Abstract
We present a deterministic HIV/AIDS model with delay. We then extend the model by adjoining terms capturing stochastic effects. The intensity of the fluctuations in the stochastic system is analytically evaluated using Fourier transform methods. We carry out simulations to assess differences in the dynamical behavior of the deterministic and stochastic models. Simulation results show that they are no significant differences in the behavior of the two models.
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Anderson, R.M., Gupta, S., May, R.M.: Potential of community-wide chemotherapy or immunotherapy to control the spread of HIV-1. Nature 350(6316), 356–359 (1991)
Horsthemke, W., Lefever, R.: Noise Induced Transitions. Springer, Berlin (1984)
Hsu Schmitz, S.F.: A mathematical model of HIV transmission in homosexuals with genetic heterogeneity. J. Theor. Med. 2, 285–296 (2000)
Hsu Schmitz, S.F.: Effects of treatment or/and vaccination on HIV transmission in homosexuals with genetic heterogeneity. Math. Biosci. 167(1), 1–18 (2000)
Gardiner, C.W.: Handbook of Stochastic Methods. Springer New York (1983)
Kurtz, T.G.: Solutions of ordinary differential equation as limits of pure jumps Markov process. J. Appl. Probab. 7, 49–58 (1970)
Kurtz, T.G.: Limit theorems for sequences of jump Markov processes approximating differential equations. J. Appl. Probab. 8, 344–356 (1971)
May, R.M., Anderson, R.M.: Transmission dynamics of HIV infection. Nature 326, 137–142 (1987)
May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1974)
Mukandavire, Z., Garira, W., Chiyaka, C.: Asymptotic properties of an HIV/AIDS model with a time delay. J. Math. Anal. Appl. 330(2), 916–933 (2007)
Nisbet, R.M., Gurney, W.S.C.: Modelling Fluctuating Populations. Wiley-Interscience (1982)
Renshaw, E.: Modelling Biological Population in Space and Time. Cambridge University Press, Cambridge (1995)
Samanta, G.P.: The effect of random fluctuating environment on interacting species with time delay. Int. J. Math. Educ. Sci. Technol. 27, 13–21 (1996)
Samanta, G.P.: Permanence and extinction of a nonautonomous HIV/AIDS epidemic model with distributed time delay. Nonlinear Anal.: Real World Appl. 12(2), 1163–1177 (2011). doi:10.1016/j.nonrwa.2010.09.010
Samanta, G.P.: Analysis of a nonautonomous HIV/AIDS epidemic model with distributed time delay. Math. Model. Anal. 15(3), 327–347 (2010)
Samanta, G.P.: Analysis of a nonautonomous HIV/AIDS model. Math. Model. Nat. Phenomena 5(6), 70–95 (2010)
Turelli, M.: Stochastic community theory: a partially guided tour. In: Hallam, T.G., Levin, S. (eds.) Mathematical Ecology. Springer, Berlin (1986)
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Mukandavire, Z., Das, P., Chiyaka, C. et al. HIV/AIDS Model with Delay and the Effects of Stochasticity. J Math Model Algor 10, 181–191 (2011). https://doi.org/10.1007/s10852-010-9148-7
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DOI: https://doi.org/10.1007/s10852-010-9148-7