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  <front>
    <journal-meta>
      <journal-id journal-id-type="pmc">pnas</journal-id>
      <journal-id journal-id-type="pubmed">Proc Natl Acad Sci U S A</journal-id>
      <journal-id journal-id-type="publisher">PNAS</journal-id>
      <issn>0027-8424</issn>
      <publisher>
        <publisher-name>The National Academy of Sciences</publisher-name>
      </publisher>
    </journal-meta>
    <article-meta>
      <article-id pub-id-type="publisher-id">181325198</article-id>
      <article-id pub-id-type="publisher-id">3251</article-id>
      <article-id pub-id-type="doi">10.1073/pnas.181325198</article-id>
      <article-id pub-id-type="other">jPNAS.v98.i18.pg10214</article-id>
      <article-id pub-id-type="pmid">11517319</article-id>
      <article-categories>
        <subj-group>
          <subject>Physical Sciences</subject>
          <subj-group>
            <subject>Applied Mathematics</subject>
          </subj-group>
        </subj-group>
        <subj-group>
          <subject>Biological Sciences</subject>
          <subj-group>
            <subject>Genetics</subject>
          </subj-group>
        </subj-group>
      </article-categories>
      <title-group>
        <article-title>The coreceptor mutation CCR5&#x0394;32 influences the dynamics of HIV
          epidemics and is selected for by HIV</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <name>
            <surname>Sullivan</surname>
            <given-names>Amy D.</given-names>
          </name>
          <xref ref-type="author-notes" rid="FN150">&#x002A;</xref>
          <xref ref-type="aff" rid="aff-1"/>
        </contrib>
        <contrib contrib-type="author">
          <name>
            <surname>Wigginton</surname>
            <given-names>Janis</given-names>
          </name>
          <xref ref-type="aff" rid="aff-1"/>
        </contrib>
        <contrib contrib-type="author">
          <name>
            <surname>Kirschner</surname>
            <given-names>Denise</given-names>
          </name>
          <xref ref-type="author-notes" rid="FN151">&#x2020;</xref>
          <xref ref-type="aff" rid="aff-1"/>
        </contrib>
      </contrib-group>
      <aff id="aff-1">Department of Microbiology and Immunology, University of Michigan Medical
        School, Ann Arbor, MI 48109-0620</aff>
      <author-notes>
        <fn id="FN150">
          <p>&#x002A; Present address: Centers for Disease Control and Prevention Epidemiology
            Program Office, State Branch Oregon Health Division, 800 NE Oregon Street, Suite 772,
            Portland, OR 97232.</p>
        </fn>
        <fn id="FN151">
          <p>&#x2020; To whom reprint requests should be addressed. E-mail:
              <email>kirschne@umich.edu</email>.</p>
        </fn>
        <fn fn-type="com">
          <p>Communicated by Avner Friedman, University of Minnesota, Minneapolis, MN</p>
        </fn>
      </author-notes>
      <pub-date date-type="pub" publication-format="print" iso-8601-date="2001-08-28">
        <day>28</day>
        <month>8</month>
        <year>2001</year>
      </pub-date>
      <pub-date date-type="pub" publication-format="electronic" iso-8601-date="2001-08-21">
        <day>21</day>
        <month>8</month>
        <year>2001</year>
      </pub-date>
      <volume>98</volume>
      <issue>18</issue>
      <fpage>10214</fpage>
      <lpage>10219</lpage>
      <history>
        <date date-type="received" iso-8601-date="2000-05-30">
          <day>30</day>
          <month>5</month>
          <year>2000</year>
        </date>
        <date date-type="accepted" iso-8601-date="2001-06-27">
          <day>27</day>
          <month>6</month>
          <year>2001</year>
        </date>
      </history>
      <permissions>
        <copyright-statement>Copyright &#x00A9; 2001, The National Academy of
          Sciences</copyright-statement>
        <copyright-year>2001</copyright-year>
      </permissions>
      <abstract>
        <p>We explore the impact of a host genetic factor on heterosexual HIV epidemics by using a
          deterministic mathematical model. A protective allele unequally distributed across
          populations is exemplified in our models by the 32-bp deletion in the host-cell chemokine
          receptor CCR5, CCR5&#x0394;32. Individuals homozygous for CCR5&#x0394;32 are protected
          against HIV infection whereas those heterozygous for CCR5&#x0394;32 have lower pre-AIDS
          viral loads and delayed progression to AIDS. CCR5&#x0394;32 may limit HIV spread by
          decreasing the probability of both risk of infection and infectiousness. In this work, we
          characterize epidemic HIV within three dynamic subpopulations: CCR5&#x002F;CCR5
          (homozygous, wild type), CCR5&#x002F;CCR5&#x0394;32 (heterozygous), and
          CCR5&#x0394;32&#x002F;CCR5&#x0394;32 (homozygous, mutant). Our results indicate that
          prevalence of HIV&#x002F;AIDS is greater in populations lacking the CCR5&#x0394;32 alleles
          (homozygous wild types only) as compared with populations that include people heterozygous
          or homozygous for CCR5&#x0394;32. Also, we show that HIV can provide selective pressure
          for CCR5&#x0394;32, increasing the frequency of this allele.</p>
      </abstract>
    </article-meta>
  </front>
  <body>
    <p>Nineteen million people have died of AIDS since the discoa12=b22+c32very of HIV in the 1980s.
      In 1999 alone, 5.4 million people were newly infected with HIV (ref. <xref ref-type="bibr"
        rid="B1">1</xref> and <ext-link ext-link-type="url"
        xmlns:xlink="http://www.w3.org/1999/xlink"
        xlink:href="http://www.unaids.org/epidemicupdate/report/Epireport.html"
        >http://www.unaids.org/epidemicupdate/report/Epireport.html</ext-link>). (For brevity, HIV-1
      is referred to as HIV in this paper.) Sub-Saharan Africa has been hardest hit, with more than
      20&#x0025; of the general population HIV-positive in some countries (<xref ref-type="bibr"
        rid="B2">2</xref>, <xref ref-type="bibr" rid="B3">3</xref>). In comparison, heterosexual
      epidemics in developed, market-economy countries have not reached such severe levels. Factors
      contributing to the severity of the epidemic in economically developing countries abound,
      including economic, health, and social differences such as high levels of sexually transmitted
      diseases and a lack of prevention programs. However, the staggering rate at which the epidemic
      has spread in sub-Saharan Africa has not been adequately explained. The rate and severity of
      this epidemic also could indicate a greater underlying susceptibility to HIV attributable not
      only to sexually transmitted disease, economics, etc., but also to other more ubiquitous
      factors such as host genetics (<xref ref-type="bibr" rid="B4">4</xref>, <xref ref-type="bibr"
        rid="B5">5</xref>).</p>
    <p>To exemplify the contribution of such a host genetic factor to HIV prevalence trends, we
      consider a well-characterized 32-bp deletion in the host-cell chemokine receptor CCR5,
      CCR5&#x0394;32. When HIV binds to host cells, it uses the CD4 receptor on the surface of host
      immune cells together with a coreceptor, mainly the CCR5 and CXCR4 chemokine receptors (<xref
        ref-type="bibr" rid="B6">6</xref>). Homozygous mutations for this 32-bp deletion offer
      almost complete protection from HIV infection, and heterozygous mutations are associated with
      lower pre-AIDS viral loads and delayed progression to AIDS (<xref ref-type="bibr" rid="B7"
        >7</xref>&#x2013;<xref ref-type="bibr" rid="B14">14</xref>). CCR5&#x0394;32 generally is
      found in populations of European descent, with allelic frequencies ranging from 0 to 0.29
        (<xref ref-type="bibr" rid="B13">13</xref>). African and Asian populations studied outside
      the United States or Europe appear to lack the CCR5&#x0394;32 allele, with an allelic
      frequency of almost zero (<xref ref-type="bibr" rid="B5">5</xref>, <xref ref-type="bibr"
        rid="B13">13</xref>). Thus, to understand the effects of a protective allele, we use a
      mathematical model to track prevalence of HIV in populations with or without CCR5&#x0394;32
      heterozygous and homozygous people and also to follow the CCR5&#x0394;32 allelic
      frequency.</p>
    <p>We hypothesize that CCR5&#x0394;32 limits epidemic HIV by decreasing infection rates, and we
      evaluate the relative contributions to this by the probability of infection and duration of
      infectivity. To capture HIV infection as a chronic infectious disease together with vertical
      transmission occurring in untreated mothers, we model a dynamic population (i.e., populations
      that vary in growth rates because of fluctuations in birth or death rates) based on realistic
      demographic characteristics (<xref ref-type="bibr" rid="B18">18</xref>). This scenario also
      allows tracking of the allelic frequencies over time. This work considers how a specific host
      genetic factor affecting HIV infectivity and viremia at the individual level might influence
      the epidemic in a dynamic population and how HIV exerts selective pressure, altering the
      frequency of this mutant allele.</p>
    <p>CCR5 is a host-cell chemokine receptor, which is also used as a coreceptor by R5 strains of
      HIV that are generally acquired during sexual transmission (<xref ref-type="bibr" rid="B6"
        >6</xref>, <xref ref-type="bibr" rid="B19">19</xref>&#x2013;<xref ref-type="bibr" rid="B25"
        >25</xref>). As infection progresses to AIDS the virus expands its repertoire of potential
      coreceptors to include other CC-family and CXC-family receptors in roughly 50&#x0025; of
      patients (<xref ref-type="bibr" rid="B19">19</xref>, <xref ref-type="bibr" rid="B26"
      >26</xref>, <xref ref-type="bibr" rid="B27">27</xref>). CCR5&#x0394;32 was identified in
      HIV-resistant people (<xref ref-type="bibr" rid="B28">28</xref>). Benefits to individuals from
      the mutation in this allele are as follows. Persons homozygous for the CCR5&#x0394;32 mutation
      are almost nonexistent in HIV-infected populations (<xref ref-type="bibr" rid="B11">11</xref>,
        <xref ref-type="bibr" rid="B12">12</xref>) (see ref. <xref ref-type="bibr" rid="B13"
        >13</xref> for review). Persons heterozygous for the mutant allele (CCR5 W/&#x0394;32) tend
      to have lower pre-AIDS viral loads. Aside from the beneficial effects that lower viral loads
      may have for individuals, there is also an altruistic effect, as transmission rates are
      reduced for individuals with low viral loads (as compared with, for example, AZT and other
      studies; ref. <xref ref-type="bibr" rid="B29">29</xref>). Finally, individuals heterozygous
      for the mutant allele (CCR5 W/&#x0394;32) also have a slower progression to AIDS than those
      homozygous for the wild-type allele (CCR5 W/W) (<xref ref-type="bibr" rid="B7"
        >7</xref>&#x2013;<xref ref-type="bibr" rid="B10">10</xref>), remaining in the population 2
      years longer, on average. Interestingly, the dearth of information on HIV disease progression
      in people homozygous for the CCR5&#x0394;32 allele (CCR5 &#x0394;32/&#x0394;32) stems from the
      rarity of HIV infection in this group (<xref ref-type="bibr" rid="B4">4</xref>, <xref
        ref-type="bibr" rid="B12">12</xref>, <xref ref-type="bibr" rid="B28">28</xref>). However, in
      case reports of HIV-infected CCR5 &#x0394;32/&#x0394;32 homozygotes, a rapid decline in
        CD4<sup>&#x002B;</sup> T cells and a high viremia are observed, likely because of initial
      infection with a more aggressive viral strain (such as X4 or R5X4) (<xref ref-type="bibr"
        rid="B30">30</xref>).</p>
    <sec>
      <title>The Model</title>
      <p>Because we are most concerned with understanding the severity of the epidemic in developing
        countries where the majority of infection is heterosexual, we consider a purely heterosexual
        model. To model the effects of the allele in the population, we examine the rate of HIV
        spread by using an enhanced susceptible-infected-AIDS model of epidemic HIV (for review see
        ref. <xref ref-type="bibr" rid="B31">31</xref>). Our model compares two population
        scenarios: a CCR5 wild-type population and one with CCR5&#x0394;32 heterozygotes and
        homozygotes in addition to the wild type. To model the scenario where there are only
        wild-type individuals present in the population (i.e., CCR5 W/W), we track the sexually
        active susceptibles at time <italic>t</italic> &#x005B;<italic>S<sub>i,j</sub>
        </italic>(<italic>t</italic>)&#x005D;, where <italic>i</italic> = 1 refers to genotype (CCR5
        W/W only in this case) and <italic>j</italic> is either the male or female subpopulation. We
        also track those who are HIV-positive at time <italic>t</italic> not yet having AIDS in
            <italic>I<sub>i,j,k</sub>
        </italic>(<italic>t</italic>) where <italic>k</italic> refers to stage of HIV infection
        &#x005B;primary (<italic>A</italic>) or asymptomatic (<italic>B</italic>)&#x005D;. The total
        number of individuals with AIDS at time <italic>t</italic> are tracked in
          <italic>A</italic>(<italic>t</italic>). The source population are children, &#x03C7;<sub>
          <italic>i,j</italic>
        </sub>(<italic>t</italic>), who mature into the sexually active population at time
          <italic>t</italic> (Fig. <xref ref-type="fig" rid="F1">1</xref>, Table <xref
          ref-type="table" rid="T1">1</xref>). We compare the model of a population lacking the
        CCR5&#x0394;32 allele to a demographically similar population with a high frequency of the
        allele. When genetic heterogeneity is included, male and female subpopulations are each
        further divided into three distinct genotypic groups, yielding six susceptible
        subpopulations, &#x005B;<italic>S<sub>i,j</sub>
        </italic>(<italic>t</italic>), where <italic>i</italic> ranges from 1 to 3, where 1 =
        CCR5W/W; 2 = CCR5 W/&#x0394;32; 3 = CCR5 &#x0394;32/&#x0394;32&#x005D;. The infected
        classes, <italic>I<sub>i,j,k</sub>
        </italic>(<italic>t</italic>), also increase in number to account for these new genotype
        compartments. In both settings we assume there is no treatment available and no knowledge of
        HIV status by people in the early acute and middle asymptomatic stages (both conditions
        exist in much of sub-Saharan Africa). In addition, we assume that sexual mixing in the
        population occurs randomly with respect to genotype and HIV disease status, all HIV-infected
        people eventually progress to AIDS, and no barrier contraceptives are used. These last
        assumptions reflect both economic and social conditions. </p>
      <fig id="F1">
        <label>Figure 1</label>
        <caption>
          <p>A schematic representation of the basic compartmental HIV epidemic model. The
            criss-cross lines indicate the sexual mixing between different compartments. Each of
            these interactions has a positive probability of taking place; they also incorporate
            individual rates of transmission indicated as &#x03BB;, but in full notation is &#x03BB;<sub>
              <italic>&#x00EE;</italic>,<italic>&#xEB30;</italic>,<italic>&#xEA50;</italic>&#x2192;<italic>i</italic>,<italic>j</italic>,</sub>
            where <italic>i</italic>,<italic>j</italic>,<italic>k</italic> is the phenotype of the
            infected partner and <italic>&#x00EE;</italic>,<italic>&#xEB30;</italic> is the
            phenotype of the susceptible partner. Also shown are the different rates of disease
            progression, &#x03B3;<sub>
              <italic>i</italic>,<italic>j</italic>,<italic>k</italic>
            </sub>, that vary according to genotype, gender, and stage. Thus, the interactions
            between different genotypes, genders, and stages are associated with a unique
            probability of HIV infection. M, male; F, female.</p>
        </caption>
        <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="pq1813251001"/>
      </fig>
      <table-wrap id="T1">
        <label>Table 1</label>
        <caption>
          <p>Children's genotype</p>
        </caption>
        <table>
          <tr>
            <th>Parents</th>
            <th colspan="4">Mother</th>
          </tr>
          <tr>
            <td colspan="5">
              <hr/>
            </td>
          </tr>
          <tr>
            <td>Father</td>
            <td/>
            <td>W&#x002F;W</td>
            <td>W&#x002F;&#x0394;32</td>
            <td>&#x0394;32&#x002F;&#x0394;32</td>
          </tr>
          <tr>
            <td/>
            <td>W&#x002F;W</td>
            <td>&#x03C7;<sub>1,<italic>j</italic>
              </sub>1,<italic>j</italic>
            </td>
            <td>&#x03C7;<sub>1,<italic>j</italic>
              </sub>1,<italic>j</italic>, &#x03C7;<sub>2,<italic>j</italic>
              </sub>2,<italic>j</italic>
            </td>
            <td>&#x03C7;<sub>2,<italic>j</italic>
              </sub>2,<italic>j</italic>
            </td>
          </tr>
          <tr>
            <td/>
            <td>W&#x002F;&#x0394;32</td>
            <td>&#x03C7;<sub>1,<italic>j</italic>
              </sub>1,<italic>j</italic>, &#x03C7;<sub>2,<italic>j</italic>
              </sub>2,<italic>j</italic>
            </td>
            <td>&#x03C7;<sub>1,<italic>j</italic>
              </sub>1,<italic>j</italic>, &#x03C7;<sub>2,<italic>j</italic>
              </sub>2,<italic>j</italic>, &#x03C7;<sub>3,<italic>j</italic>
              </sub>3,<italic>j</italic>
            </td>
            <td>&#x03C7;<sub>2,<italic>j</italic>
              </sub>2,<italic>j</italic>, &#x03C7;<sub>3,<italic>j</italic>
              </sub>3,<italic>j</italic>
            </td>
          </tr>
          <tr>
            <td/>
            <td>&#x0394;32&#x002F;&#x0394;32</td>
            <td>&#x03C7;<sub>2,<italic>j</italic>
              </sub>2,<italic>j</italic>
            </td>
            <td>&#x03C7;<sub>2,<italic>j</italic>
              </sub>2,<italic>j</italic>, &#x03C7;<sub>3,<italic>j</italic>
              </sub>3,<italic>j</italic>
            </td>
            <td>&#x03C7;<sub>3,<italic>j</italic>
              </sub>3,<italic>j</italic>
            </td>
          </tr>
        </table>
        <table-wrap-foot>
          <fn>
            <p>&#x03C7;<sub>1,<italic>j</italic>
              </sub>1,<italic>j</italic> = wild-type children; (W&#x002F;W);
                  &#x03C7;<sub>2,<italic>j</italic>
              </sub>2,<italic>j</italic> = heterozygous children (W&#x002F;&#x0394;32);
                  &#x03C7;<sub>3,<italic>j</italic>
              </sub>3,<italic>j</italic> = homozygous children (&#x0394;32&#x002F;&#x0394;32) of
              gender <italic>j</italic>. Children's genotypes are determined by using Mendelian
              inheritance patterns.</p>
          </fn>
        </table-wrap-foot>
      </table-wrap>
      <sec>
        <title>Parameter Estimates for the Model.</title>
        <p>Estimates for rates that govern the interactions depicted in Fig. <xref ref-type="fig"
            rid="F1">1</xref> were <mml:math>
            <mml:mrow>
              <mml:msup>
                <mml:mi>a</mml:mi>
                <mml:mn>2</mml:mn>
              </mml:msup>
              <mml:mo>=</mml:mo>
              <mml:msup>
                <mml:mi>b</mml:mi>
                <mml:mn>2</mml:mn>
              </mml:msup>
              <mml:mo>+</mml:mo>
              <mml:msup>
                <mml:mi>c</mml:mi>
                <mml:mn>2</mml:mn>
              </mml:msup>
            </mml:mrow>
          </mml:math> derived from the extensive literature on HIV. Our parameters and their
          estimates are summarized in Tables <xref ref-type="table" rid="T2">2</xref>&#x2013;<xref
            ref-type="table" rid="T4">4</xref>. The general form of the equations describing the
          rates of transition between population classes as depicted in Fig. <xref ref-type="fig"
            rid="F1">1</xref> are summarized as follows: <disp-formula id="E1">
            <tex-math id="M1">\documentclass[12pt]{minimal} \usepackage{wasysym}
              \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb}
              \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs}
              \DeclareFontFamily{T1}{linotext}{} \DeclareFontShape{T1}{linotext}{m}{n} {
              &#x003C;-&#x003E; linotext }{} \DeclareSymbolFont{linotext}{T1}{linotext}{m}{n}
              \DeclareSymbolFontAlphabet{\mathLINOTEXT}{linotext} \begin{document} $$
              \frac{dS_{i,j}(t)}{dt}={\chi}_{i,j}(t)-{\mu}_{j}S_{i,j}(t)-{\lambda}_{\hat
              {\imath},\hat {},\hat {k}{\rightarrow}i,j}S_{i,j}(t), $$ \end{document} </tex-math>
          </disp-formula>
          <disp-formula id="E2">
            <tex-math id="M2">\documentclass[12pt]{minimal} \usepackage{wasysym}
              \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb}
              \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs}
              \DeclareFontFamily{T1}{linotext}{} \DeclareFontShape{T1}{linotext}{m}{n} {
              &#x003C;-&#x003E; linotext }{} \DeclareSymbolFont{linotext}{T1}{linotext}{m}{n}
              \DeclareSymbolFontAlphabet{\mathLINOTEXT}{linotext} \begin{document} $$
              \hspace{1em}\hspace{1em}\hspace{.167em}\frac{dI_{i,j,A}(t)}{dt}={\lambda}_{\hat
              {\imath},\hat {},\hat
              {k}{\rightarrow}i,j}S_{i,j}(t)-{\mu}_{j}I_{i,j,A}(t)-{\gamma}_{i,j,A}I_{i,j,A}(t), $$
              \end{document} </tex-math>
          </disp-formula>
          <disp-formula id="E3">
            <tex-math id="M3">\documentclass[12pt]{minimal} \usepackage{wasysym}
              \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb}
              \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs}
              \DeclareFontFamily{T1}{linotext}{} \DeclareFontShape{T1}{linotext}{m}{n} {
              &#x003C;-&#x003E; linotext }{} \DeclareSymbolFont{linotext}{T1}{linotext}{m}{n}
              \DeclareSymbolFontAlphabet{\mathLINOTEXT}{linotext} \begin{document} $$
              \frac{dI_{i,j,B}(t)}{dt}={\gamma}_{i,j,A}I_{i,j,A}(t)-{\mu}_{j}I_{i,j,B}(t)-{\gamma}_{i,j,B}I_{i,j,B}(t),
              $$ \end{document} </tex-math>
          </disp-formula>
          <disp-formula id="E4">
            <tex-math id="M4">\documentclass[12pt]{minimal} \usepackage{wasysym}
              \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb}
              \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs}
              \DeclareFontFamily{T1}{linotext}{} \DeclareFontShape{T1}{linotext}{m}{n} {
              &#x003C;-&#x003E; linotext }{} \DeclareSymbolFont{linotext}{T1}{linotext}{m}{n}
              \DeclareSymbolFontAlphabet{\mathLINOTEXT}{linotext} \begin{document} $$
              \frac{dA(t)}{dt}={\gamma}_{i,j,B} \left( { \,\substack{ ^{3} \\ {\sum} \\ _{i=1} }\,
              }I_{i,F,B}(t)+I_{i,M,B}(t) \right) -{\mu}_{A}A(t)-{\delta}A(t), $$ \end{document}
            </tex-math>
          </disp-formula> where, in addition to previously defined populations and rates (with
            <italic>i</italic> equals genotype, <italic>j</italic> equals gender, and
            <italic>k</italic> equals stage of infection, either <italic>A</italic> or
            <italic>B</italic>), &#x03BC;<sub>
            <italic>j</italic>
          </sub>, represents the non-AIDS (natural) death rate for males and females respectively,
          and &#x03BC;<sub>A</sub> is estimated by the average (&#x03BC;<sub>F</sub> &#x002B;
            &#x03BC;<sub>M</sub>/2). This approximation allows us to simplify the model (only one
          AIDS compartment) without compromising the results, as most people with AIDS die of AIDS
            (&#x03B4;<sub>AIDS</sub>) and very few of other causes (&#x03BC;<sub>A</sub>). These
          estimates include values that affect infectivity (&#x03BB;<sub>
            <italic>&#x00EE;</italic>,<italic>&#xEB30;</italic>,<italic>&#xEA50;</italic>&#x2192;<italic>i</italic>,<italic>j</italic>
          </sub>), transmission (&#x03B2;<sub>
            <italic>&#x00EE;</italic>,<italic>&#xEB30;</italic>,<italic>&#xEA50;</italic>&#x2192;<italic>i</italic>,<italic>j</italic>
          </sub>), and disease progression (&#x03B3;<sub>
            <italic>i</italic>
          </sub>
          <sub>,</sub>
          <sub>
            <italic>j</italic>
          </sub>
          <sub>,</sub>
          <sub>
            <italic>k</italic>
          </sub>) where the
            <italic>&#x00EE;</italic>,<italic>&#xEB30;</italic>,<italic>&#xEA50;</italic> notation
          represents the genotype, gender, and stage of infection of the infected partner, and
            <italic>j</italic> &#x2260; <italic>&#xEB30;</italic>. </p>
        <table-wrap id="T2">
          <label>Table 2</label>
          <caption>
            <p>Transmission probabilities</p>
          </caption>
          <table>
            <tr>
              <th rowspan="3">HIV-infected partner (&#x00EE;&#x0131;&#x0131;&#x005E;&#x005E;,
                &#xEB30;&#xE2D4;&#xE2D4;&#x005E;&#x005E;, &#xEA50;<italic>k</italic>
                <italic>k</italic>&#x005E;&#x005E;)</th>
              <th colspan="4">Susceptible partner (<italic>i</italic>, <italic>j</italic>)</th>
            </tr>
            <tr>
              <td colspan="4">
                <hr/>
              </td>
            </tr>
            <tr>
              <th>(&#xEB30;&#xE2D4;&#xE2D4;&#x005E;&#x005E; to <italic>j</italic>)</th>
              <th>W&#x002F;W</th>
              <th>W&#x002F;&#x0394;32</th>
              <th>&#x0394;32&#x002F;&#x0394;32 </th>
            </tr>
            <tr>
              <td colspan="5">
                <hr/>
              </td>
            </tr>
            <tr>
              <td>Acute&#x002F;primary</td>
            </tr>
            <tr>
              <td>&#x2003;W&#x002F;W or &#x0394;32&#x002F;&#x0394;32</td>
              <td>M to F</td>
              <td>0.040</td>
              <td>0.040</td>
              <td>0.00040 </td>
            </tr>
            <tr>
              <td/>
              <td>F to M</td>
              <td>0.020</td>
              <td>0.020</td>
              <td>0.00020 </td>
            </tr>
            <tr>
              <td>&#x2003;W&#x002F;&#x0394;32</td>
              <td>M to F</td>
              <td>0.030</td>
              <td>0.030</td>
              <td>0.00030 </td>
            </tr>
            <tr>
              <td/>
              <td>F to M</td>
              <td>0.015</td>
              <td>0.015</td>
              <td>0.00015 </td>
            </tr>
            <tr>
              <td>Asymptomatic </td>
            </tr>
            <tr>
              <td>&#x2003;W&#x002F;W or &#x0394;32&#x002F;&#x0394;32</td>
              <td>M to F</td>
              <td>0.0010</td>
              <td>0.0010</td>
              <td>10 &#x00D7; 10<sup>&#x2212;6</sup>
              </td>
            </tr>
            <tr>
              <td/>
              <td>F to M</td>
              <td>0.0005</td>
              <td>0.0005</td>
              <td>5 &#x00D7; 10<sup>&#x2212;6</sup>
              </td>
            </tr>
            <tr>
              <td>&#x2003;W&#x002F;&#x0394;32</td>
              <td>M to F</td>
              <td>0.0005</td>
              <td>0.0005</td>
              <td>5 &#x00D7; 10<sup>&#x2212;6</sup>
              </td>
            </tr>
            <tr>
              <td/>
              <td>F to M</td>
              <td>0.00025</td>
              <td>0.00025</td>
              <td>2.5 &#x00D7; 10<sup>&#x2212;6</sup>
              </td>
            </tr>
          </table>
          <table-wrap-foot>
            <fn>
              <p>Listed are the different transmission probabilities
                    (&#x03B2;<sub>&#x00EE;&#x0131;&#x0131;&#x005E;&#x005E;,&#xEB30;&#xE2D4;&#xE2D4;&#x005E;&#x005E;,&#xEA50;<italic>k</italic>
                  <italic>k</italic>&#x005E;&#x005E;&#x2192;<italic>i</italic>,<italic>j</italic>
                </sub>) for random sexual mixing between persons where <italic>i</italic>,
                  <italic>j</italic>, <italic>k</italic> is the phenotype of the infected partner
                and <italic>i</italic>, <italic>j</italic> is the phenotype of the susceptible
                partner. M, male; F, female.</p>
            </fn>
          </table-wrap-foot>
        </table-wrap>
        <table-wrap id="T3">
          <label>Table 3</label>
          <caption>
            <p>Progression rates</p>
          </caption>
          <table>
            <tr>
              <th>Genotype</th>
              <th>Disease stage</th>
              <th>Males&#x002F;females </th>
            </tr>
            <tr>
              <td colspan="3">
                <hr/>
              </td>
            </tr>
            <tr>
              <td>W&#x002F;W</td>
              <td>A</td>
              <td>3.5</td>
            </tr>
            <tr>
              <td/>
              <td>B</td>
              <td>0.16667 </td>
            </tr>
            <tr>
              <td>W&#x002F;&#x0394;32</td>
              <td>A</td>
              <td>3.5 </td>
            </tr>
            <tr>
              <td/>
              <td>B</td>
              <td>0.125</td>
            </tr>
            <tr>
              <td>&#x0394;32&#x002F;&#x0394;32</td>
              <td>A</td>
              <td>3.5 </td>
            </tr>
            <tr>
              <td/>
              <td>B</td>
              <td>0.16667</td>
            </tr>
          </table>
          <table-wrap-foot>
            <fn>
              <p>Shown are the rates of progression, &#x03B3;<sub>
                  <italic>i</italic>,<italic>j</italic>,<italic>k</italic>
                </sub>
                <italic>i</italic>,<italic>j</italic>,<italic>k</italic> reflecting the different
                rates at which persons progress through different stages of disease by genotype,
                gender, and disease stage.</p>
            </fn>
          </table-wrap-foot>
        </table-wrap>
        <table-wrap id="T4">
          <label>Table 4</label>
          <caption>
            <p>Parameter values</p>
          </caption>
          <table>
            <tr>
              <th>Parameter</th>
              <th>Definition</th>
              <th>Value</th>
            </tr>
            <tr>
              <td colspan="3">
                <hr/>
              </td>
            </tr>
            <tr>
              <td>&#x03BC;<sub>
                  <italic>F</italic>
                </sub>
                <italic>F</italic>, &#x03BC;<sub>
                  <italic>M</italic>
                </sub>
                <italic>M</italic>
              </td>
              <td align="left">All-cause mortality for adult females (males)</td>
              <td align="left">0.015 (0.016) per year</td>
            </tr>
            <tr>
              <td>&#x03BC;<sub>&#x03C7;</sub>&#x03C7;</td>
              <td align="left">All-cause childhood mortality (&#x003C;15 years of age)</td>
              <td align="left">0.01 per year</td>
            </tr>
            <tr>
              <td>
                <italic>B</italic>
                <sub>
                  <italic>r</italic>
                </sub>
                <italic>r</italic>
              </td>
              <td align="left">Birthrate</td>
              <td align="left">0.25 per woman per year</td>
            </tr>
            <tr>
              <td>
                <italic>SA</italic>
                <sub>
                  <italic>F</italic>
                </sub>
                <italic>F</italic>
              </td>
              <td align="left">Percent females acquiring new partners (sexual activity)</td>
              <td align="left">10&#x0025;</td>
            </tr>
            <tr>
              <td>
                <italic>SA</italic>
                <sub>
                  <italic>M</italic>
                </sub>
                <italic>M</italic>
              </td>
              <td align="left">Percent males acquiring new partners (sexual activity)</td>
              <td align="left">25&#x0025;</td>
            </tr>
            <tr>
              <td>
                <italic>m</italic>
                <sub>
                  <italic>F</italic>
                </sub>
                <italic>F</italic>(&#x03C2;<inline-formula>
                  <tex-math id="M5">\documentclass[12pt]{minimal} \usepackage{wasysym}
                    \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb}
                    \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs}
                    \DeclareFontFamily{T1}{linotext}{} \DeclareFontShape{T1}{linotext}{m}{n} {
                    &#x003C;-&#x003E; linotext }{} \DeclareSymbolFont{linotext}{T1}{linotext}{m}{n}
                    \DeclareSymbolFontAlphabet{\mathLINOTEXT}{linotext} \begin{document} $$
                    {\mathrm{_{{F}}^{{2}}}} $$ \end{document} </tex-math>
                </inline-formula>)</td>
              <td align="left">Mean (variance) no. of new partners for females</td>
              <td align="left">1.8 (1.2) per year</td>
            </tr>
            <tr>
              <td>&#x03C2;<inline-formula>
                  <tex-math id="M6">\documentclass[12pt]{minimal} \usepackage{wasysym}
                    \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb}
                    \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs}
                    \DeclareFontFamily{T1}{linotext}{} \DeclareFontShape{T1}{linotext}{m}{n} {
                    &#x003C;-&#x003E; linotext }{} \DeclareSymbolFont{linotext}{T1}{linotext}{m}{n}
                    \DeclareSymbolFontAlphabet{\mathLINOTEXT}{linotext} \begin{document} $$
                    {\mathrm{_{{M}}^{{2}}}} $$ \end{document} </tex-math>
                </inline-formula>
              </td>
              <td align="left">Variance in no. of new partners for males</td>
              <td align="left">5.5 per year </td>
            </tr>
            <tr>
              <td>1 &#x2212; <italic>p</italic>
                <sub>
                  <italic>v</italic>
                </sub>
                <italic>v</italic>
              </td>
              <td align="left">Probability of vertical transmission</td>
              <td align="left">0.30 per birth</td>
            </tr>
            <tr>
              <td>
                <italic>I</italic>
                <sub>
                  <italic>i</italic>,<italic>j</italic>,<italic>k</italic>
                </sub>
                <italic>i</italic>,<italic>j</italic>,<italic>k</italic>(0)</td>
              <td align="left">Initial total population HIV-positive</td>
              <td align="left">0.50&#x0025; </td>
            </tr>
            <tr>
              <td>&#x03C7;<sub>
                  <italic>i</italic>,<italic>j</italic>
                </sub>
                <italic>i</italic>,<italic>j</italic>(0)</td>
              <td align="left">Initial total children in population (&#x003C;15 years of age)</td>
              <td align="left">45&#x0025;</td>
            </tr>
            <tr>
              <td>
                <italic>W</italic>&#x002F;<italic>W</italic> (0)</td>
              <td align="left">Initial total wild types
                  (<italic>W</italic>&#x002F;<italic>W</italic>) in population</td>
              <td align="left">80&#x0025; </td>
            </tr>
            <tr>
              <td>
                <italic>W</italic>&#x002F;&#x0394;32(0)</td>
              <td align="left">Initial total heterozygotes (<italic>W</italic>&#x002F;&#x0394;32) in
                population</td>
              <td align="left">19&#x0025;</td>
            </tr>
            <tr>
              <td>&#x0394;32&#x002F;&#x0394;32(0)</td>
              <td align="left">Initial total homozygotes (&#x0394;32&#x002F;&#x0394;32) in
                population</td>
              <td align="left">1&#x0025;</td>
            </tr>
            <tr>
              <td>
                <italic>r</italic>
                <sub>
                  <italic>M</italic>
                </sub>
                <italic>M</italic>(<italic>r</italic>
                <sub>
                  <italic>F</italic>
                </sub>
                <italic>F</italic>)</td>
              <td align="left">Initial percent males (females) in total population</td>
              <td align="left">49&#x0025; (51&#x0025;)</td>
            </tr>
            <tr>
              <td>&#x03D5;<sub>
                  <italic>F</italic>
                </sub>
                <italic>F</italic>, &#x03D5;<sub>
                  <italic>M</italic>
                </sub>
                <italic>M</italic>
              </td>
              <td align="left">Number of sexual contacts a female (male) has</td>
              <td align="left">30 (24) per partner</td>
            </tr>
            <tr>
              <td>&#x025B;<sub>
                  <italic>i</italic>,<italic>j</italic>,<italic>k</italic>
                </sub>
                <italic>i</italic>,<italic>j</italic>,<italic>k</italic>
              </td>
              <td align="left">&#x0025; effect of mutation on transmission rates (see Table <xref
                  ref-type="table" rid="T2">2</xref>)</td>
              <td align="left">0 &#x003C; &#x025B;<sub>
                  <italic>i</italic>,<italic>j</italic>,<italic>k</italic>
                </sub>
                <italic>i</italic>,<italic>j</italic>,<italic>k</italic> &#x003C; 1</td>
            </tr>
            <tr>
              <td>&#x03B4;</td>
              <td align="left">Death rate for AIDS population</td>
              <td align="left">1.0 per year </td>
            </tr>
            <tr>
              <td>
                <italic>q</italic>
              </td>
              <td align="left">Allelic frequency of &#x0394;32 allele</td>
              <td align="left">0.105573</td>
            </tr>
          </table>
          <table-wrap-foot>
            <fn>
              <p>Shown are the parameter values for parameters other than the transmission
                probabilities (Table <xref ref-type="table" rid="T2">2</xref>) and the progression
                rates (Table <xref ref-type="table" rid="T3">3</xref>). Each were estimated from
                data as described in text.</p>
            </fn>
          </table-wrap-foot>
        </table-wrap>
        <p>The effects of the CCR5 W/&#x0394;32 and CCR5 &#x0394;32/&#x0394;32 genotypes are
          included in our model through both the per-capita probabilities of infection, &#x03BB;<sub>
            <italic>&#x00EE;</italic>,<italic>&#xEB30;</italic>,<italic>&#xEA50;</italic>&#x2192;<italic>i</italic>,<italic>j</italic>
          </sub>, and the progression rates, &#x03B3;<sub>
            <italic>i</italic>
          </sub>
          <sub>,</sub>
          <sub>
            <italic>j</italic>
          </sub>
          <sub>,</sub>
          <sub>
            <italic>k</italic>
          </sub>. The infectivity coefficients, &#x03BB;<sub>
            <italic>&#x00EE;</italic>,<italic>&#xEB30;</italic>,<italic>&#xEA50;</italic>&#x2192;<italic>i</italic>,<italic>j</italic>
          </sub>, are calculated for each population subgroup based on the following: likelihood of
          HIV transmission in a sexual encounter between a susceptible and an infected
              (&#x03B2;<sub>&#x00EE;&#x0131;&#x0131;&#x005E;&#x005E;,<italic>j</italic>,&#xEA50;<italic>k</italic>
            <italic>k</italic>&#x005E;&#x005E;&#x2192;<italic>i</italic>,<italic>j</italic>
          </sub>) person; formation of new partnerships (<italic>c</italic>
          <sub>
            <italic>j</italic>
          </sub>
          <italic>j</italic>); number of contacts in a given partnership (&#x03D5;<sub>
            <italic>j</italic>
          </sub>); and probability of encountering an infected individual (<italic>I</italic>
          <sub>
            <italic>&#x00EE;</italic>,<italic>&#xEB30;</italic>,<italic>&#xEA50;</italic>
          </sub>/<italic>N</italic>
          <sub>
            <italic>&#xEB30;</italic>
          </sub>). The formula representing this probability of infection is <disp-formula>
            <tex-math id="M7">\documentclass[12pt]{minimal} \usepackage{wasysym}
              \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb}
              \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs}
              \DeclareFontFamily{T1}{linotext}{} \DeclareFontShape{T1}{linotext}{m}{n} {
              &#x003C;-&#x003E; linotext }{} \DeclareSymbolFont{linotext}{T1}{linotext}{m}{n}
              \DeclareSymbolFontAlphabet{\mathLINOTEXT}{linotext} \begin{document} $$
              {\lambda}_{\hat {i},\hat {j},\hat
              {k}{\rightarrow}i,j}=\frac{C_{j}{\cdot}{\phi}_{j}}{N_{\hat {j}}}\hspace{.167em} \left[
              { \,\substack{ \\ {\sum} \\ _{\hat {i},\hat {k}} }\, }{\beta}_{\hat {i},\hat {j},\hat
              {k}{\rightarrow}i,j}{\cdot}I_{\hat {i},\hat {j},\hat {k}} \right] , $$ \end{document}
            </tex-math>
          </disp-formula> where <italic>j</italic> &#x2260; <italic>&#xEB30;</italic> is either male
          or female. <italic>N</italic>
          <sub>
            <italic>&#xEB30;</italic>
          </sub> represents the total population of gender <italic>&#xEB30;</italic> (this does not
          include those with AIDS in the simulations).</p>
        <p>The average rate of partner acquisition, <italic>c<sub>j</sub>
          </italic>, includes the mean plus the variance to mean ratio of the relevant distribution
          of partner-change rates to capture the small number of high-risk people:
              <italic>c<sub>j</sub>
          </italic> = <italic>m<sub>j</sub>
          </italic> &#x002B; (&#x03C2;<inline-formula>
            <tex-math id="M8">\documentclass[12pt]{minimal} \usepackage{wasysym}
              \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb}
              \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs}
              \DeclareFontFamily{T1}{linotext}{} \DeclareFontShape{T1}{linotext}{m}{n} {
              &#x003C;-&#x003E; linotext }{} \DeclareSymbolFont{linotext}{T1}{linotext}{m}{n}
              \DeclareSymbolFontAlphabet{\mathLINOTEXT}{linotext} \begin{document} $$
              {\mathrm{_{{\mathit{j}}}^{2}}} $$ \end{document} </tex-math>
          </inline-formula>/<italic>m</italic>
          <sub>j</sub>) where the mean (<italic>m<sub>j</sub>
          </italic>) and variance (&#x03C2;<inline-formula>
            <tex-math id="M9">\documentclass[12pt]{minimal} \usepackage{wasysym}
              \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb}
              \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs}
              \DeclareFontFamily{T1}{linotext}{} \DeclareFontShape{T1}{linotext}{m}{n} {
              &#x003C;-&#x003E; linotext }{} \DeclareSymbolFont{linotext}{T1}{linotext}{m}{n}
              \DeclareSymbolFontAlphabet{\mathLINOTEXT}{linotext} \begin{document} $$
              {\mathrm{_{{\mathit{j}}}^{2}}} $$ \end{document} </tex-math>
          </inline-formula>) are annual figures for new partnerships only (<xref ref-type="bibr"
            rid="B32">32</xref>). These means are estimated from Ugandan data for the number of
          heterosexual partners in the past year (<xref ref-type="bibr" rid="B33">33</xref>) and the
          number of nonregular heterosexual partners (i.e., spouses or long-term partners) in the
          past year (<xref ref-type="bibr" rid="B34">34</xref>). In these sexual activity surveys,
          men invariably have more new partnerships; thus, we assumed that they would have fewer
          average contacts per partnership than women (a higher rate of new partner acquisition
          means fewer sexual contacts with a given partner; ref. <xref ref-type="bibr" rid="B35"
            >35</xref>). To incorporate this assumption in our model, the male contacts/partnership, &#x03D5;<sub>
            <italic>M</italic>
          </sub>, was reduced by 20&#x0025;. In a given population, the numbers of heterosexual
          interactions must equate between males and females. The balancing equation applied here is
            <italic>SA</italic>
          <sub>F</sub>&#x00B7;<italic>m</italic>
          <sub>F</sub>&#x00B7;<italic>N</italic>
          <sub>F</sub> = <italic>SA</italic>
          <sub>M</sub>&#x00B7;<italic>m</italic>
          <sub>M</sub>&#x00B7;<italic>N</italic>
          <sub>M</sub>, where <italic>SA<sub>j</sub>
          </italic> are the percent sexually active and <italic>N<sub>j</sub>
          </italic> are the total in the populations for gender <italic>j</italic>. To specify
          changes in partner acquisition, we apply a male flexibility mechanism, holding the female
          rate of acquisition constant and allowing the male rates to vary (<xref ref-type="bibr"
            rid="B36">36</xref>, <xref ref-type="bibr" rid="B37">37</xref>).</p>
        <sec>
          <title>Transmission probabilities.</title>
          <p>The effect of a genetic factor in a model of HIV transmission can be included by
            reducing the transmission coefficient. The probabilities of transmission per contact
            with an infected partner,
                &#x03B2;<sub>&#x00EE;&#x0131;&#x0131;&#x005E;&#x005E;,&#xEB30;&#xE2D4;&#xE2D4;&#x005E;&#x005E;,&#xEA50;<italic>k</italic>
              <italic>k</italic>&#x005E;&#x005E;&#x2192;<italic>i</italic>,<italic>j</italic>
            </sub>, have been estimated in the literature (see ref. <xref ref-type="bibr" rid="B38"
              >38</xref> for estimates in minimally treated groups). We want to capture a decreased
            risk in transmission based on genotype (ref. <xref ref-type="bibr" rid="B39">39</xref>,
            Table <xref ref-type="table" rid="T2">2</xref>). No studies have directly evaluated
            differences in infectivity between HIV-infected CCR5 W/&#x0394;32 heterozygotes and
            HIV-infected CCR5 wild types. Thus, we base estimates for reduced transmission on
            studies of groups with various HIV serum viral loads (<xref ref-type="bibr" rid="B40"
              >40</xref>), HTLV-I/II viral loads (<xref ref-type="bibr" rid="B41">41</xref>), and a
            study of the effect of AZT treatment on transmission (<xref ref-type="bibr" rid="B29"
              >29</xref>). We decrease transmission probabilities for infecting
            CCR5&#x0394;32/&#x0394;32 persons by 100-fold to reflect the rarity of infections in
            these persons. However, we assume that infected CCR5&#x0394;32/&#x0394;32 homozygotes
            can infect susceptibles at a rate similar to CCR5W/W homozygotes, as the former
            generally have high viremias (ref. <xref ref-type="bibr" rid="B30">30</xref>, Table
              <xref ref-type="table" rid="T2">2</xref>). We also assume that male-to-female
            transmission is twice as efficient as female-to-male transmission (up to a 9-fold
            difference has been reported; ref. <xref ref-type="bibr" rid="B42">42</xref>) (ref.
              <xref ref-type="bibr" rid="B43">43</xref>, Table <xref ref-type="table" rid="T2"
              >2</xref>).</p>
          <p>Given the assumption of no treatment, the high burden of disease in people with AIDS is
            assumed to greatly limit their sexual activity. Our initial model excludes people with
            AIDS from the sexually active groups. Subsequently, we allow persons with AIDS to be
            sexually active, fixing their transmission rates (&#x03B2;<sub>AIDS</sub>) to be the
            same across all CCR5 genotypes, and lower than transmission rates for primary-stage
            infection (as the viral burden on average is not as high as during the acute phase), and
            larger than transmission rates for asymptomatic-stage infection (as the viral burden
            characteristically increases during the end stage of disease).</p>
        </sec>
        <sec>
          <title>Disease progression.</title>
          <p>We assume three stages of HIV infection: primary (acute, stage A), asymptomatic HIV
            (stage B), and AIDS. The rates of transition through the first two stages are denoted by &#x03B3;<sub>
              <italic>i</italic>,<italic>j</italic>,<italic>k</italic>
            </sub>
            <italic>i</italic>,<italic>j</italic>,<italic>k</italic>, where <italic>i</italic>
            represents genotype, <italic>j</italic> is male/female, and <italic>k</italic>
            represents either stage A or stage B. Transition rates through each of these stages are
            assumed to be inversely proportional to the duration of that stage; however, other
            distributions are possible (<xref ref-type="bibr" rid="B31">31</xref>, <xref
              ref-type="bibr" rid="B44">44</xref>, <xref ref-type="bibr" rid="B45">45</xref>).
            Although viral loads generally peak in the first 2 months of infection, steady-state
            viral loads are established several months beyond this (<xref ref-type="bibr" rid="B46"
              >46</xref>). For group A, the primary HIV-infecteds, duration is assumed to be 3.5
            months. Based on results from European cohort studies (<xref ref-type="bibr" rid="B7"
              >7</xref>&#x2013;<xref ref-type="bibr" rid="B10">10</xref>), the beneficial effects of
            the CCR5 W/&#x0394;32 genotype are observed mainly in the asymptomatic years of HIV
            infection; &#x2248;7 years after seroconversion survival rates appear to be quite
            similar between heterozygous and homozygous individuals. We also assume that
            CCR5&#x0394;32/&#x0394;32-infected individuals and wild-type individuals progress
            similarly, and that men and women progress through each disease stage at the same rate.
            Given these observations, and that survival after infection may be shorter in untreated
            populations, we choose the duration time in stage B to be 6 years for wild-type
            individuals and 8 years for heterozygous individuals. Transition through AIDS,
              &#x03B4;<sub>AIDS</sub>, is inversely proportional to the duration of AIDS. We
            estimate this value to be 1 year for the time from onset of AIDS to death. The
            progression rates are summarized in Table <xref ref-type="table" rid="T3">3</xref>.</p>
        </sec>
      </sec>
      <sec>
        <title>Demographic Setting.</title>
        <p>Demographic parameters are based on data from Malawi, Zimbabwe, and Botswana (<xref
            ref-type="bibr" rid="B3">3</xref>, <xref ref-type="bibr" rid="B47">47</xref>). Estimated
          birth and child mortality rates are used to calculate the annual numbers of children (&#x03C7;<sub>
            <italic>i</italic>,<italic>j</italic>
          </sub>
          <italic>i</italic>,<italic>j</italic>) maturing into the potentially sexually active,
          susceptible group at the age of 15 years (<xref ref-type="bibr" rid="B3">3</xref>). For
          example, in the case where the mother is CCR5 wild type and the father is CCR5 wild type
          or heterozygous, the number of CCR5 W/W children is calculated as follows
            &#x005B;<italic>s</italic>uppressing (<italic>t</italic>) notation&#x005D;:
              &#x03C7;<sub>1,<italic>j</italic>
          </sub>1,<italic>j</italic> = <disp-formula>
            <tex-math id="M10">\documentclass[12pt]{minimal} \usepackage{wasysym}
              \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb}
              \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs}
              \DeclareFontFamily{T1}{linotext}{} \DeclareFontShape{T1}{linotext}{m}{n} {
              &#x003C;-&#x003E; linotext }{} \DeclareSymbolFont{linotext}{T1}{linotext}{m}{n}
              \DeclareSymbolFontAlphabet{\mathLINOTEXT}{linotext} \begin{document} $$
              B_{r}\hspace{.167em}{ \,\substack{ \\ {\sum} \\ _{k} }\, } \left[
              S_{1,F}\frac{(S_{1,M}+I_{1,M,k})}{N_{M}}+ \left[
              (0.5)S_{1,F}\frac{(S_{2,M}+I_{2,M,k})}{N_{M}} \right] + \right $$ \end{document}
            </tex-math>
          </disp-formula>
          <disp-formula>
            <tex-math id="M11">\documentclass[12pt]{minimal} \usepackage{wasysym}
              \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb}
              \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs}
              \DeclareFontFamily{T1}{linotext}{} \DeclareFontShape{T1}{linotext}{m}{n} {
              &#x003C;-&#x003E; linotext }{} \DeclareSymbolFont{linotext}{T1}{linotext}{m}{n}
              \DeclareSymbolFontAlphabet{\mathLINOTEXT}{linotext} \begin{document} $$ p_{v} \left
              \left( \frac{(I_{1,F,k}(S_{1,M}+I_{1,M,k}))}{N_{M}}+ \left[
              (0.5)I_{1,F,k}\frac{(S_{2,M}+I_{2,M,k})}{N_{M}} \right] \right) \right]
              ,\hspace{.167em} $$ \end{document} </tex-math>
          </disp-formula> where the probability of HIV vertical transmission, 1 &#x2212;
              <italic>p<sub>v</sub>
          </italic>, and the birthrate, <italic>B<sub>r</sub>
          </italic>, are both included in the equations together with the Mendelian inheritance
          values as presented in Table <xref ref-type="table" rid="T1">1</xref>. The generalized
          version of this equation (i.e., &#x03C7;<sub>
            <italic>i</italic>,<italic>j</italic>
          </sub>
          <italic>i</italic>,<italic>j</italic>) can account for six categories of children
          (including gender and genotype). We assume that all children of all genotypes are at risk,
          although we can relax this condition if data become available to support vertical
          protection (e.g., ref. <xref ref-type="bibr" rid="B48">48</xref>). All infected children
          are assumed to die before age 15. Before entering the susceptible group at age 15, there
          is additional loss because of mortality from all non-AIDS causes occurring less than 15
          years of age at a rate of &#x03BC;<sub>&#x03C7;</sub>&#x03C7; &#x00D7; &#x03C7;<sub>
            <italic>i</italic>,<italic>j</italic>
          </sub>
          <italic>i</italic>,<italic>j</italic> (where &#x03BC;<sub>&#x03C7;</sub> is the mortality
          under 15 years of age). Children then enter the population as susceptibles at an annual
          rate, &#x03C2;<sub>
            <italic>j</italic>
          </sub>
          <italic>j</italic> &#x00D7; &#x03C7;<sub>
            <italic>i</italic>,<italic>j</italic>
          </sub>
          <italic>i</italic>,<italic>j</italic>/15, where &#x03C2;<sub>
            <italic>j</italic>
          </sub> distributes the children 51&#x0025; females and 49&#x0025; males. All parameters
          and their values are summarized in Table <xref ref-type="table" rid="T4">4</xref>.</p>
      </sec>
    </sec>
    <sec>
      <title>Prevalence of HIV</title>
      <sec>
        <title>Demographics and Model Validation.</title>
        <p>The model was validated by using parameters estimated from available demographic data.
          Simulations were run in the absence of HIV infection to compare the model with known
          population growth rates. Infection was subsequently introduced with an initial low HIV
          prevalence of 0.5&#x0025; to capture early epidemic behavior.</p>
        <p>In deciding on our initial values for parameters during infection, we use Joint United
          Nations Programme on HIV&#x002F;AIDS national prevalence data for Malawi, Zimbabwe, and
          Botswana. Nationwide seroprevalence of HIV in these countries varies from
          &#x2248;11&#x0025; to over 20&#x0025; (<xref ref-type="bibr" rid="B3">3</xref>), although
          there may be considerable variation within given subpopulations (<xref ref-type="bibr"
            rid="B2">2</xref>, <xref ref-type="bibr" rid="B49">49</xref>).</p>
        <p>In the absence of HIV infection, the annual percent population growth rate in the model
          is &#x2248;2.5&#x0025;, predicting the present-day values for an average of sub-Saharan
          African cities (data not shown). To validate the model with HIV infection, we compare our
          simulation of the HIV epidemic to existing prevalence data for Kenya and Mozambique
            (<ext-link ext-link-type="url" xmlns:xlink="http://www.w3.org/1999/xlink"
            xlink:href="http://www.who.int/emc-hiv/fact-sheets/pdfs/kenya.pdf"
            >http://www.who.int/emc-hiv/fact-sheets/pdfs/kenya.pdf</ext-link> and ref. <xref
            ref-type="bibr" rid="B51">51</xref>). Prevalence data collected from these countries
          follow similar trajectories to those predicted by our model (Fig. <xref ref-type="fig"
            rid="F2">2</xref>). </p>
        <fig id="F2">
          <label>Figure 2</label>
          <caption>
            <p>Model simulation of HIV infection in a population lacking the protective
              CCR5&#x0394;32 allele compared with national data from Kenya (healthy adults) and
              Mozambique (blood donors, ref. <xref ref-type="bibr" rid="B17">17</xref>). The
              simulated population incorporates parameter estimates from sub-Saharan African
              demographics. Note the two outlier points from the Mozambique data were likely caused
              by underreporting in the early stages of the epidemic.</p>
          </caption>
          <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="pq1813251002"/>
        </fig>
      </sec>
      <sec>
        <title>Effects of the Allele on Prevalence.</title>
        <p>After validating the model in the wild type-only population, both CCR5&#x0394;32
          heterozygous and homozygous people are included. Parameter values for HIV transmission,
          duration of illness, and numbers of contacts per partner are assumed to be the same within
          both settings. We then calculate HIV/AIDS prevalence among adults for total HIV/AIDS
          cases.</p>
        <p>Although CCR5&#x0394;32/&#x0394;32 homozygosity is rarely seen in HIV-positive
          populations (prevalence ranges between 0 and 0.004&#x0025;), 1&#x2013;20&#x0025; of people
          in HIV-negative populations of European descent are homozygous. Thus, to evaluate the
          potential impact of CCR5&#x0394;32, we estimate there are 19&#x0025; CCR5 W/&#x0394;32
          heterozygous and 1&#x0025; CCR5 &#x0394;32/&#x0394;32 homozygous people in our population.
          These values are in Hardy-Weinberg equilibrium with an allelic frequency of the mutation
          as 0.105573.</p>
        <p>Fig. <xref ref-type="fig" rid="F3">3</xref> shows the prevalence of HIV in two
          populations: one lacking the mutant CCR5 allele and another carrying that allele. In the
          population lacking the protective mutation, prevalence increases logarithmically for the
          first 35 years of the epidemic, reaching 18&#x0025; before leveling off. </p>
        <fig id="F3">
          <label>Figure 3</label>
          <caption>
            <p>Prevalence of HIV/AIDS in the adult population as predicted by the model. The top
              curve (&#x25CB;) indicates prevalence in a population lacking the protective allele.
              We compare that to a population with 19&#x0025; heterozygous and 1&#x0025; homozygous
              for the allele (implying an allelic frequency of 0.105573. Confidence interval bands
              (light gray) are shown around the median simulation (&#xE80B;) providing a range of
              uncertainty in evaluating parameters for the effect of the mutation on the infectivity
              and the duration of asymptomatic HIV for heterozygotes.</p>
          </caption>
          <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="pq1813251003"/>
        </fig>
        <p>In contrast, when a proportion of the population carries the CCR5&#x0394;32 allele, the
          epidemic increases more slowly, but still logarithmically, for the first 50 years, and
          HIV/AIDS prevalence reaches &#x2248;12&#x0025; (Fig. <xref ref-type="fig" rid="F3"
            >3</xref>). Prevalence begins to decline slowly after 70 years.</p>
        <p>In the above simulations we assume that people with AIDS are not sexually active.
          However, when these individuals are included in the sexually active population the
          severity of the epidemic increases considerably (data not shown). Consistent with our
          initial simulations, prevalences are still relatively lower in the presence of the CCR5
          mutation.</p>
        <p>Because some parameters (e.g., rate constants) are difficult to estimate based on
          available data, we implement an uncertainty analysis to assess the variability in the
          model outcomes caused by any inaccuracies in estimates of the parameter values with regard
          to the effect of the allelic mutation. For these analyses we use Latin hypercube sampling,
          as described in refs. <xref ref-type="bibr" rid="B52">52</xref>&#x2013;<xref
            ref-type="bibr" rid="B56">56</xref>, Our uncertainty and sensitivity analyses focus on
          infectivity vs. duration of infectiousness. To this end, we assess the effects on the
          dynamics of the epidemic for a range of values of the parameters governing transmission
          and progression rates:
              &#x03B2;<sub>&#x00EE;&#x0131;&#x0131;&#x005E;&#x005E;,&#xEB30;&#xE2D4;&#xE2D4;&#x005E;&#x005E;,&#xEA50;<italic>k</italic>
            <italic>k</italic>&#x005E;&#x005E;&#x2192;<italic>i</italic>,<italic>j</italic>
          </sub> and &#x03B3;<sub>
            <italic>i</italic>,<italic>j</italic>,<italic>k</italic>
          </sub>
          <italic>i</italic>,<italic>j</italic>,<italic>k</italic>. All other parameters are held
          constant. These results are presented as an interval band about the average simulation for
          the population carrying the CCR5&#x0394;32 allele (Fig. <xref ref-type="fig" rid="F3"
            >3</xref>). Although there is variability in the model outcomes, the analysis indicates
          that the overall model predictions are consistent for a wide range of transmission and
          progression rates. Further, most of the variation observed in the outcome is because of
          the transmission rates for both heterosexual males and females in the primary stage of
          infection (&#x03B2;<sub>2,M,A</sub>
          <sub>&#x2192;</sub>
          <sub>
            <italic>i</italic>
          </sub>
          <sub>,F</sub>, &#x03B2;<sub>2,F,A</sub>
          <sub>&#x2192;</sub>
          <sub>
            <italic>i</italic>
          </sub>
          <sub>,M</sub>). As mentioned above, we assume lower viral loads correlate with reduced
          infectivity; thus, the reduction in viral load in heterozygotes has a major influence on
          disease spread.</p>
      </sec>
    </sec>
    <sec>
      <title>HIV Induces Selective Pressure on Genotype Frequency</title>
      <p>To observe changes in the frequency of the CCR5&#x0394;32 allele in a setting with HIV
        infection as compared with the Hardy-Weinberg equilibrium in the absence of HIV, we follow
        changes in the total number of CCR5&#x0394;32 heterozygotes and homozygotes over 1,000 years
        (Fig. <xref ref-type="fig" rid="F4">4</xref>). We initially perform simulations in the
        absence of HIV infection as a negative control to show there is not significant selection of
        the allele in the absence of infection. To determine how long it would take for the allelic
        frequency to reach present-day levels (e.g., <italic>q</italic> = 0.105573), we initiate
        this simulation for 1,000 years with a very small allelic frequency (<italic>q</italic> =
        0.00105). In the absence of HIV, the allelic frequency is maintained in equilibrium as shown
        by the constant proportions of CCR5&#x0394;32 heterozygotes and homozygotes (Fig. <xref
          ref-type="fig" rid="F4">4</xref>, solid lines). The selection for CCR5&#x0394;32 in the
        presence of HIV is seen in comparison (Fig. <xref ref-type="fig" rid="F4">4</xref>, dashed
        lines). We expand the time frame of this simulation to 2,000 years to view the point at
        which the frequency reaches present levels (where <italic>q</italic> &#x223C;0.105573 at
        year = 1200). Note that the allelic frequency increases for &#x2248;1,600 years before
        leveling off. </p>
      <fig id="F4">
        <label>Figure 4</label>
        <caption>
          <p>Effects of HIV-1 on selection of the CCR5&#x0394;32 allele. The Hardy-Weinberg
            equilibrium level is represented in the no-infection simulation (solid lines) for each
            population. Divergence from the original Hardy-Weinberg equilibrium is shown to occur in
            the simulations that include HIV infection (dashed lines). Fraction of the total
            subpopulations are presented: (<italic>A</italic>) wild types (W/W),
            (<italic>B</italic>) heterozygotes (W/&#x0394;32), and (<italic>C</italic>) homozygotes
            (&#x0394;32/&#x0394;32). Note that we initiate this simulation with a much lower allelic
            frequency (0.00105) than used in the rest of the study to better exemplify the actual
            selective effect over a 1,000-year time scale. (<italic>D</italic>) The allelic
            selection effect over a 2,000-year time scale.</p>
        </caption>
        <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="pq1813251004"/>
      </fig>
    </sec>
    <sec sec-type="discussion">
      <title>Discussion</title>
      <p>This study illustrates how populations can differ in susceptibility to epidemic HIV/AIDS
        depending on a ubiquitous attribute such as a prevailing genotype. We have examined
        heterosexual HIV epidemics by using mathematical models to assess HIV transmission in
        dynamic populations either with or without CCR5&#x0394;32 heterozygous and homozygous
        persons. The most susceptible population lacks the protective mutation in CCR5. In less
        susceptible populations, the majority of persons carrying the CCR5&#x0394;32 allele are
        heterozygotes. We explore the hypothesis that lower viral loads (CCR5&#x0394;32
        heterozygotes) or resistance to infection (CCR5&#x0394;32 homozygotes) observed in persons
        with this coreceptor mutation ultimately can influence HIV epidemic trends. Two contrasting
        influences of the protective CCR5 allele are conceivable: it may limit the epidemic by
        decreasing the probability of infection because of lower viral loads in infected
        heterozygotes, or it may exacerbate the epidemic by extending the time that infectious
        individuals remain in the sexually active population. Our results strongly suggest the
        former. Thus, the absence of this allele in Africa could explain the severity of HIV disease
        as compared with populations where the allele is present.</p>
      <p>We also observed that HIV can provide selective pressure for the CCR5&#x0394;32 allele
        within a population, increasing the allelic frequency. Other influences may have
        additionally selected for this allele. Infectious diseases such as plague and small pox have
        been postulated to select for CCR5&#x0394;32 (<xref ref-type="bibr" rid="B57">57</xref>,
          <xref ref-type="bibr" rid="B58">58</xref>). For plague, relatively high levels of
        CCR5&#x0394;32 are believed to have arisen within &#x2248;4,000 years, accounting for the
        prevalence of the mutation only in populations of European descent. Smallpox virus uses the
        CC-coreceptor, indicating that direct selection for mutations in CCR5 may have offered
        resistance to smallpox. Given the differences in the epidemic rates of plague (<xref
          ref-type="bibr" rid="B59">59</xref>), smallpox, and HIV, it is difficult to directly
        compare our results to these findings. However, our model suggests that the CCR5&#x0394;32
        mutation could have reached its present allelic frequency in Northern Europe within this
        time frame if selected for by a disease with virulence patterns similar to HIV. Our results
        further support the idea that HIV has been only recently introduced as a pathogen into
        African populations, as the frequency of the protective allele is almost zero, and our model
        predicts that selection of the mutant allele in this population by HIV alone takes at least
        1,000 years. This prediction is distinct from the frequency of the CCR5&#x0394;32 allele in
        European populations, where pathogens that may have influenced its frequency (e.g.,
          <italic>Yersinia pestis</italic>) have been present for much longer.</p>
      <p>Two mathematical models have considered the role of parasite and host genetic heterogeneity
        with regard to susceptibility to another pathogen, namely malaria (<xref ref-type="bibr"
          rid="B60">60</xref>, <xref ref-type="bibr" rid="B61">61</xref>). In each it was determined
        that heterogeneity of host resistance facilitates the maintenance of diversity in parasite
        virulence. Given our underlying interest in the coevolution of pathogen and host, we focus
        on changes in a host protective mutation, holding the virulence of the pathogen constant
        over time.</p>
      <p>Even within our focus on host protective mutations, numerous genetic factors, beneficial or
        detrimental, could potentially influence epidemics. Other genetically determined host
        factors affecting HIV susceptibility and disease progression include a CCR5 A/A to G/G
        promoter polymorphism (<xref ref-type="bibr" rid="B62">62</xref>), a CCR2 point mutation
          (<xref ref-type="bibr" rid="B11">11</xref>, <xref ref-type="bibr" rid="B63">63</xref>),
        and a mutation in the CXCR4 ligand (<xref ref-type="bibr" rid="B64">64</xref>). The CCR2b
        mutation, CCR264I, is found in linkage with at least one CCR5 promoter polymorphism (<xref
          ref-type="bibr" rid="B65">65</xref>) and is prevalent in populations where CCR5&#x0394;32
        is nonexistent, such as sub-Saharan Africa (<xref ref-type="bibr" rid="B63">63</xref>).
        However, as none of these mutations have been consistently shown to be as protective as the
        CCR5&#x0394;32 allele, we simplified our model to incorporate only the effect of
        CCR5&#x0394;32. Subsequent models could be constructed from our model to account for the
        complexity of multiple protective alleles. It is interesting to note that our model predicts
        that even if CCR264I is present at high frequencies in Africa, its protective effects may
        not augment the lack of a protective allele such as CCR5&#x0394;32.</p>
      <p>Although our models demonstrate that genetic factors can contribute to the high prevalence
        of HIV in sub-Saharan Africa, demographic factors are also clearly important in this region.
        Our models explicitly incorporated such factors, for example, lack of treatment
        availability. Additional factors were implicitly controlled for by varying only the presence
        of the CCR5&#x0394;32 allele. More complex models eventually could include interactions with
        infectious diseases that serve as cofactors in HIV transmission. The role of high sexually
        transmitted disease prevalences in HIV infection has long been discussed, especially in
        relation to core populations (<xref ref-type="bibr" rid="B15">15</xref>, <xref
          ref-type="bibr" rid="B50">50</xref>, <xref ref-type="bibr" rid="B66">66</xref>). Malaria,
        too, might influence HIV transmission, as it is associated with transient increases in semen
        HIV viral loads and thus could increase the susceptibility of the population to epidemic HIV
          (<xref ref-type="bibr" rid="B16">16</xref>).</p>
      <p>In assessing the HIV/AIDS epidemic, considerable attention has been paid to the influence
        of core groups in driving sexually transmitted disease epidemics. Our results also highlight
        how characteristics more uniformly distributed in a population can affect susceptibility. We
        observed that the genotypic profile of a population affects its susceptibility to epidemic
        HIV/AIDS. Additional studies are needed to better characterize the influence of these
        genetic determinants on HIV transmission, as they may be crucial in estimating the severity
        of the epidemic in some populations. This information can influence the design of treatment
        strategies as well as point to the urgency for education and prevention programs.</p>
    </sec>
  </body>
  <back>
    <ack>
      <p>We thank Mark Krosky, Katia Koelle, and Kevin Chung for programming and technical
        assistance. We also thank Drs. V. J. DiRita, P. Kazanjian, and S. M. Blower for helpful
        comments and discussions. We thank the reviewers for extremely insightful comments.</p>
    </ack>
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