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.


34C60 Qualitative investigation and simulation of ordinary differential equation models
92C60 Medical epidemiology
34D20 Stability of solutions to ordinary differential equations
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
[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
[3] Elaiw, A.M., Global properties of a class of HIV models, Nonlinear anal. RWA, 11, 2253-2263, (2010) · Zbl 1197.34073
[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
[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
[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.; van den Driessche, P., 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.; Gvan Griensven, G.J., 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.; Longini, I.M., 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
[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
[13] Lin, X.; Hethcote, H.W.; van den Driessche, P., An epidemiological model for HIV/AIDS with proportional recruitment, Math. biosci., 118, 181-195, (1993) · Zbl 0793.92011
[14] Longini, I.M.; Clark, W.S.; Haber, M.; Horsburgh, R., The stages of HIV infection: waiting times and infection transmission probabilities, (), 111-137
[15] McCluskey, C.C., A model of HIV/AIDS with staged progression and amelioration, Math. biosci., 181, 1-16, (2003) · Zbl 1008.92032
[16] Perelson, A.; Nelson, P., Mathematical analysis of HIV-1 dynamics in vivo, SIAM rev., 41, 3-44, (1999) · Zbl 1078.92502
[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
[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
[19] Hethcote, H.W.; Van Ark, J.W., ()
[20] Anderson, R.M.; May, R.M., Infectious diseases of humans: dynamics and control, (1992), Oxford University Press Oxford
[21] Diekmann, O.; Heesterbeek, J.A.P.; Metz, J.A.J., On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations, J. math. biol., 28, 365-382, (1990) · Zbl 0726.92018
[22] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci., 180, 29-48, (2002) · Zbl 1015.92036
[23] LaSalle, J.P., ()
[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
[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
[26] Iggidr, A.; Mbang, J.; Sallet, G.; Tewa, J.J., Multi-compartment models, Discrete contin. dyn. syst. supp., 506-519, (2007) · Zbl 1163.34366
[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
[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
[30] Capasso, V., ()
[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
[33] Horn, R.A.; Johnson, C.R., Topics in matrix analysis, (1991), Cambridge University Press Cambridge · Zbl 0729.15001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.