zbMATH — the first resource for mathematics

Stability analysis of an HIV/AIDS epidemic model with treatment. (English) Zbl 1162.92035
Summary: An HIV/AIDS epidemic model with treatment is investigated. The model allows for some infected individuals to move from the symptomatic phase to the asymptomatic phase by all sorts of treatment methods. We first establish the ODE treatment model with two infective stages. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are completely determined by the basic reproduction number \(\mathfrak R_{0}\). If \(\mathfrak R_{0}\leq 1\), the disease-free equilibrium is globally stable, whereas the unique infected equilibrium is globally asymptotically stable if \(\mathfrak R_{0}>1\). Then, we introduce a discrete time delay to the model to describe the time from the start of treatment in the symptomatic stage until treatment effects become visible. The effect of the time delay on the stability of the endemically infected equilibrium is investigated. Moreover, the delay model exhibits Hopf bifurcations by using the delay as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the results.

92D30 Epidemiology
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34D23 Global stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
Full Text: DOI
[1] Levy, J.A., Pathogenesis of human immunodeficiency virus infection, Microbiol. rev., 57, 183-289, (1993)
[2] Stoddart, C.A.; Reyes, R.A., Models of HIV-1 disease: A review of current status, Drug discovery today: disease models, 3, 1, 113-119, (2006)
[3] Anderson, R.M., The role of mathematical models in the study of HIV transmission and the epidemiology of AIDS, J. aids., 1, 241-256, (1988)
[4] Anderson, R.M.; Medly, G.F.; May, R.M.; Johnson, A.M., 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
[5] May, R.M.; Anderson, R.M., Transmission dynamics of HIV infection, Nature, 326, 137-142, (1987)
[6] Bachar, M.; Dorfmayr, A., HIV treatment models with time delay, C. R. biologies, 327, 983-994, (2004)
[7] Blower, S., Calculating the consequences: HAART and risky sex, Aids, 15, 1309-1310, (2001)
[8] Connell McCluskey, C., A model of HIV/AIDS with staged progression and amelioration, Math. biosci., 181, 1-16, (2003) · Zbl 1008.92032
[9] Hethcote, H.W.; Van Ark, J.W., Modelling HIV transmission and AIDS in the united states, Lect. notes biomath., vol. 95, (1992), Springer Berlin · Zbl 0805.92026
[10] Hsieh, Y.-H.; Chen, C.H., Modelling the social dynamics of a sex industry: its implications for spread of HIV/AIDS, Bull. math. biol., 66, 143-166, (2004) · Zbl 1334.92404
[11] Leenheer, P.D.; Smith, H.L., Virus dynamics: A global analysis, SIAM J. appl. math., 63, 1313-1327, (2003) · Zbl 1035.34045
[12] Perelson, A.S.; Nelson, P.W., Mathematical analysis of HIV-1 dynamics in vivo, SIAM rev., 41, 1, 3-44, (1999) · Zbl 1078.92502
[13] Wang, L.; Li, M.Y., Mathematical analysis of the global dynamics of a model for HIV infection of \(C D 4^+\) T-cells, Math. biosci., 200, 44-57, (2006) · Zbl 1086.92035
[14] Wang, K.; Wang, W.; Liu, X., Viral infection model with periodic lytic immune response, Chaos solitons fractals, 28, 1, 90-99, (2006) · Zbl 1079.92048
[15] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press London · Zbl 0777.34002
[16] Ma, Z.; Zhou, Y.; Wang, W.; Jin, Z., Mathematical models and dynamics of infectious diseases, (2004), China Science Press Beijing
[17] Hethcote, H.W.; Lewis, M.A.; van den Driessche, P., An epidemiological model with a delay and a nonlinear incidence rate, J. math. biol., 27, 49-64, (1989) · Zbl 0714.92021
[18] Culshaw, R.V.; Ruan, S., A delay-differential equation model of HIV infection of CD4+ T-cells, Math. biosci., 165, 27-39, (2000) · Zbl 0981.92009
[19] 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
[20] Arino, J.; McCluskey, C.C.; van den Driessche, P., Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. appl. math., 64, 260-276, (2003) · Zbl 1034.92025
[21] McCluskey, C.C.; van den Driessche, P., Global analysis of two tuberculosis models, J. dyn. diff. equ., 16, 139-166, (2004) · Zbl 1056.92052
[22] LaSalle, J.P., The stability of dynamical systems, Regional conference series in applied mathematics, (1976), SIAM Philadelphia, PA · Zbl 0364.93002
[23] Thieme, R.H., Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. math. anal., 24, 407-435, (1993) · Zbl 0774.34030
[24] Busenberg, S.; van den Driessche, P., Analysis of a disease transmission model in a population with varying size, J. math. biol., 28, 257-270, (1990) · Zbl 0725.92021
[25] Moghadas, S.M.; Gumel, A.B., A mathematical study of a model for childhood diseases with non-permanent immunity, J. comput. appl. math., 157, 347-363, (2003) · Zbl 1017.92031
[26] Li, J.; Ma, Z., Qualitative analyses of SIS epidemic model with vaccination varying total population, Math. comput. modelling, 35, 1235-1243, (2002) · Zbl 1045.92039
[27] Dieudonne, J., Foundations of modern analysis, (1960), Academic Press New York · Zbl 0100.04201
[28] Erbe, L.H.; Rao, V.S.H.; Freedman, H., Three-species food chain models with mutual interference and time delays, Math. biosci., 80, 57-80, (1986) · Zbl 0592.92024
[29] Cooke, K.L.; van den Driessche, P., On zeroes of some transcendental equations, Funkcial. ekvac., 29, 77-90, (1986) · Zbl 0603.34069
[30] Hale, J.K., Theory of functional differential equations, (1997), Springer New York · Zbl 0189.39904
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.