Spatial pattern in an epidemic system with cross-diffusion of the susceptible

Journal of Biological Systems

Volumn 17, Issue 1, 2009, Pages 141-152

  Sun, Gui Quan ^a   Jin, Zhen ^a   Liu, Quan Xing ^a   Li, Li ^a  

^a NORTH UNIVERSITY OF CHINA   (China)

Author keywords

Cross diffusion; Epidemic model; Pattern formation

Indexed keywords

EID: 65249093105     PISSN: 02183390     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218339009002843     Document Type: Article

References (30)

1. Shulgin B, Stone L, Aguar Z, Pulse vaccination strategy in the SIR epidemic model, Bull Math Biol 60:1123-1148, 1998.
2. Mena-Lorca J, Hechcote HW, Dynamic models of infectious diseases as regulatiors of population sizes, J Math Biol 30:693-716, 1969.
3. Keeling MJ, Woolhouse MEJ, Shaw DJ, Matthews L, Chase-Topping M, Haydon DT, Cornell SJ, Kappey J, Wilesmith J, Grenfell BT, Dynamics of the 2001 UK foot and mouth epidemic: Stochastic dispersal in a heterogeneous landscape, Science 294:813-817, 2001.
4. Smith DL, Lucey B, Waller LA, Childs JE, Real LA, Predicting the spatial dynamics of rabies epidemics on heterogeneous landscapes, Proc Natl Acad Sci USA 99:3668-3672, 2002.
5. Keeling MJ,Woolhouse MEJ, May RM, Davies G, Grenfell BT, Modelling vaccination strategies against foot-and-mouth disease, Nature 421:136-142, 2003.
6. Hufnagel L, Brockmann D, Geisel T, Forecast and control of epidemics in a globalized world, Proc Natl Acad Sci USA 101:15124-15129, 2004.
7. Pickett STA, Cadenasso ML, Landscape ecology: Spatial heterogeneity in ecological systems, Science 269:331-334, 1995.
8. Sun G, Jin Z, Liu Q-X, Li L, Pattern formation in a spatial S-I model with non-linear incidence rates, J Stat Mech-Theory E, P11011, 2007.
9. Ostfeld RS, Glass GE, Keesing F, Spatial epidemiology: An emerging (or re-emerging) discipline, Trends Ecol Evol 20:328-336, 2005.
10. Sun G-Q, Jin Z, Liu Q-X, Li L, Emergence of strange spatial pattern in a spatial epidemic model, Chinese Physic Lett 25:2296-2299, 2008.
11. Li L, Jin Z, Sun G-Q, Spatial pattern of an epidemic model with cross-diffusion, Chinese Physic Lett 25:3500-3503, 2008.
12. Liu Q-X, Jin Z, Formation of spatial patterns in an epidemic model with constant removal rate of the infectives, J Stat Mech-Theory E, P05002, 2007.
13. Cruickshank I, Gurney WSC, Veitch AR, The characteristics of epidemics and invasions with thresholds, Theor Pop Biol 56:279-292, 1999.
14. Chung JM, Peacock-Lopez E, Bifurcation diagrams and turing patterns in a chemical self-replicating reaction-diffusion system with cross-diffusion, J Chem Phys 127:174903, 2007.
15. Epstein IR, Pojman JA, Introduction to Nonlinear Chemical Dynamics, Oscillations, Waves, Patterns, and Chaos, Oxford, New York, 1998.
16. Gray P, Scott SK, Chemical Oscillations and Instabilities, Oxford, Oxford, 1990.
17. Edelstein-Keshet L, Mathematical Models in Biology, Random, New York, 1988.
18. Turing AM, The chemical basis of morphogenesis, Philos Trans R Soc London B 237:37-72, 1952.
19. Strogatz SH, Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, MA, 1994.
20. Sun G-Q, Jin Z, Liu Q-X, Li L, Chaos induced by breakup of waves in a spatial epidemic model with nonlinear incidence rate, J Stat Mech-Theory E, P08011, 2008.
21. Ouyang Q, Pattern Formation in a Reaction-Diffusion Systems, Scientific and Technological Education Publishing House, Shanghai (in Chinese), 2000.
22. Baurmann M, Gross T, Feudel U, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J Theor Biol 245:220-229, 2007.
23. Leppnen T, Phd thesis, Helsinki University of Technology, Finland, 2004.
24. He D, Stone L, Spatio-temporal synchronization of recurrent epidemics, Pro Roy Soc London 270:1519-1526, 2003.
25. Tilman D, Kareiva P, Spatial Ecology, Princeton University Press, 1997.
26. Rohani P, Earn DJD, Grenfell BT, Opposite patterns of synchrony in sympatric disease metapopulations, Science 286:968-971, 1999.
27. Dieckmann U, Law R, Metz J, The Geometry of Ecological Interactions, Cambridge University Press, 2000.
28. Grenfell BT, Bjornstad ON, Kappey J, Traveling waves and spatial hierarchies in measles epidemics, Nature 414:716-723, 2001.
29. Cummings DA, Irizarry RA, Huang NE, Endy TP, Nisalak A, Ungchusak K, Burke DS, Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand, Nature 427:344-347, 2004.
30. Vecchio A, Primavera L, Carbone V, Periodic and aperiodic traveling pulses in population dynamics: An example from the occurrence of epidemic infections, Phys Rev E 73:031913, 2006.