Global Stability of an SIS Epidemic Model with Age of Vaccination

Original Research


In this paper, an SIS epidemic model with age of vaccination is investigated. Asymptotic smoothness of the semi-flow is proved. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state is discussed. It is shown that if the basic reproduction number is greater than unity, the system is permanent. By constructing two Lyapunov functionals, it is proved that the endemic steady state is globally asymptotically stable if the basic reproduction number is greater than unity, and sufficient conditions are derived for the global asymptotic stability of the disease-free steady state. Numerical simulations are given to illustrate the asymptotic stabilities of the disease-free steady state and endemic state.


SIS epidemic model Vaccine-age Asymptotic smoothness Uniform persistence Global stability 



This work was supported by the National Natural Science Foundation of China (No. 11371368), the Natural Science Foundation of Hebei Province (Nos. A2014506015, A2016506002), the Science Foundation of Shijiazhuang Mechanical Engineering College (Nos. YJJXM 13008, JCYJ14011)


  1. 1.
    Elbasha, E., Dasbach, E., Insinga, R.: Model for assessing human papillomavirus vaccination strategies. Emerg. Infect. Dis. 13, 28–41 (2007)CrossRefGoogle Scholar
  2. 2.
    Xu, R.: Global stability of a delayed epidemic model with latent period and vaccination strategy. Appl. Math. Model. 36, 5293–5300 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cai, L.M., Li, X.Z.: Analysis of a SEIV epidemic model with a nonlinear incidence rate. Appl. Math. Model. 33, 2919–2926 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Wendelboe, A.M., Van Rie, A., Salmaso, S., Englund, J.A.: Duration of immunity against pertussis after natural infection or vaccination. Pediatr. Infect. Dis. J. 24(5 Suppl.), 58–61 (2005)CrossRefGoogle Scholar
  5. 5.
    Chaves, S.S., Gargiullo, P., Zhang, J.X., Civen, R., Guris, D., Mascola, L., Seward, J.F.: Loss of vaccine-induced immunity to varicella over time. N. Engl. J. Med. 356(11), 1121–1129 (2007)CrossRefGoogle Scholar
  6. 6.
    Li, J., Ma, Z.: Global analysis of SIS epidemic models with variable total population size. Math. Comput. Modell. 39, 1231–1242 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Francis, D.P., Feorino, P.M., Broderson, J.R., Mccluer, H.M., Getchell, J.P., Mcgrath, C.R., Swenson, B., Mcdugal, J.S., Palmer, E.L., Harrison, A.K., et al.: Infection of chimpanzees with lymphadenopathy-associated virus. Lancet 2, 1276–1277 (1984)CrossRefGoogle Scholar
  8. 8.
    Lange, J.M., Paul, D.A., Huisman, H.G., de Wolf, F., van den Berg, H., et al.: Persistent HIV antigenaemia and decline of HIV core antibodies associated with transition to AIDS. Br. Med. J. 293, 1459–1462 (1986)CrossRefGoogle Scholar
  9. 9.
    Hoppensteadt, F.: An age-dependent epidemic model. J. Franklin Inst. 297, 325–338 (1974)CrossRefMATHGoogle Scholar
  10. 10.
    Iannelli, M., Martcheva, M., Li, X.Z.: Strain replacement in an epidemic model with super-infection and perfect vaccination. Math. Biosci. 195, 23–46 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Li, X.Z., Wang, J., Ghosh, M.: Stability and bifurcation of an SIVS epidemic model with treatment and age of vaccination. Appl. Math. Model. 34, 437–450 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Liu, L., Wang, J., Liu, X.: Global stability of an SEIR epidemic model with age-dependent latency and relapse. Nonlinear Anal. Real World Appl. 24, 18–35 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Wang, J., Zhang, R., Kuniya, T.: The stablity analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes. J. Biol. Dyn. 9(1), 73–101 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hale, J.K.: Functional Differential Equations. Springer, Berlin (1971)CrossRefMATHGoogle Scholar
  15. 15.
    Hale, J.K., Waltman, P.: Persistence in infinite-dimensional systems. SIAM J. Math. Anal. 20(2), 388–395 (1989)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Webb, G.F.: Theory of Nonlinear Age-dependent Population Dynamics. Marcel Dekker, New York (1985)MATHGoogle Scholar
  17. 17.
    Iannelli, M.: Mathematical Theory of Age-Structured Population Dynamics. Giardini Editori e Stampatori, Pisa (1994)Google Scholar
  18. 18.
    Adams, R.A., Fournier, J.J.: Sobolev Spaces. Academic Press, New York (2003)MATHGoogle Scholar
  19. 19.
    Magal, P.: Compact attractors for time periodic age-structured population models. Electron. J. Differ. Equ. 65, 1–35 (2001)MathSciNetMATHGoogle Scholar
  20. 20.
    Magal, P., Zhao, X.-Q.: Global attractors and steady states for uniformly persistent dynamical systems. SIAM J. Math. Anal. 37(1), 251–275 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2018

Authors and Affiliations

  1. 1.Institute of Applied MathematicsShijiazhuang Mechanical Engineering CollegeShijiazhuangPeople’s Republic of China

Personalised recommendations