American Institute of Mathematical Sciences

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

Mitigation of epidemics in contact networks through optimal contact adaptation

Mina Youssef 1, and  Caterina Scoglio 1,

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.
Keywords: Spread of epidemics, network-based approach, behavioral responses., adaptive contact networks, optimal control of epidemics.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
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
References:
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D. Gao and S. Ruan, An SIS patch model with variable transmission coefficients, Mathematical Biosciences, 232 (2011), 110-115. doi: 10.1016/j.mbs.2011.05.001.  Google Scholar

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[20]

M. Hashemian, W. Qian, K. G. Stanley and N. D. Osgood, Temporal aggregation impacts on epidemiological simulations employing microcontact data, BMC Medical Informatics and Decision Making, 12 (2012), 132. doi: 10.1186/1472-6947-12-132.  Google Scholar

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C. Jiang and M. Dong, Optimal measures for SARS epidemics outbreaks, IEEE Intelligent Control and Automation WCICA, (2006). Google Scholar

[22]

M. H. R. Khouzani, S. Sarkar and E. Altman, Optimal control of epidemic evolution, Proceedings of IEEE INFOCOM 2011, Shanghai, China, (2011). doi: 10.1109/INFCOM.2011.5934963.  Google Scholar

[23]

I. Kiss, D. Green and R. Kao, The effect of network mixing patterns on epidemic dynamics and the efficacy of disease contact tracing, Journal of the Royal Society Interface, 5 (2008), 791-799. doi: 10.1098/rsif.2007.1272.  Google Scholar

[24]

C. Lagorio et al., Quarantine generated phase transition in epidemic spreading, Phys. Rev. E, 83 (2011), 026102,. Google Scholar

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A. Marathe, B. Lewis, J. Chen and S. Eubank, Sensitivity of household transmission to household contact structure and size, PLoS ONE, 6 (2011), e22461. doi: 10.1371/journal.pone.0022461.  Google Scholar

[26]

V. Marceau et al., Adaptive networks: Coevolution of disease and topology, Phys. Rev. E, 82 (2010), 036116, doi: 10.1103/PhysRevE.82.036116.  Google Scholar

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M. L. N. Mbah and C. A. Gilligan, Resource allocation for epidemic control in metapopulations, PLoS ONE, 6 (2011), e24577. Google Scholar

[28]

M. L. N. Mbah and C. A. Gilligan, Optimization of control strategies for epidemics in heterogeneous populations with symmetric and asymmetric transmission, Journal of Theoretical Biology, 262 (2010), 757-763. doi: 10.1016/j.jtbi.2009.11.001.  Google Scholar

[29]

P. V. Mieghem, J. S. Omic and R. E. Kooij, Virus spread in networks, IEEE/ACM Transaction on Networking, 17 (2009), 1-14. Google Scholar

[30]

Y. Moreno, R. Pastor-Satorras and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks, Eur. Phys. J. B, 26 (2002), 521-529. doi: 10.1140/epjb/e20020122.  Google Scholar

[31]

M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003), 167-256. doi: 10.1137/S003614450342480.  Google Scholar

[32]

L. S. Pontryagin et al., The mathematical theory of optimal processes, Interscience, 4 (1962). Google Scholar

[33]

B. Prakash, H. Tong, N. Valler, M. Faloutsos and C. Faloutsos, Virus propagation on time-varying networks: theory and immunization algorithms, ECML-PKDD 2010, Barcelona, Spain 2010. doi: 10.1007/978-3-642-15939-8_7.  Google Scholar

[34]

O. Prosper, O. Saucedo, D. Thompson, G. Torres-Garcia, X. Wang and C. Castillo-Chavez, Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza, 8 (2011), 141-170. doi: 10.3934/mbe.2011.8.141.  Google Scholar

[35]

T. Reluga, Game theory of social distancing in response to an epidemic, PLOS Computational Biology, 6 (2010), e1000793. doi: 10.1371/journal.pcbi.1000793.  Google Scholar

[36]

T. Reluga and A. Galvani, A general approach to population games with application to vaccination, Mathematical Biosciences, 230 (2011), 67-78. doi: 10.1016/j.mbs.2011.01.003.  Google Scholar

[37]

R. E. Rowthorn, R. Laxminarayan and C. A. Gilligan, Optimal control of epidemics in metapopulations, Interface Journal of the Royal Society, 6 (2009), 1135-1144. doi: 10.1098/rsif.2008.0402.  Google Scholar

[38]

C. Scoglio et al., Efficient mitigation strategies for epidemics in rural regions, PLoS ONE, 5 (2010), e11569. Google Scholar

[39]

J. Stehle, N. Voirin, A. Barrat, C. Cattuto, V. Colizza, L. Isella, C. Regis, J-F. Pinton, N. Khanafer, W. Van den Broeck and P. Vanhems, Simulation of an SEIR infectious disease model on the dynamic contact network of conference attendees, BMC Med, 9 (2011). doi: 10.1186/1741-7015-9-87.  Google Scholar

[40]

S. Towers and Z. Feng, Pandemic H1N1 influenza: Predicting the course of a pandemic and assessing the efficacy of the planned vaccination programme in the United States, Euro Surveillance, 14 (2009) . Google Scholar

[41]

E. Volz and L. A. Meyers, Susceptible-infected-recovered epidemics in dynamic contact networks, Proceedings of the Royal Society B, 274 (2011), 2925-2934. doi: 10.1098/rspb.2007.1159.  Google Scholar

[42]

M. Youssef and C. Scoglio, An individual-based approach to SIR epidemics in contact networks, JTB: Journal of Theoretical Biology, Elsevier, 283 (2011), 136-144. doi: 10.1016/j.jtbi.2011.05.029.  Google Scholar

show all references

References:
[1]

M. Ajelli et al., Comparing large-scale computational approaches to epidemic modeling: agent-based versus structured metapopulation models, BMC Infectious Diseases, 10 (2010), 190. Google Scholar

[2]

P. Bajardi et al., Modeling vaccination campaigns and the Fall/Winter 2009 activity of the new A(H1N1) influenza in the Northern Hemisphere, Emerging Health Threats Journal, 2 (2009), e11. doi: 10.3134/ehtj.09.011.  Google Scholar

[3]

A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512. doi: 10.1126/science.286.5439.509.  Google Scholar

[4]

C. Barrett, K. Bisset, S. Eubank, X. Feng and M. Marathe, EpiSimdemics: An efficient and scalable framework for simulating the spread of infectious disease on large social networks, in "Proceedings of SuperComputing 08 International Conference for High Performance Computing," Networking Storage and Analysis. Austin, Texas, November 15-21, (2008). Google Scholar

[5]

C. Barrett, K. Bisset, J. Leidig, A. Marathe and M. Marathe, An integrated modeling environment to study the co-evolution of networks, individual behavior, and epidemics, AI Magazine, 31 (2009), 75-87. Google Scholar

[6]

H. Behncke, Optimal control of deterministic epidemics, Optimal Control Applications and Methods, 21 (2000), 269-285. doi: 10.1002/oca.678.  Google Scholar

[7]

M. Bondes, D. Keenan, A. Leidner and P. Rohani, Higher disease prevalence can induce greater sociality: A game theoretic coevolutionary model, The Society for the Study of Evolution: International Journal of Organic Evolution, 59 (2005). Google Scholar

[8]

V. L. Brown and K. A. J. White, The role of optimal control in assessing the most cost-effective implementation of a vaccination programme: HPV as a case study, Mathematical Biosciences, 231 (2011), 126-134. doi: 10.1016/j.mbs.2011.02.009.  Google Scholar

[9]

V. Colizza, A. Barrat, M. Barthelemy and A. Vespignani, Epidemic predictability in meta-population models with heterogeneous couplings: The impact of disease parameter values, Int. J. Bifurcation and Chaos, 17 (2007), 2491-2500. doi: 10.1142/S0218127407018567.  Google Scholar

[10]

D. J. Daley and J. Gani, "Epidemic Modelling: An Introduction," Cambridge, Studies in Mathematical Biology, 1999. doi: 10.1017/CBO9780511608834.  Google Scholar

[11]

F. Darabi, F. N. Chowdhury and C. M. Scoglio, On the existence of a threshold for preventive behavioral responses to suppress epidemic spreading, Scientific Reports 2, Article number 632, 2012 Google Scholar

[12]

W. Dong, K. Heller and A. Pentland, Modeling infection with multi-agent dynamics, in "Social Computing, Behavioral-Cultural Modeling And Prediction," (Ed. S. Yang), Berlin Heidelberg, Springer, 7227 (2012), 172-179. doi: 10.1007/978-3-642-29047-3_21.  Google Scholar

[13]

S. Eubank et al., Modelling disease outbreaks in realistic urban social networks, Nature, 429 (2004), 180-184. Google Scholar

[14]

E. P. Fenichel et al., Adaptive human behavior in epidemiological models, Proceedings of the National Academy of Sciences, 108 (2011), 6306-6311. Google Scholar

[15]

G. A. Forster and C. A. Gilligan, Optimizing the control of disease infestations at the landscape scale, Proceedings of the National Academy of Sciences, 104 (2007), 4984-4989. doi: 10.1073/pnas.0607900104.  Google Scholar

[16]

D. Gao and S. Ruan, An SIS patch model with variable transmission coefficients, Mathematical Biosciences, 232 (2011), 110-115. doi: 10.1016/j.mbs.2011.05.001.  Google Scholar

[17]

T. Germann, K. Kadau, I. Longini and C. Macken, Mitigation strategies for pandemic influenza in the United States, Proceedings of the National Academy of Sciences, 103 (2006), 5935-5940. doi: 10.1073/pnas.0601266103.  Google Scholar

[18]

T. Gross and B. Blasius, Adaptive coevolutionary networks: A review, Journal of the Royal Society Interface, 5 (2010), e11569. doi: 10.1098/rsif.2007.1229.  Google Scholar

[19]

T. Gross, C. J. D. D'Lima and B. Blasius, Epidemic dynamics on an adaptive network, Phys. Rev. Lett., 96 (2006), 208701. doi: 10.1103/PhysRevLett.96.208701.  Google Scholar

[20]

M. Hashemian, W. Qian, K. G. Stanley and N. D. Osgood, Temporal aggregation impacts on epidemiological simulations employing microcontact data, BMC Medical Informatics and Decision Making, 12 (2012), 132. doi: 10.1186/1472-6947-12-132.  Google Scholar

[21]

C. Jiang and M. Dong, Optimal measures for SARS epidemics outbreaks, IEEE Intelligent Control and Automation WCICA, (2006). Google Scholar

[22]

M. H. R. Khouzani, S. Sarkar and E. Altman, Optimal control of epidemic evolution, Proceedings of IEEE INFOCOM 2011, Shanghai, China, (2011). doi: 10.1109/INFCOM.2011.5934963.  Google Scholar

[23]

I. Kiss, D. Green and R. Kao, The effect of network mixing patterns on epidemic dynamics and the efficacy of disease contact tracing, Journal of the Royal Society Interface, 5 (2008), 791-799. doi: 10.1098/rsif.2007.1272.  Google Scholar

[24]

C. Lagorio et al., Quarantine generated phase transition in epidemic spreading, Phys. Rev. E, 83 (2011), 026102,. Google Scholar

[25]

A. Marathe, B. Lewis, J. Chen and S. Eubank, Sensitivity of household transmission to household contact structure and size, PLoS ONE, 6 (2011), e22461. doi: 10.1371/journal.pone.0022461.  Google Scholar

[26]

V. Marceau et al., Adaptive networks: Coevolution of disease and topology, Phys. Rev. E, 82 (2010), 036116, doi: 10.1103/PhysRevE.82.036116.  Google Scholar

[27]

M. L. N. Mbah and C. A. Gilligan, Resource allocation for epidemic control in metapopulations, PLoS ONE, 6 (2011), e24577. Google Scholar

[28]

M. L. N. Mbah and C. A. Gilligan, Optimization of control strategies for epidemics in heterogeneous populations with symmetric and asymmetric transmission, Journal of Theoretical Biology, 262 (2010), 757-763. doi: 10.1016/j.jtbi.2009.11.001.  Google Scholar

[29]

P. V. Mieghem, J. S. Omic and R. E. Kooij, Virus spread in networks, IEEE/ACM Transaction on Networking, 17 (2009), 1-14. Google Scholar

[30]

Y. Moreno, R. Pastor-Satorras and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks, Eur. Phys. J. B, 26 (2002), 521-529. doi: 10.1140/epjb/e20020122.  Google Scholar

[31]

M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003), 167-256. doi: 10.1137/S003614450342480.  Google Scholar

[32]

L. S. Pontryagin et al., The mathematical theory of optimal processes, Interscience, 4 (1962). Google Scholar

[33]

B. Prakash, H. Tong, N. Valler, M. Faloutsos and C. Faloutsos, Virus propagation on time-varying networks: theory and immunization algorithms, ECML-PKDD 2010, Barcelona, Spain 2010. doi: 10.1007/978-3-642-15939-8_7.  Google Scholar

[34]

O. Prosper, O. Saucedo, D. Thompson, G. Torres-Garcia, X. Wang and C. Castillo-Chavez, Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza, 8 (2011), 141-170. doi: 10.3934/mbe.2011.8.141.  Google Scholar

[35]

T. Reluga, Game theory of social distancing in response to an epidemic, PLOS Computational Biology, 6 (2010), e1000793. doi: 10.1371/journal.pcbi.1000793.  Google Scholar

[36]

T. Reluga and A. Galvani, A general approach to population games with application to vaccination, Mathematical Biosciences, 230 (2011), 67-78. doi: 10.1016/j.mbs.2011.01.003.  Google Scholar

[37]

R. E. Rowthorn, R. Laxminarayan and C. A. Gilligan, Optimal control of epidemics in metapopulations, Interface Journal of the Royal Society, 6 (2009), 1135-1144. doi: 10.1098/rsif.2008.0402.  Google Scholar

[38]

C. Scoglio et al., Efficient mitigation strategies for epidemics in rural regions, PLoS ONE, 5 (2010), e11569. Google Scholar

[39]

J. Stehle, N. Voirin, A. Barrat, C. Cattuto, V. Colizza, L. Isella, C. Regis, J-F. Pinton, N. Khanafer, W. Van den Broeck and P. Vanhems, Simulation of an SEIR infectious disease model on the dynamic contact network of conference attendees, BMC Med, 9 (2011). doi: 10.1186/1741-7015-9-87.  Google Scholar

[40]

S. Towers and Z. Feng, Pandemic H1N1 influenza: Predicting the course of a pandemic and assessing the efficacy of the planned vaccination programme in the United States, Euro Surveillance, 14 (2009) . Google Scholar

[41]

E. Volz and L. A. Meyers, Susceptible-infected-recovered epidemics in dynamic contact networks, Proceedings of the Royal Society B, 274 (2011), 2925-2934. doi: 10.1098/rspb.2007.1159.  Google Scholar

[42]

M. Youssef and C. Scoglio, An individual-based approach to SIR epidemics in contact networks, JTB: Journal of Theoretical Biology, Elsevier, 283 (2011), 136-144. doi: 10.1016/j.jtbi.2011.05.029.  Google Scholar

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