A stochastic model of AIDS and condom use. (English) Zbl 1101.92037

Dalal, Nirav; Greenhalgh, David; Mao, Xuerong

J. Math. Anal. Appl. 325, No. 1, 36-53 (2007).

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 pth moment. Our results reveal that a certain type of stochastic perturbation may help to stabilise the underlying system.

MSC:

92D30	Epidemiology
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)

Keywords:

HIV; AIDS; condom use; Brownian motion; stochastic differential equation; Itô’s formula; asymptotic stability

References:

[1]	Arnold, L., Stochastic Differential Equations: Theory and Applications (1972), Wiley: 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: Academic Press New York
[8]	Gard, T. C., Introduction to Stochastic Differential Equations (1988), Marcel Dekker: 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, (Castillo-Chavez, C., Mathematical and Statistical Approaches to AIDS Epidemiology. Mathematical and Statistical Approaches to AIDS Epidemiology, Lecture Notes in Biomath., vol. 83 (1989), Springer: Springer Berlin), 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: Kluwer Academic Dordrecht
[13]	Mao, X., Stochastic Differential Equations and Applications (1997), Ellis Horwood: 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)