A Reaction-Diffusion Lyme Disease Model with Seasonality

This paper is devoted to the study of a reaction-diffusion Lyme disease model with seasonality. In the case of a bounded habitat, we obtain a threshold result on the global stability of either disease-free or endemic periodic solution. In the case of an unbounded habitat, we establish the existence of the disease spreading speed and its coincidence with the minimal wave speed for time-periodic traveling wave solutions. We also estimate parameter values based on some published data and use them to study the Lyme disease transmission in Port Dove, Ontario. Our numerical simulations are consistent with the obtained analytic results.

[1]  Xiao-Qiang Zhao,et al.  Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments , 2008 .

[2]  Xiao-Qiang Zhao,et al.  The periodic Ross–Macdonald model with diffusion and advection , 2010 .

[3]  Nicolas Bacaër,et al.  The epidemic threshold of vector-borne diseases with seasonality , 2006, Journal of mathematical biology.

[4]  Xiao-Qiang Zhao,et al.  Spreading speeds and traveling waves for periodic evolution systems , 2006 .

[5]  Xiao-Qiang Zhao,et al.  Chain Transitivity, Attractivity, and Strong Repellors for Semidynamical Systems , 2001 .

[6]  M Bigras-Poulin,et al.  A dynamic population model to investigate effects of climate on geographic range and seasonality of the tick Ixodes scapularis. , 2005, International journal for parasitology.

[7]  Xiao-Qiang Zhao,et al.  Spatial dynamics of a nonlocal and time-delayed reaction–diffusion system , 2008 .

[8]  Xiao-Qiang Zhao,et al.  Spreading speeds and traveling waves for abstract monostable evolution systems , 2010 .

[9]  Xiao-Qiang Zhao,et al.  A Climate-Based Malaria Transmission Model with Structured Vector Population , 2010, SIAM J. Appl. Math..

[10]  M Bigras-Poulin,et al.  Vector seasonality, host infection dynamics and fitness of pathogens transmitted by the tick Ixodes scapularis , 2006, Parasitology.

[11]  B K Szymanski,et al.  Lyme disease: self-regulation and pathogen invasion. , 1998, Journal of theoretical biology.

[12]  John S Brownstein,et al.  A Dispersal Model for the Range Expansion of Blacklegged Tick (Acari: Ixodidae) , 2004, Journal of medical entomology.

[13]  S. Randolph,et al.  Epidemiological uses of a population model for the tick Rhipicephalus appendiculatus , 1999, Tropical medicine & international health : TM & IH.

[14]  Xiao-Qiang Zhao,et al.  Global dynamics of a reaction and diffusion model for Lyme disease , 2011, Journal of Mathematical Biology.

[15]  Thomas Caraco,et al.  Stage‐Structured Infection Transmission and a Spatial Epidemic: A Model for Lyme Disease , 2002, The American Naturalist.

[16]  Xiao-Qiang Zhao,et al.  Asymptotic speeds of spread and traveling waves for monotone semiflows with applications , 2007 .

[17]  Xiao-Qiang Zhao,et al.  Dynamical systems in population biology , 2003 .

[18]  Xiao-Qiang Zhao,et al.  A periodic epidemic model in a patchy environment , 2007 .

[19]  D. Waltner-Toews,et al.  Investigation of Relationships Between Temperature and Developmental Rates of Tick Ixodes scapularis (Acari: Ixodidae) in the Laboratory and Field , 2004, Journal of medical entomology.

[20]  P. Hosseini,et al.  Seasonality and the dynamics of infectious diseases. , 2006, Ecology letters.

[21]  Hal L. Smith,et al.  Abstract functional-differential equations and reaction-diffusion systems , 1990 .

[22]  Vitaly Volpert,et al.  Traveling Wave Solutions of Parabolic Systems , 1994 .