# American Institute of Mathematical Sciences

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

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

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

Received  September 2010 Revised  March 2011 Published  December 2011

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