Global dynamics of a reaction and diffusion model for Lyme disease

Journal of Mathematical Biology

Volumn 65, Issue 4, 2012, Pages 787-808

  Zhao, Xiao Qiang ^a  

^a MEMORIAL UNIVERSITY OF NEWFOUNDLAND   (Canada)

Author keywords

Global stability; Spatial model for Lyme disease; Spreading speed; Threshold dynamics; Traveling waves

Indexed keywords

ARTICLE; BASIC REPRODUCTION NUMBER; BIOLOGICAL MODEL; BORRELIA BURGDORFERI; ECOSYSTEM; GROWTH, DEVELOPMENT AND AGING; HUMAN; LYME DISEASE; MATHEMATICAL COMPUTING; MICROBIOLOGY;

BASIC REPRODUCTION NUMBER; BORRELIA BURGDORFERI; ECOSYSTEM; HUMANS; LYME DISEASE; MODELS, BIOLOGICAL; NUMERICAL ANALYSIS, COMPUTER-ASSISTED;

EID: 84866058349     PISSN: 03036812     EISSN: 14321416     Source Type: Journal    
DOI: 10.1007/s00285-011-0482-9     Document Type: Article

References (26)

1. Barbour AG, Fish D (1993) The biological and social phenomenon of Lyme disease. Science 260: 1610-1616.
2. Caraco T, Gardner G, Maniatty W, Deelman E, Szymanski BK (1998) Lyme disease: self-regulation and pathogen invasion. J Theor Biol 193: 561-575.
3. Caraco T, Glavanakov S, Chen G, Flaherty JE, Ohsumi TK, Szymanski BK (2002) Stage-structured infection transmission and a spatial epidemic: A model for Lyme disease. Am Nat 160: 348-359.
4. Fang J, Wei J, Zhao X-Q (2008) Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system. J Differ Equ 245: 2749-2770.
5. Hale JK (1988) Asymptotic behavior of dissipative systems. American Mathematical Society, Providence.
6. Hsu S-B, Zhao X-Q (2008) Spreading speeds and traveling waves for nonmonotone integrodifference equations. SIAM J Math Anal 40: 776-789.
7. Li B, Weinberger HF, Lewis MA (2005) Spreading speeds as slowest wave speeds for cooperative systems. Math Biosci 196: 82-98.
8. Liang X, Zhao X-Q (2007) Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun Pure Appl Math 60: 1-40.
9. Liang X, Yi Y, Zhao X-Q (2006) Spreading speeds and traveling waves for periodic evolution systems. J Differ Equ 231: 57-77.
10. Lou Y, Zhao X-Q (2011) Modelling malaria control by introduction of larvivorous fish. Bull Math Biol. doi: 10. 1007/s11538-011-9628-6.
11. Lui R (1989) Biological growth and spread modeled by systems of recursions. I. Mathematical theory. Math Biosci 93: 269-295.
12. Martin RH, Smith HL (1990) Abstract functional differential equations and reaction-diffusion systems. Trans Am Math Soc 321: 1-44.
13. Mollison D (1991) Dependence of epidemic and population velocities on basic parameters. Math Biosci 107: 255-287.
14. Murray JD (2002) Mathematical biology, I: an introduction. Springer, New York.
15. Royden HL (2010) Real analysis, 4th edn. Prentice Hall, Englewood Cliffs.
16. Smith HL, Zhao X-Q (2001) Robust persistence for semidynamical systems. Nonlinear Anal 47: 6169-6179.
17. Thieme HR (1979) On a class of Hammerstein integral equations. Manuscr Math 29: 49-84.
18. Thieme HR (1992) Convergence results and Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations. J Math Biol 30: 755-763.
19. Thieme HR, Zhao X-Q (2001) A non-local delayed and diffusive predator-prey model. Nonlinear Anal RWA 2: 145-160.
20. van den Bosch F, Metz JAJ, Diekmann O (1990) The velocity of spatial population expansion. J Math Biol 28: 529-565.
21. Volpert AI, Volpert VA, Volpert VA (1994) Traveling wave solutions of parabolic systems. Translations of Mathematical Monographs, AMS.
22. Wang W, Zhao X-Q (2011) A nonlocal and time-delayed reaction-diffusion model of Dengue transmission. SIAM J Appl Math 71: 147-168.
23. Weinberger HF (1982) Long-time behavior of a class of biological models. SIAM J Math Anal 13: 353-396.
24. Weinberger HF, Lewis MA, Li B (2002) Analysis of linear determinacy for spread in cooperative models. J Math Biol 45: 183-218.
25. White DJ, Chang H-G, Benach JL, Bosler EM, Meldrum SC, Means RG, Debbie JG, Birkhead GS, Morse DL (1991) The Geographic spread and temporal increase of the Lyme disease epidemic. J Am Med Assoc 266: 1230-1236.
26. Zhao X-Q (2003) Dynamical systems in population biology. Springer, New York.