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Global dynamics of a staged-progression model for HIV/AIDS with amelioration. (English) Zbl 1225.34052
Summary: We consider a mathematical model for HIV/AIDS that incorporates staged progression and amelioration. Amelioration as a result of HAART treatment is allowed to occur across any number of stages. The global dynamics are completely determined by the basic reproduction number $R_{0}$. If $R_{0}\leq 1$, then the disease-free equilibrium (DFE) is globally asymptotically stable and the disease always dies out. If $R_{0}>1$, DFE is unstable and a unique endemic equilibrium (EE) is globally asymptotically stable, and the disease persists at the endemic equilibrium. The proof of global stability utilizes a global Lyapunov function.

34C60Qualitative investigation and simulation of models (ODE)
92C60Medical epidemiology
34D20Stability of ODE
Full Text: DOI
[1] Wang, L.; Li, M. Y.: Mathematical analysis of the global dynamics of a model for HIV infection of CD4+T cells, Math. biosci. 200, 44-57 (2006) · Zbl 1086.92035 · doi:10.1016/j.mbs.2005.12.026
[2] Wasserstein-Robbins, F.: A mathematical model of HIV infection: simulating T4, T8, macrophages, antibody, and virus via specific anti-HIV response in the presence of adaptation and tropism, Bull. math. Biol. 72, 1208-1253 (2010) · Zbl 1197.92028 · doi:10.1007/s11538-009-9488-5
[3] Elaiw, A. M.: Global properties of a class of HIV models, Nonlinear anal. RWA 11, 2253-2263 (2010) · Zbl 1197.34073 · doi:10.1016/j.nonrwa.2009.07.001
[4] Wang, K.; Fan, A.; Torres, A.: Global properties of an improved hepatitis B virus model, Nonlinear anal. RWA 11, 3131-3138 (2010) · Zbl 1197.34081 · doi:10.1016/j.nonrwa.2009.11.008
[5] Vieira, I.; Cheng, R.; Harper, P.; De Senna, V.: Small world network models of the dynamics of HIV infection, Ann. oper. Res. 178, 173-200 (2010) · Zbl 1197.90294 · doi:10.1007/s10479-009-0571-y
[6] Anderson, R. M.; May, R. M.; Medley, G. F.; Johnson, A.: A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS, IMA J. Math. appl. Med. biol. 3, 229-263 (1986) · Zbl 0609.92025
[7] Feng, Z.; Thieme, H. R.: Endemic model with arbitrarily distributed periods of infection I. General theory, SIAM J. Appl. math. 61, 803-833 (2000) · Zbl 0991.92028
[8] Gumel, A. B.; Mccluskey, C. C.; Den Driessche, P. Van: Mathematical study of a staged progression HIV model with imperfect vaccine, Bull. math. Biol. 68, 2105-2128 (2006) · Zbl 1296.92124
[9] Hendriks, J. C.; Satten, G. A.; Longini, I. M.; Van Druten, H. A.; Schellekens, P. T.; Coutinho, R. A.; Griensven, G. J. Gvan: Use of immunological markers and continuous-time Markov models to estimate progression of HIV infection in homosexual men, Aids 10, 649-656 (1996)
[10] Hethcote, H. W.; Van Ark, J. W.; Jr., I. M. Longini: A simulation model of AIDS in San Francisco: I. Model formulation and parameter estimation, Math. biosci. 106, 203-222 (1991)
[11] Hyman, J. M.; Li, J.; Stanley, E. A.: The differential infectivity and staged progression models for the transmission of HIV, Math. biosci. 155, 77-109 (1999) · Zbl 0942.92030 · doi:10.1016/S0025-5564(98)10057-3
[12] Jacquez, J. A.; Simon, C. P.; Koopman, J.; Sattenspiel, L.; Perry, T.: Modelling and analyzing HIV transmission: the effect of contact patterns, Math. biosci. 92, 119-199 (1988) · Zbl 0686.92016 · doi:10.1016/0025-5564(88)90031-4
[13] Lin, X.; Hethcote, H. W.; Den Driessche, P. Van: An epidemiological model for HIV/AIDS with proportional recruitment, Math. biosci. 118, 181-195 (1993) · Zbl 0793.92011 · doi:10.1016/0025-5564(93)90051-B
[14] Longini, I. M.; Clark, W. S.; Haber, M.; Horsburgh, R.: The stages of HIV infection: waiting times and infection transmission probabilities, Lecture notes in biomath. 83, 111-137 (1989) · Zbl 0685.92012
[15] Mccluskey, C. C.: A model of HIV/AIDS with staged progression and amelioration, Math. biosci. 181, 1-16 (2003) · Zbl 1008.92032 · doi:10.1016/S0025-5564(02)00149-9
[16] Perelson, A.; Nelson, P.: Mathematical analysis of HIV-1 dynamics in vivo, SIAM rev. 41, 3-44 (1999) · Zbl 1078.92502 · doi:10.1137/S0036144598335107
[17] Thieme, H. R.; Castillo-Chavez, C.: How May infection-age dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. math. 53, 1447-1479 (1992) · Zbl 0811.92021 · doi:10.1137/0153068
[18] Guo, H.; Li, M. Y.: Global dynamics of a staged progression model with amelioration for infectious diseases, J. biol. Syst. 2, 154-168 (2008) · Zbl 1140.92020 · doi:10.1080/17513750802120877
[19] Hethcote, H. W.; Van Ark, J. W.: Modeling HIV transmission and AIDS in the united states, Lecture notes in biomath. 95 (1992) · Zbl 0805.92026
[20] Anderson, R. M.; May, R. M.: Infectious diseases of humans: dynamics and control, (1992)
[21] Diekmann, O.; Heesterbeek, J. A. P.; Metz, J. A. J.: On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. math. Biol. 28, 365-382 (1990) · Zbl 0726.92018 · doi:10.1007/BF00178324
[22] Den Driessche, P. Van; Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci. 180, 29-48 (2002) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6
[23] Lasalle, J. P.: The stability of dynamical systems, Regional conference series in applied mathematics (1976) · Zbl 0364.93002
[24] Guo, H.; Li, M. Y.: Global dynamics of a staged progression model for infectious diseases, Math. biosci. Eng. 3, 513-525 (2006) · Zbl 1092.92040 · doi:10.3934/mbe.2006.3.513
[25] Li, M. Y.; Graef, J. R.; Wang, L.; Karsai, J.: Global dynamics of a SEIR model with varying total population size, Math. biosci. 160, 191-213 (1999) · Zbl 0974.92029 · doi:10.1016/S0025-5564(99)00030-9
[26] Iggidr, A.; Mbang, J.; Sallet, G.; Tewa, J. J.: Multi-compartment models, Discrete contin. Dyn. syst. Supp., 506-519 (2007) · Zbl 1163.34366 · http://www.aimsciences.org/journals/redirecting.jsp?paperID=2858
[27] Freedman, H. I.; So, J. W. -H.: Global stability and persistence of simple food chains, Math. biosci. 76, 69-86 (1985) · Zbl 0572.92025 · doi:10.1016/0025-5564(85)90047-1
[28] Goh, B. S.: Global stability in a class of prey--predator models, Bull. math. Biol. 40, 525-533 (1978) · Zbl 0378.92009
[29] Hsu, S. -B.: Limiting behavior for competing species, SIAM J. Appl. math. 34, 760-763 (1978) · Zbl 0381.92014 · doi:10.1137/0134064
[30] Capasso, V.: Mathematical structures of epidemic systems, Lecture notes in biomath. 97 (1993) · Zbl 0798.92024
[31] Guo, H.; Li, M. Y.; Shuai, Z.: Global stability of the endemic equilibrium of multigroup SIR epidemic model, Canad. appl. Math. quart. 14, 259-284 (2006) · Zbl 1148.34039
[32] Korobinikov, A.; Maini, P. K.: A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. biosci. Eng. 1, 57-60 (2004) · Zbl 1062.92061 · doi:10.3934/mbe.2004.1.57
[33] Horn, R. A.; Johnson, C. R.: Topics in matrix analysis, (1991) · Zbl 0729.15001