Stability of steady states and existence of travelling waves in a vector-disease model

In this paper, a host-vector model is considered for a disease without immunity in which the current density of infectious vectors is related to the number of infectious hosts at earlier times. Spatial spread in a region is modelled in the partial integro-differential equation by a diffusion term. For the general model, we first study the stability of the steady states using the contracting-convex-sets technique. When the spatial variable is one dimensional and the delay kernel assumes some special form, we establish the existence of travelling wave solutions by using the linear chain trick and the geometric singular perturbation method.

[1]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[2]  G. Reuter,et al.  Deterministic epidemic waves , 1976, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  J. Radcliffe.  The initial geographical spread of host-vector and carrier-borne epidemics , 1973, Journal of Applied Probability.

[4]  J. Carr,et al.  Deterministic epidemic waves of critical velocity , 1977, Mathematical Proceedings of the Cambridge Philosophical Society.

[5]  R. Volz,et al.  Global asymptotic stability of a periodic solution to an epidemic model , 1982, Journal of mathematical biology.

[6]  M. Assunta Pozio.  Some conditions for global asymptotic stability of equilibria of integrodifferential equations , 1983 .

[7]  D. Griffiths.  A bivariate birth-death process which approximates to the spread of a disease involving a vector , 1972, Journal of Applied Probability.

[8]  J. Medlock,et al.  Spreading disease: integro-differential equations old and new. , 2003, Mathematical biosciences.

[9]  Kenneth L. Cooke,et al.  Stability analysis for a vector disease model , 1979 .

[10]  O. Diekmann,et al.  Thresholds and travelling waves for the geographical spread of infection , 1978, Journal of mathematical biology.

[11]  Linda Rass,et al.  Spatial deterministic epidemics , 2003 .

[12]  Jianhong Wu.  Theory and Applications of Partial Functional Differential Equations , 1996 .

[13]  M. Pozio,et al.  Behaviour of solutions of some abstract functional differential equations and application to predator—prey dynamics☆ , 1980 .

[14]  Xiao-Qiang Zhao,et al.  Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models , 2003 .

[15]  Horst R. Thieme,et al.  Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations. , 1979 .

[16]  S. Ruan.  REACTION-DIFFUSION EQUATIONS WITH INFINITE DELAY , 2005 .

[17]  H. Thieme.  A model for the spatial spread of an epidemic , 1977, Journal of mathematical biology.

[18]  P. Marcati,et al.  Global asymptotic stability for a vector disease model with spatial spread , 1980, Journal of mathematical biology.

[19]  S. A. Gourley.  Travelling fronts in the diffusive Nicholson's blowflies equation with distributed delays , 2000 .

[20]  Mark Bartlett,et al.  Deterministic and Stochastic Models for Recurrent Epidemics , 1956 .

[21]  Stephen A. Gourley,et al.  Travelling fronts for the KPP equation with spatio-temporal delay , 2002 .

[22]  Stavros Busenberg,et al.  Periodic Solutions of a Periodic Nonlinear Delay Differential Equation , 1978 .

[23]  W. D. Stirling,et al.  Wave solutions for the deterministic host-vector epidemic , 1982, Mathematical Proceedings of the Cambridge Philosophical Society.

[24]  L. P. Lounibos,et al.  Invasions by insect vectors of human disease. , 2002, Annual review of entomology.

[25]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[26]  D. Aronson,et al.  Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation , 1975 .

[27]  Mark A. J. Chaplain,et al.  Travelling fronts in a food-limited population model with time delay , 2002, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[28]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .

[29]  A. Barbour.  The uniqueness of Atkinson and Reuter's epidemic waves , 1977, Mathematical Proceedings of the Cambridge Philosophical Society.