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Permanence and extinction of a nonautonomous HIV/AIDS epidemic model with distributed time delay. (English) Zbl 1203.92051
Summary: We have considered a nonautonomous stage-structured HIV/AIDS epidemic model having two stages of the period of infection according to the developing progress of infection before AIDS defined in, with varying total population size and distributed time delay to become infectious. The infected persons in the different stages have different abilities of transmitting the disease. By all kinds of treatment methods, some people with the symptomatic stages can be transformed into asymptomatic stages. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using the inequality analytical technique. We have obtained an explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values ${R}_{0}$ and ${R}^{*}$ and further obtained that the disease will be permanent when ${R}_{0}>1$ and the disease will be going extinct when ${R}^{*}<1$. By the Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. Also, we have observed that the time delay decreases the lower bounds of the infective and full-blown AIDS group. Computer simulations are carried out to explain the analytical findings. The aim of the analysis of this model is to identify the parameters of interest for further study, with a view to informing and assisting policy-makers in targeting prevention and treatment resources for maximum effectiveness.
##### MSC:
 92C60 Medical epidemiology 34K20 Stability theory of functional-differential equations 34K25 Asymptotic theory of functional-differential equations 92D30 Epidemiology 65C20 Models (numerical methods)
##### References:
 [1] UNAIDS, WHO, 2007 AIDS epidemic update, December 2007. [2] Centers for Disease Control and Prevention, HIV and Its Transmission, Divisions of HIV/AIDS Prevention, 2003. [3] Lipman, M. C. I.; Baker, R. W.; Johnson, M. A.: An atlas of differential diagnosis in HIV disease, (2003) [4] Morgan, D.; Mahe, C.; Mayanja, B.; Okongo, J. M.; Lubega, R.; Whitworth, J. A.: HIV-1 infection in rural africa: is there a difference in median time to AIDS and survival compared with that in industrialized countries?, Aids 16, 597-632 (2002) [5] Stoddart, C. A.; Reyes, R. A.: Models of HIV-1 disease: a review of current status, Drug discovery today: disease models 3, No. 1, 113-119 (2006) [6] Cai, L. M.; Li, X.; Ghosh, M.; Guo, B.: Stability of an HIV/AIDS epidemic model with treatment, J. comput. Appl. math. 229, 313-323 (2009) · Zbl 1162.92035 · doi:10.1016/j.cam.2008.10.067 [7] 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) [8] 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 [9] May, R. M.; Anderson, R. M.: Transmission dynamics of HIV infection, Nature 326, 137-142 (1987) [10] Bachar, M.; Dorfmayr, A.: HIV treatment models with time delay, CR biologies 327, 983-994 (2004) [11] Blower, S.: Calculating the consequences: HAART and risky sex, Aids 15, 1309-1310 (2001) [12] Mccluskey, C. Connell: A model of HIV/AIDS with staged progression and amelioration, Math. biosci. 181, 1-16 (2003) · Zbl 1008.92032 · doi:10.1016/S0025-5564(02)00149-9 [13] Elaiw, A. M.: Global properties of a class of HIV models, Nonlinear anal.: RWA 11, 2253-2263 (2010) · Zbl 1197.34073 · doi:10.1016/j.nonrwa.2009.07.001 [14] Hethcote, H. W.; Van Ark, J. W.: Modelling HIV transmission and AIDS in the united states, Lect. notes biomath. 95 (1992) · Zbl 0805.92026 [15] 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) [16] Leenheer, P. D.; Smith, H. L.: Virus dynamics: a global analysis, SIAM J. Appl. math. 63, 1313-1327 (2003) · Zbl 1035.34045 · doi:10.1137/S0036139902406905 [17] Mukandavire, Z.; Chiyaka, C.; Garira, W.; Musuka, G.: Mathematical analysis of a sex-structured HIV/AIDS model with a discrete time delay, Nonlinear anal. 71, 1082-1093 (2009) · Zbl 1178.34103 · doi:10.1016/j.na.2008.11.026 [18] Naresh, R.; Sharma, D.; Tripathi, A.: Modelling the effect of tuberculosis on the spread of HIV infection in a population with density-dependent birth and death rate, Math. comput. Modelling 50, 1154-1166 (2009) · Zbl 1185.92078 · doi:10.1016/j.mcm.2009.05.033 [19] Nyabadza, F.; Chiyaka, C.; Mukandavire, Z.; Hovemusekwa, S. D.: Analysis of an HIV/AIDS model with public-health information campaigns and individual withdrawal, J. biol. Syst. 18, 357-375 (2010) [20] Perelson, A. S.; Nelson, P. W.: Mathematical analysis of HIV-1 dynamics in vivo, SIAM rev. 41, No. 1, 3-44 (1999) · Zbl 1078.92502 · doi:10.1137/S0036144598335107 [21] Wang, L.; Chen, L.; Nieto, J. J.: The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear anal.: RWA 11, 1374-1386 (2010) · Zbl 1188.93038 · doi:10.1016/j.nonrwa.2009.02.027 [22] 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 · doi:10.1016/j.mbs.2005.12.026 [23] Wang, K.; Wang, W.; Liu, X.: Viral infection model with periodic lytic immune response, Chaos solitons fractals 28, No. 1, 90-99 (2006) · Zbl 1079.92048 · doi:10.1016/j.chaos.2005.05.003 [24] Yan, P.: Impulsive SUI epidemic model for HIV/AIDS with chronological age and infection age, J. theoret. Biol. 265, 177-184 (2010) [25] Yu, Y.; Nieto, J. J.; Torres, A.; Wang, K.: A viral infection model with a nonlinear infection rate, Boundary value problems 2009 (2009) · Zbl 1187.34062 · doi:10.1155/2009/958016 [26] Busenberg, S.; Cooke, K.: Vertically transmitted diseases, (1993) · Zbl 0837.92021 [27] Cushing, J. M.: Integrodifferential equations and delay models in population dynamics, (1977) · Zbl 0363.92014 [28] Gopalsamy, K.: Stability and oscillations in delay-differential equations of population dynamics, (1992) · Zbl 0752.34039 [29] Kuang, Y.: Delay-differential equations with applications in population dynamics, (1993) · Zbl 0777.34002 [30] Perelson, A. S.; Neumann, A. U.; Markowitz, M.; Leonard, J. M.; Ho, D. D.: HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science 271, 1582-1586 (1996) [31] Herz, A. V. M.; Bonhoeffer, S.; Anderson, R. M.; May, R. M.; Nowak, M. A.: Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay, Proc. natl. Acad. sci. USA 93, 7247-7251 (1996) [32] 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 · doi:10.1016/S0025-5564(00)00006-7 [33] Herzong, G.; Redheffer, R.: Nonautonomous SEIRS and thron models for epidemiology and cell biology, Nonlinear anal.: RWA 5, 33-44 (2004) · Zbl 1067.92053 · doi:10.1016/S1468-1218(02)00075-5 [34] Thieme, H. R.: Uniform weak implies uniform strong persistence for non-autonomous semiflows, Proc. am. Math. soci. 127, 2395-2403 (1999) · Zbl 0918.34053 · doi:10.1090/S0002-9939-99-05034-0 [35] Thieme, H. R.: Uniform persistence and permanence for nonautonomous semiflows in population biology, Math. biosci. 166, 173-201 (2000) · Zbl 0970.37061 · doi:10.1016/S0025-5564(00)00018-3 [36] Zhang, T.; Teng, Z.: On a nonautonomous SEIRS model in epidemiology, Bull. math. Biol. 69, 2537-2559 (2007) [37] Zhang, T.; Teng, Z.: Permanence and extinction for a nonautonomous SIRS epidemic model with time delay, Appl. math. Model. 33, 1058-1071 (2009) · Zbl 1168.34358 · doi:10.1016/j.apm.2007.12.020 [38] Hale, J. K.: Theory of functional differential equations, (1977) [39] Teng, Z.; Chen, L.: The positive periodic solutions of periodic Kolmogorov type systems with delays, Acta math. Appl. sin. 22, 446-456 (1999) · Zbl 0976.34063 [40] Anderson, R. M.; May, R. M.: Population biology of infectious diseases. Part I, Nature 280, 361-367 (1979) [41] Capasso, V.: Mathematical structures of epidemic systems, Lectures notes in biomathematics 97 (1993) · Zbl 0798.92024 [42] Diekmann, O.; Heesterbeek, J. A. P.: Mathematical epidemiology of infectious diseases: model building analysis, and interpretation, (2000) [43] Kermark, M. D.; Mckendrick, A. G.: Contributions to the mathematical theory of epidemics. Part I, Proc. R. Soc. A 115, No. 5, 700-721 (1927) [44] Ma, Z.; Zhou, Y.; Wang, W.; Jin, Z.: Mathematical modelling and research of epidemic dynamical systems, (2004) [45] Meng, X.; Chen, L.; Cheng, H.: Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination, Appl. math. Comput. 186, 516-529 (2007) · Zbl 1111.92049 · doi:10.1016/j.amc.2006.07.124