×

Stability analysis of a non-linear HIV/AIDS epidemic model with vaccination and antiretroviral therapy. (English) Zbl 1427.92057

Summary: In this paper, we like to propose and analyze a non-linear HIV/AIDS epidemic model with vaccination and antiretroviral therapy. For our convenient study we have divided the total populations into five classes such as susceptible class, unaware HIV infected class, aware HIV infected class, pre AIDS class and AIDS class respectively. For our present purpose we have taken only the disease spread through horizontal transmission into consideration. In this paper we have tried to develop a nonlinear HIV/AIDS mathematical model to study the transmission dynamics of HIV at four compartments of the populations with vaccination and antiretroviral therapy and to prove the positivity and boundedness of its solutions. In this paper we have added a treatment procedure i.e. antiretroviral therapy and tried to find out its effect. We have also analyzed the stability behaviour of the system. Finally we have found that vaccination and antiretroviral therapy is an effective way to control the disease transmission. The mathematical model solved numerically by using an iterative numerical recipe which supports the theoretical or analytical results.

MSC:

92C60 Medical epidemiology
34D23 Global stability of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: Link

References:

[1] UNAIDS Report on the Global AIDS Epidemic, HIV estimates with uncertainty bounds, 1990 - 2012, (2013) www.unaids.com, Last Accessed:, 1st January 2014.
[2] Centers for Disease Control, Pneumocystis Pneumonia-Los Angeles, Morbidity and Mortality Weekly Report 30, (1981) 250-252.
[3] Centers for Disease Control, : Update on acquired immune deficiency syndrome (AIDS)-United States, Morbidity and Mortality Weekly Report 31 (1982) 507-514.
[4] J. Coffin, A. Hasse, J. A. Levy, L., Montagnier, S., Oroszlan : Human immunodeficiency viruses, Science, 232, (1986) 697.
[5] Annual report : published by Department of AIDS Control, Ministry of Health and Family Welfare, Government of India , 2013.
[6] L. M. Cai, X. Z. Li : Stability analysis of an HIV/AIDS Epidemic Model with Treatment, Journal of Computational and Applied Mathematics 229(2009) 313-323. · Zbl 1162.92035
[7] L. M. Cai, S. L. Guo : Analysis of an Extended HIV/AIDS Epidemic Model with Treatment, Applied Mathematics and Computation 236, (2014) 621-627. · Zbl 1334.92226
[8] H. -F. Huo, L. -X., Feng : Global Stability for an HIV/AIDS Epidemic Model with Different Latent Stages and Treatment, Applied Mathematical Modeling 37 (2013) 1480-1489. · Zbl 1351.34044
[9] A. M. ELaiw, Global Properties of a Class of HIV Models, Nnolinear Analysis: Real World Applications 11 (2010) 2253-2263. · Zbl 1197.34073
[10] D. M. Xiao, S. G., Ruan : Global Analysis of an Epidemic Model with Non-Monotone Incidence Rate, Mathematical Biosciences 208 (2007) 419-429. · Zbl 1119.92042
[11] R. M. Anderson, The role of mathematical models in the study of HIV transmission and the epidemiology of AIDS, J. AIDS 1 (1988) 241-256.
[12] S. Busenberg, K. Cooke, H. Ying-Hen : A model for HIV in Asia, Math. Biosci. 128 (1995) 185-210. · Zbl 0833.92016
[13] O. Diekmann, P.J. A. Heesterbeek, J. A.J. Metz, On the definition and the basic reproduction ratio R0in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28 (1990) 365-382. · Zbl 0726.92018
[14] K. Dietz, On the transmission dynamics of HIV, Math. Biosci. 90 (1988) 397-414. · Zbl 0651.92019
[15] Y. -H. Hsieh, C. H. Chen, : Modeling the social dynamics of a sex industry: Its implications for spread of HIV/AIDS, Bull. Math. Biol. 66 (2004) 143-166. · Zbl 1334.92404
[16] National AIDS Control Organization Country Scenario AIDS, Published by NACO, Ministry of Health, Government of India, NewDelhi, 2004.
[17] Ram Naresh, Agraj Tripathi, Sandip Omar, : Modelling the spread of AIDS epidemic with vertical transmission, Applied Mathematics and Computation, 178, (2006) 262-272. · Zbl 1096.92038
[18] R. O. Simwa, G.P., Pokhariyal, : A dynamical model for stage-specific HIV incidences with application to SubSaharan Africa, Applied Mathematics and Computation, Elsevier 6 (2003) 14. · Zbl 1026.92039
[19] S. Issa, E. S. Massawe, O. D. Makinde, Modelling the effect of screening on the spread of HIV infection in a Homogeneous population with infective immigrants, Scientific Research and Essays (SRE) (2011) 4397-4405.
[20] J. S., Montaner, R. Hogg, E. Wood, T. Kerr, M. Tyndall, The case for expanding access to highly active antiretroviral
[21] W Cascarilla Novi, Dwi Lestari, Local Stability of AIDS Epidemic Model Through Treatment and Vertical Transmission with Time Delay, Journal of Physics 693 (01) (2010).
[22] J. P. LaSalle, The Stability of Dynamical Systems, in: Regional Conference Series inApplied Mathematics, SIAM, Philadelphia, PA. , 1976. · Zbl 0364.93002
[23] J. Tewa, J. S. Dimi, S. Bowong, Lyapunov function for a dengue disease transmission model, Chaos, Solitons and Fractals 39 (2009) 936-941. · Zbl 1197.34099
[24] L. X. F. , Huo, Global stability of an epidemic model with incomplete treatment and vaccination, Discret. Dyn. Nat Soci (2012), 530267, (2012) pages.14. · Zbl 1244.93120
[25] R. Naresh, A. Tripathi, D. Sharma, : Modelling and analysis of the spread of AIDS epidemic with immigration of HIV infection, Math. Comput. Model 49 (5-6), (2009) 880-892. · Zbl 1165.34377
[26] Hai-Feng, Huo, Chen. Rui, Wang. Xun-Yang : Modelling and stability of HIV/AIDS epidemic model with treatment, Applie Mathematical Modelling 40, (2016) 6550-6559. · Zbl 1465.92119
[27] Defang, Liu, Bochu, Wang, A novel time delayed HIV/AIDS model with vaccination and antiretroviral therapy and its stability analysis, Applie Mathematical Modelling 37, (2013) 4608-4625. · Zbl 1426.92079
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.