A mathematical model of HTLV-I infection with two time delays
In this paper, we include two time delays in a mathematical model for the CD8$^+$ cytotoxicT lymphocytes (CTLs) response to the Human T-cell leukaemia virus type I (HTLV-I) infection,where one is the intracellular infection delay and the other is the immune delay to account for aseries of immunolog...
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AIMS Press
2014-12-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2015.12.431 |
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| author | Xuejuan Lu Lulu Hui Shengqiang Liu Jia Li |
| author_facet | Xuejuan Lu Lulu Hui Shengqiang Liu Jia Li |
| author_sort | Xuejuan Lu |
| collection | DOAJ |
| description | In this paper, we include two time delays in a mathematical model for the CD8$^+$ cytotoxicT lymphocytes (CTLs) response to the Human T-cell leukaemia virus type I (HTLV-I) infection,where one is the intracellular infection delay and the other is the immune delay to account for aseries of immunological events leading to the CTL response. We show that the global dynamicsof the model system are determined by two threshold values $R_0$, the correspondingreproductive number of a viral infection, and $R_1$, the corresponding reproductive numberof a CTL response, respectively. If $R_0<1$, the infection-free equilibrium is globallyasymptotically stable, and the HTLV-I viruses are cleared. If $R_1 < 1 < R_0$, the immune-freeequilibrium is globally asymptotically stable, and the HTLV-I infection is chronic but with nopersistent CTL response. If $1 < R_1$, a unique HAM/TSP equilibrium exists, and the HTLV-Iinfection becomes chronic with a persistent CTL response. Moreover, we show that the immunedelay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations. Our numericalsimulations suggest that if $1 < R_1$, an increase of the intracellular delay may stabilize theHAM/TSP equilibrium while the immune delay can destabilize it. If both delays increase, thestability of the HAM/TSP equilibrium may generate rich dynamics combining the ``stabilizing"effects from the intracellular delay with those ``destabilizing" influences from immune delay. |
| format | Article |
| id | doaj-art-4e1098ccd7de4db6a65d7576b4a836c5 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2014-12-01 |
| publisher | AIMS Press |
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| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-4e1098ccd7de4db6a65d7576b4a836c52025-01-24T02:31:53ZengAIMS PressMathematical Biosciences and Engineering1551-00182014-12-0112343144910.3934/mbe.2015.12.431A mathematical model of HTLV-I infection with two time delaysXuejuan Lu0Lulu Hui1Shengqiang Liu2Jia Li3Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 3041#, 2 Yi-Kuang street, Harbin, 150080Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 3041#, 2 Yi-Kuang street, Harbin, 150080Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin, 150080Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899In this paper, we include two time delays in a mathematical model for the CD8$^+$ cytotoxicT lymphocytes (CTLs) response to the Human T-cell leukaemia virus type I (HTLV-I) infection,where one is the intracellular infection delay and the other is the immune delay to account for aseries of immunological events leading to the CTL response. We show that the global dynamicsof the model system are determined by two threshold values $R_0$, the correspondingreproductive number of a viral infection, and $R_1$, the corresponding reproductive numberof a CTL response, respectively. If $R_0<1$, the infection-free equilibrium is globallyasymptotically stable, and the HTLV-I viruses are cleared. If $R_1 < 1 < R_0$, the immune-freeequilibrium is globally asymptotically stable, and the HTLV-I infection is chronic but with nopersistent CTL response. If $1 < R_1$, a unique HAM/TSP equilibrium exists, and the HTLV-Iinfection becomes chronic with a persistent CTL response. Moreover, we show that the immunedelay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations. Our numericalsimulations suggest that if $1 < R_1$, an increase of the intracellular delay may stabilize theHAM/TSP equilibrium while the immune delay can destabilize it. If both delays increase, thestability of the HAM/TSP equilibrium may generate rich dynamics combining the ``stabilizing"effects from the intracellular delay with those ``destabilizing" influences from immune delay.https://www.aimspress.com/article/doi/10.3934/mbe.2015.12.431hopf bifurcation.epidemic thresholdhtlv-i infectionlyapunov functionaltime delay |
| spellingShingle | Xuejuan Lu Lulu Hui Shengqiang Liu Jia Li A mathematical model of HTLV-I infection with two time delays Mathematical Biosciences and Engineering hopf bifurcation. epidemic threshold htlv-i infection lyapunov functional time delay |
| title | A mathematical model of HTLV-I infection with two time delays |
| title_full | A mathematical model of HTLV-I infection with two time delays |
| title_fullStr | A mathematical model of HTLV-I infection with two time delays |
| title_full_unstemmed | A mathematical model of HTLV-I infection with two time delays |
| title_short | A mathematical model of HTLV-I infection with two time delays |
| title_sort | mathematical model of htlv i infection with two time delays |
| topic | hopf bifurcation. epidemic threshold htlv-i infection lyapunov functional time delay |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2015.12.431 |
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