                        <times/>
                        <ci>alpha_2</ci>
                        <ci>I_2</ci>
                     </apply>
                     <apply>
                        <times/>
                        <ci>d</ci>
                        <ci>R_2</ci>
                     </apply>
                  </apply>
                  <apply>
                     <times/>
                     <ci>g</ci>
                     <ci>R_2</ci>
                  </apply>
               </apply>
               <apply>
                  <times/>
                  <ci>beta_1</ci>
                  <ci>R_2</ci>
                  <ci>P_1</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component xmlns="http://www.cellml.org/cellml/1.0#" name="R12">
      <variable name="R_12" units="dimensionless" initial_value="0" public_interface="out"/>
      <variable name="alpha_2" units="first_order_rate_constant" public_interface="in"/>
      <variable name="alpha_1" units="first_order_rate_constant" public_interface="in"/>
      <variable name="d" units="first_order_rate_constant" public_interface="in"/>
      <variable name="g" units="first_order_rate_constant" public_interface="in"/>
      <variable name="I_21" units="dimensionless" public_interface="in"/>
      <variable name="I_12" units="dimensionless" public_interface="in"/>
      <variable name="time" units="day" public_interface="in"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>R_12</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <minus/>
                  <apply>
                     <plus/>
                     <apply>
                        <times/>
                        <ci>alpha_2</ci>
                        <ci>I_12</ci>
                     </apply>
                     <apply>
                        <times/>
                        <ci>alpha_1</ci>
                        <ci>I_21</ci>
                     </apply>
                  </apply>
                  <apply>
                     <times/>
                     <ci>d</ci>
                     <ci>R_12</ci>
                  </apply>
               </apply>
               <apply>
                  <times/>
                  <ci>g</ci>
                  <ci>R_12</ci>
               </apply>
            </apply>
         </apply>
      </math>
   </component>
   <component xmlns="http://www.cellml.org/cellml/1.0#" name="P1">
      <variable name="P_1" units="dimensionless" initial_value="1" public_interface="out"/>
      <variable name="k_1" units="first_order_rate_constant" initial_value="1"/>
      <variable name="u" units="first_order_rate_constant" public_interface="in"/>
      <variable name="I_1" units="dimensionless" public_interface="in"/>
      <variable name="I_21" units="dimensionless" public_interface="in"/>
      <variable name="time" units="day" public_interface="in"/>
      <variable name="log_P1" units="dimensionless"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>P_1</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <times/>
                  <ci>k_1</ci>
                  <apply>
                     <plus/>
                     <ci>I_1</ci>
                     <ci>I_21</ci>
                  </apply>
               </apply>
               <apply>
                  <times/>
                  <ci>u</ci>
                  <ci>P_1</ci>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>log_P1</ci>
            <apply>
               <log/>
               <ci>P_1</ci>
            </apply>
         </apply>
      </math>
   </component>
   <component xmlns="http://www.cellml.org/cellml/1.0#" name="P2">
      <variable name="P_2" units="dimensionless" initial_value="1" public_interface="out"/>
      <variable name="k_2" units="first_order_rate_constant" initial_value="1"/>
      <variable name="u" units="first_order_rate_constant" public_interface="in"/>
      <variable name="I_2" units="dimensionless" public_interface="in"/>
      <variable name="I_12" units="dimensionless" public_interface="in"/>
      <variable name="time" units="day" public_interface="in"/>
      <variable name="log_P2" units="dimensionless"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <apply>
               <diff/>
               <bvar>
                  <ci>time</ci>
               </bvar>
               <ci>P_2</ci>
            </apply>
            <apply>
               <minus/>
               <apply>
                  <times/>
                  <ci>k_2</ci>
                  <apply>
                     <plus/>
                     <ci>I_2</ci>
                     <ci>I_12</ci>
                  </apply>
               </apply>
               <apply>
                  <times/>
                  <ci>u</ci>
                  <ci>P_2</ci>
               </apply>
            </apply>
         </apply>
         <apply>
            <eq/>
            <ci>log_P2</ci>
            <apply>
               <log/>
               <ci>P_2</ci>
            </apply>
         </apply>
      </math>
   </component>
   <component xmlns="http://www.cellml.org/cellml/1.0#" name="memory_duration">
      <variable name="G" units="dimensionless"/>
      <variable name="g" units="first_order_rate_constant" initial_value="0.01" public_interface="out"/>
      <math xmlns="http://www.w3.org/1998/Math/MathML">
         <apply>
            <eq/>
            <ci>G</ci>
            <apply>
               <divide/>
               <cn cellml:units="first_order_rate_constant">1</cn>
               <ci>g</ci>
            </apply>
         </apply>
      </math>
   </component>
   <component xmlns="http://www.cellml.org/cellml/1.0#" name="kinetic_parameters">
      <variable name="beta_1" units="first_order_rate_constant" initial_value="1" public_interface="out"/>
      <variable name="alpha_1" units="first_order_rate_constant" initial_value="0.1" public_interface="out"/>
      <variable name="beta_2" units="first_order_rate_constant" initial_value="1" public_interface="out"/>
      <variable name="alpha_2" units="first_order_rate_constant" initial_value="0.1" public_interface="out"/>
      <variable name="a_1" units="first_order_rate_constant" initial_value="0.03" public_interface="out"/>
      <variable name="a_2" units="first_order_rate_constant" initial_value="1" public_interface="out"/>
      <variable name="d" units="first_order_rate_constant" initial_value="0.01" public_interface="out"/>
      <variable name="u" units="first_order_rate_constant" initial_value="0.5" public_interface="out"/>
   </component>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="S" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I1" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I2" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I12" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I21" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R1" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R2" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R12" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="P1" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="P2" component_2="environment"/>
      <map_variables variable_1="time" variable_2="time"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="S" component_2="P1"/>
      <map_variables variable_1="P_1" variable_2="P_1"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="S" component_2="P2"/>
      <map_variables variable_1="P_2" variable_2="P_2"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="S" component_2="R1"/>
      <map_variables variable_1="R_1" variable_2="R_1"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="S" component_2="R2"/>
      <map_variables variable_1="R_2" variable_2="R_2"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="S" component_2="R12"/>
      <map_variables variable_1="R_12" variable_2="R_12"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I1" component_2="S"/>
      <map_variables variable_1="S" variable_2="S"/>
      <map_variables variable_1="I_1" variable_2="I_1"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I2" component_2="S"/>
      <map_variables variable_1="S" variable_2="S"/>
      <map_variables variable_1="I_2" variable_2="I_2"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I12" component_2="S"/>
      <map_variables variable_1="I_12" variable_2="I_12"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I21" component_2="S"/>
      <map_variables variable_1="I_21" variable_2="I_21"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I1" component_2="P1"/>
      <map_variables variable_1="I_1" variable_2="I_1"/>
      <map_variables variable_1="P_1" variable_2="P_1"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I2" component_2="P2"/>
      <map_variables variable_1="I_2" variable_2="I_2"/>
      <map_variables variable_1="P_2" variable_2="P_2"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I21" component_2="P1"/>
      <map_variables variable_1="I_21" variable_2="I_21"/>
      <map_variables variable_1="P_1" variable_2="P_1"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I12" component_2="P2"/>
      <map_variables variable_1="I_12" variable_2="I_12"/>
      <map_variables variable_1="P_2" variable_2="P_2"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R1" component_2="I1"/>
      <map_variables variable_1="I_1" variable_2="I_1"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R1" component_2="I12"/>
      <map_variables variable_1="R_1" variable_2="R_1"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R2" component_2="I21"/>
      <map_variables variable_1="R_2" variable_2="R_2"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R2" component_2="I2"/>
      <map_variables variable_1="I_2" variable_2="I_2"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R1" component_2="P2"/>
      <map_variables variable_1="P_2" variable_2="P_2"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R2" component_2="P1"/>
      <map_variables variable_1="P_1" variable_2="P_1"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R12" component_2="I21"/>
      <map_variables variable_1="I_21" variable_2="I_21"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R12" component_2="I12"/>
      <map_variables variable_1="I_12" variable_2="I_12"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="S" component_2="kinetic_parameters"/>
      <map_variables variable_1="d" variable_2="d"/>
      <map_variables variable_1="beta_1" variable_2="beta_1"/>
      <map_variables variable_1="beta_2" variable_2="beta_2"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I1" component_2="kinetic_parameters"/>
      <map_variables variable_1="alpha_1" variable_2="alpha_1"/>
      <map_variables variable_1="a_1" variable_2="a_1"/>
      <map_variables variable_1="beta_1" variable_2="beta_1"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I2" component_2="kinetic_parameters"/>
      <map_variables variable_1="alpha_2" variable_2="alpha_2"/>
      <map_variables variable_1="a_2" variable_2="a_2"/>
      <map_variables variable_1="beta_2" variable_2="beta_2"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I21" component_2="kinetic_parameters"/>
      <map_variables variable_1="alpha_1" variable_2="alpha_1"/>
      <map_variables variable_1="a_1" variable_2="a_1"/>
      <map_variables variable_1="beta_1" variable_2="beta_1"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="I12" component_2="kinetic_parameters"/>
      <map_variables variable_1="alpha_2" variable_2="alpha_2"/>
      <map_variables variable_1="a_2" variable_2="a_2"/>
      <map_variables variable_1="beta_2" variable_2="beta_2"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R1" component_2="kinetic_parameters"/>
      <map_variables variable_1="alpha_1" variable_2="alpha_1"/>
      <map_variables variable_1="d" variable_2="d"/>
      <map_variables variable_1="beta_2" variable_2="beta_2"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R2" component_2="kinetic_parameters"/>
      <map_variables variable_1="alpha_2" variable_2="alpha_2"/>
      <map_variables variable_1="beta_1" variable_2="beta_1"/>
      <map_variables variable_1="d" variable_2="d"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R12" component_2="kinetic_parameters"/>
      <map_variables variable_1="alpha_1" variable_2="alpha_1"/>
      <map_variables variable_1="alpha_2" variable_2="alpha_2"/>
      <map_variables variable_1="d" variable_2="d"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="P1" component_2="kinetic_parameters"/>
      <map_variables variable_1="u" variable_2="u"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="P2" component_2="kinetic_parameters"/>
      <map_variables variable_1="u" variable_2="u"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="S" component_2="memory_duration"/>
      <map_variables variable_1="g" variable_2="g"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R1" component_2="memory_duration"/>
      <map_variables variable_1="g" variable_2="g"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R2" component_2="memory_duration"/>
      <map_variables variable_1="g" variable_2="g"/>
   </connection>
   <connection xmlns="http://www.cellml.org/cellml/1.0#">
      <map_components component_1="R12" component_2="memory_duration"/>
      <map_variables variable_1="g" variable_2="g"/>
   </connection>
</model>