Mathematical Biosciences and Engineering (MBE)

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

Pages: 97 - 110, Volume 9, Issue 1, January 2012      doi:10.3934/mbe.2012.9.97

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Zhenyuan Guo - College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China (email)
Lihong Huang - College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China (email)
Xingfu Zou - Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada (email)

Abstract: 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.

Keywords:  Infectious disease, SIR model, discontinuous treatment, stability, generalized Lyapunov method, convergence in finite time.
Mathematics Subject Classification:  Primary: 92D30; Secondary:34C23.

Received: September 2010;      Accepted: March 2011;      Available Online: December 2011.