Impact of discontinuous treatments on disease dynamics in an SIR
epidemic model
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1.
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082
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2.
Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7
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Received:
01 September 2010
Accepted:
29 June 2018
Published:
01 December 2011
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MSC :
Primary: 92D30; Secondary:34C23.
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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 SIRepidemic model[J]. Mathematical Biosciences and Engineering, 2012, 9(1): 97-110. doi: 10.3934/mbe.2012.9.97
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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.
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