Nonlinear Dynamics

, Volume 77, Issue 1–2, pp 31–40 | Cite as

Pattern dynamics of an epidemic model with nonlinear incidence rate

  • Tao Wang
Original Paper


All species live in space, and the research of spatial diseases can be used to control infectious diseases. As a result, it is more realistic to study the spatial pattern of epidemic models with space and time. In this paper, spatial dynamics of an epidemic model with nonlinear incidence rate is investigated. We find that there are different types of stationary patterns by amplitude equations and numerical simulations. The obtained results may well explain the distribution of disease observed in the real world and provide some insights on disease control.


Nonlinear incidence Spatial epidemic model Amplitudes equation Pattern selection 


  1. 1.
    Murray, J.D.: Mathematical Biology, 3rd edn. Springer, Berlin (1993)CrossRefMATHGoogle Scholar
  2. 2.
    Anderson, R.M., May, R.M.: Infectious Disease of Humans: Dynamics and Control. Oxford University Press, Oxford (1992)Google Scholar
  3. 3.
    Castillo-Chavez, C., Yakubu, A.-A.: Mathematical Approaches for Emerging and Re-emerging Infectious Diseases: An Introduction. Springer, Berlin (2002)Google Scholar
  4. 4.
    Riley, S., Fraser, C., Donnelly, C.A., Ghani, A.C., Abu-Raddad, L.J., Hedley, A.J., Leung, G.M., Ho, L.M., Lam, T.H., Thach, T.Q., Chau, P., Chan, K.P., Lo, S.V., Leung, P.Y., Tsang, T., Ho, W., Lee, K.H., Lau, E.M., Ferguson, N.M., Anderson, R.M.: Transmission dynamics of the etiological agent of sars in Hong Kong: impact of public health interventions. Science 300, 1961–1966 (2003)CrossRefGoogle Scholar
  5. 5.
    Fraser, C., et al.: Pandemic potential of a strain of influenza A(H1N1): early findings. Science 324, 1557–1561 (2009)Google Scholar
  6. 6.
    Hiroshi, N., Wilson, N., Baker, M.G.: Estimating the reproduction number of the novel influenza A virus (H1N1) in a Southern Hemisphere setting: preliminary estimate in New Zealand. J. N. Z. Med. Assoc. 122, 73–77 (2009)Google Scholar
  7. 7.
    Benjamin, J.C., et al.: Comparative epidemiology of human infections with avian influenza A H7N9 and H5N1 viruses in China: a population-based study of laboratoryconfirmed cases. Lancet 382, 129–137 (2013)CrossRefGoogle Scholar
  8. 8.
    Chen, Y., Liang, W., Yang, S., Wu, N.: Human infections with the emerging avian influenza A H7N9 virus from wet market poultry: clinical analysis and characterisation of viral genome. Lancet 381, 1916–1925 (2013)CrossRefGoogle Scholar
  9. 9.
    Li, A.-W.: Impact of noise on pattern formation in a predator–prey model. Nonlinear Dyn. 66, 689–694 (2011)Google Scholar
  10. 10.
    Liu, P.-P., Xue, Y.: Spatiotemporal dynamics of a predator–prey model. Nonlinear Dyn. 69, 71–77 (2012)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Sun, G.-Q., Zhang, G., Jin, Z., Li, L.: Predator cannibalism can give rise to regular spatial pattern in a predator–prey system. Nonlinear Dyn. 58, 75–84 (2009)CrossRefMATHGoogle Scholar
  12. 12.
    Sun, G.-Q.: Pattern formation of an epidemic model with diffusion. Nonlinear Dyn. 69, 1097–1104 (2012)CrossRefGoogle Scholar
  13. 13.
    Hassell, M.P., Comins, H.N., May, R.M.: Species coexistence and self-organizing spatial dynamics. Nature 370, 290–292 (1994)CrossRefGoogle Scholar
  14. 14.
    Ballegooijen, W.M., Boerlijst, M.C.: Emergent trade-of and selection for outbreak frequency in spatial epidemics. Proc. Natl. Acad. Sci. USA 101, 18246–18250 (2004)CrossRefGoogle Scholar
  15. 15.
    Gubler, D.J.: Epidemic dengue/dengue hemorrhagic fever as a public health, social and economic problem in the 21st century. Trends Microbiol. 10, 100–103 (2002)CrossRefGoogle Scholar
  16. 16.
    Liu, Q.-X., Jin, Z.: Formation of spatial patterns in an epidemic model with constant removal rate of the infectives. J. Stat. Mech. 5, P05002 (2007)Google Scholar
  17. 17.
    Sun, G., Jin, Z., Liu, Q.-X., Li, L.: Pattern formation in a spatial S-I model with non-linear incidence rate. J. Stat. Mech. 11, P11011 (2007)CrossRefGoogle Scholar
  18. 18.
    Li, L., Jin, Z., Sun, G.-Q.: Traveling pattern induced by migration in an epidemic model. J. Biol. Syst. 17, 319–328 (2009)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Sun, G.-Q., Jin, Z., Liu, Q.-X., Li, L.: Spatial pattern in an epidemic system with cross-diffusion of the susceptible. J. Biol. Syst. 17, 141–152 (2009)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Sun, G.-Q., Li, L., Jin, Z., Zhang, Z.-K., Zhou, T.: Pattern dynamics in a spatial predator–prey system with Allee effect. Abs. Appl. Anal. 2013, 921879 (2013)MathSciNetGoogle Scholar
  21. 21.
    Sun, G.-Q., Li, L., Zhang, Z.-K.: Spatial dynamics of a vegetation model in an arid flat environment. Nonlinear Dyn. 73, 2207–2219 (2013)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Ouyang, Q.: Pattern Formation in Reaction–Diffusion Systems. Sci-Tech Education Publishing House, Shanghai (2000) Google Scholar
  23. 23.
    Sun, G.-Q., Zhang, J., Song, L.-P., Jin, Z., Li, B.-L.: Pattern formation of a spatial predator–prey system. Appl. Math. Comput. 218, 11151–11162 (2012)Google Scholar
  24. 24.
    Sun, G.-Q., Jin, Z., Liu, Q.-X., Li, L.: Dynamical complexity of a spatial predator–prey model with migration. Ecol. Model. 219, 248–255 (2008)CrossRefGoogle Scholar
  25. 25.
    Keeling, M.J., Rohani, P.: Modeling Infectious Diseases in Humans and Animals. University Press, Princeton (2007)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsShihezi UniversityShiheziPeople’s Republic of China

Personalised recommendations