Stability analysis of an HIV/AIDS epidemic model with treatment
An HIV/AIDS epidemic model with treatment is investigated. The model allows for some infected individuals to move from the symptomatic phase to the asymptomatic phase by all sorts of treatment methods. We first establish the ODE treatment model with two infective stages. Mathematical analyses establ...
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| Veröffentlicht in: | Journal of computational and applied mathematics 2009-07, Vol.229 (1), p.313-323 |
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| creator | Cai, Liming Li, Xuezhi Ghosh, Mini Guo, Baozhu |
| description | An HIV/AIDS epidemic model with treatment is investigated. The model allows for some infected individuals to move from the symptomatic phase to the asymptomatic phase by all sorts of treatment methods. We first establish the ODE treatment model with two infective stages. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are completely determined by the basic reproduction number
ℜ
0
. If
ℜ
0
≤
1
, the disease-free equilibrium is globally stable, whereas the unique infected equilibrium is globally asymptotically stable if
ℜ
0
>
1
. Then, we introduce a discrete time delay to the model to describe the time from the start of treatment in the symptomatic stage until treatment effects become visible. The effect of the time delay on the stability of the endemically infected equilibrium is investigated. Moreover, the delay model exhibits Hopf bifurcations by using the delay as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the results. |
| doi_str_mv | 10.1016/j.cam.2008.10.067 |
| format | Article |
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ℜ
0
. If
ℜ
0
≤
1
, the disease-free equilibrium is globally stable, whereas the unique infected equilibrium is globally asymptotically stable if
ℜ
0
>
1
. Then, we introduce a discrete time delay to the model to describe the time from the start of treatment in the symptomatic stage until treatment effects become visible. The effect of the time delay on the stability of the endemically infected equilibrium is investigated. Moreover, the delay model exhibits Hopf bifurcations by using the delay as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the results.</description><identifier>ISSN: 0377-0427</identifier><identifier>EISSN: 1879-1778</identifier><identifier>DOI: 10.1016/j.cam.2008.10.067</identifier><identifier>CODEN: JCAMDI</identifier><language>eng</language><publisher>Kidlington: Elsevier B.V</publisher><subject>Basic reproduction number ; Exact sciences and technology ; Global analysis, analysis on manifolds ; Global stability ; HIV/AIDS ; Hopf bifurcation ; Human immunodeficiency virus ; Mathematical analysis ; Mathematics ; Numerical analysis ; Numerical analysis. Scientific computation ; Ordinary differential equations ; Sciences and techniques of general use ; Time delay ; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds</subject><ispartof>Journal of computational and applied mathematics, 2009-07, Vol.229 (1), p.313-323</ispartof><rights>2008 Elsevier B.V.</rights><rights>2009 INIST-CNRS</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c421t-aaf68b95c27955bec0db62741f474e1655ea1bf8c7c784c40417b19e0be05aa03</citedby><cites>FETCH-LOGICAL-c421t-aaf68b95c27955bec0db62741f474e1655ea1bf8c7c784c40417b19e0be05aa03</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/S0377042708005815$$EHTML$$P50$$Gelsevier$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,3536,27903,27904,65309</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=21544815$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Cai, Liming</creatorcontrib><creatorcontrib>Li, Xuezhi</creatorcontrib><creatorcontrib>Ghosh, Mini</creatorcontrib><creatorcontrib>Guo, Baozhu</creatorcontrib><title>Stability analysis of an HIV/AIDS epidemic model with treatment</title><title>Journal of computational and applied mathematics</title><description>An HIV/AIDS epidemic model with treatment is investigated. The model allows for some infected individuals to move from the symptomatic phase to the asymptomatic phase by all sorts of treatment methods. We first establish the ODE treatment model with two infective stages. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are completely determined by the basic reproduction number
ℜ
0
. If
ℜ
0
≤
1
, the disease-free equilibrium is globally stable, whereas the unique infected equilibrium is globally asymptotically stable if
ℜ
0
>
1
. Then, we introduce a discrete time delay to the model to describe the time from the start of treatment in the symptomatic stage until treatment effects become visible. The effect of the time delay on the stability of the endemically infected equilibrium is investigated. Moreover, the delay model exhibits Hopf bifurcations by using the delay as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the results.</description><subject>Basic reproduction number</subject><subject>Exact sciences and technology</subject><subject>Global analysis, analysis on manifolds</subject><subject>Global stability</subject><subject>HIV/AIDS</subject><subject>Hopf bifurcation</subject><subject>Human immunodeficiency virus</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Numerical analysis</subject><subject>Numerical analysis. Scientific computation</subject><subject>Ordinary differential equations</subject><subject>Sciences and techniques of general use</subject><subject>Time delay</subject><subject>Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds</subject><issn>0377-0427</issn><issn>1879-1778</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2009</creationdate><recordtype>article</recordtype><recordid>eNqFkE1LxDAQhoMouK7-AG-9qKeuk3aStHgQ8XNB8LDqNaTpFLP0Y02isv_eLise9TQfPO8MPIwdc5hx4PJ8ObOmm2UAxTjPQKodNuGFKlOuVLHLJpArlQJmap8dhLAEAFlynLDLRTSVa11cJ6Y37Tq4kAzN2CcP89fzq_nNIqGVq6lzNumGmtrky8W3JHoysaM-HrK9xrSBjn7qlL3c3T5fP6SPT_fz66vH1GLGY2pMI4uqFDZTpRAVWagrmSnkDSokLoUgw6umsMqqAi0CclXxkqAiEMZAPmVn27srP7x_UIi6c8FS25qeho-gS8glosBsJE__JHNEWeYo_wUzEDIXagPyLWj9EIKnRq-864xfaw56Y18v9Whfb-xvVqP9MXPyc9wEa9rGm9668BvMuEAsuBi5iy1Ho7xPR14H66i3VDtPNup6cH98-Qbc05fs</recordid><startdate>20090701</startdate><enddate>20090701</enddate><creator>Cai, Liming</creator><creator>Li, Xuezhi</creator><creator>Ghosh, Mini</creator><creator>Guo, Baozhu</creator><general>Elsevier B.V</general><general>Elsevier</general><scope>6I.</scope><scope>AAFTH</scope><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7U9</scope><scope>H94</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20090701</creationdate><title>Stability analysis of an HIV/AIDS epidemic model with treatment</title><author>Cai, Liming ; Li, Xuezhi ; Ghosh, Mini ; Guo, Baozhu</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c421t-aaf68b95c27955bec0db62741f474e1655ea1bf8c7c784c40417b19e0be05aa03</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2009</creationdate><topic>Basic reproduction number</topic><topic>Exact sciences and technology</topic><topic>Global analysis, analysis on manifolds</topic><topic>Global stability</topic><topic>HIV/AIDS</topic><topic>Hopf bifurcation</topic><topic>Human immunodeficiency virus</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Numerical analysis</topic><topic>Numerical analysis. Scientific computation</topic><topic>Ordinary differential equations</topic><topic>Sciences and techniques of general use</topic><topic>Time delay</topic><topic>Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Cai, Liming</creatorcontrib><creatorcontrib>Li, Xuezhi</creatorcontrib><creatorcontrib>Ghosh, Mini</creatorcontrib><creatorcontrib>Guo, Baozhu</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Virology and AIDS Abstracts</collection><collection>AIDS and Cancer Research Abstracts</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Journal of computational and applied mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Cai, Liming</au><au>Li, Xuezhi</au><au>Ghosh, Mini</au><au>Guo, Baozhu</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Stability analysis of an HIV/AIDS epidemic model with treatment</atitle><jtitle>Journal of computational and applied mathematics</jtitle><date>2009-07-01</date><risdate>2009</risdate><volume>229</volume><issue>1</issue><spage>313</spage><epage>323</epage><pages>313-323</pages><issn>0377-0427</issn><eissn>1879-1778</eissn><coden>JCAMDI</coden><abstract>An HIV/AIDS epidemic model with treatment is investigated. The model allows for some infected individuals to move from the symptomatic phase to the asymptomatic phase by all sorts of treatment methods. We first establish the ODE treatment model with two infective stages. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are completely determined by the basic reproduction number
ℜ
0
. If
ℜ
0
≤
1
, the disease-free equilibrium is globally stable, whereas the unique infected equilibrium is globally asymptotically stable if
ℜ
0
>
1
. Then, we introduce a discrete time delay to the model to describe the time from the start of treatment in the symptomatic stage until treatment effects become visible. The effect of the time delay on the stability of the endemically infected equilibrium is investigated. Moreover, the delay model exhibits Hopf bifurcations by using the delay as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the results.</abstract><cop>Kidlington</cop><pub>Elsevier B.V</pub><doi>10.1016/j.cam.2008.10.067</doi><tpages>11</tpages><oa>free_for_read</oa></addata></record> |
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| source | Elsevier ScienceDirect Journals; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | Basic reproduction number Exact sciences and technology Global analysis, analysis on manifolds Global stability HIV/AIDS Hopf bifurcation Human immunodeficiency virus Mathematical analysis Mathematics Numerical analysis Numerical analysis. Scientific computation Ordinary differential equations Sciences and techniques of general use Time delay Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds |
| title | Stability analysis of an HIV/AIDS epidemic model with treatment |
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