Dynamics of an \({ SVEIRS}\) Epidemic Model with Vaccination and Saturated Incidence Rate

  • Kunwer Singh MathurEmail author
  • Prakash Narayan
Original Paper


Measles and influenza are two major diseases–caused an epidemic in India. Therefore, in this paper, a \({ SVEIRS}\) epidemic mathematical model for measles and influenza is proposed and analyzed, where pre and post vaccinations are considered as control strategies with waning natural, vaccine-induced immunity and saturation incidence rate. The dissection of the proposed model is conferred in terms of the associated reproduction number \({\mathcal {R}}_v\), which is determined by the next-generation approach and obtained that if \({\mathcal {R}}_v\le 1\), the disease-free equilibrium exists and it is locally as well as globally asymptotically stable. Further for \({\mathcal {R}}_v> 1\), a unique endemic equilibrium exists and it is also locally as well as globally asymptotically stable under certain conditions, which shows the prevalence and persistence of the disease in the population.


Pre and post vaccinations Reproduction number Saturated incidence rate Global stability 



We are very thankful to the anonymous referees and the editor in chief for their careful reading, constructive criticisms, helpful comments, and valuable suggestions, which have helped us to improve the quality of this work significantly.


  1. 1.
    Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Kar, T., Jana, S.: A theoretical study on mathematical modelling of an infectious disease with application of optimal control. Biosystems 111(1), 37–50 (2013)CrossRefGoogle Scholar
  3. 3.
    Havelaar, A.H., Swart, A.: Impact of waning acquired immunity and asymptomatic infections on case-control studies for enteric pathogens. Epidemics 17, 56–63 (2016)CrossRefGoogle Scholar
  4. 4.
    Tian, X., Xu, R.: Asymptotic properties of a Hepatitis B virus infection model with time delay. Discrete Dyn. Nat. Soc. 2010, 1–21 (2010)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Truscott, J., Webb, C., Gilligan, C.: Asymptotic analysis of an epidemic model with primary and secondary infection. Bull. Math. Biol. 59(6), 1101–1123 (1997)CrossRefGoogle Scholar
  6. 6.
    Robinson, M., Stilianakis, N.I.: A model for the emergence of drug resistance in the presence of asymptomatic infections. Math. Biosci. 243(2), 163–177 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Martcheva, M.: An Introduction to Mathematical Epidemiology, vol. 61. Springer, New York (2015)zbMATHGoogle Scholar
  8. 8.
    Hsu, S.B., Hsieh, Y.H.: On the role of asymptomatic infection in transmission dynamics of infectious diseases. Bull. Math. Biol. 70(1), 134–155 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Elbasha, E., Podder, C., Gumel, A.: Analyzing the dynamics of an SIRS vaccination model with waning natural and vaccine-induced immunity. Nonlinear Anal. Real World Appl. 12(5), 2692–2705 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Sun, C., Hsieh, Y.H.: Global analysis of an SEIR model with varying population size and vaccination. Appl. Math. Model. 34(10), 2685–2697 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Samadder, A., Ghosh, K., Chaudhuri, K.: A mathematical model of epidemiology in presence of vaccination for the spread of contagious diseases transmitting without vector. World J. Model. Simul. 9(3), 192–200 (2013)Google Scholar
  12. 12.
    Liu, D., Wang, B.: A novel time delayed HIV/AIDS model with vaccination & antiretroviral therapy and its stability analysis. Appl. Math. Model. 37(7), 4608–4625 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Edmunds, W.J., Gay, N.J., Kretzschmar, M., Pebody, R., Wachmann, H.: The pre-vaccination epidemiology of measles, mumps and rubella in europe: implications for modelling studies. Epidemiol. Infect. 125(3), 635–650 (2000)CrossRefGoogle Scholar
  14. 14.
    Manfredi, P., Cleur, E.M., Williams, J.R., Salmaso, S., Degli Atti, M.C.: The pre-vaccination regional epidemiological landscape of measles in italy: contact patterns, effort needed for eradication, and comparison with other regions of europe. Popul. Health Metrics 3(1), 1–16 (2005)CrossRefGoogle Scholar
  15. 15.
    Sahu, G.P., Dhar, J.: Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate. Appl. Math. Model. 36(3), 908–923 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cai, L.M., Li, X.Z.: Analysis of a SEIV epidemic model with a nonlinear incidence rate. Appl. Math. Model. 33(7), 2919–2926 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Schenzle, D.: An age-structured model of pre- and post-vaccination measles transmission. Math. Med. Biol. J. IMA 1(2), 169–191 (1984)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Harris, R.C., Sumner, T., Knight, G.M., White, R.G.: Systematic review of mathematical models exploring the epidemiological impact of future TB vaccines. Hum. Vaccines Immunother 12(11), 2813–2832 (2016)CrossRefGoogle Scholar
  19. 19.
    Kermark, M., Mckendrick, A.: Contributions to the mathematical theory of epidemics, part I. Proc. R. Soc. Lond. A 115, 700–721 (1927)CrossRefGoogle Scholar
  20. 20.
    Misra, A., Sharma, A., Shukla, J.: Stability analysis and optimal control of an epidemic model with awareness programs by media. Biosystems 138, 53–62 (2015)CrossRefGoogle Scholar
  21. 21.
    Cai, L., Li, X., Ghosh, M., Guo, B.: Stability analysis of an HIV/AIDS epidemic model with treatment. J. Comput. Appl. Math. 229(1), 313–323 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wilson, E.B., Worcester, J.: The law of mass action in epidemiology. Proc. Natl. Acad. Sci. 31(1), 24–34 (1945)CrossRefGoogle Scholar
  23. 23.
    Wilson, E.B., Worcester, J.: The law of mass action in epidemiology II. Proc. Natl. Acad. Sci. 31(4), 109–116 (1945)CrossRefGoogle Scholar
  24. 24.
    Tian, B., Yuan, R.: Traveling waves for a diffusive SEIR epidemic model with non-local reaction and with standard incidences. Nonlinear Anal. Real World Appl. 37, 162–181 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zhang, J., Jia, J., Song, X.: Analysis of an SEIR Epidemic Model with Saturated Incidence and Saturated Treatment Function. Sci. World J. 2014, 1–11 (2014)Google Scholar
  26. 26.
    Capasso, V., Serio, G.: A generalization of the Kermack-McKendrick deterministic epidemic model. Math. Biosci. 42(1–2), 43–61 (1978)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Liu, X., Yang, L.: Stability analysis of an SEIQV epidemic model with saturated incidence rate. Nonlinear Anal. Real World Appl. 13(6), 2671–2679 (2012)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wang, J., Jiang, Q.: Analysis of an SIS epidemic model with treatment. Adv. Differ. Equ. 2014(1), 246 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Longini Jr., I.M., Halloran, M.E., Nizam, A., Yang, Y.: Containing pandemic influenza with antiviral agents. Am. J. Epidemiol. 159(7), 623–633 (2004)CrossRefGoogle Scholar
  30. 30.
    Longini Jr., I.M., Nizam, A., Xu, S., Ungchusak, K., Hanshaoworakul, W., Cummings, D.A., Halloran, M.E.: Containing pandemic influenza at the source. Science 309(5737), 1083–1087 (2005)CrossRefGoogle Scholar
  31. 31.
    Germann, T.C., Kadau, K., Longini, I.M., Macken, C.A.: Mitigation strategies for pandemic influenza in the united states. Proc. Natl. Acad. Sci. 103(15), 5935–5940 (2006)CrossRefGoogle Scholar
  32. 32.
    Stilianakis, N.I., Perelson, A.S., Hayden, F.G.: Emergence of drug resistance during an influenza epidemic: insights from a mathematical model. J. Infect. Dis. 177(4), 863–873 (1998)CrossRefGoogle Scholar
  33. 33.
    Chan, P.K.: Outbreak of avian influenza a (H5N1) virus infection in Hong Kong in 1997. Clin. Infect. Dis. 34(Supplement–2), S58–S64 (2002)CrossRefGoogle Scholar
  34. 34.
    W.H.O.W. Group: Nonpharmaceutical interventions for pandemic influenza, national and community measures. Emerg. Infect. Dis. 12(1) 88–94 (2006)Google Scholar
  35. 35.
    Nafta, I., Ţurcanu, A., Braun, I., Companetz, W., Simionescu, A., Birţ, E., Florea, V.: Administration of amantadine for the prevention of Hong Kong influenza. Bull. World Health Organ. 42(3), 423–427 (1970)Google Scholar
  36. 36.
    Oker-Blom, N., Hovi, T., Leinikki, P., Palosuo, T., Pettersson, R., Suni, J.: Protection of man from natural infection with influenza A2 Hong Kong virus by amantadine: a controlled field trial. Br. Med. J. 3(5724), 676–678 (1970)CrossRefGoogle Scholar
  37. 37.
    Monto, A.S., Gunn, R.A., Bandyk, M.G., King, C.L.: Prevention of russian influenza by amantadine. J. Am.Med. Assoc. 241(10), 1003–1007 (1979)CrossRefGoogle Scholar
  38. 38.
    Pettersson, R., Hellström, P.E., Penttinen, K., Pyhälä, R., Tokola, O., Vartio, T., Visakorpi, R.: Evaluation of amantadine in the prophylaxis of influenza A (H1N1) virus infection: a controlled field trial among young adults and high-risk patients. J. Infect. Dis. 142(3), 377–383 (1980)CrossRefGoogle Scholar
  39. 39.
    Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1–2), 29–48 (2002)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Castillo-Chavez, C., Feng, Z., Huang, W.: On the computation of \({\cal{R}}_0\) and its role on global stability. In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, the IMA Volumes in Mathematics and its Applications vol. 1, pp. 229–250 (2002)CrossRefGoogle Scholar
  41. 41.
    Samsuzzoha, M., Singh, M., Lucy, D.: Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza. Appl. Math. Model. 37(3), 903–915 (2013)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Sahu, G.P., Dhar, J.: Dynamics of an SEQIHRS epidemic model with media coverage, quarantine and isolation in a community with pre-existing immunity. J. Math. Anal. Appl. 421(2), 1651–1672 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsDr. Harisingh Gour VishwavidyalayaSagarIndia

Personalised recommendations