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Mukandavire, Z.; Garira, W.; Chiyaka, C.

Asymptotic properties of an HIV/AIDS model with a time delay. (English) Zbl 1110.92043

J. Math. Anal. Appl. 330, No. 2, 916-933 (2007).
Summary: A mathematical model for HIV/AIDS with explicit incubation period is presented as a system of discrete time delay differential equations and its important mathematical features are analysed. The disease-free and endemic equilibria are found and their local stability investigated. We use the Lyapunov functional approach to show the global stability of the endemic equilibrium. Qualitative analysis of the model including positivity and boundedness of solutions, and persistence are also presented. The HIV/AIDS model is numerically analysed to asses the effects of incubation periods on the dynamics of HIV/AIDS and the demographic impact of the epidemic using the demographic and epidemiological parameters for Zimbabwe.

Cited in 46 Documents

MSC:

92D30 Epidemiology
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K25 Asymptotic theory of functional-differential equations

Keywords:

HIV/AIDS model; incubation; delay; global stability; persistence; Lyapunov functional
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\textit{Z. Mukandavire} et al., J. Math. Anal. Appl. 330, No. 2, 916--933 (2007; Zbl 1110.92043)
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References:

[1] Z. Mukandavire, W. Garira, HIV/AIDS model for assessing the effects of prophylactic sterilizing vaccines, condoms and treatment with amelioration, J. Biol. Syst. 14 (3) (2006), in press · Zbl 1116.92042
[2] Genik, L.; van den Driessche, P., An epidemic model with recruitment-death demographics and discrete delays, Fields inst. commun., 21, 238-249, (1991) · Zbl 0926.92029
[3] Hyman, J.M.; Stanley, E.A., Using mathematical models to understand the AIDS epidemic, Math. biosci., 90, 415-473, (1998) · Zbl 0727.92025
[4] May, R.M.; Anderson, R.M., The transmission dynamics of human immunodeficiency virus (HIV), Philosophical trans. roy. soc. London ser. B biological sciences, 565-607, (1998)
[5] Alfonseca, M.; Martinez-Bravo, M.T.; Torrea, J.L., Mathematical models for the analysis of hepatitis B and AIDS epidemics, Simulation, ISSN: 0037-5497/00, 74, 219-226, (2000), (Simulation Councils, Technical Article) · Zbl 1027.92018
[6] Luboobi, L.S., Mathematical models for the dynamics of the AIDS epidemic, (), 76-83
[7] J.Y.T. Mugisha, The mathematical dynamics of HIV/AIDS epidemic in age-structured population, in: V.G. Masanja (Ed.), Proceedings of XI SAMSA Symposium on the Potential of Mathematical Modeling of Problems from the SAMSA Region, 1997, pp. 8-23
[8] Srinivasa, A.S.R., Mathematical modelling of AIDS epidemic in India, Current science, 84, 9, 1193-1197, (2003)
[9] Hsu Schmitz, S.F., Effects of genetic heterogeneity on HIV transmission in homosexual populations, (), 45-260 · Zbl 1023.92029
[10] Hsu Schmitz, S.F., A mathematical model of HIV transmission in homosexuals with genetic heterogeneity, J. theor. med., 2, 285-296, (2000) · Zbl 0962.92035
[11] Brauer, F., Age of infection in epidemiology models, Electron. J. differential equations (conf.), 12, 29-37, (2005) · Zbl 1081.92034
[12] Arino, J.; van den Driessche, P., Time delays in epidemic models: modeling and numerical aspects, accessed 2005 · Zbl 1131.34055
[13] Busenberg, S.; Cooke, K., Vertically transmitted diseases models and dynamics, Biomathematics, vol. 23, (1993), Springer · Zbl 0837.92021
[14] Culshaw, R.V.; Ruan, S., A delay-differential equation model of HIV infection of CD\(4^+\) T-cells, Math. biosci., 165, 27-39, (2000) · Zbl 0981.92009
[15] Dixit, N.M.; Perelson, A.S., Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay, J. theor. biol., 226, 95-109, (2003)
[16] C.T.H. Baker, G.A. Bocharov, F.A. Rihan, A report on the use of delay differential equations in numerical modelling in the biosciences, Numerical Analysis Report 343 (1999)
[17] Thieme, H.R., Global asymptotic stability in epidemic models, Lecture notes in math., vol. 1017, (1983), Springer · Zbl 0533.92023
[18] Beretta, E.; Kuang, Y., Modeling and analysis of marine bacteriophage infection with latency period, Nonl. anal. B, 2, 1, 35-74, (2001) · Zbl 1015.92049
[19] Berreta, E.; Kuang, Y., Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. math. anal., 33, 5, 1144-1165, (2002) · Zbl 1013.92034
[20] Kuang, Y.; So, J.W.-H., Analysis of a two-stage population model with space-limited recruitment, SIAM J. appl. math., 55, 6, 1675-1696, (1995) · Zbl 0847.34076
[21] Kuang, Y., Delay differential equations with applications in population biology, (1993), Academic Press New York
[22] Wendi, W.; Zhien, M., Global dynamics of an epidemic model with time delay, Nonlinear anal. real world appl., 3, 365-373, (2002) · Zbl 0998.92038
[23] Kyrychko, Y.N.; Blyuss, K.B., Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear anal. real world appl., 6, 495-507, (2005) · Zbl 1144.34374
[24] Ma, W.; Takeuchi, Y., Asymptotic properties of a delayed SIR epidemic model with density dependent birth rate, Discrete contin. dyn. syst. ser. B, 4, 3, 671-678, (2004) · Zbl 1101.92043
[25] Qui, Z.; Yu, J.; Zou, Y., The asymptotic behaviour of a chemostat model, Discrete contin. dyn. syst. ser. B, 4, 3, 721-727, (2004) · Zbl 1114.92073
[26] Brauer, F.; Castillo-Chavez, C., Mathematical models in population biology and epidemiology, (2001), Springer · Zbl 0967.92015
[27] Diekmann, O.; Heesterbeek, J.A.P.; Metz, J.A.P., On the definition and computation of the basic reproduction ratio \(\mathcal{R}_0\) in models for infectious diseases in heterogeneous populations, J. math. biol., 28, 365-382, (1990) · Zbl 0726.92018
[28] 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
[29] May, R.M.; Anderson, R.M., Transmission dynamics of HIV infection, Nature, 326, 137-142, (1987)
[30] Kribs-Zaleta, C.M.; Velasco-Hernandez, J.X., A simple vaccination model with multiple endemic states, Math. biosci., 164, 183-201, (2000) · Zbl 0954.92023
[31] Kribs-Zaleta, C.M., Structured models for heterosexual disease transmission, Math. biosci., 160, 83, (1999) · Zbl 0980.92031
[32] Thieme, H.R., Uniform persistence and permanence for nonautonomous semiflows in population biology, Math. biosci., 166, 173-201, (2000) · Zbl 0970.37061
[33] Thieme, H.R., Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. math. anal., 24, 2, 405-435, (1993) · Zbl 0774.34030
[34] Jordan, D.W.; Smith, P., Nonlinear ordinary differential equations, (1999), Oxford University Press, pp. 348-388
[35] USAID, Country profile, accessed 2005
[36] Ministry Of Health And Child Welfare Zimbabwe, CDC zimbabwe and UNAIDS, (2003), Zimbabwe National HIV and AIDS Estimates
[37] 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
[38] Anderson, R.M.; Gupta, S.; May, R.M., Potential of community-wide chemotherapy or immunotherapy to control the spread of HIV-1, Nature, 350, 350-356, (1991)
[39] Hsu Schmitz, S.F., Effects of treatment or/and vaccination on HIV transmission in homosexuals with genetic heterogeneity, Math. biosci., 167, 1-18, (2000) · Zbl 0979.92023
[40] Baggaley, R.F.; Ferguson, N.M.; Garnett, G.P., The epidemiological impact of antiretroviral use predicted by mathematical models: A review, (2005), Emerging Themes in Epidemiology
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