×

zbMATH — the first resource for mathematics

Turing patterns in a diffusive epidemic model with saturated infection force. (English) Zbl 1398.92259
Summary: In this paper, we investigate the complex dynamics of a reaction-diffusion epidemic model with a saturated infection force analytically and numerically. We give the stability of the constant positive steady-states and the nonexistence/existence of nonconstant positive steady-states of the model which shows that, if the diffusion coefficients are properly chosen, the model exhibits stationary Turing pattern as a result of diffusion. Via numerical simulations, we present the evolutionary processes that involve organism distribution and the interaction of spatially distributed infection with local diffusion, and find that the model dynamics exhibits a diffusion-controlled formation growth of holes, stripes and spots pattern replication. In the viewpoint of epidemiology, we must regulate and control the parameters in the special range to control the disease.

MSC:
92D30 Epidemiology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Organization, The top 10 causes of death, 2017, http://www.who.int/mediacentre/factsheets/fs310/en/.
[2] Artalejo, J. R.; Economou, A.; Lopezherrero, M. J., Stochastic epidemic models revisited: analysis of some continuous performance measures, J. Biol. Dyn., 6, 2, 189, (2012)
[3] Cai, Y.; Kang, Y.; Banerjee, M.; Wang, W. M., A stochastic epidemic model incorporating media coverage, Commun. Math. Sci., 14, 4, 893-910, (2016) · Zbl 1344.92155
[4] Britton, T.; Lindenstrand, D., Epidemic modelling: aspects where stochasticity matters, Math. Biosci., 222, 2, 109-116, (2009) · Zbl 1178.92046
[5] Hethcote, H. W., The mathematics of infectious diseases, SIAM Rev., 42, 4, 599-653, (2000) · Zbl 0993.92033
[6] Li, M. Y.; Smith, H. L.; Wang, L., Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., 62, 1, 58-69, (2013) · Zbl 0991.92029
[7] Ruan, S.; Wang, W. D., Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Eq., 188, 1, 135-163, (2003) · Zbl 1028.34046
[8] Ma, Z.; Zhou, Y.; Wu, J., Modeling and Dynamics of Infectious Diseases, (2009), Higher education Press; World Scientific · Zbl 1180.92081
[9] Cai, Y.; Wang, W. M., Fish-hook bifurcation branch in a spatial heterogeneous epidemic model with cross-diffusion, Nonlinear Anal. RWA, 30, 99-125, (2016) · Zbl 1339.35329
[10] Wang, W. M.; Cai, Y.; Li, J.; Gui, Z., Periodic behavior in a FIV model with seasonality as well as environment fluctuations, J. Franklin Inst., 354, 16, 7410-7428, (2017) · Zbl 1373.93315
[11] Cai, Y.; Kang, Y.; Wang, W. M., A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput., 305, 221-240, (2017)
[12] Cai, Y.; Kang, Y.; Banerjee, M.; Wang, W. M., Complex dynamics of a host-parasite model with both horizontal and vertical transmissions in a spatial heterogeneous environment, Nonlinear Anal. RWA, 40, 444-465, (2018) · Zbl 1379.92056
[13] Cai, Y.; Jiao, J.; Gui, Z.; Liu, Y.; Wang, W. M., Environmental variability in a stochastic epidemic model, Appl. Math. Comp., 329, 210-226, (2018)
[14] Wang, W. M.; Cai, Y.; Ding, Z.; Gui, Z., A stochastic differential equation SIS epidemic model incorporating Ornstein-Uhlenbeck process, Phys. A, 509, 921-936, (2018)
[15] Guo, W.; Cai, Y.; Zhang, Q.; Wang, W. M., Stochastic persistence and stationary distribution in an sis epidemic model with media coverage, Phys. A, 492, 2220-2236, (2018)
[16] Wang, W. D., Epidemic models with nonlinear infection forces, Math. Biosci. Eng., 3, 1, 267-279, (2006) · Zbl 1089.92052
[17] Zhu, H.; Campbell, S. A., Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63, 2, 636-682, (2002) · Zbl 1036.34049
[18] Ruan, S. G.; Wang, W. D., Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Eq., 188, 1, 135-163, (2001)
[19] Neuhauser, C., Mathematical challenges in spatial ecology, Not. AMS, 48, 11, (2001) · Zbl 1128.92328
[20] Hosono, Y.; Ilyas, B., Traveling wave solutions for a simple diffusive epidemic model, Math. Model. Meth. Appl. Sci., 05, 935-966, (1995) · Zbl 0836.92023
[21] Cruickshank, I.; Gurney, W. S.C.; Veitch, A. R., The characteristics of epidemics and invasions with thresholds, Theor. Popu. Biol., 56, 3, 279-292, (1999) · Zbl 0963.92034
[22] Turechek, W. W.; Madden, L. V., Spatial pattern analysis of strawberry leaf blight in perennial production systems, Phytopathology, 89, 5, 421, (1999)
[23] Peng, R.; Shi, J.; Wang, M., Stationary pattern of a ratio-dependent food chain model with diffusion, SIAM J. Appl. Math., 67, 5, 1479-1503, (2007) · Zbl 1210.35268
[24] Sun, G.-Q.; Jin, Z.; Liu, Q.-X.; Li, L., Spatial pattern in an epidemic system with cross-diffusion of the susceptible, J. Biol. Sys., 17, 1, 141-152, (2009) · Zbl 1364.92053
[25] Cai, Y.; Wang, W. M., Spatiotemporal dynamics of a reaction-diffusion epidemic model with nonlinear incidence rate, J. Stat. Mech. Theor. Exp., 2011, 2, (2011)
[26] Wang, W. M.; Lin, Y.; Wang, H.; Liu, H.; Tan, Y., Pattern selection in an epidemic model with self and cross diffusion, J. Biol. Sys., 19, 01, 19-31, (2011)
[27] Wang, W. M.; Cai, Y.; Wu, M.; Wang, K.; Li, Z., Complex dynamics of a reaction-diffusion epidemic model, Nonlinear Anal. RWA, 13, 5, 2240-2258, (2012) · Zbl 1327.92069
[28] Cai, Y.; Liu, W.; Wang, Y.; Wang, W. M., Complex dynamics of a diffusive epidemic model with strong allee effect, Nonlinear Anal. RWA, 14, 4, 1907-1920, (2013) · Zbl 1274.92009
[29] Cai, Y.; Yan, S.; Wang, H.; Lian, X.; Wang, W. M., Spatiotemporal dynamics in a reaction-diffusion epidemic model with a time-delay in transmission, Inter. J. Bifurc. Chaos, 25, 08, 1550099, (2015) · Zbl 1321.35005
[30] Turing, A. M., The chemical basis of morphogenesis, Bull. Math. Biol., 1953, 1-2, 153-197, (1990)
[31] Britton, N. F., Essential Mathematical Biology, (2009), Tsinghua University Press, Springer
[32] Holmes, E. E.; Lewis, M. A.; Banks, J. E.; Veit, R. R., Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75, 1, 17-29, (1994)
[33] Ruan, S. G.; Wang, W. D., Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Eq., 188, 1, 135-163, (2003) · Zbl 1028.34046
[34] Wu, Z.; Yin, J.; Wang, C., Elliptic parabolic equations, Proc. Math. Stat., 119, 1, 101-114, (2015)
[35] Henry, D., Geometric Theory of Semilinear Parabolic Equations, (1981), Springer-Verlag · Zbl 0456.35001
[36] Lou, Y.; Ni, W. M., Diffusion, self-diffusion cross-diffusion, J. Differ. Eq., 131, 1, 79-131, (1996) · Zbl 0867.35032
[37] Lin, C. S.; Ni, W. M.; Takagi, I., Large amplitude stationary solutions to a chemotaxis system, J. Differ. Eq., 72, 1, 1-27, (1988) · Zbl 0676.35030
[38] Shi, H.; Ruan, S., Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA J. Appl. Math., 80, 1534-1568, (2015) · Zbl 1327.35376
[39] Garvie, M. R., Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in m atlab, Bull. Math. Biol., 69, 3, 931, (2007) · Zbl 1298.92081
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.