An Epidemic Model for Tick-Borne Disease with Two Delays

We have considered an epidemic model of a tick-borne infection which has nonviraemic transmission in addition to the viremic transmission. The basic reproduction number , which is a threshold quantity for stability of equilibria, is calculated. If , then the infection-free equilibrium is globally asymptotically stable, and this is the only equilibrium. On the contrary, if , then an infection equilibrium appears which is globally asymptotically stable, when one time delay is absent. By applying a permanence theorem for infinite dimensional systems, we obtain that the disease is always present when .

[1]  O. Diekmann,et al.  On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations , 1990, Journal of mathematical biology.

[2]  P. Nuttall,et al.  Efficient transmission of tick-borne encephalitis virus between cofeeding ticks. , 1993, Journal of medical entomology.

[3]  Philip K Maini,et al.  A lyapunov function and global properties for sir and seir epidemiological models with nonlinear incidence. , 2004, Mathematical biosciences and engineering : MBE.

[4]  Andrei Korobeinikov,et al.  Global Properties of Infectious Disease Models with Nonlinear Incidence , 2007, Bulletin of mathematical biology.

[5]  P. Hudson,et al.  The role of non-viraemic transmission on the persistence and dynamics of a tick borne virus – Louping ill in red grouse ( Lagopus lagopus scoticus ) and mountain hares ( Lepus timidus ) , 2004 .

[6]  P. Hudson,et al.  Disease persistence and apparent competition in a three-host community: an empirical and analytical study of large-scale, wild populations , 2001 .

[7]  R. Levins,et al.  The role of seasonality in the dynamics of deer tick populations , 2005, Bulletin of mathematical biology.

[8]  E. Gould,et al.  Persistence and transmission of tick-borne viruses: Ixodes ricinus and louping-ill virus in red grouse populations , 1995, Parasitology.

[9]  K. Hadeler,et al.  Ticks and tick-borne diseases: a vector-host interaction model for the brown ear tick (Rhipicephalus appendiculatus) , 2000, Statistical methods in medical research.

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

[11]  Xinyu Song,et al.  GLOBAL STABILITY OF A STAGE-STRUCTURED PREDATOR-PREY SYSTEM , 2008 .

[12]  Andrei Korobeinikov,et al.  Lyapunov functions and global properties for SEIR and SEIS epidemic models. , 2004, Mathematical medicine and biology : a journal of the IMA.

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

[14]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[15]  Horst R. Thieme,et al.  Persistence under relaxed point-dissipativity (with application to an endemic model) , 1993 .

[16]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[17]  E. Gould,et al.  Ticks need not bite their red grouse hosts to infect them with louping ill virus , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[18]  M. Begon,et al.  Persistence of tick-borne virus in the presence of multiple host species: tick reservoirs and parasite mediated competition. , 1999, Journal of theoretical biology.

[19]  Lansun Chen,et al.  A MODEL FOR TWO SPECIES WITH STAGE STRUCTURE AND FEEDBACK CONTROL , 2008 .

[20]  P. Hudson,et al.  Thresholds for disease persistence in models for tick-borne infections including non-viraemic transmission, extended feeding and tick aggregation. , 2003, Journal of theoretical biology.

[21]  Xiao-Qiang Zhao,et al.  Robust persistence for semidynamical systems , 2001 .

[22]  Yasuhiro Takeuchi,et al.  Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate , 2010, Bulletin of mathematical biology.

[23]  Y. Kuang.  Delay Differential Equations: With Applications in Population Dynamics , 2012 .

[24]  C. Connell McCluskey,et al.  Complete global stability for an SIR epidemic model with delay — Distributed or discrete , 2010 .

[25]  A. Spielman,et al.  A comprehensive multiple matrix model representing the life cycle of the tick that transmits the agent of Lyme disease. , 1992, Journal of theoretical biology.

[26]  W. Marsden.  I and J , 2012 .