A class of delayed HIV-1 infection models with latently infected cells
-
摘要: 提出了一类具有潜伏感染细胞的时滞HIV-1传染病模型, 定义了基本再生数$ R_0 $, 给出了无病平衡点$ P_0 (x_0 , \, 0, \, 0, \, 0) $和慢性感染平衡点$ P^\ast (x^\ast , \, \omega ^\ast , \, y^\ast , \, v^\ast ) $的存在条件.首先利用线性化方法, 得到了无病平衡点和慢性感染平衡点的局部渐近稳定性.进一步通过构造相应的Lyapunov函数, 并结合LaSalle不变集原理, 证明了当$ R_0 \leqslant 1 $时, 无病平衡点$ P_0 (x_0 , \, 0, \, 0, \, 0) $是全局渐近稳定的; 当$ R_0 >1 $时, 慢性感染平衡点$ P^\ast (x^\ast , \, \omega ^\ast , \, y^\ast , \, v^\ast ) $是全局渐近稳定的, 但无病平衡点$ P_0 (x_0 , \, 0, \, 0, \, 0) $是不稳定的.结果表明, 模型中的潜伏感染时滞和感染时滞并不影响模型的全局稳定性, 并通过数值模拟验证了所得结论.
-
关键词:
- HIV-1传染病模型 /
- 潜伏感染细胞 /
- 时滞 /
- Lyapunov函数
Abstract: A class of delayed HIV-1 infection models with latently infected cells was proposed. The basic reproductive number $ R_0 $ was defined, and the existence conditions of disease-free equilibrium $ P_0 (x_0 , \, 0, \, 0, \, 0) $ and chronic-infection equilibrium $ P^\ast (x^\ast , \, \omega ^\ast , \, y^\ast , \, v^\ast ) $ were given. First, the local asymptotic stability of infection-free equilibrium and chronic-infection equilibrium was obtained by the linearization method. Further, by constructing the corresponding Lyapunov functions and using LaSalle's invariant principle, it was proved that when the basic reproductive number $ R_0 $ was less than or equal to unity, the infection-free equilibrium $ P_0 (x_0 , \, 0, \, 0, \, 0) $ was globally asymptotically stable; moreover, when the basic reproductive number $ R_0 $ was greater than unity, the chronic-infective equilibrium $ P^\ast (x^\ast , \, \omega ^\ast , \, y^\ast , \, v^\ast ) $ was globally asymptotically stable, but the infection-free equilibrium $ P_0 (x_0 , \, 0, \, 0, \, 0) $ was unstable. The results showed that a latently infected delay and an intracellular delay did not affect the global stability of the model, and numerical simulations were carried out to illustrate the theoretical results.-
Key words:
- HIV-1 infection model /
- latently infected cells /
- delay /
- Lyapunov function
-
-
[1] 孙起麟.艾滋病病毒感染和治疗动力学的理论研究与应用[D].北京: 北京科技大学, 2015. http://cdmd.cnki.com.cn/Article/CDMD-10008-1015615931.htm [2] 王开发, 邱志鹏, 邓国宏.病毒感染群体动力学模型分析[J].系统科学与数学, 2003, 32(4):433-443. http://d.old.wanfangdata.com.cn/Periodical/xtkxysx-zw200304001 [3] PERELSON A S, NELSON P W. Mathematical models of HIV dynamics in vivo[J]. SIAM Review, 1999, 41(1):3-44 doi: 10.1137/S0036144598335107 [4] NOWAK M A, ANDERSON R M, BOERLIJST M C, et al. HIV-1 evolution and disease progression[J], Science, 1996, 274(5289):1008-1011. doi: 10.1126/science.274.5289.1008 [5] KOROBEINIKOV A. Global properties of basic virus dynamics models[J]. Bulletin of Mathematical Biology, 2004, 66(4):879-883. doi: 10.1016/j.bulm.2004.02.001 [6] NOWAK M A, BANGHAM C R M. Population dynamics of immune responses to persistent viruses[J]. Science, 1996, 272(5258):74-79. doi: 10.1126/science.272.5258.74 [7] SONG X Y, NEUMANN A U. Global stability and periodic solution of the viral dynamics[J]. Journal of Mathematical Analysis and Applications, 2007, 329(1):281-297. doi: 10.1016/j.jmaa.2006.06.064 [8] BEDDINGTON J R. Mutual Interference Between Parasites or Predators and its Effect on Searching Efficiency[J]. Journal of Animal Ecology, 1975, 44(1):331-340. doi: 10.2307-3866/ [9] DEANGELIS D L, GOLDSTEIN R A, O'NEILL R V. A model for tropic interaction[J]. Ecology, 1975, 56(4):881-892. doi: 10.2307/1936298 [10] XU R. Global stability of an HIV-1 infection model with saturation infection and intracellular delay[J]. Journal of Mathematical Analysis and Application, 2011, 375(1):75-81. doi: 10.1016/j.jmaa.2010.08.055 [11] GUO T, LIU H H, XU C L, et al. Dynamics of a delayed HIV-1 infection model with saturation incidence rate and CTL immune response[J]. International Journal of Bifurcation and Chaos, 2016, 26(4):1-26. http://d.old.wanfangdata.com.cn/Conference/9125274 [12] BAGASRA O, POMERANTZ R J. Human immunodeficiency virus type-Ⅰ provirus is demonstrated in peripheral blood monocytes in vivo:A study utilizing an in situ polymerase chain reaction[J]. AIDS Research and Human Retroviruses, 1993, 9(1):69-76. doi: 10.1089/aid.1993.9.69 [13] PACE M J, AGOSTO L, GRAF E H. HIV reservoirs and latency models[J]. Virology, 2011, 411(2):344-354. doi: 10.1016/j.virol.2010.12.041 [14] CAPISTRÁN M A. A study of latency, reactivation and apoptosis throughout HIV pathogenesis[J]. Mathematical and Computer Modelling, 2010, 52(7/8):1011-1015. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=c1ed78d21d7c0c6842937caf8c0c6c38 [15] WANG H B, XU R, WANG Z W, et al. Global dynamics of a class of HIV-1 infection models with latently infected cells[J]. Nonlinear Analysis:Modeling and Control, 2015, 20(1):21-37. doi: 10.15388/NA.2015.1.2 [16] HALE J K, LUNEL S V. Introduction to Functional Differential Equations[M]. New York:Springer, 1993.