Mathematical Biosciences and Engineering, 2012, 9(1): 97-110. doi: 10.3934/mbe.2012.9.97.

Primary: 92D30; Secondary:34C23.

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Impact of discontinuous treatments on disease dynamics in an SIR epidemic model

1. College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082
2. Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7

We consider an SIR epidemic model with discontinuous treatment strategies. Under some reasonable assumptions on the discontinuous treatment function, we are able to determine the basic reproduction number $\mathcal{R}_0$, confirm the well-posedness of the model, describe the structure of possible equilibria as well as establish the stability/instability of the equilibria. Most interestingly, we find that in the case that an equilibrium is asymptotically stable, the convergence to the equilibrium can actually be achieved in finite time, and we can estimate this time in terms of the model parameters, initial sub-populations and the initial treatment strength. This suggests that from the view point of eliminating the disease from the host population, discontinuous treatment strategies would be superior to continuous ones. The methods we use to obtain the mathematical results are the generalized Lyapunov theory for discontinuous differential equations and some results on non-smooth analysis.
  Article Metrics

Keywords discontinuous treatment; Infectious disease; convergence in finite time.; SIR model; generalized Lyapunov method; stability

Citation: Zhenyuan Guo, Lihong Huang, Xingfu Zou. Impact of discontinuous treatments on disease dynamics in an SIR epidemic model. Mathematical Biosciences and Engineering, 2012, 9(1): 97-110. doi: 10.3934/mbe.2012.9.97


This article has been cited by

  • 1. Zuowei Cai, Lihong Huang, Periodic dynamics of delayed Lotka–Volterra competition systems with discontinuous harvesting policies via differential inclusions, Chaos, Solitons & Fractals, 2013, 54, 39, 10.1016/j.chaos.2013.05.005
  • 2. Jiafu Wang, Fengqin Zhang, Lin Wang, Equilibrium, pseudoequilibrium and sliding-mode heteroclinic orbit in a Filippov-type plant disease model, Nonlinear Analysis: Real World Applications, 2016, 31, 308, 10.1016/j.nonrwa.2016.01.017
  • 3. Zuowei Cai, Lihong Huang, Lingling Zhang, Xiaolian Hu, Dynamical behavior for a class of predator-prey system with general functional response and discontinuous harvesting policy, Mathematical Methods in the Applied Sciences, 2015, 38, 18, 4679, 10.1002/mma.3379
  • 4. Tailei Zhang, Ruini Kang, Kai Wang, Junli Liu, Global dynamics of an SEIR epidemic model with discontinuous treatment, Advances in Difference Equations, 2015, 2015, 1, 10.1186/s13662-015-0695-0
  • 5. Yilei Tang, Dongmei Xiao, Weinian Zhang, Di Zhu, Dynamics of epidemic models with asymptomatic infection and seasonal succession, Mathematical Biosciences and Engineering, 2017, 14, 5/6, 1407, 10.3934/mbe.2017073
  • 6. Daozhong Luo, Dongshu Wang, On almost periodicity of delayed predator-prey model with mutual interference and discontinuous harvesting policies, Mathematical Methods in the Applied Sciences, 2016, 39, 15, 4311, 10.1002/mma.3861
  • 7. Da-peng Gao, Nan-jing Huang, Threshold dynamics of an SEIR epidemic model with a nonlinear incidence rate and a discontinuous treatment function, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2020, 114, 1, 10.1007/s13398-019-00751-z

Reader Comments

your name: *   your email: *  

Copyright Info: 2012, , licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved