Journal of Mathematical Biology

, Volume 75, Issue 6–7, pp 1381–1409 | Cite as

Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary

  • Zhigui Lin
  • Huaiping ZhuEmail author


In this paper, a reaction–diffusion system is proposed to model the spatial spreading of West Nile virus in vector mosquitoes and host birds in North America. Transmission dynamics are based on a simplified model involving mosquitoes and birds, and the free boundary is introduced to model and explore the expanding front of the infected region. The spatial-temporal risk index \(R_0^F(t)\), which involves regional characteristic and time, is defined for the simplified reaction–diffusion model with the free boundary to compare with other related threshold values, including the usual basic reproduction number \(R_0\). Sufficient conditions for the virus to vanish or to spread are given. Our results suggest that the virus will be in a scenario of vanishing if \(R_0\le 1\), and will spread to the whole region if \(R_{0}^F(t_0)\ge 1\) for some \(t_0\ge 0\), while if \(R^F_0(0)<1<R_0\), the spreading or vanishing of the virus depends on the initial number of infected individuals, the area of the infected region, the diffusion rate and other factors. Moreover, some remarks on the basic reproduction numbers and the spreading speeds are presented and compared.


West Nile virus Vector mosquitoes Host birds Spatial spreading Reaction–diffusion systems Free boundary The basic reproduction number Risk index Spreading speeds 

Mathematics Subject Classification

Primary 35K55 35R35 Secondary 35B40 92D30 


  1. Abdelrazec A, Lenhart S, Zhu H (2014) Transmission dynamics of West Nile virus in mosquitoes and corvids and non-corvids. J Math Biol 68(6):1553–1582MathSciNetCrossRefzbMATHGoogle Scholar
  2. Allen LJS, Bolker BM, Lou Y, Nevai AL (2008) Asymptotic profiles of the steady states for an SIS epidemic reaction–diffusion model. Discrete Contin Dyn Syst Ser A 21:1–20MathSciNetCrossRefzbMATHGoogle Scholar
  3. Álvarez-Caudevilla P, López-Gómez J (2008) Asymptotic behaviour of principal eigenvalues for a class of cooperative systems. J Differ Equ 244(5):1093–1113MathSciNetCrossRefzbMATHGoogle Scholar
  4. Aronson DG, Weinberger HF (1975) Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Partial differential equations and related topics, Lecture Notes in Math., vol 446. Springer, Berlin, pp 5–49Google Scholar
  5. Bowman C, Gumel AB, Wu J, van den Driessche P, Zhu H (2005) A mathematical model for assessing control strategies against West Nile virus. Bull Math Biol 67(5):1107–1133MathSciNetCrossRefzbMATHGoogle Scholar
  6. CDC (2013) West Nile virus disease and other arboviral diseases in United States, 2012. MMWR 62:513–517Google Scholar
  7. Chen XF, Friedman A (2000) A free boundary problem arising in a model of wound healing. SIAM J Math Anal 32:778–800MathSciNetCrossRefzbMATHGoogle Scholar
  8. Cruz-Pacheco G, Esteva L, Montaño-Hirose JA, Vargas C (2005) Modelling the dynamics of West Nile virus. Bull Math Biol 67(6):1157–1172MathSciNetCrossRefzbMATHGoogle Scholar
  9. Dohm DJ, Sardelis MR, Turell MJ (2002) Experimental vertical transmission of West Nile virus by Culex pipiens (Diptera: Culicidae). J Med Entomol 39:640–644CrossRefGoogle Scholar
  10. Du YH, Guo ZM (2011) Spreading–vanishing dichotomy in the diffusive logistic model with a free boundary II. J Differ Equ 250:4336–4366MathSciNetCrossRefzbMATHGoogle Scholar
  11. Du YH, Lin ZG (2010) Spreading–vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J Math Anal 42:377–405MathSciNetCrossRefzbMATHGoogle Scholar
  12. Du YH, Lin ZG (2014) The diffusive competition model with a free boundary: invasion of a superior or inferior competitor. Discrete Contin Dyn Syst Ser B 19:3105–3132MathSciNetCrossRefzbMATHGoogle Scholar
  13. Du YH, Lou BD (2015) Spreading and vanishing in nonlinear diffusion problems with free boundaries. J Eur Math Soc 17:2673–2724MathSciNetCrossRefzbMATHGoogle Scholar
  14. Fila M, Souplet P (2001) Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem. Interfaces Free Bound 3:337–344MathSciNetzbMATHGoogle Scholar
  15. Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugen 7:335–369zbMATHGoogle Scholar
  16. Ge J, Kim KI, Lin ZG, Zhu HP (2015) A SIS reaction–diffusion–advection model in a low-risk and high-risk domain. J Differ Equ 259:5486–5509MathSciNetCrossRefzbMATHGoogle Scholar
  17. Ge J, Lei CX, Lin ZG (2017) Reproduction numbers and the expanding fronts for a diffusion–advection SIS model in heterogeneous time-periodic environment. Nonlinear Anal. Real World Appl 33:100–120MathSciNetCrossRefzbMATHGoogle Scholar
  18. Gu H, Lin ZG, Lou BD (2015) Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries. Proc Am Math Soc 143:1109–1117MathSciNetCrossRefzbMATHGoogle Scholar
  19. Huang W, Han M, Liu K (2010) Dynamics of an SIS reaction–diffusion epidemic model for disease transmission. Math Biosci Eng 7:51–66MathSciNetCrossRefzbMATHGoogle Scholar
  20. Kolmogorov AN, Petrovsky IG, Piskunov NS (1937) Ètude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull Univ Moscou Sér Internat A1:1–26 (English transl. in: Dynamics of Curved Fronts, P. Pelcé (ed.), Academic Press, 1988, 105–130)Google Scholar
  21. Ladyzenskaja OA, Solonnikov VA, Ural’ceva NN (1968) Linear and quasilinear equations of parabolic type. American Mathematical Society, ProvidenceGoogle Scholar
  22. Lei CX, Lin ZG, Zhang QY (2014) The spreading front of invasive species in favorable habitat or unfavorable habitat. J Differ Equ 257:145–166MathSciNetCrossRefzbMATHGoogle Scholar
  23. Lewis MA, Renclawowicz J, van den Driessche P (2006) Traveling waves and spread rates for a West Nile virus model. Bull Math Biol 68(1):3–23MathSciNetCrossRefzbMATHGoogle Scholar
  24. Li BT, Weinberger HF, Lewis MA (2005) Spreading speeds as slowest wave speeds for cooperative systems. Math Biosci 196:82–98MathSciNetCrossRefzbMATHGoogle Scholar
  25. Lin ZG (2007) A free boundary problem for a predator–prey model. Nonlinearity 20:1883–1892MathSciNetCrossRefzbMATHGoogle Scholar
  26. Liu RS, Shuai JP, Wu JH, Zhu HP (2006) Modeling spatial spread of West Nile virus and impact of directional dispersal of birds. Math Biosci Eng 3(1):145–160MathSciNetzbMATHGoogle Scholar
  27. López-Gómez J (1996) The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems. J Differ Equ 127(1):263–294MathSciNetCrossRefzbMATHGoogle Scholar
  28. Maidana NA, Yang HM (2009) Spatial spreading of West Nile virus described by traveling waves. J Theor Biol 258:403–417MathSciNetCrossRefGoogle Scholar
  29. Smith HL (1995) Monotone dynamical systems. American Mathematical Society, ProvidenceGoogle Scholar
  30. van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48MathSciNetCrossRefzbMATHGoogle Scholar
  31. Wang WD, Zhao X-Q (2012) Basic reproduction numbers for reaction–diffusion epidemic models. SIAM J Appl Dyn Syst 11:1652–1673MathSciNetCrossRefzbMATHGoogle Scholar
  32. Wang WD, Zhao X-Q (2015) Spatial invasion threshold of Lyme disease. SIAM J Appl Math 75:1142–1170MathSciNetCrossRefzbMATHGoogle Scholar
  33. Wonham MJ, de-Camino-Beck T, Lewis MA (2004) An epidemiological model for West Nile virus: invasion analysis and control application. Proc R Soc Lond B 271:501–507CrossRefGoogle Scholar
  34. Yu X, Zhao X-Q (2016) A nonlocal spatial model for Lyme disease. J Differ Equ 261:340–372MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of Mathematical ScienceYangzhou UniversityYangzhouChina
  2. 2.Laboratory of Mathematical Parallel Systems (LAMPS), Department of Mathematics and StatisticsYork UniversityTorontoCanada

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