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Andreas Romeyke authoredAndreas Romeyke authored
testfile.xml 189.09 KiB
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<article article-type="research-article" dtd-version="1.3" xml:lang="en"
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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 CCR5Δ32 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">*</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">†</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>* 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>† 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 © 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, CCR5Δ32. Individuals
homozygous for CCR5Δ32 are protected against HIV infection whereas those
heterozygous for CCR5Δ32 have lower pre-AIDS viral loads and delayed
progression to AIDS. CCR5Δ32 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/CCR5
(homozygous, wild type), CCR5/CCR5Δ32 (heterozygous), and
CCR5Δ32/CCR5Δ32 (homozygous, mutant). Our results indicate
that prevalence of HIV/AIDS is greater in populations lacking the
CCR5Δ32 alleles (homozygous wild types only) as compared with populations
that include people heterozygous or homozygous for CCR5Δ32. Also, we show
that HIV can provide selective pressure for CCR5Δ32, increasing the
frequency of this allele.</p>
</abstract>
</article-meta>
</front>
<body>
<p>Nineteen million people have died of AIDS since the discovery 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% 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,
CCR5Δ32. 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>–<xref ref-type="bibr" rid="B14">14</xref>).
CCR5Δ32 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
CCR5Δ32 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 CCR5Δ32 heterozygous and homozygous people and
also to follow the CCR5Δ32 allelic frequency.</p>
<p>We hypothesize that CCR5Δ32 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>–<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% 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>). CCR5Δ32 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 CCR5Δ32 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/Δ32) 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/Δ32) 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>–<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 CCR5Δ32 allele (CCR5 Δ32/Δ32) 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 Δ32/Δ32 homozygotes, a rapid
decline in CD4<sup>+</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
CCR5Δ32 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> [<italic>S<sub>i,j</sub>
</italic>(<italic>t</italic>)], 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 [primary (<italic>A</italic>) or asymptomatic
(<italic>B</italic>)]. 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, χ<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 CCR5Δ32 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, [<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/Δ32; 3 = CCR5 Δ32/Δ32]. 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 λ, but
in full notation is λ<sub>
<italic>î</italic>,<italic></italic>,<italic></italic>→<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>î</italic>,<italic></italic> is the phenotype of
the susceptible partner. Also shown are the different rates of disease
progression, γ<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"
> </graphic>
</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/W</td>
<td>W/Δ32</td>
<td>Δ32/Δ32</td>
</tr>
<tr>
<td/>
<td>W/W</td>
<td>χ<sub>1,<italic>j</italic>
</sub>1,<italic>j</italic>
</td>
<td>χ<sub>1,<italic>j</italic>
</sub>1,<italic>j</italic>, χ<sub>2,<italic>j</italic>
</sub>2,<italic>j</italic>
</td>
<td>χ<sub>2,<italic>j</italic>
</sub>2,<italic>j</italic>
</td>
</tr>
<tr>
<td/>
<td>W/Δ32</td>
<td>χ<sub>1,<italic>j</italic>
</sub>1,<italic>j</italic>, χ<sub>2,<italic>j</italic>
</sub>2,<italic>j</italic>
</td>
<td>χ<sub>1,<italic>j</italic>
</sub>1,<italic>j</italic>, χ<sub>2,<italic>j</italic>
</sub>2,<italic>j</italic>, χ<sub>3,<italic>j</italic>
</sub>3,<italic>j</italic>
</td>
<td>χ<sub>2,<italic>j</italic>
</sub>2,<italic>j</italic>, χ<sub>3,<italic>j</italic>
</sub>3,<italic>j</italic>
</td>
</tr>
<tr>
<td/>
<td>Δ32/Δ32</td>
<td>χ<sub>2,<italic>j</italic>
</sub>2,<italic>j</italic>
</td>
<td>χ<sub>2,<italic>j</italic>
</sub>2,<italic>j</italic>, χ<sub>3,<italic>j</italic>
</sub>3,<italic>j</italic>
</td>
<td>χ<sub>3,<italic>j</italic>
</sub>3,<italic>j</italic>
</td>
</tr>
</table>
<table-wrap-foot>
<fn>
<p>χ<sub>1,<italic>j</italic>
</sub>1,<italic>j</italic> = wild-type children; (W/W);
χ<sub>2,<italic>j</italic>
</sub>2,<italic>j</italic> = heterozygous children
(W/Δ32); χ<sub>3,<italic>j</italic>
</sub>3,<italic>j</italic> = homozygous children
(Δ32/Δ32) 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 derived from the extensive literature
on HIV. Our parameters and their estimates are summarized in Tables <xref
ref-type="table" rid="T2">2</xref>–<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} { <-> 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} { <-> 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} { <-> 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} { <-> 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>), μ<sub>
<italic>j</italic>
</sub>, represents the non-AIDS (natural) death rate for males and females
respectively, and μ<sub>A</sub> is estimated by the average
(μ<sub>F</sub> + μ<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 (δ<sub>AIDS</sub>) and
very few of other causes (μ<sub>A</sub>). These estimates include values
that affect infectivity (λ<sub>
<italic>î</italic>,<italic></italic>,<italic></italic>→<italic>i</italic>,<italic>j</italic>
</sub>), transmission (β<sub>
<italic>î</italic>,<italic></italic>,<italic></italic>→<italic>i</italic>,<italic>j</italic> </sub>), and disease progression (γ<sub>
<italic>i</italic>
</sub>
<sub>,</sub>
<sub>
<italic>j</italic>
</sub>
<sub>,</sub>
<sub>
<italic>k</italic>
</sub>) where the
<italic>î</italic>,<italic></italic>,<italic></italic>
notation represents the genotype, gender, and stage of infection of the infected
partner, and <italic>j</italic> ≠ <italic></italic>. </p>
<table-wrap id="T2">
<label>Table 2</label>
<caption>
<p>Transmission probabilities</p>
</caption>
<table>
<tr>
<th rowspan="3">HIV-infected partner
(îıı^^,
^^, <italic>k</italic>
<italic>k</italic>^^)</th>
<th colspan="4">Susceptible partner (<italic>i</italic>,
<italic>j</italic>)</th>
</tr>
<tr>
<td colspan="4">
<hr/>
</td>
</tr>
<tr>
<th>(^^ to
<italic>j</italic>)</th>
<th>W/W</th>
<th>W/Δ32</th>
<th>Δ32/Δ32 </th>
</tr>
<tr>
<td colspan="5">
<hr/>
</td>
</tr>
<tr>
<td>Acute/primary</td>
</tr>
<tr>
<td> W/W or Δ32/Δ32</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> W/Δ32</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> W/W or Δ32/Δ32</td>
<td>M to F</td>
<td>0.0010</td>
<td>0.0010</td>
<td>10 × 10<sup>−6</sup>
</td>
</tr>
<tr>
<td/>
<td>F to M</td>
<td>0.0005</td>
<td>0.0005</td>
<td>5 × 10<sup>−6</sup>
</td>
</tr>
<tr>
<td> W/Δ32</td>
<td>M to F</td>
<td>0.0005</td>
<td>0.0005</td>
<td>5 × 10<sup>−6</sup>
</td>
</tr>
<tr>
<td/>
<td>F to M</td>
<td>0.00025</td>
<td>0.00025</td>
<td>2.5 × 10<sup>−6</sup>
</td>
</tr>
</table>
<table-wrap-foot>
<fn>
<p>Listed are the different transmission probabilities
(β<sub>îıı^^,^^,<italic>k</italic>
<italic>k</italic>^^→<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/females </th>
</tr>
<tr>
<td colspan="3">
<hr/>
</td> </tr>
<tr>
<td>W/W</td>
<td>A</td>
<td>3.5</td>
</tr>
<tr>
<td/>
<td>B</td>
<td>0.16667 </td>
</tr>
<tr>
<td>W/Δ32</td>
<td>A</td>
<td>3.5 </td>
</tr>
<tr>
<td/>
<td>B</td>
<td>0.125</td>
</tr>
<tr>
<td>Δ32/Δ32</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, γ<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>μ<sub>
<italic>F</italic>
</sub>
<italic>F</italic>, μ<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>μ<sub>χ</sub>χ</td>
<td align="left">All-cause childhood mortality (<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%</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%</td>
</tr>
<tr>
<td>
<italic>m</italic>
<sub>
<italic>F</italic>
</sub>
<italic>F</italic>(ς<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} { <->
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>ς<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} { <->
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 − <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% </td>
</tr>
<tr>
<td>χ<sub>
<italic>i</italic>,<italic>j</italic>
</sub>
<italic>i</italic>,<italic>j</italic>(0)</td>
<td align="left">Initial total children in population (<15 years
of age)</td>
<td align="left">45%</td>
</tr>
<tr>
<td>
<italic>W</italic>/<italic>W</italic> (0)</td>
<td align="left">Initial total wild types
(<italic>W</italic>/<italic>W</italic>) in
population</td>
<td align="left">80% </td>
</tr>
<tr>
<td>
<italic>W</italic>/Δ32(0)</td>
<td align="left">Initial total heterozygotes
(<italic>W</italic>/Δ32) in population</td>
<td align="left">19%</td>
</tr>
<tr>
<td>Δ32/Δ32(0)</td>
<td align="left">Initial total homozygotes
(Δ32/Δ32) in population</td>
<td align="left">1%</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% (51%)</td>
</tr>
<tr>
<td>ϕ<sub>
<italic>F</italic>
</sub>
<italic>F</italic>, ϕ<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>ɛ<sub>
<italic>i</italic>,<italic>j</italic>,<italic>k</italic>
</sub>
<italic>i</italic>,<italic>j</italic>,<italic>k</italic>
</td>
<td align="left">% effect of mutation on transmission rates (see
Table <xref ref-type="table" rid="T2">2</xref>)</td>
<td align="left">0 < ɛ<sub>
<italic>i</italic>,<italic>j</italic>,<italic>k</italic>
</sub>
<italic>i</italic>,<italic>j</italic>,<italic>k</italic> <
1</td>
</tr>
<tr>
<td>δ</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 Δ32 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/Δ32 and CCR5 Δ32/Δ32 genotypes are
included in our model through both the per-capita probabilities of infection, λ<sub>
<italic>î</italic>,<italic></italic>,<italic></italic>→<italic>i</italic>,<italic>j</italic>
</sub>, and the progression rates, γ<sub>
<italic>i</italic>
</sub>
<sub>,</sub>
<sub>
<italic>j</italic>
</sub>
<sub>,</sub>
<sub>
<italic>k</italic>
</sub>. The infectivity coefficients, λ<sub> <italic>î</italic>,<italic></italic>,<italic></italic>→<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
(β<sub>îıı^^,<italic>j</italic>,<italic>k</italic>
<italic>k</italic>^^→<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 (ϕ<sub>
<italic>j</italic>
</sub>); and probability of encountering an infected individual
(<italic>I</italic>
<sub>
<italic>î</italic>,<italic></italic>,<italic></italic>
</sub>/<italic>N</italic>
<sub>
<italic></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} { <-> 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> ≠ <italic></italic> is
either male or female. <italic>N</italic>
<sub>
<italic></italic>
</sub> represents the total population of gender <italic></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> + (ς<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} { <-> 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 (ς<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} { <-> 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, ϕ<sub>
<italic>M</italic>
</sub>, was reduced by 20%. 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>·<italic>m</italic>
<sub>F</sub>·<italic>N</italic>
<sub>F</sub> = <italic>SA</italic>
<sub>M</sub>·<italic>m</italic>
<sub>M</sub>·<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,
β<sub>îıı^^,^^,<italic>k</italic>
<italic>k</italic>^^→<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/Δ32
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
CCR5Δ32/Δ32 persons by 100-fold to reflect the rarity of
infections in these persons. However, we assume that infected
CCR5Δ32/Δ32 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 (β<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 γ<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>–<xref ref-type="bibr" rid="B10">10</xref>), the
beneficial effects of the CCR5 W/Δ32 genotype are observed mainly in
the asymptomatic years of HIV infection; ≈7 years after
seroconversion survival rates appear to be quite similar between
heterozygous and homozygous individuals. We also assume that
CCR5Δ32/Δ32-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, δ<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 (χ<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 [<italic>s</italic>uppressing (<italic>t</italic>)
notation]: χ<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} { <-> 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} { <-> 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 −
<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., χ<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 μ<sub>χ</sub>χ × χ<sub>
<italic>i</italic>,<italic>j</italic>
</sub>
<italic>i</italic>,<italic>j</italic> (where μ<sub>χ</sub> is the
mortality under 15 years of age). Children then enter the population as
susceptibles at an annual rate, ς<sub>
<italic>j</italic>
</sub>
<italic>j</italic> × χ<sub>
<italic>i</italic>,<italic>j</italic>
</sub>
<italic>i</italic>,<italic>j</italic>/15, where ς<sub>
<italic>j</italic>
</sub> distributes the children 51% females and 49% 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% 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/AIDS national prevalence data for Malawi,
Zimbabwe, and Botswana. Nationwide seroprevalence of HIV in these countries
varies from ≈11% to over 20% (<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 ≈2.5%, 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
CCR5Δ32 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"
> </graphic>
</fig>
</sec>
<sec> <title>Effects of the Allele on Prevalence.</title>
<p>After validating the model in the wild type-only population, both CCR5Δ32
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 CCR5Δ32/Δ32 homozygosity is rarely seen in HIV-positive
populations (prevalence ranges between 0 and 0.004%), 1–20%
of people in HIV-negative populations of European descent are homozygous. Thus,
to evaluate the potential impact of CCR5Δ32, we estimate there are
19% CCR5 W/Δ32 heterozygous and 1% CCR5
Δ32/Δ32 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%
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 (○) indicates prevalence in a population lacking
the protective allele. We compare that to a population with 19%
heterozygous and 1% homozygous for the allele (implying an
allelic frequency of 0.105573. Confidence interval bands (light gray)
are shown around the median simulation () 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"
> </graphic>
</fig>
<p>In contrast, when a proportion of the population carries the CCR5Δ32
allele, the epidemic increases more slowly, but still logarithmically, for the
first 50 years, and HIV/AIDS prevalence reaches ≈12% (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>–<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:
β<sub>îıı^^,^^,<italic>k</italic>
<italic>k</italic>^^→<italic>i</italic>,<italic>j</italic>
</sub> and γ<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 CCR5Δ32 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
(β<sub>2,M,A</sub>
<sub>→</sub> <sub>
<italic>i</italic>
</sub>
<sub>,F</sub>, β<sub>2,F,A</sub>
<sub>→</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 CCR5Δ32 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 CCR5Δ32 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 CCR5Δ32 heterozygotes and homozygotes
(Fig. <xref ref-type="fig" rid="F4">4</xref>, solid lines). The selection for
CCR5Δ32 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> ∼0.105573 at year = 1200). Note that
the allelic frequency increases for ≈1,600 years before leveling off. </p>
<fig id="F4">
<label>Figure 4</label>
<caption>
<p>Effects of HIV-1 on selection of the CCR5Δ32 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/Δ32), and (<italic>C</italic>) homozygotes
(Δ32/Δ32). 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"
> </graphic>
</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 CCR5Δ32
heterozygous and homozygous persons. The most susceptible population lacks the
protective mutation in CCR5. In less susceptible populations, the majority of
persons carrying the CCR5Δ32 allele are heterozygotes. We explore the
hypothesis that lower viral loads (CCR5Δ32 heterozygotes) or resistance to
infection (CCR5Δ32 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 CCR5Δ32
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 CCR5Δ32 (<xref ref-type="bibr"
rid="B57">57</xref>, <xref ref-type="bibr" rid="B58">58</xref>). For plague,
relatively high levels of CCR5Δ32 are believed to have arisen within
≈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 CCR5Δ32
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 CCR5Δ32 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 CCR5Δ32 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 CCR5Δ32 allele, we simplified our model to incorporate only the effect
of CCR5Δ32. 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
CCR5Δ32.</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 CCR5Δ32 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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