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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 852698, 10 pages
Stationary Patterns of a Cross-Diffusion Epidemic Model
1School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China
2College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China
3Department of Applied Mathematics, Shanghai Finance University, Shanghai 201209, China
4College of Physics and Electronic Information Engineering, Wenzhou University, Wenzhou 325035, China
Received 16 September 2013; Accepted 3 October 2013
Academic Editor: Carlo Bianca
Copyright © 2013 Yongli Cai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- W. O. Kermack and A. G. McKendrick, “Contributions to the mathematical theory of epidemics-I,” Bulletin of Mathematical Biology, vol. 53, no. 1-2, pp. 33–55, 1991.
- Z. Ma, Y. Zhou, and J. Wu, Modeling and Dynamics of Infectious Diseases, Higher Education Press, 2009.
- E. E. Holmes, M. A. Lewis, J. E. Banks, and R. R. Veit, “Partial differential equations in ecology: spatial interactions and population dynamics,” Ecology, vol. 75, no. 1, pp. 17–29, 1994.
- A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, vol. 14 of Interdisciplinary Applied Mathematics, Springer, New York, NY, USA, 2nd edition, 2001.
- C. Neuhauser, “Mathematical challenges in spatial ecology,” Notices of the American Mathematical Society, vol. 48, no. 11, pp. 1304–1314, 2001.
- B. T. Grenfell, O. N. Bjørnstad, and J. Kappey, “Travelling waves and spatial hierarchies in measles epidemics,” Nature, vol. 414, no. 6865, pp. 716–723, 2001.
- J. D. Murray, Mathematical Biology. II, vol. 18 of Interdisciplinary Applied Mathematics, Springer, New York, NY, USA, 3rd edition, 2003.
- L. Rass and J. Radcliffe, Spatial Deterministic Epidemics, vol. 102 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 2003.
- R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Chichester, UK, 2003.
- A. L. Lloyd and V. A. A. Jansen, “Spatiotemporal dynamics of epidemics: synchrony in metapopulation models,” Mathematical Biosciences, vol. 188, pp. 1–16, 2004.
- W. M. Van Ballegooijen and M. C. Boerlijst, “Emergent trade-offs and selection for outbreak frequency in spatial epidemics,” Proceedings of the National Academy of Sciences of the United States of America, vol. 101, no. 52, pp. 18246–18250, 2004.
- G. Mulone, B. Straughan, and W. Wang, “Stability of epidemic models with evolution,” Studies in Applied Mathematics, vol. 118, no. 2, pp. 117–132, 2007.
- H. Malchow, S. V. Petrovskii, and E. Venturino, Spatiotemporal Patterns in Ecology Andepidemiology-Theory, Models, and Simulation, Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2008.
- R. K. Upadhyay, N. Kumari, and V. S. H. Rao, “Modeling the spread of bird flu and predicting outbreak diversity,” Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1638–1648, 2008.
- K. Wang, W. Wang, and S. Song, “Dynamics of an HBV model with diffusion and delay,” Journal of Theoretical Biology, vol. 253, no. 1, pp. 36–44, 2008.
- G.-Q. Sun, Z. Jin, Q.-X. Liu, and L. Li, “Spatial pattern in an epidemic system with cross-diffusion of the susceptible,” Journal of Biological Systems, vol. 17, no. 1, pp. 141–152, 2009.
- M. Bendahmane and M. Saad, “Mathematical analysis and pattern formation for a partial immune system modeling the spread of an epidemic disease,” Acta Applicandae Mathematicae, vol. 115, no. 1, pp. 17–42, 2011.
- Y. Cai and W. Wang, “Spatiotemporal dynamics of a reaction-diffusion epidemic model with nonlinear incidence rate,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2011, no. 2, Article ID P02025, 2011.
- W. Wang, Y. Lin, H. Wang, H. Liu, and Y. Tan, “Pattern selection in an epidemic model with self and cross diffusion,” Journal of Biological Systems, vol. 19, no. 1, pp. 19–31, 2011.
- W. Wang, Y. Cai, M. Wu, K. Wang, and Z. Li, “Complex dynamics of a reaction-diffusion epidemic model,” Nonlinear Analysis: Real World Applications, vol. 13, no. 5, pp. 2240–2258, 2012.
- Y. Cai, W. Liu, Y. Wang, and W. Wang, “Complex dynamics of a diffusive epidemic model with strong Allee effect,” Nonlinear Analysis: Real World Applications, vol. 14, no. 4, pp. 1907–1920, 2013.
- Y. Lou and W.-M. Ni, “Diffusion, self-diffusion and cross-diffusion,” Journal of Differential Equations, vol. 131, no. 1, pp. 79–131, 1996.
- E. H. Kerner, “Further considerations on the statistical mechanics of biological associations,” The Bulletin of Mathematical Biophysics, vol. 21, no. 2, pp. 217–255, 1959.
- N. Shigesada, K. Kawasaki, and E. Teramoto, “Spatial segregation of interacting species,” Journal of Theoretical Biology, vol. 79, no. 1, pp. 83–99, 1979.
- M. Wang, “Non-constant positive steady states of the Sel'kov model,” Journal of Differential Equations, vol. 190, no. 2, pp. 600–620, 2003.
- P. Y. H. Pang and M. Wang, “Strategy and stationary pattern in a three-species predator-prey model,” Journal of Differential Equations, vol. 200, no. 2, pp. 245–273, 2004.
- M. Wang, “Stationary patterns of strongly coupled prey-predator models,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 484–505, 2004.
- M. Wang, “Stationary patterns caused by cross-diffusion for a three-species prey-predator model,” Computers & Mathematics with Applications, vol. 52, no. 5, pp. 707–720, 2006.
- R. Peng, J. Shi, and M. Wang, “Stationary pattern of a ratio-dependent food chain model with diffusion,” SIAM Journal on Applied Mathematics, vol. 67, no. 5, pp. 1479–1503, 2007.
- R. Peng, J. Shi, and M. Wang, “On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law,” Nonlinearity, vol. 21, no. 7, pp. 1471–1488, 2008.
- R. Peng and J. Shi, “Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: strong interaction case,” Journal of Differential Equations, vol. 247, no. 3, pp. 866–886, 2009.
- F. Yi, J. Wei, and J. Shi, “Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system,” Journal of Differential Equations, vol. 246, no. 5, pp. 1944–1977, 2009.
- B. Li and M. Wang, “Stationary patterns of the stage-structured predator-prey model with diffusion and cross-diffusion,” Mathematical and Computer Modelling, vol. 54, no. 5-6, pp. 1380–1393, 2011.
- S. A. Levin and L. A. Segel, “Pattern generation in space and aspect,” SIAM Review, vol. 27, no. 1, pp. 45–67, 1985.
- M. Barthélemy, A. Barrat, R. Pastor-Satorras, and A. Vespignani, “Dynamical patterns of epidemic outbreaks in complex heterogeneous networks,” Journal of Theoretical Biology, vol. 235, no. 2, pp. 275–288, 2005.
- V. Colizza and A. Vespignani, “Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: theory and simulations,” Journal of Theoretical Biology, vol. 251, no. 3, pp. 450–467, 2008.
- O. G. Jepps, C. Bianca, and L. Rondoni, “Onset of diffusive behavior in confined transport systems,” Chaos, vol. 18, no. 1, Article ID 013127, 2008.
- J. R. Cannon and D. J. Galiffa, “An epidemiology model suggested by yellow fever,” Mathematical Methods in the Applied Sciences, vol. 35, no. 2, pp. 196–206, 2012.
- K. P. Das and J. Chattopadhyay, “Role of environmental disturbance in an eco-epidemiological model with disease from external source,” Mathematical Methods in the Applied Sciences, vol. 35, no. 6, pp. 659–675, 2012.
- C. Bianca, “Existence of stationary solutions in kinetic models with gaussian thermostats,” Mathematical Methods in the Applied Sciences, vol. 36, no. 13, pp. 1768–1775, 2013.
- K. I. Kim, Z. Lin, and Q. Zhang, “An SIR epidemic model with free boundary,” Nonlinear Analysis: Real World Applications, vol. 14, no. 5, pp. 1992–2001, 2013.
- Z. Liu, “Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates,” Nonlinear Analysis: Real World Applications, vol. 14, no. 3, pp. 1286–1299, 2013.
- M. Qiao, A. Liu, and U. Foryś, “Qualitative analysis of the SICR epidemic model with impulsive vaccinations,” Mathematical Methods in the Applied Sciences, vol. 36, no. 6, pp. 695–706, 2013.
- L. Wang, Z. Teng, and H. Jiang, “Global attractivity of a discrete SIRS epidemic model with standard incidence rate,” Mathematical Methods in the Applied Sciences, vol. 36, no. 5, pp. 601–619, 2013.
- S. Wang, W. Liu, Z. Guo, and W. Wang, “Traveling wave solutions in a reaction-diffusion epidemic model,” Abstract and Applied Analysis, vol. 2013, Article ID 216913, 13 pages, 2013.
- A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.
- M. Kirane, “Global bounds and asymptotics for a system of reaction-diffusion equations,” Journal of Mathematical Analysis and Applications, vol. 138, no. 2, pp. 328–342, 1989.
- D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981.
- L. Melkemi, A. Z. Mokrane, and A. Youkana, “On the uniform boundedness of the solutions of systems of reaction-diffusion equations,” Electronic Journal of Qualitative Theory of Differential Equations, no. 24, pp. 1–10, 2005.
- E. H. Daddiouaissa, “Existence of global solutions for a system of reaction-diffusion equations having a triangular matrix,” Electronic Journal of Differential Equations, vol. 2008, no. 141, pp. 1–9, 2008.
- L. Nirenberg, Topics in Nonlinear Functional Analysis, vol. 6, AMS Bookstore, 2001.
- C.-S. Lin, W.-M. Ni, and I. Takagi, “Large amplitude stationary solutions to a chemotaxis system,” Journal of Differential Equations, vol. 72, no. 1, pp. 1–27, 1988.
- D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224, Springer, Berlin, Germany, 2nd edition, 1983.
- A. M. Turing, “The chemical basis of morphogenesis,” Philosophical Transactions of The Royal Society of London B, vol. 237, no. 641, pp. 37–72, 1952.