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
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.

[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.

[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.

[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).

[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.

[6]

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

[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).

[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.

[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.

[10]

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

[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

[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.

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[24]

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

[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.

[26]

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

[27]

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

[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.

[29]

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

[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.

[31]

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

[32]

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

[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.

[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.

[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.

[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.

[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.

[38]

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

[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.

[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) .

[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.

[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.

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.

[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.

[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.

[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).

[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.

[6]

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

[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).

[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.

[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.

[10]

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

[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

[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.

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[24]

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

[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.

[26]

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

[27]

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

[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.

[29]

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

[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.

[31]

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

[32]

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

[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.

[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.

[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.

[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.

[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.

[38]

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

[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.

[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) .

[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.

[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.

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