Impact of delay on HIV-1 dynamics of fighting a virus withanother virus
In this paper, we propose a mathematical model for HIV-1 infectionwith intracellular delay. The model examines a viral-therapy for controllinginfections through recombining HIV-1 virus with a genetically modifiedvirus. For this model, the basic reproduction number $\mathcal{R}_0$are identified and i...
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AIMS Press
2014-05-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2014.11.1181 |
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| author | Yun Tian Yu Bai Pei Yu |
| author_facet | Yun Tian Yu Bai Pei Yu |
| author_sort | Yun Tian |
| collection | DOAJ |
| description | In this paper, we propose a mathematical model for HIV-1 infectionwith intracellular delay. The model examines a viral-therapy for controllinginfections through recombining HIV-1 virus with a genetically modifiedvirus. For this model, the basic reproduction number $\mathcal{R}_0$are identified and its threshold properties are discussed.When $\mathcal{R}_0 < 1$, the infection-free equilibrium $E_0$ is globallyasymptotically stable. When $\mathcal{R}_0 > 1$, $E_0$ becomesunstable and there occurs the single-infection equilibrium $E_s$,and $E_0$ and $E_s$ exchange their stability at the transcriticalpoint $\mathcal{R}_0 =1$.If $1< \mathcal{R}_0 < R_1$, where $R_1$ is a positive constant explicitlydepending on the model parameters, $E_s$ is globally asymptotically stable,while when $\mathcal{R}_0 > R_1$, $E_s$ loses itsstability to the double-infection equilibrium $E_d$.There exist a constant $R_2$ such that $E_d$ is asymptoticallystable if $R_1<\mathcal R_0 < R_2$, and $E_s$ and $E_d$ exchange theirstability at the transcritical point $\mathcal{R}_0 =R_1$.We use one numerical exampleto determine the largest range of $\mathcal R_0$ for the localstability of $E_d$ and existence of Hopf bifurcation. Some simulationsare performed to support the theoretical results.These results show that the delay plays animportant role in determining the dynamic behaviour of the system.In the normal range of values, the delay may change the dynamic behaviourquantitatively, such as greatly reducing the amplitudes of oscillations,or even qualitatively changes the dynamical behaviour such as revokingoscillating solutions to equilibrium solutions.This suggests that the delay is a very importantfact which should not be missed in HIV-1 modelling. |
| format | Article |
| id | doaj-art-3d05cea528d849afb611b570253a97be |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2014-05-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-3d05cea528d849afb611b570253a97be2025-01-24T02:28:54ZengAIMS PressMathematical Biosciences and Engineering1551-00182014-05-011151181119810.3934/mbe.2014.11.1181Impact of delay on HIV-1 dynamics of fighting a virus withanother virusYun Tian0Yu Bai1Pei Yu2Department of Applied Mathematics, Western University, London, Ontario N6A 5B7Department of Applied Mathematics, Western University, London, Ontario N6A 5B7Department of Applied Mathematics, Western University, London, Ontario N6A 5B7In this paper, we propose a mathematical model for HIV-1 infectionwith intracellular delay. The model examines a viral-therapy for controllinginfections through recombining HIV-1 virus with a genetically modifiedvirus. For this model, the basic reproduction number $\mathcal{R}_0$are identified and its threshold properties are discussed.When $\mathcal{R}_0 < 1$, the infection-free equilibrium $E_0$ is globallyasymptotically stable. When $\mathcal{R}_0 > 1$, $E_0$ becomesunstable and there occurs the single-infection equilibrium $E_s$,and $E_0$ and $E_s$ exchange their stability at the transcriticalpoint $\mathcal{R}_0 =1$.If $1< \mathcal{R}_0 < R_1$, where $R_1$ is a positive constant explicitlydepending on the model parameters, $E_s$ is globally asymptotically stable,while when $\mathcal{R}_0 > R_1$, $E_s$ loses itsstability to the double-infection equilibrium $E_d$.There exist a constant $R_2$ such that $E_d$ is asymptoticallystable if $R_1<\mathcal R_0 < R_2$, and $E_s$ and $E_d$ exchange theirstability at the transcritical point $\mathcal{R}_0 =R_1$.We use one numerical exampleto determine the largest range of $\mathcal R_0$ for the localstability of $E_d$ and existence of Hopf bifurcation. Some simulationsare performed to support the theoretical results.These results show that the delay plays animportant role in determining the dynamic behaviour of the system.In the normal range of values, the delay may change the dynamic behaviourquantitatively, such as greatly reducing the amplitudes of oscillations,or even qualitatively changes the dynamical behaviour such as revokingoscillating solutions to equilibrium solutions.This suggests that the delay is a very importantfact which should not be missed in HIV-1 modelling.https://www.aimspress.com/article/doi/10.3934/mbe.2014.11.1181hopf bifurcationrecombinant virushiv-1 modellyapunov functionlasalle invariance principle.delayglobal stability |
| spellingShingle | Yun Tian Yu Bai Pei Yu Impact of delay on HIV-1 dynamics of fighting a virus withanother virus Mathematical Biosciences and Engineering hopf bifurcation recombinant virus hiv-1 model lyapunov function lasalle invariance principle. delay global stability |
| title | Impact of delay on HIV-1 dynamics of fighting a virus withanother virus |
| title_full | Impact of delay on HIV-1 dynamics of fighting a virus withanother virus |
| title_fullStr | Impact of delay on HIV-1 dynamics of fighting a virus withanother virus |
| title_full_unstemmed | Impact of delay on HIV-1 dynamics of fighting a virus withanother virus |
| title_short | Impact of delay on HIV-1 dynamics of fighting a virus withanother virus |
| title_sort | impact of delay on hiv 1 dynamics of fighting a virus withanother virus |
| topic | hopf bifurcation recombinant virus hiv-1 model lyapunov function lasalle invariance principle. delay global stability |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2014.11.1181 |
| work_keys_str_mv | AT yuntian impactofdelayonhiv1dynamicsoffightingaviruswithanothervirus AT yubai impactofdelayonhiv1dynamicsoffightingaviruswithanothervirus AT peiyu impactofdelayonhiv1dynamicsoffightingaviruswithanothervirus |