Krstić, Marija The effect of stochastic perturbation on a nonlinear delay malaria epidemic model. (English) Zbl 1245.60058 Math. Comput. Simul. 82, No. 4, 558-569 (2011). Summary: The subject of this paper is the stochastic epidemic malaria model with time delay, described by the system of the Itô stochastic functional delay equations. We center such a system around the endemic equilibrium state and, by the Lyapunov functional method, we obtain sufficient conditions for model parameters, as well as for time delays within which we can claim the asymptotical mean square stability and stability in probability. Finally, we present an example to show the compatibility of our mathematical results with the stochastic delay malaria model with quantities which are reliable data, as well as an example which shows that introduction of environmental noise annuls Hopf Bifurcation of the corresponding deterministic model. Cited in 9 Documents MSC: 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 92D25 Population dynamics (general) 93E15 Stochastic stability in control theory Keywords:endemic equilibrium; human population; incubation period; mosquito population; stability PDFBibTeX XMLCite \textit{M. Krstić}, Math. Comput. Simul. 82, No. 4, 558--569 (2011; Zbl 1245.60058) Full Text: DOI References: [1] Abbas, S.; Bahuguna, D.; Banerjee, M., Effect of stochastic perturbation on a two species competitive model, Nonlinear Analysis: Hibrid Systems, 3, 195-206 (2009) · Zbl 1192.34097 [2] Beretta, E.; Kolmanovskii, V.; Shaikhet, L., Stability of epidemic model with time delays influenced by stochastic perturbations, Mathematics and Computers in Simulation, 45, 269-277 (1998) · Zbl 1017.92504 [3] Carletti, M., Mean-square stability of a stochastic model for bacteriophage infection with time delays, Mathematical Biosciences, 210, 395-414 (2007) · Zbl 1134.92036 [4] Carletti, M., On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Mathematical Biosciences, 175, 117-131 (2002) · Zbl 0987.92027 [5] Chitnis, N.; Cushing, J. M.; Hyman, J. M., Bifurcation analysis of a mathematical model for malaria transmission, Siam Journal on Applied Mathematics, 67, 1, 24-45 (2006) · Zbl 1107.92047 [6] en.wikipedia.org/wiki/Malaria; en.wikipedia.org/wiki/Malaria [7] Kloeden, P. E.; Platen, E., Numerical Solution of Stochastic Differential Equations (1995), Springer: Springer Berlin · Zbl 0858.65148 [8] Kolmanovskii, V. Z.; Nosov, V. R., Stability of Functional Differential Equations (1986), Academic Press, Inc.: Academic Press, Inc. London [9] Ruana, S.; Xiaob, D.; Beierc, J. C., On the delayed Ross-Macdonald model for malaria transmission, Bulletin of Mathematical Biology, 70, 1098-1114 (2008) · Zbl 1142.92040 [10] Saker, S. H., Stability and Hopf bifurcations of nonlinear delay malaria epidemic model, Nonlinear Analysis: Real World Applications, 11, 784-799 (2010) · Zbl 1181.37121 [11] Tumwiine, J.; Mugisha, J. Y.T.; Luboobi, L. S., A host-vector model for malaria with infective immigrants, Journal of Mathematical Analysis and Applications, 361, 139-149 (2010) · Zbl 1176.92045 [12] Wei, H.-M.; Li, X.-Z.; Martcheva, M., An epidemic model of a vector-borne disease with direct transmission and time delay, Journal of Mathematical Analysis and Applications, 342, 895-908 (2008) · Zbl 1146.34059 [13] Yu, J.; Jiang, D.; Shi, N., Global stability of two-group SIR model with random perturbation, Journal of Mathematical Analysis and Applications, 360, 235-244 (2009) · Zbl 1184.34064 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.