Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2013, Article ID 470646, 10 pages
http://dx.doi.org/10.1155/2013/470646
Research Article

Positive Periodic Solutions of an Epidemic Model with Seasonality

1Complex Sciences Center, Shanxi University, Taiyuan 030006, China
2School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China
3Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China
4Department of Applied Mathematics, Xidian University, Xi’an, Shaanxi 710071, China
5Institute of Information Economy, Hangzhou Normal University, Hangzhou 310036, China
6Web Sciences Center, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China
7Department of Physics, University of Fribourg, Chemin du Musée 3, 1700 Fribourg, Switzerland

Received 17 August 2013; Accepted 12 September 2013

Academic Editors: P. Leach, O. D. Makinde, and Y. Xia

Copyright © 2013 Gui-Quan Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. K. Dietz and J. A. P. Heesterbeek, “Bernoulli was ahead of modern epidemiology,” Nature, vol. 408, no. 6812, pp. 513–514, 2000. View at Google Scholar · View at Scopus
  2. K. Dietz and J. A. P. Heesterbeek, “Daniel Bernoulli's epidemiological model revisited,” Mathematical Biosciences, vol. 180, pp. 1–21, 2002. View at Publisher · View at Google Scholar · View at Scopus
  3. R. Ross, “An application of the theory of probabilities to the study of a priori pathometry (part I),” Proceedings of the Royal Society A, vol. 92, no. 638, pp. 204–230, 1916. View at Publisher · View at Google Scholar
  4. R. Ross and H. P. Hudson, “An application of the theory of probabilities to the study of a priori pathometry (part II),” Proceedings of the Royal Society A, vol. 93, no. 650, pp. 212–225, 1917. View at Publisher · View at Google Scholar
  5. W. O. Kermack and A. G. McKendrick, “A contribution to the mathematical theory of epidemics (part I),” Proceedings of the Royal Society A, vol. 115, no. 772, pp. 700–721, 1927. View at Publisher · View at Google Scholar
  6. W. O. Kermack and A. G. McKendrick, “A contribution to the mathematical theory of epidemics (part II),” Proceedings of the Royal Society A, vol. 138, no. 834, pp. 55–83, 1932. View at Publisher · View at Google Scholar
  7. A. Tsoularis and J. Wallace, “Analysis of logistic growth models,” Mathematical Biosciences, vol. 179, no. 1, pp. 21–55, 2002. View at Publisher · View at Google Scholar · View at Scopus
  8. I. Nåsell, “On the quasi-stationary distribution of the stochastic logistic epidemic,” Mathematical Biosciences, vol. 156, no. 1-2, pp. 21–40, 1999. View at Publisher · View at Google Scholar · View at Scopus
  9. H. Fujikawa, A. Kai, and S. Morozumi, “A new logistic model for Escherichia coli growth at constant and dynamic temperatures,” Food Microbiology, vol. 21, no. 5, pp. 501–509, 2004. View at Publisher · View at Google Scholar · View at Scopus
  10. L. Berezansky and E. Braverman, “Oscillation properties of a logistic equation with several delays,” Journal of Mathematical Analysis and Applications, vol. 247, no. 1, pp. 110–125, 2000. View at Google Scholar · View at Scopus
  11. M. Tabata, N. Eshima, and I. Takagi, “The nonlinear integro-partial differential equation describing the logistic growth of human population with migration,” Applied Mathematics and Computation, vol. 98, pp. 169–183, 1999. View at Publisher · View at Google Scholar
  12. L. Korobenko and E. Braverman, “On logistic models with a carrying capacity dependent diffusion: stability of equilibria and coexistence with a regularly diffusing population,” Nonlinear Analysis: Real World Applications, vol. 13, no. 6, pp. 2648–2658, 2012. View at Publisher · View at Google Scholar · View at Scopus
  13. Y. Muroya, “Global attractivity for discrete models of nonautonomous logistic equations,” Computers and Mathematics with Applications, vol. 53, no. 7, pp. 1059–1073, 2007. View at Publisher · View at Google Scholar · View at Scopus
  14. S. Invernizzi and K. Terpin, “A generalized logistic model for photosynthetic growth,” Ecological Modelling, vol. 94, no. 2-3, pp. 231–242, 1997. View at Publisher · View at Google Scholar · View at Scopus
  15. Z. Min, W. Bang-Jun, and J. Feng, “Coalmining cities' economic growth mechanism and sustainable development analysis based on logistic dynamics model,” Procedia Earth and Planetary Science, vol. 1, no. 1, pp. 1737–1743, 2009. View at Publisher · View at Google Scholar
  16. L. I. Aniţa, S. Aniţa, and V. Arnǎutu, “Global behavior for an age-dependent population model with logistic term and periodic vital rates,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 368–379, 2008. View at Publisher · View at Google Scholar · View at Scopus
  17. Z. A. Banaszak, X. Q. Tang, S. C. Wang, and M. B. Zaremba, “Logistics models in flexible manufacturing,” Computers in Industry, vol. 43, no. 3, pp. 237–248, 2000. View at Publisher · View at Google Scholar · View at Scopus
  18. S. Brianzoni, C. Mammana, and E. Michetti, “Nonlinear dynamics in a business-cycle model with logistic population growth,” Chaos, Solitons and Fractals, vol. 40, no. 2, pp. 717–730, 2009. View at Publisher · View at Google Scholar · View at Scopus
  19. W. P. London and J. A. Yorke, “Recurrent outbreaks of measles, chickenpox and mumps. I. Seasonal variation in contact rates,” The American Journal of Epidemiology, vol. 98, no. 6, pp. 453–468, 1973. View at Google Scholar · View at Scopus
  20. S. F. Dowell, “Seasonal variation in host susceptibility and cycles of certain infectious diseases,” Emerging Infectious Diseases, vol. 7, no. 3, pp. 369–374, 2001. View at Google Scholar · View at Scopus
  21. O. N. Bjørnstad, B. F. Finkenstädt, and B. T. Grenfell, “Dynamics of measles epidemics: estimating scaling of transmission rates using a Time series SIR model,” Ecological Monographs, vol. 72, no. 2, pp. 169–184, 2002. View at Google Scholar · View at Scopus
  22. J. Zhang, Z. Jin, G. Q. Sun, X. Sun, and S. Ruan, “Modeling seasonal rabies epidemics in China,” Bulletin of Mathematical Biology, vol. 74, no. 5, pp. 1226–1251, 2012. View at Publisher · View at Google Scholar · View at Scopus
  23. I. B. Schwartz, “Small amplitude, long period outbreaks in seasonally driven epidemics,” Journal of Mathematical Biology, vol. 30, no. 5, pp. 473–491, 1992. View at Google Scholar · View at Scopus
  24. I. B. Schwartz and H. L. Smith, “Infinite subharmonic bifurcation in an SEIR epidemic model,” Journal of Mathematical Biology, vol. 18, no. 3, pp. 233–253, 1983. View at Google Scholar · View at Scopus
  25. H. L. Smith, “Multiple stable subharmonics for a periodic epidemic model,” Journal of Mathematical Biology, vol. 17, no. 2, pp. 179–190, 1983. View at Google Scholar · View at Scopus
  26. J. Dushoff, J. B. Plotkin, S. A. Levin, and D. J. D. Earn, “Dynamical resonance can account for seasonality of influenza epidemics,” Proceedings of the National Academy of Sciences of the United States of America, vol. 101, no. 48, pp. 16915–16916, 2004. View at Publisher · View at Google Scholar · View at Scopus
  27. J. L. Ma and Z. Ma, “Epidemic threshold conditions for seasonally forced SEIR models,” Mathematical Biosciences and Engineering, vol. 3, no. 1, pp. 161–172, 2006. View at Google Scholar · View at Scopus
  28. D. J. D. Earn, P. Rohani, B. M. Bolker, and B. T. Grenfell, “A simple model for complex dynamical transitions in epidemics,” Science, vol. 287, no. 5453, pp. 667–670, 2000. View at Publisher · View at Google Scholar · View at Scopus
  29. M. J. Keeling, P. Rohani, and B. T. Grenfell, “Seasonally forced disease dynamics explored as switching between attractors,” Physica D, vol. 148, no. 3-4, pp. 317–335, 2001. View at Publisher · View at Google Scholar · View at Scopus
  30. J. L. Aron and I. B. Schwartz, “Seasonality and period-doubling bifurcations in an epidemic model,” Journal of Theoretical Biology, vol. 110, no. 4, pp. 665–679, 1984. View at Google Scholar · View at Scopus
  31. N. C. Grassly and C. Fraser, “Seasonal infectious disease epidemiology,” Proceedings of the Royal Society B, vol. 273, no. 1600, pp. 2541–2550, 2006. View at Publisher · View at Google Scholar · View at Scopus
  32. J. Zhang, Z. Jin, G. Q. Sun, T. Zhou, and S. Ruan, “Analysis of rabies in China: transmission dynamics and control,” PLoS ONE, vol. 6, no. 7, Article ID e20891, 2011. View at Google Scholar · View at Scopus
  33. O. Diekmann, J. A. P. Heesterbeek, and J. A. Metz, “On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations,” Journal of mathematical biology, vol. 28, no. 4, pp. 365–382, 1990. View at Google Scholar · View at Scopus
  34. O. Diekmann, J. A. P. Heesterbeek, and M. G. Roberts, “The construction of next-generation matrices for compartmental epidemic models,” Journal of the Royal Society Interface, vol. 7, no. 47, pp. 873–885, 2010. View at Publisher · View at Google Scholar · View at Scopus
  35. P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29–48, 2002. View at Publisher · View at Google Scholar · View at Scopus
  36. F. Berezovsky, G. Karev, B. J. Song, and C. C. Chavez, “A simple epidemic model with surprising dynamics,” Mathematical Biosciences and Engineering, vol. 2, pp. 133–152, 2005. View at Google Scholar
  37. J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, Academic Press, New York, NY, USA, 1961.
  38. E. A. Barbashin, Introduction to the Theory of Stability, Wolters-Noordhoff, Groningen, The Netherlands, 1970.
  39. D. Schenzle, “An age-structured model of pre- and post-vaccination measles transmission,” Mathematical Medicine and Biology, vol. 1, no. 2, pp. 169–191, 1984. View at Publisher · View at Google Scholar · View at Scopus
  40. N. Bacaër and S. Guernaoui, “The epidemic threshold of vector-borne diseases with seasonality: the case of cutaneous leishmaniasis in Chichaoua, Morocco,” Journal of Mathematical Biology, vol. 53, no. 3, pp. 421–436, 2006. View at Publisher · View at Google Scholar · View at Scopus
  41. W. D. Wang and X. Q. Zhao, “Threshold dynamics for compartmental epidemic models in periodic environments,” Journal of Dynamics and Differential Equations, vol. 20, no. 3, pp. 699–717, 2008. View at Publisher · View at Google Scholar · View at Scopus
  42. F. Zhang and X. Q. Zhao, “A periodic epidemic model in a patchy environment,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 496–516, 2007. View at Publisher · View at Google Scholar · View at Scopus
  43. H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, New York, NY, USA, 1995.
  44. H. R. Thieme, “Convergence results and a Poincaré-Bendixson trichotomy for asymptotically automous differential equations,” Journal of Mathematical Biology, vol. 30, pp. 755–763, 1992. View at Google Scholar
  45. L. Perko, Differential Equations and Dynamical Systems, Springer, New York, NY, USA, 2000.
  46. X. Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, NY, USA, 2003.
  47. Z. G. Bai and Y. C. Zhou, “Threshold dynamics of a bacillary dysentery model with seasonal fluctuation,” Discrete and Continuous Dynamical Systems B, vol. 15, no. 1, pp. 1–14, 2011. View at Publisher · View at Google Scholar · View at Scopus
  48. Z. G. Bai, Y. C. Zhou, and T. L. Zhang, “Existence of multiple periodic solutions for an SIR model with seasonality,” Nonlinear Analysis: Theory, Methods and Applications, vol. 74, no. 11, pp. 3548–3555, 2011. View at Publisher · View at Google Scholar · View at Scopus
  49. N. Bacaër, “Approximation of the basic reproduction number R0 for vector-borne diseases with a periodic vector population,” Bulletin of Mathematical Biology, vol. 69, no. 3, pp. 1067–1091, 2007. View at Publisher · View at Google Scholar · View at Scopus
  50. N. Bacaër and X. Abdurahman, “Resonance of the epidemic threshold in a periodic environment,” Journal of Mathematical Biology, vol. 57, no. 5, pp. 649–673, 2008. View at Publisher · View at Google Scholar · View at Scopus
  51. N. Bacaër, “Periodic matrix population models: growth rate, basic reproduction number, and entropy,” Bulletin of Mathematical Biology, vol. 71, no. 7, pp. 1781–1792, 2009. View at Publisher · View at Google Scholar · View at Scopus
  52. N. Bacaër and E. H. A. Dads, “Genealogy with seasonality, the basic reproduction number, and the influenza pandemic,” Journal of Mathematical Biology, vol. 62, no. 5, pp. 741–762, 2011. View at Publisher · View at Google Scholar · View at Scopus
  53. L. J. Liu, X. Q. Zhao, and Y. C. Zhou, “A tuberculosis model with seasonality,” Bulletin of Mathematical Biology, vol. 72, no. 4, pp. 931–952, 2010. View at Publisher · View at Google Scholar · View at Scopus
  54. J. L. Liu, “Threshold dynamics for a HFMD epidemic model with periodic transmission rate,” Nonlinear Dynamics, vol. 64, no. 1-2, pp. 89–95, 2011. View at Publisher · View at Google Scholar · View at Scopus
  55. Y. Nakata and T. Kuniya, “Global dynamics of a class of SEIRS epidemic models in a periodic environment,” Journal of Mathematical Analysis and Applications, vol. 363, no. 1, pp. 230–237, 2010. View at Publisher · View at Google Scholar · View at Scopus
  56. B. G. Williams and C. Dye, “Infectious disease persistence when transmission varies seasonally,” Mathematical Biosciences, vol. 145, no. 1, pp. 77–88, 1997. View at Publisher · View at Google Scholar · View at Scopus
  57. I. A. Moneim, “The effect of using different types of periodic contact rate on the behaviour of infectious diseases: a simulation study,” Computers in Biology and Medicine, vol. 37, no. 11, pp. 1582–1590, 2007. View at Publisher · View at Google Scholar · View at Scopus
  58. C. L. Wesley and L. J. S. Allen, “The basic reproduction number in epidemic models with periodic demographics,” Journal of Biological Dynamics, vol. 3, no. 2-3, pp. 116–129, 2009. View at Publisher · View at Google Scholar
  59. D. Greenhalgh and I. A. Moneim, “SIRS epidemic model and simulations using different types of seasonal contact rate,” Systems Analysis Modelling Simulation, vol. 43, no. 5, pp. 573–600, 2003. View at Publisher · View at Google Scholar · View at Scopus