## 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 Probabilistic models, generic numerical methods in probability and statistics
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### 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), CRC Press-Parthenon Publishers, pp. 22-27 [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, 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 [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, C.R. biologies, 327, 983-994, (2004) [11] Blower, S., Calculating the consequences: HAART and risky sex, Aids, 15, 1309-1310, (2001) [12] Connell McCluskey, C., A model of HIV/AIDS with staged progression and amelioration, Math. biosci., 181, 1-16, (2003) · Zbl 1008.92032 [13] Elaiw, A.M., Global properties of a class of HIV models, Nonlinear anal.: RWA, 11, 2253-2263, (2010) · Zbl 1197.34073 [14] Hethcote, H.W.; Van Ark, J.W., () [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) · Zbl 1334.92404 [16] Leenheer, P.D.; Smith, H.L., Virus dynamics: a global analysis, SIAM. J. appl. math., 63, 1313-1327, (2003) · Zbl 1035.34045 [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 [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 [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) · Zbl 1342.92269 [20] Perelson, A.S.; Nelson, P.W., Mathematical analysis of HIV-1 dynamics in vivo, SIAM rev., 41, 1, 3-44, (1999) · Zbl 1078.92502 [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 [22] 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 [23] 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 [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), Article ID 958016, 19 pages · Zbl 1187.34062 [26] Busenberg, S.; Cooke, K., Vertically transmitted diseases, (1993), Springer Berlin · Zbl 0837.92021 [27] Cushing, J.M., Integrodifferential equations and delay models in population dynamics, (1977), Spring Heidelberg · Zbl 0363.92014 [28] Gopalsamy, K., Stability and oscillations in delay-differential equations of population dynamics, (1992), Kluwer Dordrecht · Zbl 0752.34039 [29] Kuang, Y., Delay-differential equations with applications in population dynamics, (1993), Academic Press New York · 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 $$C D 4^+ T$$-cells, Math. biosci., 165, 27-39, (2000) · Zbl 0981.92009 [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 [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 [35] Thieme, H.R., Uniform persistence and permanence for nonautonomous semiflows in population biology, Math. biosci., 166, 173-201, (2000) · Zbl 0970.37061 [36] Zhang, T.; Teng, Z., On a nonautonomous SEIRS model in epidemiology, Bull. math. biol., 69, 2537-2559, (2007) · Zbl 1245.34040 [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 [38] Hale, J.K., Theory of functional differential equations, (1977), Springer-Verlag New York [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., () [42] Diekmann, O.; Heesterbeek, J.A.P., Mathematical epidemiology of infectious diseases: model building analysis, and interpretation, (2000), John Wiley and Sons Ltd Chichester, New York · Zbl 0997.92505 [43] Kermark, M.D.; Mckendrick, A.G., Contributions to the mathematical theory of epidemics. part I, Proc. R. soc. A, 115, 5, 700-721, (1927) · JFM 53.0517.01 [44] Ma, Z.; Zhou, Y.; Wang, W.; Jin, Z., Mathematical modelling and research of epidemic dynamical systems, (2004), Science Press Beijing [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
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