Differential Equations and Dynamical Systems

, Volume 27, Issue 1–3, pp 299–312 | Cite as

Dynamical Model of Epidemic Along with Time Delay; Holling Type II Incidence Rate and Monod–Haldane Type Treatment Rate

  • Abhishek Kumar
  • NilamEmail author
Original Research


The present study aims to control the infectious diseases and epidemics in the human population. Therefore, in the present article, we have proposed a delayed SIR epidemic model along with Holling type II incidence rate and treatment rate as Monod–Haldane type. Model stability has been established in the three regions of the basic reproduction number \( {\text{R}}_{0} \) i.e. \( {\text{R}}_{0} \) equals to one, greater than one and less than one. The model is locally asymptotically stable for disease-free equilibrium \( {\text{Q}} \) when the basic reproduction number \( {\text{R}}_{0} \) is less than one (\( {\text{R}}_{0} < 1) \) and unstable when \( {\text{R}}_{0} > 1 \) for time lag \( \tau \ge 0 \). We investigated the stability of the model for disease-free equilibrium at \( {\text{R}}_{0} \) equals to one using central manifold theory. Using center manifold theory, we proved that at \( {\text{R}}_{0} = 1 \), disease-free equilibrium changes its stability from stable to unstable. We also investigated the stability for endemic equilibrium \( {\text{Q}}^{ *} \) for time lag \( \tau \ge 0 \). Further, numerical simulations are presented to exemplify the analytical studies.


Epidemic SIR model Delay differential equation Monod–Haldane type treatment rate Holling type II incidence rate Stability Center manifold theory 

Mathematics Subject Classification

34D20 92B05 37M05 



The authors would like to gratefully acknowledge Delhi Technological University, Delhi, India for providing financial support to carry out this research work. Authors are also grateful to the anonymous referees for their valuable reviews and suggestions which improved the quality of the paper.


  1. 1.
    Gumel, A.B., Connell Mccluskey, 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
  2. 2.
    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(1), 1850055 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dubey, B., Patara, A., Srivastava, P.K., Dubey, U.S.: Modelling and analysis of a SEIR model with different types of nonlinear treatment rates. J. Biol. Syst. 21(3), 1350023 (2013)CrossRefzbMATHGoogle Scholar
  4. 4.
    Dubey, B., Dubey, P., Dubey, Uma S.: Dynamics of a SIR model with nonlinear incidence rate and treatment rate. Appl. Appl. Maths. 2(2), 718–737 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Capasso, V., Serio, G.: A generalization of the Kermack–Mckendrick deterministic epidemic model. Math. Biosci. 42, 41–61 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Castillo-Chavez, C., Song, B.: Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1, 361–404 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Baek, H., Do, Y., Saito, Y.: Analysis of an impulsive predator–prey system with Monod-Haldane functional response and seasonal effects. Math. Prob. Eng. 29, Article Id 543187 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Andrews, J.F.: A mathematical model for the continuous culture of microorganisms untilizing inhibitory substrates. Biotechnol. Bioeng. 10, 707–723 (1968)CrossRefGoogle Scholar
  9. 9.
    Hattaf, K., Yousfi, N.: Mathematical model of influenza A (H1N1) infection. Adv. Std. Biol. 1, 383–390 (2009)Google Scholar
  10. 10.
    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
  11. 11.
    Dubey, P., Dubey, B., Dubey, U.S.: An SIR model with nonlinear incidence rate and Holling type III treatment rate. Appl. Anal. Biol. Phys. Sci. 186, 63–81 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Driessche, P.V.D., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartment models of disease transmission, math. Bioscience 180, 29–48 (2002)zbMATHGoogle Scholar
  13. 13.
    Xu, R., Ma, Z.: Stability of a delayed SIRS epidemic model with a nonlinear incidence rate. Chaos Solut. Fract. 41, 2319–2325 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sahani, S.K., Yashi: Analysis of a delayed HIV infection model, International workshop on a computational intelligence (IWCI) Dhaka, Bangladesh, pp. 246–251. (2016)
  15. 15.
    Ruan, S., Wei, J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two. Dyn. Contin. Discret. Impuls. Syst. Ser. A Math. Anal. 10, 863–874 (2003)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Sastry, S.: Analysis, Stability and Control. Springer, New York (1999)zbMATHGoogle Scholar
  17. 17.
    Tipsri, S., Chinviriyasit, W.: Stability analysis of SEIR model with saturated incidence and time delay. Int. J. Appl. Phys. Math. 4(1), 42–45 (2014)CrossRefzbMATHGoogle Scholar
  18. 18.
    Sokol, W., Howell, J.A.: Kineties of phenol oxidation by ashed cell. Biotechnol. Bioeng. 23, 2039–2049 (1980)CrossRefGoogle Scholar
  19. 19.
    Wang, X.: A simple proof of Descartes’s rule of signs. Am. Math. Mon. (2004). Google Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsDelhi Technological UniversityDelhiIndia

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