A Theoretical Model for the Transmission Dynamics of the Buruli Ulcer with Saturated Treatment
Identifieur interne : 002326 ( Pmc/Curation ); précédent : 002325; suivant : 002327A Theoretical Model for the Transmission Dynamics of the Buruli Ulcer with Saturated Treatment
Auteurs : Ebenezer Bonyah [Ghana] ; Isaac Dontwi [Ghana] ; Farai Nyabadza [Afrique du Sud]Source :
- Computational and Mathematical Methods in Medicine [ 1748-670X ] ; 2014.
Abstract
The management of the Buruli ulcer (BU) in Africa is often accompanied by limited resources, delays in treatment, and macilent capacity in medical facilities. These challenges limit the number of infected individuals that access medical facilities. While most of the mathematical models with treatment assume a treatment function proportional to the number of infected individuals, in settings with such limitations, this assumption may not be valid. To capture these challenges, a mathematical model of the Buruli ulcer with a saturated treatment function is developed and studied. The model is a coupled system of two submodels for the human population and the environment. We examine the stability of the submodels and carry out numerical simulations. The model analysis is carried out in terms of the reproduction number of the submodel of environmental dynamics. The dynamics of the human population submodel, are found to occur at the steady states of the submodel of environmental dynamics. Sensitivity analysis is carried out on the model parameters and it is observed that the BU epidemic is driven by the dynamics of the environment. The model suggests that more effort should be focused on environmental management. The paper is concluded by discussing the public implications of the results.
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DOI: 10.1155/2014/576039
PubMed: 25214885
PubMed Central: 4158283
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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">A Theoretical Model for the Transmission Dynamics of the Buruli Ulcer with Saturated Treatment</title><author><name sortKey="Bonyah, Ebenezer" sort="Bonyah, Ebenezer" uniqKey="Bonyah E" first="Ebenezer" last="Bonyah">Ebenezer Bonyah</name><affiliation wicri:level="1"><nlm:aff id="I1">Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana</nlm:aff><country xml:lang="fr">Ghana</country><wicri:regionArea>Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi</wicri:regionArea></affiliation></author><author><name sortKey="Dontwi, Isaac" sort="Dontwi, Isaac" uniqKey="Dontwi I" first="Isaac" last="Dontwi">Isaac Dontwi</name><affiliation wicri:level="1"><nlm:aff id="I1">Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana</nlm:aff><country xml:lang="fr">Ghana</country><wicri:regionArea>Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi</wicri:regionArea></affiliation></author><author><name sortKey="Nyabadza, Farai" sort="Nyabadza, Farai" uniqKey="Nyabadza F" first="Farai" last="Nyabadza">Farai Nyabadza</name><affiliation wicri:level="1"><nlm:aff id="I2">Department of Mathematical Science, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa</nlm:aff><country xml:lang="fr">Afrique du Sud</country><wicri:regionArea>Department of Mathematical Science, University of Stellenbosch, Private Bag X1, Matieland 7602</wicri:regionArea></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">25214885</idno><idno type="pmc">4158283</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4158283</idno><idno type="RBID">PMC:4158283</idno><idno type="doi">10.1155/2014/576039</idno><date when="2014">2014</date><idno type="wicri:Area/Pmc/Corpus">002327</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">002327</idno><idno type="wicri:Area/Pmc/Curation">002326</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Curation">002326</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">A Theoretical Model for the Transmission Dynamics of the Buruli Ulcer with Saturated Treatment</title><author><name sortKey="Bonyah, Ebenezer" sort="Bonyah, Ebenezer" uniqKey="Bonyah E" first="Ebenezer" last="Bonyah">Ebenezer Bonyah</name><affiliation wicri:level="1"><nlm:aff id="I1">Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana</nlm:aff><country xml:lang="fr">Ghana</country><wicri:regionArea>Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi</wicri:regionArea></affiliation></author><author><name sortKey="Dontwi, Isaac" sort="Dontwi, Isaac" uniqKey="Dontwi I" first="Isaac" last="Dontwi">Isaac Dontwi</name><affiliation wicri:level="1"><nlm:aff id="I1">Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana</nlm:aff><country xml:lang="fr">Ghana</country><wicri:regionArea>Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi</wicri:regionArea></affiliation></author><author><name sortKey="Nyabadza, Farai" sort="Nyabadza, Farai" uniqKey="Nyabadza F" first="Farai" last="Nyabadza">Farai Nyabadza</name><affiliation wicri:level="1"><nlm:aff id="I2">Department of Mathematical Science, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa</nlm:aff><country xml:lang="fr">Afrique du Sud</country><wicri:regionArea>Department of Mathematical Science, University of Stellenbosch, Private Bag X1, Matieland 7602</wicri:regionArea></affiliation></author></analytic><series><title level="j">Computational and Mathematical Methods in Medicine</title><idno type="ISSN">1748-670X</idno><idno type="eISSN">1748-6718</idno><imprint><date when="2014">2014</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>The management of the Buruli ulcer (BU) in Africa is often accompanied by limited resources, delays in treatment, and macilent capacity in medical facilities. These challenges limit the number of infected individuals that access medical facilities. While most of the mathematical models with treatment assume a treatment function proportional to the number of infected individuals, in settings with such limitations, this assumption may not be valid. To capture these challenges, a mathematical model of the Buruli ulcer with a saturated treatment function is developed and studied. The model is a coupled system of two submodels for the human population and the environment. We examine the stability of the submodels and carry out numerical simulations. The model analysis is carried out in terms of the reproduction number of the submodel of environmental dynamics. The dynamics of the human population submodel, are found to occur at the steady states of the submodel of environmental dynamics. Sensitivity analysis is carried out on the model parameters and it is observed that the BU epidemic is driven by the dynamics of the environment. The model suggests that more effort should be focused on environmental management. The paper is concluded by discussing the public implications of the results.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Van Ravensway, J" uniqKey="Van Ravensway J">J van Ravensway</name></author><author><name sortKey="Benbow, Me" uniqKey="Benbow M">ME Benbow</name></author><author><name sortKey="Tsonis, Aa" uniqKey="Tsonis A">AA Tsonis</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Merritt, Rw" uniqKey="Merritt R">RW Merritt</name></author><author><name sortKey="Walker, Ed" uniqKey="Walker E">ED Walker</name></author><author><name sortKey="Small, Plc" uniqKey="Small P">PLC Small</name></author><author><name sortKey="Wallace, Jr" uniqKey="Wallace J">JR Wallace</name></author><author><name sortKey="Johnson, Pdr" uniqKey="Johnson P">PDR Johnson</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Duker, Aa" uniqKey="Duker A">AA 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<front><journal-meta><journal-id journal-id-type="nlm-ta">Comput Math Methods Med</journal-id><journal-id journal-id-type="iso-abbrev">Comput Math Methods Med</journal-id><journal-id journal-id-type="publisher-id">CMMM</journal-id><journal-title-group><journal-title>Computational and Mathematical Methods in Medicine</journal-title></journal-title-group><issn pub-type="ppub">1748-670X</issn><issn pub-type="epub">1748-6718</issn><publisher><publisher-name>Hindawi Publishing Corporation</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">25214885</article-id><article-id pub-id-type="pmc">4158283</article-id><article-id pub-id-type="doi">10.1155/2014/576039</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group></article-categories><title-group><article-title>A Theoretical Model for the Transmission Dynamics of the Buruli Ulcer with Saturated Treatment</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Bonyah</surname><given-names>Ebenezer</given-names></name><xref ref-type="aff" rid="I1"><sup>1</sup></xref></contrib><contrib contrib-type="author"><name><surname>Dontwi</surname><given-names>Isaac</given-names></name><xref ref-type="aff" rid="I1"><sup>1</sup></xref></contrib><contrib contrib-type="author"><contrib-id contrib-id-type="orcid" authenticated="false">http://orcid.org/0000-0001-8607-7372</contrib-id><name><surname>Nyabadza</surname><given-names>Farai</given-names></name><xref ref-type="aff" rid="I2"><sup>2</sup></xref><xref ref-type="corresp" rid="cor1">*</xref></contrib></contrib-group><aff id="I1"><sup>1</sup>Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana</aff><aff id="I2"><sup>2</sup>Department of Mathematical Science, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa</aff><author-notes><corresp id="cor1">*Farai Nyabadza: <email>f.nyaba@gmail.com</email></corresp><fn fn-type="other"><p>Academic Editor: Chung-Min Liao</p></fn></author-notes><pub-date pub-type="ppub"><year>2014</year></pub-date><pub-date pub-type="epub"><day>21</day><month>8</month><year>2014</year></pub-date><volume>2014</volume><elocation-id>576039</elocation-id><history><date date-type="received"><day>16</day><month>5</month><year>2014</year></date><date date-type="rev-recd"><day>5</day><month>8</month><year>2014</year></date><date date-type="accepted"><day>6</day><month>8</month><year>2014</year></date></history><permissions><copyright-statement>Copyright © 2014 Ebenezer Bonyah et al.</copyright-statement><copyright-year>2014</copyright-year><license xlink:href="https://creativecommons.org/licenses/by/3.0/"><license-p>This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.</license-p></license></permissions><abstract><p>The management of the Buruli ulcer (BU) in Africa is often accompanied by limited resources, delays in treatment, and macilent capacity in medical facilities. These challenges limit the number of infected individuals that access medical facilities. While most of the mathematical models with treatment assume a treatment function proportional to the number of infected individuals, in settings with such limitations, this assumption may not be valid. To capture these challenges, a mathematical model of the Buruli ulcer with a saturated treatment function is developed and studied. The model is a coupled system of two submodels for the human population and the environment. We examine the stability of the submodels and carry out numerical simulations. The model analysis is carried out in terms of the reproduction number of the submodel of environmental dynamics. The dynamics of the human population submodel, are found to occur at the steady states of the submodel of environmental dynamics. Sensitivity analysis is carried out on the model parameters and it is observed that the BU epidemic is driven by the dynamics of the environment. The model suggests that more effort should be focused on environmental management. The paper is concluded by discussing the public implications of the results.</p></abstract></article-meta></front><floats-group><fig id="fig1" orientation="portrait" position="float"><label>Figure 1</label><caption><p>A schematic diagram for the model.</p></caption><graphic xlink:href="CMMM2014-576039.001"></graphic></fig><fig id="fig2" orientation="portrait" position="float"><label>Figure 2</label><caption><p>The plot of the force of infection as a function of <italic>R</italic><sub><italic>T</italic></sub>. The force of infection increases linearly with the reproduction number. The human population is at risk only if <italic>R</italic><sub><italic>T</italic></sub> > 1.</p></caption><graphic xlink:href="CMMM2014-576039.002"></graphic></fig><fig id="fig3" orientation="portrait" position="float"><label>Figure 3</label><caption><p>The scatter plots for the parameters <italic>α</italic>, <italic>β</italic><sub>3</sub>, <italic>μ</italic><sub><italic>d</italic></sub>, and <italic>μ</italic><sub><italic>W</italic></sub>.</p></caption><graphic xlink:href="CMMM2014-576039.003"></graphic></fig><fig id="fig4" orientation="portrait" position="float"><label>Figure 4</label><caption><p>The tornado plots for the four parameters in the model reproduction number.</p></caption><graphic xlink:href="CMMM2014-576039.004"></graphic></fig><fig id="fig5" orientation="portrait" position="float"><label>Figure 5</label><caption><p>The phase diagrams for <italic>R</italic><sub><italic>T</italic></sub> = 0.8889 (a) and <italic>R</italic><sub><italic>T</italic></sub> = 5.3333 (b).</p></caption><graphic xlink:href="CMMM2014-576039.005"></graphic></fig><fig id="fig6" orientation="portrait" position="float"><label>Figure 6</label><caption><p>A phase diagram for the human population showing the endemic steady state. For a randomly chosen set of initial conditions, all trajectories tend to an endemic equilibrium for the following parameter values: <italic>μ</italic><sub><italic>H</italic></sub> = 0.02, <italic>θ</italic> = 0.04, Λ = 0.07, <italic>σ</italic> = 0.4, and <italic>γ</italic> = 0.7.</p></caption><graphic xlink:href="CMMM2014-576039.006"></graphic></fig><fig id="fig7" orientation="portrait" position="float"><label>Figure 7</label><caption><p>Fraction of the infected human population for <italic>R</italic><sub><italic>T</italic></sub> = 1.6492 for the parameters <italic>μ</italic><sub><italic>d</italic></sub> and <italic>μ</italic><sub><italic>W</italic></sub>. The parameter values used for the constant parameters are <italic>μ</italic><sub><italic>H</italic></sub> = 0.00045, <italic>m</italic><sub>1</sub> = 10, <italic>θ</italic> = 0.011, <italic>γ</italic> = 0.000016, <italic>β</italic><sub>3</sub> = 0.09, <italic>σ</italic> = 0.08, <inline-formula><mml:math id="M72"><mml:mover accent="true"><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mo>~</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0.4000</mml:mn></mml:math></inline-formula>, <italic>N</italic><sub><italic>H</italic></sub> = 100000, <italic>α</italic> = 0.00615, <italic>β</italic><sub>1</sub> = 0.00001, and <italic>β</italic><sub>2</sub> = 0.0000002.</p></caption><graphic xlink:href="CMMM2014-576039.007"></graphic></fig><fig id="fig8" orientation="portrait" position="float"><label>Figure 8</label><caption><p>The proportion of infected humans for the given values of <italic>β</italic><sub>2</sub> and the following parameter values: <italic>μ</italic><sub><italic>H</italic></sub> = 0.00045; <italic>m</italic><sub>1</sub> = 10; <italic>θ</italic> = 0.011; <italic>γ</italic> = 0.000016; <italic>β</italic><sub>3</sub> = 0.09; <italic>μ</italic><sub><italic>d</italic></sub> = 0.02; <italic>σ</italic> = 0.14; <italic>K</italic> = 0.4000; <italic>μ</italic><sub><italic>W</italic></sub> = 0.025; <italic>N</italic><sub><italic>H</italic></sub> = 100000; <italic>α</italic> = 0.006; and <italic>β</italic><sub>1</sub> = 0.00001.</p></caption><graphic xlink:href="CMMM2014-576039.008"></graphic></fig><fig id="fig9" orientation="portrait" position="float"><label>Figure 9</label><caption><p>A phase diagram for the infected water bugs and<italic> M. ulcerans</italic> in the environment for the same parameters presented in <xref ref-type="fig" rid="fig7">Figure 7</xref> with <italic>μ</italic><sub><italic>d</italic></sub> = 0.02 and <italic>μ</italic><sub><italic>W</italic></sub> = 0.025.</p></caption><graphic xlink:href="CMMM2014-576039.009"></graphic></fig></floats-group></pmc></record>Pour manipuler ce document sous Unix (Dilib)
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