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Rao, Feng; Wang, Weiming; Li, Zhibin

Stability analysis of an epidemic model with diffusion and stochastic perturbation. (English) Zbl 1243.92045

Commun. Nonlinear Sci. Numer. Simul. 17, No. 6, 2551-2563 (2012).
Summary: We investigate the stability of an epidemic model with diffusion and stochastic perturbation. We first show both the local and global stability of the endemic equilibrium of the deterministic epidemic model by analyzing the corresponding characteristic equation and Lyapunov function. Second, for the corresponding reaction-diffusion epidemic model, we present conditions for global asymptotical stability of the endemic equilibrium. We carry out an analytical study for the stochastic model in detail and find out the conditions for the asymptotic stability of the endemic equilibrium in the mean sense. Furthermore, we perform a series of numerical simulations to illustrate our mathematical findings.

Cited in 12 Documents

MSC:

92D30 Epidemiology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
34D20 Stability of solutions to ordinary differential equations
93E15 Stochastic stability in control theory
34D23 Global stability of solutions to ordinary differential equations

Keywords:

Lyapunov function; Itô formula
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\textit{F. Rao} et al., Commun. Nonlinear Sci. Numer. Simul. 17, No. 6, 2551--2563 (2012; Zbl 1243.92045)
Full Text: DOI

References:

[1] Anderson, R.; May, R., Population biology of infectious dieases, Part I Nature, 280, 361-367 (1979)
[2] Ma, J.; Ma, Z., Epidemic threshold conditions for seasonally forced SEIR models, Math Biosci Eng, 3, 1, 161-172 (2006) · Zbl 1089.92048
[4] Kermack, W.; McKendrick, A., Contributions to the mathematical theory of epidemics (Part I), Proc Roy Soc A, 115, 700-721 (1927) · JFM 53.0517.01
[5] Beretta, E.; Hara, T.; Ma, W.; Takeuchi, Y., Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal, 47, 6, 4107-4115 (2001) · Zbl 1042.34585
[6] Hsu, S.; Roeger, L., The final size of a SARS epidemic model without quarantine, J Math Anal Appl, 333, 2, 557-566 (2007) · Zbl 1113.92055
[7] Meng, X.; Chen, L., The dynamics of a new SIR epidemic model concerning pulse vaccination strategy, Appl Math Comput, 197, 2, 528-597 (2008) · Zbl 1131.92056
[8] Roy, M.; Holt, R., Effects of predation on host-pathogen dynamics in SIR models, Theor Popul Biol, 73, 3, 319-331 (2008) · Zbl 1209.92054
[9] Shi, R.; Jiang, X.; Chen, L., The effect of impulsive vaccination on an SIR epidemic model, Appl Math Comput, 212, 2, 305-311 (2009) · Zbl 1186.92040
[10] Wei, H.; Li, X.; Martcheva, M., An epidemic model of a vector-borne disease with direct transmission and time delay, J Math Anal Appl, 342, 2, 895-908 (2008) · Zbl 1146.34059
[11] Zhang, Z.; Peng, J., A SIRS epidemic model with infection-age dependence, J Math Anal Appl, 331, 2, 1396-1414 (2007) · Zbl 1116.35024
[12] Zhang, F.; Li, Z.; Zhang, F., Global stability of an SIR epidemic model with constant infectious period, Appl Math Comput, 199, 1, 285-291 (2008) · Zbl 1136.92336
[13] Zhang, T.; Teng, Z., Permanence and extinction for a nonautonomous SIRS epidemic model with time delay, Appl Math Model, 33, 2, 1058-1071 (2009) · Zbl 1168.34358
[14] Holmes, E.; Lewis, M.; Banks, J.; Veit, R., Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75, 17-29 (1994)
[15] Murray, J., Mathematical biology (2003), Springer: Springer New York
[16] Rass, L.; Radcliffe, J., Spatial deterministic epidemics (2003), American Mathematical Society · Zbl 1018.92028
[17] Cantrell, R.; Cosner, C., Spatial ecology via reaction-diffusion equations (2003), Wiley · Zbl 1059.92051
[18] Wang, K.; Wang, W.; Song, S., Dynamics of an HBV model with diffusion and delay, J Theor Biol, 253, 1, 36-44 (2008) · Zbl 1398.92257
[19] Mulone, G.; Straughan, B.; Wang, W., Stability of epidemic models with evolution, Studies Appl Math, 118, 2, 117-132 (2007)
[20] Capasso, V.; Liddo, A., Asymptotic behaviour of reaction-diffusion systems in population and epidemic models, The role of cross diffusion, J Math Biol, 32, 453-463 (1994) · Zbl 0802.35077
[21] Bandyopadhyay, M.; Chattopadhyay, J., Ratio-dependent predator-prey model: effect of environmental fluctuation and stability, Nonlinearity, 18, 2, 913-936 (2005) · Zbl 1078.34035
[22] Gard, T., Introduction to stochastic differential equations (1988), Dekker: Dekker New York · Zbl 0628.60064
[23] May, R., Stability and complexity in model ecosystems (2001), Princeton University Press: Princeton University Press NJ
[24] Dalal, N.; Greenhalgh, D.; Mao, X., A stochastic model for internal HIV dynamics, J Math Anal Appl, 341, 2, 1084-1101 (2008) · Zbl 1132.92015
[25] Mao, X., Stochastic stabilisation and destabilisation, Systems Control Lett, 23, 279-290 (1994)
[26] Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process Appl, 97, 1, 95-110 (2002) · Zbl 1058.60046
[27] Berettaa, E.; Kolmanovskiib, V.; Shaikhet, L., Stability of epidemic model with time delays influenced by stochastic perturbations, Math Comput Simu, 45, 3-4, 269-277 (1998) · Zbl 1017.92504
[28] Tornatore, E.; Buccellato, S.; Vetro, P., Stability of a stochastic SIR system, Phys A, 354, 111-126 (2005)
[29] Yu, J.; Jiang, D.; Shi, N., Global stability of two-group SIR model with random perturbation, J Math Anal Appl, 360, 1, 235-244 (2009) · Zbl 1184.34064
[30] Meng, X., Stability of a novel stochastic epidemic model with double epidemic hypothesis, Appl Math Comput, 217, 2, 506-515 (2010) · Zbl 1198.92041
[31] Jiang, D.; Ji, C.; Shi, N.; Yu, J., The long time behavior of DI SIR epidemic model with stochastic perturbation, J Math Anal Appl, 372, 1, 162-180 (2010) · Zbl 1194.92053
[32] Carletti, M.; Burrage, K.; Burrage, P., Numerical simulation of stochastic ordinary differential equations in biomathematical modelling, Math Comput Simulat, 64, 271-277 (2004) · Zbl 1039.65005
[33] Xiao, D.; Ruan, S., Global analysis of an epidemic model with nonmonotone incidence rate, Math Biosci, 208, 2, 419-429 (2007) · Zbl 1119.92042
[34] Arcuri, P.; Murray, J., Pattern sensitivity to boundary and initial conditions in reaction-diffusion models, J Math Biol, 24, 2, 141-165 (1986) · Zbl 0595.92001
[35] Wang, W.; Lin, Y.; Zhang, L.; Rao, F.; Tan, Y., Complex patterns in a predator-prey model with self and cross-diffusion, Commun Nonlinear Sci Numer Simulat, 16, 2006-2015 (2011) · Zbl 1221.35423
[36] Henry, D., Geometric theory of semilinear parabolic equations, (Lecture Notes in Mathematics (1981), Springer-Verlag) · Zbl 0456.35001
[37] Øksendal, B., Stochastic differential equations: an introduction with applications (1995), Springer · Zbl 0841.60037
[38] Liu, M.; Wang, K., Global stability of a nonlinear stochastic predator-prey system with Beddington-DeAngelis functional response, Commun Nonlinear Sci Numer Simulat, 16, 3, 1114-1121 (2011) · Zbl 1221.34152
[39] Luo, Q.; Mao, X., Stochastic population dynamics under regime switching, J Math Anal Appl, 334, 1, 69-84 (2007) · Zbl 1113.92052
[40] Lv, J.; Wang, K., Asymptotic properties of a stochastic predator-prey system with Holling II functional response, Commun Nonlinear Sci Numer Simulat, 16, 10, 4037-4048 (2011) · Zbl 1218.92072
[41] Afanasev, V.; Kolmanowskii, V.; Nosov, V., Mathematical theory of control systems design (1996), Kluwer: Kluwer Dordrecht
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