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A stochastic model of AIDS and condom use. (English) Zbl 1101.92037
Summary: We introduce stochasticity into a model of AIDS and condom use via the technique of parameter perturbation which is standard in stochastic population modelling. We show that the model established in this paper possesses non-negative solutions as desired in any population dynamics. We also carry out a detailed analysis on asymptotic stability both in probability one and in p th moment. Our results reveal that a certain type of stochastic perturbation may help to stabilise the underlying system.

92D30 Epidemiology
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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[1] Arnold, L., Stochastic differential equations: theory and applications, (1972), Wiley New York
[2] Bahar, A.; Mao, X., Stochastic delay lotka – volterra model, J. math. anal. appl., 292, 364-380, (2004) · Zbl 1043.92034
[3] Blythe, S.P.; Anderson, R.M., Distributed incubation and infections periods in models of transmission dynamics of human immunodeficiency virus (HIV), IMA J. math. appl. med. biol., 5, 1-19, (1988) · Zbl 0686.92015
[4] Blythe, S.P.; Anderson, R.M., Variable infectiousness in HIV transmission models, IMA J. math. appl. med. biol., 5, 181-200, (1988) · Zbl 0655.92021
[5] Blythe, S.P.; Anderson, R.M., Heterogenous sexual activity models of HIV transmission in male homosexual populations, IMA J. math. appl. med. biol., 5, 237-260, (1988) · Zbl 0688.92010
[6] Braumann, C.A., Variable effort harvesting models in random environments: generalization to density-dependent noise intensities, Math. biosci., 177-178, 229-245, (2002) · Zbl 1003.92027
[7] Friedman, A., Stochastic differential equations and their applications, (1976), Academic Press New York
[8] Gard, T.C., Introduction to stochastic differential equations, (1988), Marcel Dekker New York · Zbl 0682.92018
[9] Greenhalgh, D.; Doyle, M.; Lewis, F., A mathematical model of AIDS and condom use, IMA J. math. appl. med. biol., 18, 225-262, (2001) · Zbl 0998.92032
[10] Jacquez, J.A.; Simon, C.P.; Koopman, J.S., Structured mixing: heterogenous mixing by the definition of activity groups, (), 301-315
[11] Jacquez, J.A.; Simon, C.P.; Koopman, J.S., The reproduction number in deterministic models of contagious disease, Comments theor. biol., 2, 159-209, (1988)
[12] Lipster, R.S.; Shiryayev, A.N., Theory of martingales, (1986), Kluwer Academic Dordrecht
[13] Mao, X., Stochastic differential equations and applications, (1997), Ellis Horwood Chichester · Zbl 0874.60050
[14] Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic process. appl., 97, 95-110, (2002) · Zbl 1058.60046
[15] Mao, X.; Yuan, C.; Zou, J., Stochastic differential delay equations of population dynamics, J. math. anal. appl., 304, 296-320, (2005) · Zbl 1062.92055
[16] NIAID, the relationship between the human immunodeficiency virus and the acquired immunodeficiency syndrome, (1995)
[17] UNAIDS, report on the global AIDS epidemic, (2004)
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