Startseite Technik Dynamic Behavior of an SIR Epidemic Model along with Time Delay; Crowley–Martin Type Incidence Rate and Holling Type II Treatment Rate
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Dynamic Behavior of an SIR Epidemic Model along with Time Delay; Crowley–Martin Type Incidence Rate and Holling Type II Treatment Rate

  • Abhishek Kumar und Nilam EMAIL logo
Veröffentlicht/Copyright: 13. August 2019
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Abstract

In this article, we propose and analyze a time-delayed susceptible–infected–recovered (SIR) mathematical model with nonlinear incidence rate and nonlinear treatment rate for the control of infectious diseases and epidemics. The incidence rate of infection is considered as Crowley–Martin functional type and the treatment rate is considered as Holling functional type II. The stability of the model is investigated for the disease-free equilibrium (DFE) and endemic equilibrium (EE) points. From the mathematical analysis of the model, we prove that the model is locally asymptotically stable for DFE when the basic reproduction number R0 is less than unity (R0<1) and unstable when R0 is greater than unity (R0>1) for time lag τ0. The stability behavior of the model for DFE at R0=1 is investigated using Castillo-Chavez and Song theorem, which shows that the model exhibits forward bifurcation at R0=1. We investigate the stability of the EE for time lag τ0. We also discussed the Hopf bifurcation of EE numerically. Global stability of the model equilibria is also discussed. Furthermore, the model has been simulated numerically to exemplify analytical studies.

MSC 2010: 34D20; 37M05; 92B05

Acknowledgements

The authors are gratefully acknowledged to Delhi Technological University for providing financial support for this research. The authors are thankful to the handling editor and anonymous referees for their critical reviews and useful suggestions to enhance the paper.

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Received: 2018-01-23
Accepted: 2019-07-22
Published Online: 2019-08-13
Published in Print: 2019-11-18

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 5.1.2026 von https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2018-0208/pdf
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