Stability Analysis of a Delayed HIV/AIDS Epidemic Model with Treatment and Vertical Transmission

Abstract

A delayed HIV/AIDS epidemic model with treatment and vertical transmission is investigated. The model allows some infected individuals to move from the symptomatic phase to the asymptomatic phase; next generation of infected individuals may be infected and it will take them some time to get maturity and infect others. Mathematical analysis shows that the global dynamics of the spread of the HIV/AIDS are completely determined by the basic reproduction number R0 for our model. If R0 < 1 then disease free equilibrium is globally asymptotically stable, whereas the unique infected equilibrium is globally asymptotically stable if R0 > 1.

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Osman, Z. and Abdurahman, X. (2015) Stability Analysis of a Delayed HIV/AIDS Epidemic Model with Treatment and Vertical Transmission. Applied Mathematics, 6, 1781-1789. doi: 10.4236/am.2015.610158.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] Cai, L.M. and Li, X.Z. (2009) Stability Analysis of an HIV/AIDS Epidemic Model with Treatment. Journal of Computational and Applied Mathematics, 229, 313-323. http://dx.doi.org/10.1016/j.cam.2008.10.067 [2] Cai, L.M. and Guo, S.L. (2014) Analysis of an Extended HIV/AIDS Epidemic Model with Treatment. Applied Mathematics and Computation, 236, 621-627. http://dx.doi.org/10.1016/j.amc.2014.02.078 [3] Huo, H.-F. and Feng, L.-X. (2013) Global Stability for an HIV/AIDS Epidemic Model with Different Latent Stages and Treatment. Applied Mathematical Modeling, 37, 1480-1489. http://dx.doi.org/10.1016/j.apm.2012.04.013 [4] Elaiw, A.M. (2010) Global Properties of a Class of HIV Models. Nonlinear Analysis: Real World Applications, 11, 2253-2263. http://dx.doi.org/10.1016/j.nonrwa.2009.07.001 [5] Xiao, D.M. and Ruan, S.G. (2007) Global Analysis of an Epidemic Model with Non-Monotone Incidence Rate. Mathematical Biosciences, 208, 419-429. http://dx.doi.org/10.1016/j.mbs.2006.09.025 [6] Naresh, R., Tripathi, A. and Omar, S. (2006) Modeling the Spread of AIDS Epidemic with Vertical Transmission. Applied Mathematics and Computation, 178, 262-272. http://dx.doi.org/10.1016/j.amc.2005.11.041 [7] d’Onofrio, A. (2005) On Pulse Vaccination Strategy in the SIR Epidemic Model with Vertical Transmission. Applied Mathematics Letters, 18, 729-732. http://dx.doi.org/10.1016/j.aml.2004.05.012 [8] Li, M.Y. and Smith, H.L. (2001) Global Dynamics of an SEIR Epidemic Model with Vertical Transmission. SIAM Journal of Applied Mathematics, 62, 58-69. http://dx.doi.org/10.1137/S0036139999359860 [9] Naresh, R. and Sharma, D. (2011) An HIV/AIDS Model with Vertical Transmission and Time Delay. World Journal of Modeling and Simulation, 7, 230-240. [10] Liu, J.L. and Zhang, T.L. (2012) Global Stability for Delay SIR Epidemic Model with Vertical Transmission. Open Journal of Applied Sciences, 2, 1-4. http://dx.doi.org/10.4236/ojapps.2012.24b001 [11] Gumel, A.B., McCluskey, C.C. and van den Driessche, P. (2006) Mathematical Study of a Staged-Progression HIV Model with Imperfect Vaccine. Bulletin of Mathematical Biology, 68, 2105-2128. http://dx.doi.org/10.1007/s11538-006-9095-7 [12] Cai, L.M., Fang, B. and Li, X.Z. (2014) A Note of a Staged Progression HIV Model with Imperfect Vaccine. Applied Mathematics and Computation, 234, 412-416. http://dx.doi.org/10.1016/j.amc.2014.01.179 [13] McCluskey, C.C. and vanden Driessche, P. (2004) Global Analysis of Two Tuberculosis Models. Journal of Dynamics and Differential Equations, 16, 139-166. http://dx.doi.org/10.1023/B:JODY.0000041283.66784.3e [14] LaSalle, J.P. (1976) The Stability of Dynamical Systems. In: Regional Conference Series in Applied Mathematics. SIAM, Philadelphia.