# American Institute of Mathematical Sciences

2013, 10(5&6): 1691-1701. doi: 10.3934/mbe.2013.10.1691

## Optimal isolation strategies of emerging infectious diseases with limited resources

 1 School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China 2 Centre for Disease Modeling, York University, 4700 Keele Street, Toronto, ON M3J1P3 3 School of Information Science and Engineering, Central South University, Changsha, Hunan 410083, China

Received  September 2012 Revised  May 2013 Published  August 2013

A classical deterministic SIR model is modified to take into account of limited resources for diagnostic confirmation/medical isolation. We show that this modification leads to four different scenarios (instead of three scenarios in comparison with the SIR model) for optimal isolation strategies, and obtain analytic solutions for the optimal control problem that minimize the outbreak size under the assumption of limited resources for isolation. These solutions and their corresponding optimal control policies are derived explicitly in terms of initial conditions, model parameters and resources for isolation (such as the number of intensive care units). With sufficient resources, the optimal control strategy is the normal Bang-Bang control. However, with limited resources the optimal control strategy requires to switch to time-variant isolation at an optimal rate proportional to the ratio of isolated cases over the entire infected population once the maximum capacity is reached.
Citation: Yinggao Zhou, Jianhong Wu, Min Wu. Optimal isolation strategies of emerging infectious diseases with limited resources. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1691-1701. doi: 10.3934/mbe.2013.10.1691
##### References:
 [1] A. Abakuks, "Optimal Policies for Epidemics," D. Phil. Thesis, Univ. Of Sussex, 1972. [2] A. Abakuks, An optimal isolation policy for an epidemic, J. Appl. Probability, 10 (1973), 247-262. doi: 10.2307/3212343. [3] A. Abakuks, Optimal immunization policies for epidemics, Adv. Appl. Probability, 6 (1974), 494-511. doi: 10.2307/1426230. [4] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford Science Publications, Oxford, 1991. [5] H. Behncke, Optimal control of deterministic epidemics, Opt. Control Appl. Methods, 21 (2000), 269-285. doi: 10.1002/oca.678. [6] M. C. J. Bootsma and N. M. Ferguson, The effect of public health measures on the 1918 influenza pandemic in U.S. cities, PNAS, 104 (2007), 7588-7593. doi: 10.1073/pnas.0611071104. [7] C. B. Bridges, M. J. Kuehnert and C. B. Hall, Transmission of influenza: Implications for control in health care settings, Clin. Infect. Dis., 37 (2003), 1094-1101. [8] E. Bryson and Y. Ho, "Applied Optimal Control-Optimization, Estimation, and Control," Taylor & Francis, New York, London, 1975. [9] F. Carrat, J. Luong and H. Lao, A.-V. Sallé, C. Lajaunie and H. Wackernagel, A 'smallworld-like' model for comparing interventions aimed at preventing and controlling influenza pandemics, BMC Medicine, 4 (2006). [10] S. Cauchemez, F. Carrat, C. Viboud, A. J. Valleron and P. Y. Boëlle, A Bayesian MCMC approach to study transmission of influenza: Application to household longitudinal data, Stat. Med., 23 (2004), 3469-3487. doi: 10.1002/sim.1912. [11] G. Chowell, C. E. Ammon, N. W. Hengartner and J. M. Hyman, Transmission dynamics of the great influenza pandemic of 1918 in Geneva, Switzerland: Assessing the effects of hypothetical intervention, J. Theor. Bio., 241 (2005), 193-204. doi: 10.1016/j.jtbi.2005.11.026. [12] G. Chowell, H. Nishiura and L. M. A. Bettencourt, Comparative estimation of the reproduction number for pandemic influenza from daily case notification data, J. R. Soc. Interface, 4 (2006), 155-166. doi: 10.1098/rsif.2006.0161. [13] D. Clancy, Optimal intervention for epidemic models with general infection and removal rate functions, J. Math. Biol., 39 (1999), 309-331. doi: 10.1007/s002850050193. [14] B. Cooper, Poxy models and rash decisions, PNAS, 103 (2006), 12221-12222. doi: 10.1073/pnas.0605502103. [15] J. Dushoff, J. B. Plotkin, S. A. Levin and D. J. D. Earn, Dynamical resonance can account for seasonality of influenza epidemics, PNAS, 101 (2004), 16915-16916. doi: 10.1073/pnas.0407293101. [16] B. D. Elderd, V. M. Dukic and G. Dwyer, Uncertainty in predictions of diseases spread and public health responses to bioterrorism and emerging diseases, PNAS, 103 (2006), 15639-15697. doi: 10.1073/pnas.0600816103. [17] N. M. Ferguson, D. A. T. Cummings and C. Fraser, et al., Strategies for mitigating an influenza pandemic, Nature, 442 (2006), 448-452. doi: 10.1038/nature04795. [18] H. P. Geering, "Optimal Control with Engineering Applications," Springer-Verlag, Berlin-Heidelberg, 2007. [19] R. J. Glass, L. M. Glass and W. E. Beyeler, et al., Targeted social distancing design for pandemic influenza, Emerg. Infect. Dis., 12 (2006), 1671-1681. [20] E. Hansen and T. Day, Optimal control of epidemics with limited resources, J. Math. Bio., 62 (2011), 423-451. doi: 10.1007/s00285-010-0341-0. [21] D. Hull, "Optimal Control Theory for Applications," Mechanical Engineering Series, Springer-Verlag, New York, 2003. [22] E. H. Kaplan, D. L. Craft and L. M. Wein, Emergency response to a smallpox attack: The case for mass vaccination, PNAS, 99 (2002), 10935-10940. doi: 10.1073/pnas.162282799. [23] F. Lin, K. Muthuraman and M. Lawley, An optimal control theory approach to non-pharmaceutical interventions, BMC Infect. Dis., 10 (2010), 1-13. doi: 10.1186/1471-2334-10-32. [24] M. Lipsitch, T. Cohen and B. Cooper, et al., Transmission dynamics of Severe Acute Respiratory Syndrome, Science, 300 (2003), 1966-1970. doi: 10.1126/science.1086616. [25] I. M. Longini, M. E. Halloran and A. Nizam, et al., Containing pandemic influenza with antiviral agents, Am. J. Epidemiol., 159 (2004), 623-633. doi: 10.1093/aje/kwh092. [26] I. M. Longini, A. Nizam and S. Xu, et al., Containing pandemic influenza at the source, Science, 309 (2005), 1083-1087. doi: 10.1126/science.1115717. [27] H. Markel, A. M. Stern and J. A. Navarro, et al., Nonpharmaceutical influenza mitigation strategies, US communities, 1918-1920 pandemic, Emerg. Infect. Dis., 12 (2006), 1961-1964. doi: 10.3201/eid1212.060506. [28] C. E. Mills, J. M. Robins and M. Lipsitch, Transmissibility of 1918 pandemic influenza, Nature, 432 (2004), 904-906. doi: 10.1038/nature03063. [29] R. Morton and K. H. Wickwire, On the optimal control of a deterministic epidemic, Adv. Appl. Probability, 6 (1974), 622-635. doi: 10.2307/1426183. [30] S. Riley, C. Fraser and C. A. Donnelly, et al., Transmission dynamics of the etiological agent of SARS in Hong Kong: Impact of public health intervention, Science, 300 (2003), 1961-1966. doi: 10.1126/science.1086478. [31] S. Riley and N. M. Ferguson, Smallpox transmission and control: Spatial dynamics in Great Britain, PNAS, 103 (2007), 12637-12642. doi: 10.1073/pnas.0510873103. [32] S. P. Sethi and P. W. Staats, Optimal control of some simple deterministic epidemic models, J. Oper. Res. Soc., 29 (1978), 129-136. doi: 10.2307/3009792. [33] S. P. Sethi, Optimal quarantine programmes for controlling an epidemic spread, J. Oper. Res. Soc., 29 (1978), 265-268. doi: 10.2307/3009454. [34] T. Wai, G. Turinici and A. Danchin, A double epidemic model for the SARS propagation, BMC Infect. Dis., 3 (2003), 1-16. [35] H. J. Wearing, P. Rohani and M. J. Keeling, Appropriate models for the management of infectious diseases, PLoS Medicine, 2 (2005), 1532-1540. doi: 10.1371/journal.pmed.0020174. [36] R. J. Webby and R. G. Webster, Are we ready for pandemic influenza, Science, 302 (2003), 1519-1522. doi: 10.1126/science.1090350. [37] K. H. Wickwire, Optimal isolation policies for deterministic and stochastic epidemics, Math. Biosci., 26 (1975), 325-346. doi: 10.1016/0025-5564(75)90020-6. [38] J. T. Wu, S. Riley and C. Fraser, et al., Reducing the impact of the next influenza pandemic using household-based public health interventions, PLoS Medicine, 3 (2006), 1532-1540. doi: 10.1371/journal.pmed.0030361.

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##### References:
 [1] A. Abakuks, "Optimal Policies for Epidemics," D. Phil. Thesis, Univ. Of Sussex, 1972. [2] A. Abakuks, An optimal isolation policy for an epidemic, J. Appl. Probability, 10 (1973), 247-262. doi: 10.2307/3212343. [3] A. Abakuks, Optimal immunization policies for epidemics, Adv. Appl. Probability, 6 (1974), 494-511. doi: 10.2307/1426230. [4] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford Science Publications, Oxford, 1991. [5] H. Behncke, Optimal control of deterministic epidemics, Opt. Control Appl. Methods, 21 (2000), 269-285. doi: 10.1002/oca.678. [6] M. C. J. Bootsma and N. M. Ferguson, The effect of public health measures on the 1918 influenza pandemic in U.S. cities, PNAS, 104 (2007), 7588-7593. doi: 10.1073/pnas.0611071104. [7] C. B. Bridges, M. J. Kuehnert and C. B. Hall, Transmission of influenza: Implications for control in health care settings, Clin. Infect. Dis., 37 (2003), 1094-1101. [8] E. Bryson and Y. Ho, "Applied Optimal Control-Optimization, Estimation, and Control," Taylor & Francis, New York, London, 1975. [9] F. Carrat, J. Luong and H. Lao, A.-V. Sallé, C. Lajaunie and H. Wackernagel, A 'smallworld-like' model for comparing interventions aimed at preventing and controlling influenza pandemics, BMC Medicine, 4 (2006). [10] S. Cauchemez, F. Carrat, C. Viboud, A. J. Valleron and P. Y. Boëlle, A Bayesian MCMC approach to study transmission of influenza: Application to household longitudinal data, Stat. Med., 23 (2004), 3469-3487. doi: 10.1002/sim.1912. [11] G. Chowell, C. E. Ammon, N. W. Hengartner and J. M. Hyman, Transmission dynamics of the great influenza pandemic of 1918 in Geneva, Switzerland: Assessing the effects of hypothetical intervention, J. Theor. Bio., 241 (2005), 193-204. doi: 10.1016/j.jtbi.2005.11.026. [12] G. Chowell, H. Nishiura and L. M. A. Bettencourt, Comparative estimation of the reproduction number for pandemic influenza from daily case notification data, J. R. Soc. Interface, 4 (2006), 155-166. doi: 10.1098/rsif.2006.0161. [13] D. Clancy, Optimal intervention for epidemic models with general infection and removal rate functions, J. Math. Biol., 39 (1999), 309-331. doi: 10.1007/s002850050193. [14] B. Cooper, Poxy models and rash decisions, PNAS, 103 (2006), 12221-12222. doi: 10.1073/pnas.0605502103. [15] J. Dushoff, J. B. Plotkin, S. A. Levin and D. J. D. Earn, Dynamical resonance can account for seasonality of influenza epidemics, PNAS, 101 (2004), 16915-16916. doi: 10.1073/pnas.0407293101. [16] B. D. Elderd, V. M. Dukic and G. Dwyer, Uncertainty in predictions of diseases spread and public health responses to bioterrorism and emerging diseases, PNAS, 103 (2006), 15639-15697. doi: 10.1073/pnas.0600816103. [17] N. M. Ferguson, D. A. T. Cummings and C. Fraser, et al., Strategies for mitigating an influenza pandemic, Nature, 442 (2006), 448-452. doi: 10.1038/nature04795. [18] H. P. Geering, "Optimal Control with Engineering Applications," Springer-Verlag, Berlin-Heidelberg, 2007. [19] R. J. Glass, L. M. Glass and W. E. Beyeler, et al., Targeted social distancing design for pandemic influenza, Emerg. Infect. Dis., 12 (2006), 1671-1681. [20] E. Hansen and T. Day, Optimal control of epidemics with limited resources, J. Math. Bio., 62 (2011), 423-451. doi: 10.1007/s00285-010-0341-0. [21] D. Hull, "Optimal Control Theory for Applications," Mechanical Engineering Series, Springer-Verlag, New York, 2003. [22] E. H. Kaplan, D. L. Craft and L. M. Wein, Emergency response to a smallpox attack: The case for mass vaccination, PNAS, 99 (2002), 10935-10940. doi: 10.1073/pnas.162282799. [23] F. Lin, K. Muthuraman and M. Lawley, An optimal control theory approach to non-pharmaceutical interventions, BMC Infect. Dis., 10 (2010), 1-13. doi: 10.1186/1471-2334-10-32. [24] M. Lipsitch, T. Cohen and B. Cooper, et al., Transmission dynamics of Severe Acute Respiratory Syndrome, Science, 300 (2003), 1966-1970. doi: 10.1126/science.1086616. [25] I. M. Longini, M. E. Halloran and A. Nizam, et al., Containing pandemic influenza with antiviral agents, Am. J. Epidemiol., 159 (2004), 623-633. doi: 10.1093/aje/kwh092. [26] I. M. Longini, A. Nizam and S. Xu, et al., Containing pandemic influenza at the source, Science, 309 (2005), 1083-1087. doi: 10.1126/science.1115717. [27] H. Markel, A. M. Stern and J. A. Navarro, et al., Nonpharmaceutical influenza mitigation strategies, US communities, 1918-1920 pandemic, Emerg. Infect. Dis., 12 (2006), 1961-1964. doi: 10.3201/eid1212.060506. [28] C. E. Mills, J. M. Robins and M. Lipsitch, Transmissibility of 1918 pandemic influenza, Nature, 432 (2004), 904-906. doi: 10.1038/nature03063. [29] R. Morton and K. H. Wickwire, On the optimal control of a deterministic epidemic, Adv. Appl. Probability, 6 (1974), 622-635. doi: 10.2307/1426183. [30] S. Riley, C. Fraser and C. A. Donnelly, et al., Transmission dynamics of the etiological agent of SARS in Hong Kong: Impact of public health intervention, Science, 300 (2003), 1961-1966. doi: 10.1126/science.1086478. [31] S. Riley and N. M. Ferguson, Smallpox transmission and control: Spatial dynamics in Great Britain, PNAS, 103 (2007), 12637-12642. doi: 10.1073/pnas.0510873103. [32] S. P. Sethi and P. W. Staats, Optimal control of some simple deterministic epidemic models, J. Oper. Res. Soc., 29 (1978), 129-136. doi: 10.2307/3009792. [33] S. P. Sethi, Optimal quarantine programmes for controlling an epidemic spread, J. Oper. Res. Soc., 29 (1978), 265-268. doi: 10.2307/3009454. [34] T. Wai, G. Turinici and A. Danchin, A double epidemic model for the SARS propagation, BMC Infect. Dis., 3 (2003), 1-16. [35] H. J. Wearing, P. Rohani and M. J. Keeling, Appropriate models for the management of infectious diseases, PLoS Medicine, 2 (2005), 1532-1540. doi: 10.1371/journal.pmed.0020174. [36] R. J. Webby and R. G. Webster, Are we ready for pandemic influenza, Science, 302 (2003), 1519-1522. doi: 10.1126/science.1090350. [37] K. H. Wickwire, Optimal isolation policies for deterministic and stochastic epidemics, Math. Biosci., 26 (1975), 325-346. doi: 10.1016/0025-5564(75)90020-6. [38] J. T. Wu, S. Riley and C. Fraser, et al., Reducing the impact of the next influenza pandemic using household-based public health interventions, PLoS Medicine, 3 (2006), 1532-1540. doi: 10.1371/journal.pmed.0030361.
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