# 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:
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Dis., 10 (2010), 1. doi: 10.1186/1471-2334-10-32. Google Scholar [24] M. Lipsitch, T. Cohen and B. Cooper, et al., Transmission dynamics of Severe Acute Respiratory Syndrome,, Science, 300 (2003), 1966. doi: 10.1126/science.1086616. Google Scholar [25] I. M. Longini, M. E. Halloran and A. Nizam, et al., Containing pandemic influenza with antiviral agents,, Am. J. Epidemiol., 159 (2004), 623. doi: 10.1093/aje/kwh092. Google Scholar [26] I. M. Longini, A. Nizam and S. Xu, et al., Containing pandemic influenza at the source,, Science, 309 (2005), 1083. doi: 10.1126/science.1115717. Google Scholar [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. doi: 10.3201/eid1212.060506. Google Scholar [28] C. E. Mills, J. M. Robins and M. Lipsitch, Transmissibility of 1918 pandemic influenza,, Nature, 432 (2004), 904. doi: 10.1038/nature03063. Google Scholar [29] R. Morton and K. H. Wickwire, On the optimal control of a deterministic epidemic,, Adv. Appl. Probability, 6 (1974), 622. doi: 10.2307/1426183. Google Scholar [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. doi: 10.1126/science.1086478. Google Scholar [31] S. Riley and N. M. Ferguson, Smallpox transmission and control: Spatial dynamics in Great Britain,, PNAS, 103 (2007), 12637. doi: 10.1073/pnas.0510873103. Google Scholar [32] S. P. Sethi and P. W. Staats, Optimal control of some simple deterministic epidemic models,, J. Oper. Res. Soc., 29 (1978), 129. doi: 10.2307/3009792. Google Scholar [33] S. P. Sethi, Optimal quarantine programmes for controlling an epidemic spread,, J. Oper. Res. Soc., 29 (1978), 265. doi: 10.2307/3009454. Google Scholar [34] T. Wai, G. Turinici and A. Danchin, A double epidemic model for the SARS propagation,, BMC Infect. Dis., 3 (2003), 1. Google Scholar [35] H. J. Wearing, P. Rohani and M. J. Keeling, Appropriate models for the management of infectious diseases,, PLoS Medicine, 2 (2005), 1532. doi: 10.1371/journal.pmed.0020174. Google Scholar [36] R. J. Webby and R. G. Webster, Are we ready for pandemic influenza,, Science, 302 (2003), 1519. doi: 10.1126/science.1090350. Google Scholar [37] K. H. Wickwire, Optimal isolation policies for deterministic and stochastic epidemics,, Math. Biosci., 26 (1975), 325. doi: 10.1016/0025-5564(75)90020-6. Google Scholar [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. doi: 10.1371/journal.pmed.0030361. Google Scholar

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##### References:
 [1] A. Abakuks, "Optimal Policies for Epidemics,", D. Phil. Thesis, (1972). Google Scholar [2] A. Abakuks, An optimal isolation policy for an epidemic,, J. Appl. Probability, 10 (1973), 247. doi: 10.2307/3212343. Google Scholar [3] A. Abakuks, Optimal immunization policies for epidemics,, Adv. Appl. Probability, 6 (1974), 494. doi: 10.2307/1426230. Google Scholar [4] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford Science Publications, (1991). Google Scholar [5] H. Behncke, Optimal control of deterministic epidemics,, Opt. Control Appl. Methods, 21 (2000), 269. doi: 10.1002/oca.678. Google Scholar [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. doi: 10.1073/pnas.0611071104. Google Scholar [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. Google Scholar [8] E. Bryson and Y. Ho, "Applied Optimal Control-Optimization, Estimation, and Control,", Taylor & Francis, (1975). Google Scholar [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). Google Scholar [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. doi: 10.1002/sim.1912. Google Scholar [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. doi: 10.1016/j.jtbi.2005.11.026. Google Scholar [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. doi: 10.1098/rsif.2006.0161. Google Scholar [13] D. Clancy, Optimal intervention for epidemic models with general infection and removal rate functions,, J. Math. Biol., 39 (1999), 309. doi: 10.1007/s002850050193. Google Scholar [14] B. Cooper, Poxy models and rash decisions,, PNAS, 103 (2006), 12221. doi: 10.1073/pnas.0605502103. Google Scholar [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. doi: 10.1073/pnas.0407293101. Google Scholar [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. doi: 10.1073/pnas.0600816103. Google Scholar [17] N. M. Ferguson, D. A. T. Cummings and C. Fraser, et al., Strategies for mitigating an influenza pandemic,, Nature, 442 (2006), 448. doi: 10.1038/nature04795. Google Scholar [18] H. P. Geering, "Optimal Control with Engineering Applications,", Springer-Verlag, (2007). Google Scholar [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. Google Scholar [20] E. Hansen and T. Day, Optimal control of epidemics with limited resources,, J. Math. Bio., 62 (2011), 423. doi: 10.1007/s00285-010-0341-0. Google Scholar [21] D. Hull, "Optimal Control Theory for Applications,", Mechanical Engineering Series, (2003). Google Scholar [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. doi: 10.1073/pnas.162282799. Google Scholar [23] F. Lin, K. Muthuraman and M. Lawley, An optimal control theory approach to non-pharmaceutical interventions,, BMC Infect. Dis., 10 (2010), 1. doi: 10.1186/1471-2334-10-32. Google Scholar [24] M. Lipsitch, T. Cohen and B. Cooper, et al., Transmission dynamics of Severe Acute Respiratory Syndrome,, Science, 300 (2003), 1966. doi: 10.1126/science.1086616. Google Scholar [25] I. M. Longini, M. E. Halloran and A. Nizam, et al., Containing pandemic influenza with antiviral agents,, Am. J. Epidemiol., 159 (2004), 623. doi: 10.1093/aje/kwh092. Google Scholar [26] I. M. Longini, A. Nizam and S. Xu, et al., Containing pandemic influenza at the source,, Science, 309 (2005), 1083. doi: 10.1126/science.1115717. Google Scholar [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. doi: 10.3201/eid1212.060506. Google Scholar [28] C. E. Mills, J. M. Robins and M. Lipsitch, Transmissibility of 1918 pandemic influenza,, Nature, 432 (2004), 904. doi: 10.1038/nature03063. Google Scholar [29] R. Morton and K. H. Wickwire, On the optimal control of a deterministic epidemic,, Adv. Appl. Probability, 6 (1974), 622. doi: 10.2307/1426183. Google Scholar [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. doi: 10.1126/science.1086478. Google Scholar [31] S. Riley and N. M. Ferguson, Smallpox transmission and control: Spatial dynamics in Great Britain,, PNAS, 103 (2007), 12637. doi: 10.1073/pnas.0510873103. Google Scholar [32] S. P. Sethi and P. W. Staats, Optimal control of some simple deterministic epidemic models,, J. Oper. Res. Soc., 29 (1978), 129. doi: 10.2307/3009792. Google Scholar [33] S. P. Sethi, Optimal quarantine programmes for controlling an epidemic spread,, J. Oper. Res. Soc., 29 (1978), 265. doi: 10.2307/3009454. Google Scholar [34] T. Wai, G. Turinici and A. Danchin, A double epidemic model for the SARS propagation,, BMC Infect. Dis., 3 (2003), 1. Google Scholar [35] H. J. Wearing, P. Rohani and M. J. Keeling, Appropriate models for the management of infectious diseases,, PLoS Medicine, 2 (2005), 1532. doi: 10.1371/journal.pmed.0020174. Google Scholar [36] R. J. Webby and R. G. Webster, Are we ready for pandemic influenza,, Science, 302 (2003), 1519. doi: 10.1126/science.1090350. Google Scholar [37] K. H. Wickwire, Optimal isolation policies for deterministic and stochastic epidemics,, Math. Biosci., 26 (1975), 325. doi: 10.1016/0025-5564(75)90020-6. Google Scholar [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. doi: 10.1371/journal.pmed.0030361. Google Scholar
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