Differential Equations and Dynamical Systems

, Volume 27, Issue 1–3, pp 299–312

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

• Abhishek Kumar
• Nilam
Original Research

## Abstract

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.

## Keywords

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

## Notes

### Acknowledgements

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.

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