# American Institute of Mathematical Sciences

2013, 10(4): 1227-1251. doi: 10.3934/mbe.2013.10.1227

## Mitigation of epidemics in contact networks through optimal contact adaptation

 1 K-State Epicenter, Department of Electrical and Computer Engineering, Kansas State University, 2061 Rathbone Hall, Manhattan, KS 66506-5204, United States, United States

Received  September 2012 Revised  March 2013 Published  June 2013

This paper presents an optimal control problem formulation to minimize the total number of infection cases during the spread of susceptible-infected-recovered SIR epidemics in contact networks. In the new approach, contact weighted are reduced among nodes and a global minimum contact level is preserved in the network. In addition, the infection cost and the cost associated with the contact reduction are linearly combined in a single objective function. Hence, the optimal control formulation addresses the tradeoff between minimization of total infection cases and minimization of contact weights reduction. Using Pontryagin theorem, the obtained solution is a unique candidate representing the dynamical weighted contact network. To find the near-optimal solution in a decentralized way, we propose two heuristics based on Bang-Bang control function and on a piecewise nonlinear control function, respectively. We perform extensive simulations to evaluate the two heuristics on different networks. Our results show that the piecewise nonlinear control function outperforms the well-known Bang-Bang control function in minimizing both the total number of infection cases and the reduction of contact weights. Finally, our results show awareness of the infection level at which the mitigation strategies are effectively applied to the contact weights.
Citation: Mina Youssef, Caterina Scoglio. Mitigation of epidemics in contact networks through optimal contact adaptation. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1227-1251. doi: 10.3934/mbe.2013.10.1227
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