• Chen, Xiangyong ;
  • Li, Chunji ;
  • Lu, Jufang ;
  • Jing, Yuanwei
  • Received : 2011.01.04
  • Published : 2012.05.31


This paper is estimating the domain of attraction for a class of susceptible-exposed-infectious-recovered (SEIR) epidemic dynamic models by using sum of squares optimization. First, the stability is analyzed for the equilibriums of SEIR model, and the domain of attraction in the endemic equilibrium is estimated by using sum of squares optimization. Finally, a numerical example is examined.


domain of attraction;SOS optimization;SEIR epidemic model


  1. F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New York, 2001.
  2. G. Chesi, Estimating the domain of attraction for non-polynomial systems via LMI optimization, Automatica. 45 (2009), 1536-1541.
  3. H. Guo, M. Li, and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canad. Appl. Math. Quart. 14 (2006), no. 3, 259-284.
  4. O. Hachicho, A novel LMI-based optimization algorithm for the guaranteed estimation of the domain of attraction using rational Lyapunov functions, J. Franklin Inst. 344 (2007), no. 5, 535-552.
  5. O. Hachicho and B. Tibken, Estimating domains of attraction of a class of nonlinear dynamical systems with LMI methods based on the theory of moments, In Proc. CDC. Las Vegas, Nevada. 2002, 3150-3155.
  6. W. Hahn, Stability of Motion, Springer Verlag, Berlin, 1967.
  7. Z. Jarvis-Wloszek, Lyapunov based analysis and controller synthesis for polynomial systems using sum-of squares optimization, Ph. D. Thesis. dissertation. Pasadena, CA. 2003.
  8. Z. Jarvis-Wloszek, R. Feeley, W. Tan, K. Sun, and A. Packard, Some controls applica- tions of sum of squares programming, In Proc. CDC. Hawaii, USA (2003), 4676-4681.
  9. Y. W. Jing, X. Y. Chen, C. J. Li, et al., Domain of attraction estimation for SIRS epidemic models via Sum-of-Square Optimization, August 28 - September 2, in Proc. IFAC WC. Milano, Italy. 2011.
  10. H. Khalil, Nonlinear Systems, 3rd ed. Prentice Hall, 2002.
  11. C. Li, C. Ryoo, N. Li, and L. Cao, Estimating the domain of attraction via moment matrices, Bull. Korean Math. Soc. 46 (2009), no. 6, 1237-1248.
  12. Z. Ma, Y. Zhou, and W. Wang, Mathematical Modeling and Research of Epidemic Dynamical Systems, Beijing, Science Press, 2004.
  13. O. Makinde, A domain decomposition approach to a SIR epidemic model with constant vaccination strategy, Appl. Math. Comput. 184 (2007), no. 2, 842-848.
  14. L. G. Matallana, A. M. Blanco, and J. A. Bandoni, Estimation of domains of attraction in epidemiological models with constant removal rates of infected individuals, 16th argentine bioengineering congress and the 5th conference of clinical engineering, Journal of Physics: Conference Series. 90 (2007), Art. no. 012052.
  15. L. G. Matallana, A. M. Blanco, and J. A. Bandoni, Estimation of domains of attraction: a global optimization approach, Math. Comput. Modelling 52 (2010), no. 3-4, 574-585.
  16. P. A. Parrilo, Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization, Ph. D. dissertation. Pasadena, CA. 2000.
  17. S. Prajna, A. Papachristodoulou, P. Seiler, and P. A. Parrilo, SOSTOOLS: Sum of squares optimization toolbox for MATLAB, Available from http://www.cds.caltech. edu/sostools and, 2002.
  18. S. Prajna, A. Papachristodoulou, P. Seiler, et al. New developments in sum of squares optimization and SOSTOOLS, In Proc. ACC. Boston, USA (2004), 5608-5611.
  19. W. Tan, Nonlinear control analysis and synthesis using sum-of-squares programming, Ph. D. dissertation. UC, Berkeley. 2003.
  20. W. Tan and A. Packard, Stability region analysis using polynomial and composite polynomial Lyapunov functions and sum-of-squares programming, IEEE Trans. Automat. Control 53 (2008), no. 2, 565-571.
  21. U. Topcu, A. Packard, and P. Seiler. Local stability analysis using simulations and sum- of-squares programming, Automatica J. IFAC 44 (2008), no. 10, 2669-2675.
  22. U. Topcu, A. Packard, P. Seiler, and T. Wheeler. Stability region analysis using simulations and sum-of-squares programming, In Proc. ACC. New York, USA. (2007), 6009- 6014.
  23. G. Zaman, Y. Kang, and I. Jung, Stability analysis and optimal vaccination of an SIR epidemic model, Biosystems. 93 (2008), 240-249.
  24. J. Zhang, J. Li, and Z. Ma, Global dynamics of an SEIR epidemic model with immigration of different compartments, Acta Math. Sci. Ser. B Engl. Ed. 26 (2006), no. 3, 551-567.

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Supported by : National Science Foundation of China