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A deterministic time-delayed SIR epidemic model: mathematical modeling and analysis

Abstract

In this paper, a deterministic model for transmission of an epidemic has been proposed by dividing the total population into three subclasses, namely susceptible, infectious and recovered. The incidence rate of infection is taken as a nonlinear functional along with time delay, and treatment rate of infected is considered as Holling type III functional. We have structured a deterministic transmission model of the epidemic taking into account the factors that affect the epidemic transmission such as social and natural factors, inhibitory effects and numerous control measures. The delayed model has been analyzed mathematically for two equilibria, namely disease-free equilibrium (DFE) and endemic equilibrium. It is found that DFE is locally and globally asymptotically stable when the basic reproduction number \( (R_{0} ) \) is less than unity. It has also been shown that the delayed system for DFE at \( R_{0} = 1 \) is linearly neutrally stable. The existence of an endemic equilibrium has been shown and found that under some conditions, endemic equilibrium is locally asymptotically stable, and is globally asymptotically stable when \( R_{0} > 1 \). Further, the endemic equilibrium exhibits Hopf bifurcation under some conditions. Finally, an undelayed system has been analyzed, and it is shown that at \( R_{0} = 1 \), DFE exhibits a forward bifurcation.

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Acknowledgements

The authors acknowledged Delhi Technological University for providing the monetary help for this research. The authors thank the handling editor and anonymous reviewers for their careful reading of our manuscript and their insightful comments and suggestions.

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Appendix

Appendix

  1. 1.

    The linearized matrix corresponding to the infection-free equilibrium \( Q(\frac{A}{\mu },0) \) is

    $$ J_{1} = \left( {\begin{array}{*{20}c} { - \mu } & { - \frac{\alpha A}{\mu }{\text{e}}^{ - \lambda \tau } } \\ 0 & {\frac{\alpha A}{\mu }{\text{e}}^{ - \lambda \tau } - (\mu + \sigma + \theta )} \\ \end{array} } \right) $$
  2. 2.

    The linearized matrix corresponding to the endemic equilibrium is

    $$ J_{2} = \left( {\begin{array}{*{20}c} { - \mu - \frac{{\alpha Y^{*} }}{{1 + \beta Y^{*} }}} & { - \frac{{\alpha X^{*} }}{{\left( {1 + \beta Y^{*} } \right)^{2} }}{\text{e}}^{ - \lambda \tau } } \\ {\frac{{\alpha Y^{*} }}{{1 + \beta Y^{*} }}} & {\frac{{\alpha X^{*} }}{{\left( {1 + \beta Y^{*} } \right)^{2} }}{\text{e}}^{ - \lambda \tau } - (\mu + \sigma + \theta ) - \frac{{2\omega Y^{*} }}{{\left( {1 + \varphi Y^{{*^{2} }} } \right)^{2} }}} \\ \end{array} } \right) $$

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Kumar, A., Goel, K. & Nilam A deterministic time-delayed SIR epidemic model: mathematical modeling and analysis. Theory Biosci. 139, 67–76 (2020). https://doi.org/10.1007/s12064-019-00300-7

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Keywords

  • Epidemic
  • Delay SIR model
  • Holling type III treatment rate
  • Nonlinear incidence rate
  • Stability
  • Bifurcation