Phase transition in spatial epidemics using cellular automata with noise

Ecological Research

Volumn 26, Issue 2, 2011, Pages 333-340

  Sun, Gui Quan ^a,b   Jin, Zhen ^a   Song, Li Peng ^a   Chakraborty, Amit ^b   Li, Bai Lian ^a,b  

^a NORTH UNIVERSITY OF CHINA   (China)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Cellular automata; Coherence resonance; Epidemiology; Extinction; Noise

Indexed keywords

CELLULAR AUTOMATON; EPIDEMIC; EPIDEMIOLOGY; NOISE; PHASE TRANSITION; RESONANCE; SPATIOTEMPORAL ANALYSIS; STOCHASTICITY;

EID: 79952753894     PISSN: 09123814     EISSN: 14401703     Source Type: Journal    
DOI: 10.1007/s11284-010-0789-9     Document Type: Article

References (71)

1. Alonso D, McKane AJ, Pascual M (2007) Stochastic amplification in epidemics. J R Soc Interface 4: 575-582.
2. Anderson RM, May RM (1992) Infectious disease of humans: dynamics and control. Oxford University Press, Oxford.
3. Bartlett MS (1957) Measles periodicity and community size. J R Stat Soc Ser A 120: 48-70.
4. Bartlett MS (1960) The critical community size for measles in the United States. J R Stat Soc Ser A 123: 37-44.
5. Bauch CT, Earn DJD (2003a) Interepidemic intervals in forced and unforced seir models. Fields Inst Commun 36: 33-44.
6. Bauch CT, Earn DJD (2003b) Transients and attractors in epidemics. Proc R Soc B 270: 1573-1578.
7. Blasius B, Huppert A, Stone L (1999) Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399: 354-355.
8. Boland RP, Galla T, McKane AJ (2008) How limit cycles and quasi-cycles are related in systems with intrinsic noise. J Stat Mech P09001.
9. Castillo-Chavez C, Yakubu A-A (2002) Mathematical approaches for emerging and re-emerging infectious diseases: an introduction. Springer, Berlin.
10. Dieckmann U, Law R, Metz JAJ (2000) Geometry of ecological interactions simplifying spatial complexity. Cambridge University Press, Cambridge.
11. Durrett R, Levin SA (1994a) The importance of being discrete (and spatial). Theor Popul Biol 46: 363-394.
12. Durrett R, Levin SA (1994b) Stochastic spatial models-a user guide to ecological applications. Philos Trans R Soc Lond B 343: 329-350.
13. Earn DJD, Rohani P, Bolker BM, Grenfell BT (2000) A simple model for complex dynamical transitions in epidemics. Science 287: 667-670.
14. Foley P (1994) Predicting extinction times from environmental stochasticity and carrying capacity. Conserv Biol 8: 124-137.
15. Gammaitoni L, Häggi P, Jung P, Marchesoni F (1998) Stochastic resonance. Rev Mod Phys 70: 223-287.
16. Grenfell BT, Bjønstad ON, Kappey J (2001) Traveling waves and spatial hierarchies in measles epidemics. Nature 414: 716-723.
17. He D, Stone L (2003) Spatio-temporal synchronization of recurrent epidemics. Proc R Soc B 270: 1519-1526.
18. Hilker FM, Malchow H, Langlais M, Petrovskii SV (2006) Oscillations and waves in a virally infected plankton system. Part II: Transition from lysogeny to lysis. Ecol Complex 3: 200-208.
19. Johst K, Wissel C (1997) Extinction risk in a temporally correlated fluctuating environment. Theor Popul Biol 52: 91-100.
20. Keeling MJ, Grenfell BT (1997) Disease extinction and community size: modeling the persistence of measles. Science 275: 65-67.
21. Keeling MJ, Rohani P, Grenfell BT (2001a) Seasonally forced disease dynamics explained by switching between attractors. Phys D 148: 317-335.
22. Keeling MJ, Woolhouse MEJ, May RM, Davies G, Grenfell BT (2003) Modelling vaccination strategies against foot-and-mouth disease. Nature 421: 136-142.
23. Keeling MJ, Woolhouse MEJ, Shaw DJ, Matthews L, Chase-Topping M, Haydon DT, Cornell SJ, Kappey J, Wilesmith J, Grenfell BT (2001b) UK foot and mouth epidemic: stochastic dispersal in a heterogeneous landscape. Science 294: 813-817.
24. Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc B 115: 700-721.
25. Lee S, Neiman A, Kim S (1998) Coherence resonance in a hodgkin-huxley neuron. Phys Rev E 57 (3): 3292-3297.
26. Lindner B, Schimansky-Geier L (1999) Analytical approach to the stochastic Fitzhugh-Nagumo system and coherence resonance. Phys Rev E 60 (6): 7270-7276.
27. Lion S, van Baalen M (2008) Self-structuring in spatial evolutionary ecology. Ecol Lett 11: 277-285.
28. Liu Q-X, Wang R-H, Jin Z (2009) Persistence, extinction and spatio-temporal synchronization of SIRS spatial models. J Stat Mech P07007.
29. MacArther RH (1958) Population ecology of some warblers of northeastern coniferous forests. Ecology 39(4): 599-619.
30. Malchow H, Petrovskii SV, Venturino E (2008) Spatiotemporal patterns in ecology and epidemiology: theory, models, and simulations. Chapman and Hall, CRC, London.
31. Malchowa H, Hilker FM, Petrovskii SV, Brauer K (2004) Oscillations and waves in a virally infected plankton system. Part I: The lysogenic stage. Ecol Complex 1: 211-223.
32. Mode CJ, Jacobsen ME (1987) A study of the impact of environmental stochasticity on extinction probabilities by Monte Carlo integration. Math Biosci 83: 105-125.
33. Murray JD (1993) Mathematical biology. Springer, Berlin.
34. Nasell I (1999) On the quasi-stationary distribution of the stochastic logistic epidemic. Math Biosci 156: 21-40.
35. Nasell I (2001) Extinction and quasi-stationarity in the verhulst logistic model. J Theor Biol 211: 11-27.
36. Neiman A, Saparin P, Stone L (1997) Coherence resonance at noisy precursors of bifurcations in nonlinear dynamical systems. Phys Rev E 56 (1): 270-273.
37. Neiman A, Silchenko A, Anishchenko V, Schimansky-Geier L (1998) Stochastic resonance: noise-enhanced phase coherence. Phys Rev E 58 (6): 7118-7125.
38. Nguyen HTH, Rohani P (2008) Noise, nonlinearity and seasonality: the epidemics of whooping cough revisited. J R Soc Interface 5: 403-413.
39. Ostfeld RS, Glass GE, Keesing F (2005) Spatial epidemiology: an emerging (or re-emerging) discipline. Trends Ecol Evol 20: 328-336.
40. Perc M (2005a) Noise-induced spatial periodicity in excitable chemical media. Chem Phys Lett 410: 49-53.
41. Perc M (2005b) Persistency of noise-induced spatial periodicity in excitable media. Europhys Lett 72: 712-718.
42. Perc M (2005c) Spatial coherence resonance in excitable media. Phys Rev E 72: 016207.
43. Perc M (2005d) Spatial decoherence induced by small-world connectivity in excitable media. New J Phys 7: 252.
44. Perc M (2006) Double resonance in cooperation induced by noise and network variation for an evolutionary prisoner's dilemma. New J Phys 8: 183.
45. Perc M (2007) Transition from gaussian to levy distributions of stochastic payoff variations in the spatial prisoner's dilemma game. Phys Rev E 75: 022101.
46. Perc M (2008) Coherence resonance in a spatial prisoner's dilemma game. New J Phys 8: 22.
47. Perc M, Szolnoki A (2007) Noise-guided evolution within cyclical interactions. New J Phys 9: 267.
48. Perc M, Szolnoki A (2008) Social diversity and promotion of cooperation in the spatial prisoner's dilemma game. Phys Rev E 77: 011904.
49. Perc M, Szolnoki A (2010) Coevolutionary games: a mini review. BioSystems 99: 109-125.
50. Perc M, Szolnoki A, Szabo G (2007) Cyclical interactions with alliance-specific heterogeneous invasion rates. Phys Rev E 75: 052102.
51. Petrovskii S, Li B-L, Malchow H (2004) Transition to spatiotemporal chaos can resolve the paradox of enrichment. Ecol Complex 1: 37-47.
52. Pikovsky AS, Kurths J (1997) Coherence resonance in a noise-driven excitable system. Phys Rev Lett 78: 775-778.
53. Rand DA, Wilson HB (1991) Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics. Proc R Soc B 246: 179-184.
54. Reichenbach T, Mobilia M, Frey E (2008) Self-organization of mobile populations in cyclic competition. J Theor Biol 254: 368-383.
55. Ripa J, Heino M (1999) Linear analysis solves two puzzles in population dynamics: the route to extinction and extinction in coloured environments. Ecol Lett 2: 219-222.
56. Rohani P, Earn DJ, Grenfell BT (1999) Opposite patterns of synchrony in sympatric disease metapopulations. Science 286: 968-971.
57. Rohani P, Keeling MJ, Grenfell BT (2002) The interplay between noise and determinism in childhood diseases. Am Nat 159: 469-481.
58. Roughgarden J (1975) A simple model for population dynamics in stochastic environments. Am Nat 109: 713-736.
59. Roy M, Pascual M (2006) On representing network heterogeneities in the incidence rate of simple epidemic models. Ecol Complex 3: 80-90.
60. Shirley MDF, Rushton SP (2005) The impacts of network topology on disease spread. Ecol Complex 2: 287-299.
61. Sidorowich JJ (1992) Repellors attract attention. Nature 355: 584-585.
62. Simöes M, da Gama MMT, Nunes A (2007) Stochastic fluctuations in epidemics on networks. J R Soc Interface 5: 555-566.
63. Smith DL, Lucey B, Waller LA, Childs JE, Real LA (2002) Predicting the spatial dynamics of rabies epidemics on heterogeneous landscapes. Proc Natl Acad Sci USA 99: 3668-3672.
64. Sun G-Q, Jin Z, Liu Q-X, Li L (2008) Chaos induced by breakup of waves in a spatial epidemic model with nonlinear incidence rate. J Stat Mech Theory E, P0811.
65. Sun G-Q, Li L, Jin Z, Li B-L (2010a) Effect of noise on the pattern formation in an epidemic model. Numer Methods Partial Differ Equ 26: 1168-1179.
66. Sun G-Q, Liu Q-X, Jin Z, Chakraborty A, Li B-L (2010b) Inuence of infection rate and migration on extinction of disease in spatial epidemics. J Theor Biol 264: 95-103.
67. van Ballegooijen WM, Boerlijst MC (2004) Emergent trade-offs and selection for outbreak frequency in spatial epidemics. Proc Natl Acad Sci USA 101: 18246-18250.
68. Viboud C, Bjornstad ON, Smith DL, Simonsen L, Miller MA, Grenfell BT (2006) Synchrony, waves, and spatial hierarchies in the spread of influenza. Science 312: 447-451.
69. Wichmann MC, Jeltsch F, Dean W, Moloney K, Wissel C (2003a) Implications of climate change for the persistence of raptors in arid savanna. Oikos 102: 186-202.
70. Wichmann MC, Johst K, Moloney KA, Wissel C, Jeltsch F (2003b) Extinction risk in periodically fluctuating environments. Ecol Model 167: 221-231.
71. Zhou C, Kurths J, Hu B (2001) Array-enhanced coherence resonance: nontrivial effects of heterogeneity and spatial independence of noise. Phys Rev Lett 87(9): 098101.