Analysis and Control of an SEIR Epidemic System with Nonlinear Transmission Rate

Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 421)

Introduction

Mathematical models describing the population dynamics of infectious diseases have been playing an important role in better understanding epidemiological patterns and disease control for a long time. In order to predict the spread of infectious disease among regions, many epidemic models have been proposed and analyzed in recent years (see [7, 4, 22, 25, 36, 19, 20]). However, most of the literature researched on epidemic systems (see [36, 24, 8, 13]) assumes that the disease incubation is negligible that, once infected, each susceptible individual (in class S) becomes infectious instantaneously (in class I) and later recovers (in class R) with a permanent or temporary acquired immunity. The model based on these assumptions is customarily called an SIR (susceptible-infectious-recovered) or SIRS (susceptible-infectious-recovered-susceptible) system (see [10, 9]). Many diseases such as measles, severe acute respiratory syndromes (SARS), and so on, however, incubate inside the hosts for a period of time before the hosts become infectious. So the systems that are more general than SIR or SIRS types need to be studied to investigate the role of incubation in disease transmission. We may assume that a susceptible individual first goes through a latent period (and is said to become exposed or in the class E) after infection before becoming infectious. Thus, the resulting models are of SEIR (susceptible-exposed-infectious-recovered) or SEIRS (susceptible-exposed-infectious-recovered-susceptible) types, respectively, depending on whether the acquired immunity is permanent or not.

Keywords

Measle Smallpox 

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References

  1. 1.
    Broer, H., Naudot, V., Roussarie, R., Saleh, K.: Dynamics of a predator-prey model with non- monotonic response function. Disc. Cont. Dyn. Syst. A 18, 221–251 (2007)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Broer, H., Simo, C., Vitolo, R.: Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance bubble. Phy. D 237, 1773–1799 (2008)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Broer, H., Simo, C., Vitolo, R.: The Hopf-saddle-node bifurcation for fixed points of 3D- diffeomorphisms: The Arnol’d resonance web. Bul. Bel. Math. Soc. Ste. 15, 769–787 (2008)MathSciNetMATHGoogle Scholar
  4. 4.
    Chen, L.S., Chen, J.: Nonlinear Biologic Dynamic Systems. Science Press, Beijing (1993)Google Scholar
  5. 5.
    Cooke, K.L., Driessche, P.V.: Analysis of an SEIRS epidemic model with two delays. J. Math. Bio. 35, 240–260 (1996)MATHCrossRefGoogle Scholar
  6. 6.
    Dai, L.: Singular Control Systems. Springer, Heidelberg (1998)Google Scholar
  7. 7.
    Fan, M., Michael, Y.L., Wang, K.: Global stability of an SEIS epidemic model with recruitment and a varying total population size. Math. Bio. 170, 199–208 (2001)MATHCrossRefGoogle Scholar
  8. 8.
    Ghoshal, G., Sander, L.M., Sokolov, I.M.: SIS epidemics with household structure: The self-consistent field method. Math. Bio. 190, 71–85 (2004)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Glendinning, P., Perry, L.P.: Melnikov analysis of chaos in a simple epidemiological model. J. Math. Biol. 35, 359–373 (1997)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Greenhalgh, D., Khan, Q.J.A., Lewis, F.I.: Hopf bifurcation in two SIRS density dependent epidemic models. Math. Comp. Model. 39, 1261–1283 (2004)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Greenhalgh, D.: Hopf bifurcation in epidemic models with a latent period and non-permanent immunity. Math. Compu. Model 25, 85–107 (1997)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations. Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983)MATHGoogle Scholar
  13. 13.
    Hilker, F.M., Michel, L., Petrovskii, S.V., Malchow, H.: A diffusive SI model with Allee effect and application to FIV. Math. Bio. 206, 61–80 (2007)MATHCrossRefGoogle Scholar
  14. 14.
    Isidori, A.: Nonlinear Control System. Springer, Berlin (1985)Google Scholar
  15. 15.
    Jia, Q.: Hyperchaos generated from the Lorenz chaotic system and its control. Phy. Lett. A 366, 217–222 (2007)MATHCrossRefGoogle Scholar
  16. 16.
    Jiang, M.J., Chen, C.L., Chen, C.K.: Sliding mode control of hyperchaos in Rossler systems. Chaos Soli. Frac. 14, 1465–1476 (2002)CrossRefGoogle Scholar
  17. 17.
    Jyi, M., Chen, C.L., Chen, C.K.: Sliding mode control of hyperchaos in Rossler systems. Chaos Soli. Frac. 14, 1465–1476 (2002)MATHCrossRefGoogle Scholar
  18. 18.
    Kamo, M., Sasaki, A.: The effect of cross-immunity and seasonal forcing in a multi-strain epidemic model. Phys. D 165, 228–241 (2002)MATHCrossRefGoogle Scholar
  19. 19.
    Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. London A 115, 700–721 (1927)MATHCrossRefGoogle Scholar
  20. 20.
    Kot, M.: Elements of Mathematical Biology. Cambridge University Press, Cambridge (2001)Google Scholar
  21. 21.
    Kuznetsov, Y.A., Piccardi, C.: Bifurcation analysis of periodic SEIR and SIR epidemic models. Math. Bio. 32, 109–121 (1994)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Li, X.Z., Gupur, G., Zhu, G.T.: Threshold and stability results for an age-structured SEIR epidemic model. Comp. Math. Appl. 42, 883–907 (2001)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Liu, W.M., Hethcote, H.W., Levin, S.A.: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Bio. 25, 359–380 (1987)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Lu, Z.H., Liu, X.N., Chen, L.S.: Hopf bifurcation of nonlinear incidence rates SIR epidemiological models with stage structure. Comm. Nonl. Sci. Num. Sim. 6, 205–209 (2001)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    May, R.M., Oster, G.F.: Bifurcation and dynamic complexity in simple ecological models. Amer. Nat. 110, 573–599 (1976)CrossRefGoogle Scholar
  26. 26.
    Michael, Y.L., Graef, J.R., Wang, L.C., Karsai, J.: Global dynamics of an SEIR model with varying total population size. Math. Bio. 160, 191–213 (1999)MATHCrossRefGoogle Scholar
  27. 27.
    Olsen, L.F., Schaffer, W.M.: Chaos versus periodicity: Alternative hypotheses for childhood epidemics. Science 249, 499–504 (1990)CrossRefGoogle Scholar
  28. 28.
    Rafikov, M., Balthazar, J.M.: On control and synchronization in chaotic and hyperchaotic systems via linear feedback control. Commu. Non. Sci. Num. Sim. 13, 1246–1255 (2008)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Rosehart, W.D., Canizares, C.A.: Bifurcation analysis of various power system models. Elec. Pow. Ene. Syst. 21, 171–182 (1999)CrossRefGoogle Scholar
  30. 30.
    Sun, C.J., Lin, Y.P., Tang, S.P.: Global stability for a special SEIR epidemic model with nonlinear incidence rates. Chaos Soli. Frac. 33, 290–297 (2007)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Venkatasubramanian, V., Schattler, H., Zaborszky, J.: Analysis of local bifurcation mechanisms in large differential-algebraic systems such as the power system. In: Proc. 32nd Conf. Deci. Cont., vol. 4, pp. 3727–3733 (1993)Google Scholar
  32. 32.
    Xu, W.B., Liu, H.L., Yu, J.Y., Zhu, G.T.: Stability results for an age-structured SEIR epidemic model. J. Sys. Sci. Inf. 3, 635–642 (2005)Google Scholar
  33. 33.
    Wang, J., Chen, C.: Nonlinear control of differential algebraic model in power systems. Proc. CSEE 21, 15–18 (2001)MATHGoogle Scholar
  34. 34.
    Yan, Z.Y., Yu, D.: Hyperchaos synchronization and control on a new hyperchaotic attractor. Chaos Soli. Frac. 35, 333–345 (2008)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Yau, H.T., Yan, J.J.: Robust controlling hyperchaos of the Rossler system subject to input nonlinearities by using sliding mode control. Chaos Soli. Frac. 33, 1767–1776 (2007)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Zeng, G.Z., Chen, L.S., Sun, L.H.: Complexity of an SIRS epidemic dynamics model with impulsive vaccination control. Chaos Soli. Frac. 26, 495–505 (2005)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Zhang, H., Ma, X.K., Li, M., Zou, J.K.: Controlling and tracking hyperchaotic Rossler system via active backstepping design. Chaos Soli. Frac. 26, 353–361 (2005)MATHCrossRefGoogle Scholar
  38. 38.
    Zhang, J.S.: Economy cybernetics of singular systems. Tsinghua University Press, Beijing (1990)Google Scholar
  39. 39.
    Zhang, Y., Zhang, Q.L., Zhao, L.C., Liu, P.Y.: Tracking control of chaos in singular biological economy systems. J. Nor. Uni. 28, 157–164 (2007)MathSciNetGoogle Scholar
  40. 40.
    Zhang, Y., Zhang, Q.L.: Chaotic control based on descriptor bioeconomic systems. Cont. Deci. 22, 445–452 (2007)Google Scholar
  41. 41.
    Zhou, X.B., Wu, Y., Li, Y., Xue, H.Q.: Adaptive control and synchronization of a novel hyperchaotic system with uncertain parameters. Appl. Math. Compu. 203, 80–85 (2008)MathSciNetMATHCrossRefGoogle Scholar

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© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.College of SciencesNortheastern UniversityShenyangChina, People’s Republic

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