SeMA Journal

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A short study of an SIR model with inclusion of an alert class, two explicit nonlinear incidence rates and saturated treatment rate

  • Abhishek Kumar
  • NilamEmail author
  • Raj Kishor


In this paper, we present a susceptible–alert–infected–recovered (SAIR) epidemic model with the consideration of two explicit saturated incidence rates and Holling functional type II treatment rate. Awareness about the epidemic may play a vital role in the control of the spread of an epidemic. Hence, an alert compartment has been incorporated into the model. It strives us to take two incidence rates: one from the susceptible class to infected class and another from alert class to infected class. Holling functional type II treatment rate has been introduced to capture the effects of resource limitation in treating infectives. The model has a disease-free equilibrium (DFE), which is locally asymptotically stable when \( R_{0} < 1 \). Using the center manifold theory, we show that DFE exhibits the forward bifurcation at \( R_{0} = 1 \). Stability of the endemic equilibrium has also been analyzed and discussed. Numerical simulations have been done by MATLAB 2012b and the outcomes have been discussed with the help of graphs in the paper.


Epidemic SAIR model Saturated treatment rate Basic reproduction number Center manifold theory Stability 

Mathematics Subject Classification

34D20 92B05 37M05 



The authors are thankful to Delhi Technological University, Delhi, for monetary support for this research. The authors also gratefully acknowledge the handling editor and anonymous reviewers for their valuable suggestions which enhance the quality of the paper.


  1. 1.
    Alexander, M.E., Bowman, C., Moghadas, S.M., Summers, R., Gumel, A.B., Sahai, B.M.: A vaccination model for transmission dynamics of influenza. SIAM J. Appl. Dyn. Syst. 3(4), 503–524 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Buonomo, B., d’Onofrio, A., Lacitignola, D.: Global stability of an SIR epidemic model with information dependent vaccination. Math. Biosci. 216(1), 9–16 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Capasso, V., Serio, G.: A generalization of the Kermack–Mckendrick deterministic epidemic model. Math. Biosci. 42, 41–61 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Castillo-Chavez, C., Song, B.: Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1, 361–404 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Driessche, P.V.D., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartment models of disease transmission. Math. Biosci. 180, 29–48 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dubey, B., Patara, A., Srivastava, P.K., Dubey, U.S.: Modeling and analysis of a SEIR model with different types of nonlinear treatment rates. J. Biol. Syst. 21(3), 1350023 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Goel, K., Nilam, : A mathematical and numerical study of a SIR epidemic model with time delay, nonlinear incidence and treatment rates. Theory Biosci. (2019). Google Scholar
  8. 8.
    Gumel, A.B., Mccluskey, C.C., Watmough, J.: An SVEIR model for assessing the potential impact of an imperfect anti-SARS vaccine. Math. Biosci. Eng. 3, 485–494 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hattaf, K., Lashari, A.A., Louartassi, Y., Yousfi, N.: A delayed SIR epidemic model with general incidence rate. Electron. J. Qual. Theory Differ. Equ. 3, 1–9 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Karim, S.A.A., Razali, R.: A proposed mathematical model of influenza A, H1N1 for Malaysia. J. Appl. Sci. 11(8), 1457–1460 (2011)Google Scholar
  11. 11.
    Kumar, A., Nilam, : Stability of a time delayed SIR epidemic model along with nonlinear incidence rate and Holling type II treatment rate. Int. J. Comput. Methods 15(6), 1850055 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kumar, A., Nilam, : Dynamical model of epidemic along with time delay; Holling type II incidence rate and Monod-Haldane treatment rate. Differ. Equ. Dyn. Syst. 27(1–3), 299–312 (2019)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Li Michael, Y., Graef, J.R., Wang, L., Karsai, J.: Global dynamics of a SEIR model with varying total population size. Math. Biosci. 160, 191–213 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Li, G.H., Zhang, Y.X.: Dynamic behavior of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates. PLoS ONE 12(4), e0175789 (2017)CrossRefGoogle Scholar
  15. 15.
    Liu, W.M., Hethcote, H.W., Levin, S.A.: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25(4), 359–380 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mena-Lorca, J., Hethcote, H.W.: Dynamic models of infectious disease as regulators of population size. J. Math. Biol. 30(7), 693–716 (1992)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Sastry, S.: Analysis, Stability and Control. Springer, New York (1999)zbMATHGoogle Scholar
  18. 18.
    Wang, X.: A simple proof of Descartes’s rule of signs. Am. Math. Mon. (2004). Google Scholar
  19. 19.
    Wang, W., Ruan, S.: Bifurcation in an epidemic model with constant removal rates of the infectives. J. Math. Anal. Appl. 21, 775–793 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Xu, R., Ma, Z.: Stability of a delayed SIRS epidemic model with a nonlinear incidence rate. Chaos Solut. Fractals 41, 2319–2325 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhang, Z., Suo, S.: Qualitative analysis of a SIR epidemic model with saturated treatment rate. J. Appl. Math. Comput. 34, 177–194 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhou, L., Fan, M.: Dynamics of a SIR epidemic model with limited medical resources revisited. Nonlinear Anal. RWA 13, 312–324 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsDelhi Technological UniversityDelhiIndia
  2. 2.Raj TutorsBijnorIndia

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