Sun, Gui-Quan; Li, Li; Jin, Zhen; Li, Bai-Lian Effect of noise on the pattern formation in an epidemic model. (English) Zbl 1194.92066 Numer. Methods Partial Differ. Equations 26, No. 5, 1168-1179 (2010). Summary: We present novel numerical evidence of a complicated phenomenon controlled by noise in a spatial epidemic model. The number of the spot is decreased as the noise intensity being increased, which we show by performing a series of numerical simulations. Moreover, when the noise intensity and temporal correlation are both large enough, the model dynamics exhibits a noise controlled transition from spotted patterns to stripe growth. In addition, we show in detail the number of the spotted and stripe patterns, with the identification of a wide range of noise intensity and temporal correlations. The obtained results show that noise plays an important role in the pattern formation of the epidemic model, which may provide guidance to prevent and control the spread of disease. Cited in 12 Documents MSC: 92D30 Epidemiology 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 65C20 Probabilistic models, generic numerical methods in probability and statistics Keywords:epidemic model; noise; transition; pattern formation PDFBibTeX XMLCite \textit{G.-Q. Sun} et al., Numer. Methods Partial Differ. Equations 26, No. 5, 1168--1179 (2010; Zbl 1194.92066) Full Text: DOI References: [1] Anderson, Infectious diseases of humans (1992) [2] Diekmann, Mathematical epidemiology of infectious diseases: model building, analysis and interpretation (2000) · Zbl 0997.92505 [3] Kermack, A contribution to the mathematical theory of epidemics, Proc R Soc Edinb A 115 pp 700– (1927) · JFM 53.0517.01 [4] Mollison, Spatial contact models for ecological and epidemic spread (with discussion), J Roy Stat Soc B 39 pp 283– (1977) · Zbl 0374.60110 [5] Murray, Mathematical biology (1993) [6] Sun, Pattern formation in a spatial SI model with non-linear incidence rates, J Stat Mech 11 pp P11011– (2007) [7] Liu, Formation of spatial patterns in an epidemic model with constant removal rate of the infectives, J Stat Mech 5 pp P05002– (2007) [8] Sun, Chaos induced by breakup of waves in a spatial epidemic model with nonlinear incidence rate, J Stat Mech 8 pp P08011– (2008) [9] Grenfell, Travelling waves and spatial hierarchies in measles epidemics, Nature 414 pp 716– (2001) [10] Cummings, Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand, Nature 427 pp 344– (2004) [11] Vecchio, Periodic and aperiodic traveling pulses in population dynamics: an example from the occurrence of epidemic infections, Phys Rev E 73 pp 031913– (2006) [12] Earn, A simple model for complex dynamical transitions in epidemics, Science 287 pp 667– (2000) [13] Diekmann, Patterns in the effects of infectious diseases on population growth, J Math Biol 29 pp 539– (1991) · Zbl 0732.92024 [14] Keeling, Dynamics of the 2001 UK Foot and Mouth Epidemic: stochastic dispersal in a heterogeneous landscape, Science 294 pp 813– (2001) [15] Owen, How predation can slow, stop or reverse a prey invasion, Bull Math Biol 63 pp 655– (2001) · Zbl 1323.92181 [16] Petrovskii, Some exact solutions of a generalized Fisher equation related to the problem of biological invasion, Math Biosci 172 pp 73– (2001) [17] Kot, Dispersal data and the spread of invading organisms, Ecology 77 pp 2027– (1996) [18] Rohani, Opposite patterns of synchrony in sympatric disease metapopulations, Science 286 pp 968– (1999) [19] Rohani, The interplay between noise and determinism in childhood diseases, Am Nat 159 pp 469– (2002) [20] Bauch, Transients and attractors in epidemics, Proc R Soc B 270 pp 1573– (2003) [21] García-Ojalvo, Noise in spatially extended systems (1999) · Zbl 0938.60002 [22] Gammaitoni, Stochastic resonance, Rev Mod Phys 70 pp 223– (1998) [23] Horsthemke, Noise-induced transitions (1984) [24] Lesmes, Noise-controlled self-replicating patterns, Phys Rev Lett 91 pp 238301– (2003) [25] Liu, Dynamical behavior of epidemiological model with nonlinear incidence rate, J Math Biol 25 pp 359– (1987) [26] Liu, Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models, J Math Biol 23 pp 187– (1986) · Zbl 0582.92023 [27] Mankin, Trichotomous-noise-induced catastrophic shifts in symbiotic ecosystems, Phys Rev E 65 pp 051108– (2002) [28] Mankin, Colored-noise-induced discontinuous transitions in symbiotic ecosystems, Phys Rev E 69 pp 061106– (2004) [29] Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev 43 pp 525– (2001) · Zbl 0979.65007 [30] Wang, Resonant-pattern formation induced by additive noise in periodically forced reaction-diffusion systems, Phys Rev E 74 pp 036210– (2006) [31] Petrovskii, Transition to spatiotemporal chaos can resolve the paradox of enrichment, Ecol Complexity 1 pp 37– (2004) [32] Malchow, Spatiotemporal patterns in ecology and epidemiology: theory, models, and simulations (2008) · Zbl 1298.92004 [33] Munteanu, Pattern formation in noisy self-replicating spots, Int J Bifur Chaos 16 pp 3679– (2006) · Zbl 1113.92007 [34] Alonso, Stochastic amplification in epidemics, J R Soc Interface 4 pp 575– (2007) [35] Nguyen, Noise, nonlinearity and seasonality: the epidemics of whooping cough revisited, J R Soc Interface 5 pp 403– (2008) This reference list is based on information provided by the publisher or from digital mathematics libraries. 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