Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 639405, 22 pages
http://dx.doi.org/10.1155/2014/639405
Research Article

Dynamical Behavior and Stability Analysis in a Hybrid Epidemiological-Economic Model with Incubation

1Institute of Systems Science, Northeastern University, Shenyang 110004, China
2State Key Laboratory of Integrated Automation of Process Industry, Northeastern University, Shenyang 110004, China
3Changli Institute of Fruit Forestry, Hebei Academy of Agricultural and Forestry Sciences, Changli 066600, China
4Institute of Biotechnology, College of Life and Health Sciences, Northeastern University, Shenyang 110004, China

Received 12 January 2014; Accepted 15 April 2014; Published 12 May 2014

Academic Editor: Weiming Wang

Copyright © 2014 Chao Liu 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. R. M. Anderson and R. M. May, “Population biology of infectious diseases: part I,” Nature, vol. 280, no. 5721, pp. 361–367, 1979. View at Publisher · View at Google Scholar · View at Scopus
  2. S. A. Levin, T. G. Hallam, and J. J. Gross, Applied Mathematical Ecology, Springer, New York, NY, USA, 1990.
  3. H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599–653, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. Z. E. Ma, Y. Zhou, W. Wang, and Z. Jin, Mathematical Models and Dynamics OfInfectious Disease, Science Press, Beijing, China, 2004.
  5. K. L. Cooke, “Stability analysis for a vector disease model,” The Rocky Mountain Journal of Mathematics, vol. 9, no. 1, pp. 31–42, 1979, Conference on Deterministic Differential Equations and Stochastic Processes Models for Biological Systems (San Cristobal, N.M., 1977). View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. W. M. Liu, H. W. Hethcote, and S. A. Levin, “Dynamical behavior of epidemiological models with nonlinear incidence rates,” Journal of Mathematical Biology, vol. 25, no. 4, pp. 359–380, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. Y. Takeuchi, W. Ma, and E. Beretta, “Global asymptotic properties of a delay SIR epidemic model with finite incubation times,” Nonlinear Analysis: Theory, Methods & Applications, vol. 42, no. 6, pp. 931–947, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. E. Beretta, T. Hara, W. Ma, and Y. Takeuchi, “Global asymptotic stability of an SIR epidemic model with distributed time delay,” Nonlinear Analysis, vol. 47, pp. 4107–4115, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. C. C. McCluskey, “Global stability of an SIR epidemic model with delay and general nonlinear incidence,” Mathematical Biosciences and Engineering, vol. 7, no. 4, pp. 837–850, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. X. Zhou and J. Cui, “Stability and Hopf bifurcation of a delay eco-epidemiological model with nonlinear incidence rate,” Mathematical Modelling and Analysis, vol. 15, no. 4, pp. 547–569, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J.-J. Wang, J.-Z. Zhang, and Z. Jin, “Analysis of an SIR model with bilinear incidence rate,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 2390–2402, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Y. Enatsu, E. Messina, Y. Muroya, Y. Nakata, E. Russo, and A. Vecchio, “Stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5327–5336, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with a nonlinear incidence rate,” Journal of Differential Equations, vol. 188, no. 1, pp. 135–163, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. W. Ma, M. Song, and Y. Takeuchi, “Global stability of an SIR epidemic model with time delay,” Applied Mathematics Letters, vol. 17, no. 10, pp. 1141–1145, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. R. Xu and Z. Ma, “Global stability of a SIR epidemic model with nonlinear incidence rate and time delay,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 3175–3189, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resource, John Wiley & Sons, New York, NY, USA, 2nd edition, 1990. View at MathSciNet
  17. P. W. Dong, S. Y. Zhuang, X. H. Lin, and X. Z. Zhang, “Economic evaluation of forestay industry based on ecosystem coupling,” Mathematical and Computer Modelling, vol. 58, no. 6, pp. 1010–1017, 2013. View at Publisher · View at Google Scholar
  18. N. Bairagi, S. Chaudhuri, and J. Chattopadhyay, “Harvesting as a disease control measure in an eco-epidemiological system—a theoretical study,” Mathematical Biosciences, vol. 217, no. 2, pp. 134–144, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. Chakraborty, S. Pal, and N. Bairagi, “Dynamics of a ratio-dependent eco-epidemiological system with prey harvesting,” Nonlinear Analysis: Real World Applications, vol. 11, no. 3, pp. 1862–1877, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. R. Bhattacharyya and B. Mukhopadhyay, “On an eco-epidemiological model with prey harvesting and predator switching: local and global perspectives,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3824–3833, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. L. Zou, Z. Xiong, and Z. Shu, “The dynamics of an eco-epidemic model with distributed time delay and impulsive control strategy,” Journal of the Franklin Institute, vol. 348, no. 9, pp. 2332–2349, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. H. S. Gordon, “The economic theory of a common property resource: the fishery,” Journal of Political Economy, vol. 62, no. 2, pp. 124–142, 1954. View at Publisher · View at Google Scholar
  23. C. Liu, Q. Zhang, Y. Zhang, and X. Duan, “Bifurcation and control in a differential-algebraic harvested prey-predator model with stage structure for predator,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 18, no. 10, pp. 3159–3168, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. C. Liu, Q. Zhang, X. Zhang, and X. Duan, “Dynamical behavior in a stage-structured differential-algebraic prey-predator model with discrete time delay and harvesting,” Journal of Computational and Applied Mathematics, vol. 231, no. 2, pp. 612–625, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. X. Zhang, Q.-L. Zhang, C. Liu, and Z.-Y. Xiang, “Bifurcations of a singular prey-predator economic model with time delay and stage structure,” Chaos, Solitons & Fractals, vol. 42, no. 3, pp. 1485–1494, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. G. Zhang, B. Chen, L. Zhu, and Y. Shen, “Hopf bifurcation for a differential-algebraic biological economic system with time delay,” Applied Mathematics and Computation, vol. 218, no. 15, pp. 7717–7726, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. K. Chakraborty, M. Chakraborty, and T. K. Kar, “Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay,” Nonlinear Analysis: Hybrid Systems, vol. 5, no. 4, pp. 613–625, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. C. Liu, Q. Zhang, J. Huang, and W. Tang, “Dynamical analysis and control in a delayed differential-algebraic bio-economic model with stage structure and diffusion,” International Journal of Biomathematics, vol. 5, no. 2, pp. 1–30, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  29. S. Campbell, Singular Systems of Differential Equations, Priman, London, UK, 1980.
  30. P. Muller, “Linear mechanical descriptor systems: identification, analysis and design,” in Proceedings of the IFAC International Conference on Control of Industrial Systems, pp. 501–506, Belfort, France, May 1997.
  31. M. S. Silva and T. P. de Lima, “Looking for nonnegative solutions of a Leontief dynamic model,” Linear Algebra and its Applications, vol. 364, no. 1, pp. 281–316, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. S. Ayasun, C. O. Nwankpa, and H. G. Kwatny, “Computation of singular and singularity induced bifurcation points of differential-algebraic power system model,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 51, no. 8, pp. 1525–1538, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  33. M. Yue and R. Schlueter, “Bifurcation subsystem and its application in power system analysis,” IEEE Transactions on Power Systems, vol. 19, no. 4, pp. 1885–1893, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  34. W. Marszalek and Z. W. Trzaska, “Singularity-induced bifurcations in electrical power systems,” IEEE Transactions on Power Systems, vol. 20, no. 1, pp. 312–320, 2005. View at Publisher · View at Google Scholar · View at Scopus
  35. D. G. Luenberger, “Nonlinear descriptor systems,” Journal of Economic Dynamics & Control, vol. 1, no. 3, pp. 219–242, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. D. G. Luenberger and A. Arbel, “Singular dynamic Leontief systems,” Econometrics, vol. 45, no. 32, pp. 991–995, 1997. View at Google Scholar
  37. X. Yang, L. Chen, and J. Chen, “Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models,” Computers & Mathematics with Applications, vol. 32, no. 4, pp. 109–116, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  38. H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, USA, 2003. View at MathSciNet
  39. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99, Springer, New York, NY, USA, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  40. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191, Academic Press, New York, NY, USA, 1993. View at MathSciNet
  41. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42, Springer, New York, NY, USA, 1983. View at MathSciNet
  42. V. Venkatasubramanian, H. Schättler, and J. Zaborszky, “Local bifurcations and feasibility regions in differential-algebraic systems,” IEEE Transactions on Automatic Control, vol. 40, no. 12, pp. 1992–2013, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. M. Kot, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, UK, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. V. Venkatasubramanian, “Singularity induced bifurcation and the van der Pol oscillator,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 41, no. 11, pp. 765–769, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  45. R. E. Beardmore, “The singularity-induced bifurcation and its Kronecker normal form,” SIAM Journal on Matrix Analysis and Applications, vol. 23, no. 1, pp. 126–137, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  46. L. Yang and Y. Tang, “An improved version of the singularity-induced bifurcation theorem,” IEEE Transactions on Automatic Control, vol. 46, no. 9, pp. 1483–1486, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  47. L. Dai, Singular Control Systems, vol. 118, Springer, New York, NY, USA, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  48. H. I. Freedman and V. Sree Hari Rao, “The trade-off between mutual interference and time lags in predator-prey systems,” Bulletin of Mathematical Biology, vol. 45, no. 6, pp. 991–1004, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  49. B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41, Cambridge University Press, Cambridge, UK, 1981. View at MathSciNet