Impact of discontinuous treatments on disease dynamics in an SIRepidemic model - Zhenyuan Guo; Lihong Huang; Xingfu Zou (Mathematical Biosciences and Engineering)
We consider an SIR epidemic model withdiscontinuous treatment strategies. Under some reasonableassumptions on the discontinuous treatment function, we are able todetermine the basic reproduction number $\mathcal{R}_0$, confirm thewell-posedness of the model, describe the structure of possibleequilibria as well as establish the stability/instability of theequilibria. Most interestingly, we find that in the case that anequilibrium is asymptotically stable, the convergence to theequilibrium can actually be achieved in finite time, and wecan estimate this time in terms of the model parameters, initialsub-populations and the initial treatment strength. This suggeststhat from the view point of eliminating the disease from the hostpopulation, discontinuous treatment strategies would be superior tocontinuous ones. The methods we use to obtain the mathematicalresults are the generalized Lyapunov theory for discontinuousdifferential equations and some results on non-smooth analysis.