### Mathematical Biosciences and Engineering

2013, Issue 4: 1227-1251. doi: 10.3934/mbe.2013.10.1227

# Mitigation of epidemics in contact networks through optimal contact adaptation

• Received: 01 September 2012 Accepted: 29 June 2018 Published: 01 June 2013
• MSC : Primary: 58F15, 58F17; Secondary: 53C35.

• 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[J]. Mathematical Biosciences and Engineering, 2013, 10(4): 1227-1251. doi: 10.3934/mbe.2013.10.1227

### Related Papers:

• 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.

 [1] BMC Infectious Diseases, 10 (2010), 190. [2] Emerging Health Threats Journal, 2 (2009), e11. [3] Science, 286 (1999), 509-512. [4] in "Proceedings of SuperComputing 08 International Conference for High Performance Computing," Networking Storage and Analysis. Austin, Texas, November 15-21, (2008). [5] AI Magazine, 31 (2009), 75-87. [6] Optimal Control Applications and Methods, 21 (2000), 269-285. [7] The Society for the Study of Evolution: International Journal of Organic Evolution, 59 (2005). [8] Mathematical Biosciences, 231 (2011), 126-134. [9] Int. J. Bifurcation and Chaos, 17 (2007), 2491-2500. [10] Cambridge, Studies in Mathematical Biology, 1999. [11] Scientific Reports 2, Article number 632, 2012 [12] in "Social Computing, Behavioral-Cultural Modeling And Prediction," (Ed. S. Yang), Berlin Heidelberg, Springer, 7227 (2012), 172-179. [13] Nature, 429 (2004), 180-184. [14] Proceedings of the National Academy of Sciences, 108 (2011), 6306-6311. [15] Proceedings of the National Academy of Sciences, 104 (2007), 4984-4989. [16] Mathematical Biosciences, 232 (2011), 110-115. [17] Proceedings of the National Academy of Sciences, 103 (2006), 5935-5940. [18] Journal of the Royal Society Interface, 5 (2010), e11569. [19] Phys. Rev. Lett., 96 (2006), 208701. [20] BMC Medical Informatics and Decision Making, 12 (2012), 132. [21] IEEE Intelligent Control and Automation WCICA, (2006). [22] Proceedings of IEEE INFOCOM 2011, Shanghai, China, (2011). [23] Journal of the Royal Society Interface, 5 (2008), 791-799. [24] Phys. Rev. E, 83 (2011), 026102,. [25] PLoS ONE, 6 (2011), e22461. [26] Phys. Rev. E, 82 (2010), 036116, [27] PLoS ONE, 6 (2011), e24577. [28] Journal of Theoretical Biology, 262 (2010), 757-763. [29] IEEE/ACM Transaction on Networking, 17 (2009), 1-14. [30] Eur. Phys. J. B, 26 (2002), 521-529. [31] SIAM Review, 45 (2003), 167-256. [32] Interscience, 4 (1962). [33] ECML-PKDD 2010, Barcelona, Spain 2010. [34] 8 (2011), 141-170. [35] PLOS Computational Biology, 6 (2010), e1000793. [36] Mathematical Biosciences, 230 (2011), 67-78. [37] Interface Journal of the Royal Society, 6 (2009), 1135-1144. [38] PLoS ONE, 5 (2010), e11569. [39] BMC Med, 9 (2011). [40] Euro Surveillance, 14 (2009) . [41] Proceedings of the Royal Society B, 274 (2011), 2925-2934. [42] JTB: Journal of Theoretical Biology, Elsevier, 283 (2011), 136-144.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.285 1.3