- Author:
- Tommy Yu <tommy.yu@auckland.ac.nz>
- Date:
- 2010-08-25 18:37:17+12:00
- Desc:
- xml:base fix
- Permanent Source URI:
- https://models.physiomeproject.org/workspace/perelson_kirschner_deboer_1993/rawfile/63e5e8e2657647b24c9552db53a555b7d6ae8555/perelson_1993b.cellml
<?xml version="1.0"?>
<!-- FILE : perelson_1993b.cellml
CREATED : 12th Jan 2010
LAST MODIFIED : 13th Jan 2010
AUTHOR : Ethan Choi
MODEL STATUS : This model conforms to the CellML 1.1 Specification. Runs in OpenCell
DESCRIPTION : This file contains a CellML description of Perelson's 1993 mathematical model of the dynamics of HIV infection of CD4+ T cells--><model xmlns="http://www.cellml.org/cellml/1.0#" xmlns:cmeta="http://www.cellml.org/metadata/1.0#" cmeta:id="perelson_1993b" name="perelson_1993b">
<!-- documentation -->
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Dynamics of HIV infection of CD4+ T cells</title>
<author>
<firstname>Ethan</firstname>
<surname>Choi</surname>
<affiliation>
<shortaffil>Auckland Bioengineering Institute, The University of Auckland</shortaffil>
</affiliation>
</author>
</articleinfo>
<section id="sec_status">
<title>Model Status</title>
<para>This model has been built with the differential expressions in Perelson, Kirschner and de Boer's 1993 paper for the 2nd model (perelson_1993b); where viral infection of T-cell precursors occur replacing the source, s with the function s_V. This file is known to run in OpenCell and COR, and is set to the parameters in Table 1 of the paper. By changing the the parameters, the model will replicate all panels in figures 11. Note that in the paper, some figure diagrams are scaled logarithmically.</para>
</section>
<sect1 id="sec_structure">
<title>Model Structure</title>
<para>ABSTRACT: We examine a model for the interaction of HIV with CD4+ T cells that considers four populations: uninfected T cells, latently infected T cells, actively infected T cells, and free virus. Using this model we show that many of the puzzling quantitative features of HIV infection can be explained simply. We also consider effects of AZT on viral growth and T-cell population dynamics. The model exhibits two steady states, an uninfected state in which no virus is present and an endemically infected state, in which virus and infected T cells are present. We show that if N, the number of infectious virions produced per actively infected T cell, is less a critical value, Ncrit, then the uninfected state is the only steady state in the nonnegative orthant, and this state is stable. For N > Ncrit, the uninfected state is unstable, and the endemically infected state can be either stable, or unstable and surrounded by a stable limit cycle. Using numerical bifurcation techniques we map out the parameter regimes of these various behaviors. oscillatory behavior seems to lie outside the region of biologically realistic parameter values. When the endemically infected state is stable, it is characterized by a reduced number of T cells compared with the uninfected state. Thus T-cell depletion occurs through the establishment of a new steady state. The dynamics of the establishment of this new steady state are examined both numerically and via the quasi-steady-state approximation. We develop approximations for the dynamics at early times in which the free virus rapidly binds to T cells, during an intermediate time scale in which the virus grows exponentially, and a third time scale on which viral growth slows and the endemically infected steady state is approached. Using the quasi-steady-state approximation the model can be simplified to two ordinary differential equations the summarize much of the dynamical behavior. We compute the level of T cells in the endemically infected state and show how that level varies with the parameters in the model. The model predicts that different viral strains, characterized by generating differing numbers of infective virions within infected T cells, can cause different amounts of T-cell depletion and generate depletion at different rates. Two versions of the model are studied. In one the source of T cells from precursors is constant, whereas in the other the source of T cells decreases with viral load, mimicking the infection and killing of T-cell precursors. The latter gives more realistic predictions than the model with a constant source.</para>
<informalfigure float="0" id="fig_reaction_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>model diagram</title>
</objectinfo>
<imagedata fileref="perelson_1993.png"/>
</imageobject>
</mediaobject>
<caption>Schematic diagram of the model.</caption>
</informalfigure>
<para>The original paper reference is cited below:</para>
<para>Dynamics of HIV infection of CD4+ T cells, Perelson, Kirschner, Boer, <emphasis>Mathematical Biosciences</emphasis>, 114, 81-125. <ulink url="http://www.ncbi.nlm.nih.gov/pubmed/8096155">PubMed ID: 8096155</ulink></para>
</sect1>
</article>
</documentation>
<!-- units -->
<units name="day">
<unit multiplier="86400" units="second"/>
</units>
<units name="per_day">
<unit exponent="-1" multiplier="86400" units="second"/>
</units>
<units name="mm3">
<unit exponent="3" prefix="milli" units="metre"/>
</units>
<units name="per_mm3">
<unit exponent="-3" prefix="milli" units="metre"/>
</units>
<units name="per_day_mm3">
<unit units="per_day"/>
<unit units="per_mm3"/>
</units>
<units name="mm3_per_day">
<unit exponent="3" prefix="milli" units="metre"/>
<unit units="per_day"/>
</units>
<!-- components -->
<component name="environment">
<variable cmeta:id="environment_time" name="time" public_interface="out" units="day"/>
</component>
<component cmeta:id="uninfected" name="uninfected">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#uninfected">This component relates to the uninfected CD4+ T cell dynamics</rdf:Description>
<rdf:Description rdf:about="#T">Uninfected CD4+ T cell population size</rdf:Description>
<rdf:Description rdf:about="#s_V">rate of supply of CD4+ T cells from precursors, as a function of viral load</rdf:Description>
</rdf:RDF>
<variable name="time" public_interface="in" units="day"/>
<variable initial_value="10" name="s" units="per_day_mm3"/>
<variable initial_value="0.03" name="r" units="per_day"/>
<variable initial_value="1500" name="T_max" units="per_mm3"/>
<variable initial_value="0.02" name="mu_T" public_interface="out" units="per_day"/>
<variable initial_value="1" name="theta" units="per_mm3"/>
<variable name="k_1" public_interface="in" units="mm3_per_day"/>
<variable name="T_1" public_interface="in" units="per_mm3"/>
<variable name="T_2" public_interface="in" units="per_mm3"/>
<variable name="V" public_interface="in" units="per_mm3"/>
<variable cmeta:id="s_V" name="s_V" units="per_day_mm3"/>
<variable cmeta:id="T" initial_value="1000" name="T" public_interface="out" units="per_mm3"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="T_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci>T</ci>
</apply>
<apply>
<minus/>
<apply>
<plus/>
<apply>
<minus/>
<ci>s_V</ci>
<apply>
<times/>
<ci>mu_T</ci>
<ci>T</ci>
</apply>
</apply>
<apply>
<times/>
<ci>r</ci>
<ci>T</ci>
<apply>
<minus/>
<cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn>
<apply>
<divide/>
<apply>
<plus/>
<ci>T</ci>
<ci>T_1</ci>
<ci>T_2</ci>
</apply>
<ci>T_max</ci>
</apply>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>k_1</ci>
<ci>V</ci>
<ci>T</ci>
</apply>
</apply>
</apply>
<apply id="s_V_function">
<eq/>
<ci>s_V</ci>
<apply>
<divide/>
<apply>
<times/>
<ci>theta</ci>
<ci>s</ci>
</apply>
<apply>
<plus/>
<ci>theta</ci>
<ci>V</ci>
</apply>
</apply>
</apply>
</math>
</component>
<component cmeta:id="latently_infected" name="latently_infected">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#latently_infected">This component relates to the latently infected CD4+ helper cell dynamics</rdf:Description>
<rdf:Description rdf:about="#T_1">Latently infected CD4+ helper cell population size (T* in the paper)</rdf:Description>
</rdf:RDF>
<variable name="time" public_interface="in" units="day"/>
<variable initial_value="2.4E-5" name="k_1" public_interface="out" units="mm3_per_day"/>
<variable name="k_2" public_interface="in" units="per_day"/>
<variable name="mu_T" public_interface="in" units="per_day"/>
<variable name="V" public_interface="in" units="per_mm3"/>
<variable name="T" public_interface="in" units="per_mm3"/>
<variable cmeta:id="T_1" initial_value="0" name="T_1" public_interface="out" units="per_mm3"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="T_1_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci>T_1</ci>
</apply>
<apply>
<minus/>
<apply>
<minus/>
<apply>
<times/>
<ci>k_1</ci>
<ci>V</ci>
<ci>T</ci>
</apply>
<apply>
<times/>
<ci>mu_T</ci>
<ci>T_1</ci>
</apply>
</apply>
<apply>
<times/>
<ci>k_2</ci>
<ci>T_1</ci>
</apply>
</apply>
</apply>
</math>
</component>
<component cmeta:id="actively_infected" name="actively_infected">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#actively_infected">This component relates to the actively infected CD4+ helper cell dynamics</rdf:Description>
<rdf:Description rdf:about="#T_2">actively infected CD4+ helper cell population size (T** in the paper)</rdf:Description>
</rdf:RDF>
<variable name="time" public_interface="in" units="day"/>
<variable initial_value="3E-3" name="k_2" public_interface="out" units="per_day"/>
<variable initial_value="0.24" name="mu_b" public_interface="out" units="per_day"/>
<variable name="T_1" public_interface="in" units="per_mm3"/>
<variable cmeta:id="T_2" initial_value="0" name="T_2" public_interface="out" units="per_mm3"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="T_2_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci>T_2</ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci>k_2</ci>
<ci>T_1</ci>
</apply>
<apply>
<times/>
<ci>mu_b</ci>
<ci>T_2</ci>
</apply>
</apply>
</apply>
</math>
</component>
<component cmeta:id="free_virus_particle" name="free_virus_particle">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#free_virus_particle">This component relates to the HIV population size</rdf:Description>
<rdf:Description rdf:about="#V">concentration of free infectious viral particles</rdf:Description>
</rdf:RDF>
<variable name="time" public_interface="in" units="day"/>
<variable name="mu_b" public_interface="in" units="per_day"/>
<variable name="k_1" public_interface="in" units="mm3_per_day"/>
<variable initial_value="2.4" name="mu_V" units="per_day"/>
<variable cmeta:id="free_virus_particle_N" initial_value="1400" name="N" units="dimensionless"/>
<variable name="T" public_interface="in" units="per_mm3"/>
<variable name="T_2" public_interface="in" units="per_mm3"/>
<variable cmeta:id="V" initial_value="1.0E-3" name="V" public_interface="out" units="per_mm3"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="V_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci>V</ci>
</apply>
<apply>
<minus/>
<apply>
<minus/>
<apply>
<times/>
<ci>N</ci>
<ci>mu_b</ci>
<ci>T_2</ci>
</apply>
<apply>
<times/>
<ci>k_1</ci>
<ci>V</ci>
<ci>T</ci>
</apply>
</apply>
<apply>
<times/>
<ci>mu_V</ci>
<ci>V</ci>
</apply>
</apply>
</apply>
</math>
</component>
<component cmeta:id="T_cell_population" name="T_cell_population">
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="#T_cell_population">This component calculates the total CD4+ T cell</rdf:Description>
<rdf:Description rdf:about="#T_tot">Total CD4+ T-cell population</rdf:Description>
</rdf:RDF>
<variable name="T" public_interface="in" units="per_mm3"/>
<variable name="T_1" public_interface="in" units="per_mm3"/>
<variable name="T_2" public_interface="in" units="per_mm3"/>
<variable name="T_tot" units="per_mm3"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="T_tot_calculation">
<eq/>
<ci>T_tot</ci>
<apply>
<plus/>
<ci>T</ci>
<ci>T_1</ci>
<ci>T_2</ci>
</apply>
</apply>
</math>
</component>
<!-- connections -->
<connection>
<map_components component_1="environment" component_2="uninfected"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="environment" component_2="latently_infected"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="environment" component_2="actively_infected"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="environment" component_2="free_virus_particle"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="uninfected" component_2="latently_infected"/>
<map_variables variable_1="T" variable_2="T"/>
<map_variables variable_1="mu_T" variable_2="mu_T"/>
<map_variables variable_1="k_1" variable_2="k_1"/>
<map_variables variable_1="T_1" variable_2="T_1"/>
</connection>
<connection>
<map_components component_1="uninfected" component_2="actively_infected"/>
<map_variables variable_1="T_2" variable_2="T_2"/>
</connection>
<connection>
<map_components component_1="uninfected" component_2="free_virus_particle"/>
<map_variables variable_1="T" variable_2="T"/>
<map_variables variable_1="V" variable_2="V"/>
</connection>
<connection>
<map_components component_1="latently_infected" component_2="actively_infected"/>
<map_variables variable_1="T_1" variable_2="T_1"/>
<map_variables variable_1="k_2" variable_2="k_2"/>
</connection>
<connection>
<map_components component_1="latently_infected" component_2="free_virus_particle"/>
<map_variables variable_1="k_1" variable_2="k_1"/>
<map_variables variable_1="V" variable_2="V"/>
</connection>
<connection>
<map_components component_1="actively_infected" component_2="free_virus_particle"/>
<map_variables variable_1="mu_b" variable_2="mu_b"/>
<map_variables variable_1="T_2" variable_2="T_2"/>
</connection>
<connection>
<map_components component_1="T_cell_population" component_2="uninfected"/>
<map_variables variable_1="T" variable_2="T"/>
</connection>
<connection>
<map_components component_1="T_cell_population" component_2="latently_infected"/>
<map_variables variable_1="T_1" variable_2="T_1"/>
</connection>
<connection>
<map_components component_1="T_cell_population" component_2="actively_infected"/>
<map_variables variable_1="T_2" variable_2="T_2"/>
</connection>
<!-- metadata -->
<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
<rdf:Description rdf:about="">
<dc:title xmlns:dc="http://purl.org/dc/elements/1.1/">Dynamics of HIV infection of CD4+ T cells (Model 2)</dc:title>
<dc:creator xmlns:dc="http://purl.org/dc/elements/1.1/" rdf:parseType="Resource">
<vCard:N xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" rdf:parseType="Resource">
<vCard:Family>Choi</vCard:Family>
<vCard:Given>Ethan</vCard:Given>
</vCard:N>
<vCard:EMAIL xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" rdf:parseType="Resource">
<rdf:value>mcho099@aucklanduni.ac.nz</rdf:value>
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<vCard:Orgname>The University of Auckland</vCard:Orgname>
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</rdf:Description>
<rdf:Description rdf:about="#perelson_1993b">
<dc:title xmlns:dc="http://purl.org/dc/elements/1.1/">Dynamics of HIV infection of CD4+ T cells</dc:title>
<cmeta:comment rdf:parseType="Resource">
<rdf:value>This is the CellML description of Perelson, Kirschner, and Boer's 1993 mathematical model for the dynamics of HIV infection of CD4+ T cells</rdf:value>
<dc:creator xmlns:dc="http://purl.org/dc/elements/1.1/" rdf:parseType="Resource">
<vCard:FN xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Ethan Choi</vCard:FN>
</dc:creator>
</cmeta:comment>
<cmeta:bio_entity>CD4+ T cell</cmeta:bio_entity>
<!-- Keywords -->
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<bqs:subject_type>keyword</bqs:subject_type>
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<rdf:Bag>
<rdf:li>Immunology</rdf:li>
<rdf:li>CD4 T cell</rdf:li>
<rdf:li>HIV</rdf:li>
<rdf:li>dynamics</rdf:li>
<rdf:li>AZT</rdf:li>
</rdf:Bag>
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</dc:subject>
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<vCard:Family>Perelson</vCard:Family>
<vCard:Given>Alan</vCard:Given>
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<vCard:Given>Denise</vCard:Given>
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<vCard:N xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" rdf:parseType="Resource">
<vCard:Family>de Boer</vCard:Family>
<vCard:Given>Rob</vCard:Given>
</vCard:N>
</bqs:Person>
</rdf:li>
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</dc:creator>
<dc:title xmlns:dc="http://purl.org/dc/elements/1.1/">Dynamics of HIV infection of CD4+ T cells</dc:title>
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<dcterms:W3CDTF>1993-03</dcterms:W3CDTF>
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<dc:title xmlns:dc="http://purl.org/dc/elements/1.1/">Mathematical Biosciences</dc:title>
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<!-- OpenCell metadata -->
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