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2012, 9(1): 147-164. doi: 10.3934/mbe.2012.9.147

## Assessing the effect of non-pharmaceutical interventions on containing an emerging disease

 1 School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, China 2 Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, 210094, China

Received  March 2011 Revised  July 2011 Published  December 2011

Non-pharmaceutical interventions, such as quarantine, isolation and entry screening, are usually the primary public health measures to control the spread of an emerging infectious disease through a human population. This paper proposes a multi-regional deterministic compartmental model to assess the effectiveness and implications of non-pharmaceutical interventions. The reproduction number is determined as the spectral radius of a nonnegative matrix product. Comparisons are made using the reproduction number, epidemic peaks and cumulative number of infections and mortality as indexes. Simulation results show that quarantine of suspected cases and isolation of cases with symptom are effective in reducing disease burden for multiple regions. Using entry screening strategy leads to a moderate time delay for epidemic peaks, but is of no help for preventing an epidemic breaking out. The study further shows that isolation strategy is always the best choice in the presence or absence of stringent hygiene precautions and should be given priority in combating an emerging epidemic.
Citation: Zi Sang, Zhipeng Qiu, Xiefei Yan, Yun Zou. Assessing the effect of non-pharmaceutical interventions on containing an emerging disease. Mathematical Biosciences & Engineering, 2012, 9 (1) : 147-164. doi: 10.3934/mbe.2012.9.147
##### References:
 [1] M. E. Alexander, S. M. Moghadas, G. Röst and J. Wu, A delay differential model for pandemic influenza with antiviral treatment, Bull. Math. Biol., 70 (2008), 382-397. doi: 10.1007/s11538-007-9257-2. [2] M. E. Alexander, C. Bowman, Z. Feng, M. Gardam, S. Moghadas, G. Rost, J. Wu and P. Yan, Emergence of drug-resistance: Dynamical implications for pandemic influenza, Proc. R. Soc. B., 274 (2007), 1675-1684. doi: 10.1098/rspb.2007.0422. [3] J. Arino and P. van den Driessche, A multi-city epidemic model, Math. Popul. Studies, 10 (2003), 175-193. doi: 10.1080/08898480306720. [4] J. Arino and P. van den Driessche, The basic reproduction number in a multi-city compartmental epidemic model, in "Positive Systems" (Rome, 2003), Lect. Notes Contr. Inf. Sci., 294, Springer, Berlin, (2003), 135-142. [5] J. Arino, R. Jordan and P. van den Driessche, Quarantine in a multi-species epidemics model with spatial dynamics, Math. Biosci., 206 (2007), 46-60. doi: 10.1016/j.mbs.2005.09.002. [6] F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001. [7] C. Castillo-Chavez and A. Yakubu, Dispersal disease and life-history evolution, Math. Biosci., 173 (2001), 35-53. doi: 10.1016/S0025-5564(01)00065-7. [8] G. Chowell, P. W. Fenimore, M. A. Castillo-Garsow and C. Castillo-Chavez, SARS outbreaks in Ontario, Hong Kong and Singapore: The role of diagnosis and isolation as a control mechanism, J. Theor. Biol., 224 (2003), 1-8. doi: 10.1016/S0022-5193(03)00228-5. [9] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases Model Building, Analysis and Interpretation," Wiley Series in Mathematical and Computational Biology, Wiley, Chichester, 2000. [10] Z. Feng, Final and peak epidemic sizes for SEIR models with quarantine and isolation, Mathematical Biosciences and Engineering, 4 (2007), 675-686. doi: 10.3934/mbe.2007.4.675. [11] Z. Feng, Y. Yang, D. Xu, P. Zhang, M. McCauley and J. Glasser, Timely identification of optimal control strategies for emerging infectious diseases, J. Theor. Biol., 259 (2009), 165-171. doi: 10.1016/j.jtbi.2009.03.006. [12] J. Glasser, D. Taneri, Z. Feng, J. H. Chuang, P. Tull, W. Thompson, M. M. McCauley and J. Alexander, Evaluation of targeted influenza vaccination strategies via population modeling, PLoS One, 5 (2010), e12777. doi: 10.1371/journal.pone.0012777. [13] A. B. Gumel, S. Ruan, T. Day, J. Watmough, F. Brauer, P. van den Driessche, D. Gabrielson, C. Bowman, M. E. Alexander, S. Ardal, J. Wu and B. M. Sahai, Modeling strategies for controlling SARS outbreaks, Proc. R. Soc. B., 271 (2004), 2223-2232. doi: 10.1098/rspb.2004.2800. [14] H. W. Hethcote, A thousand and one epidemic models, in "Frontiers in Mathematical Biology" (ed. S. A. Levin), Lecture Notes in Biomathematics, 100, Springer-Verlag, Berlin-Heidelberg-New York, (1994), 504-515. [15] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [16] M. Lipsitch, T. Cohen, M. Muray and B. R. Levin, Antiviral resistance and the control of pandemic influenza, PLoS Med., 4 (2007), 111-120. [17] J. O. Lloyd-Smith, A. P. Galvani and W. M. Getz, Curtailing transmission of severe acute respiratory syndrome within a community and its hospital, Proc. R. Soc. B., 170 (2003), 1979-1989. doi: 10.1098/rspb.2003.2481. [18] I. M. Longini, M. E. Halloran, A. Nizam and Y. Yang, Containing pandemic influenza with antiviral agents, Am. J. Epidemiol., 159 (2004), 623-633. doi: 10.1093/aje/kwh092. [19] J. M. McCaw and J. McVernon, Prophylaxis or treatment? Optimal use of an antiviral stockpile during an influenza pandemic, Math. Biosci., 209 (2007), 336-360. doi: 10.1016/j.mbs.2007.02.003. [20] C. N. Podder, A. B. Gumel, C. Bowman and R. G. McLeod, Mathematical study of the impact of quarantine, isolation and vaccination in curtailing an epidemic, Journal of Biological Sciences, 15 (2007), 1-18. [21] Z. Qiu and Z. Feng, Transmission dynamics of an influenza model with vaccination and antiviral treatment, Bull. Math. Biol., 72 (2010), 1-33. doi: 10.1007/s11538-009-9435-5. [22] R. R. Regoes and S. Bonhoeffer, Emergence of drug-resistant influenza virus: Population dynamical considerations, Science, 312 (2006), 389-391. doi: 10.1126/science.1122947. [23] S. Ruan, W. Wang and S. A. Levin, The effect of global travel on the spread of SARS, Math. Biosci. Eng., 3 (2006), 205-218. doi: 10.3934/mbe.2006.3.205. [24] A. R. Tuite, D. N. Fisman, J. C. Kwong and A. L. Greer, Optimal pandemic influenza vaccine allocation strategies for the Canadian population, PLoS One, 5 (2010), e10520. doi: 10.1371/journal.pone.0010520. [25] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [26] W. Wang and X. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001. [27] P. Yan and Z. Feng, Variability order of the latent and the infectious periods in a deterministic SEIR epidemic model and evaluation of control effectiveness, Math. Biosci., 224 (2010), 43-52.

show all references

##### References:
 [1] M. E. Alexander, S. M. Moghadas, G. Röst and J. Wu, A delay differential model for pandemic influenza with antiviral treatment, Bull. Math. Biol., 70 (2008), 382-397. doi: 10.1007/s11538-007-9257-2. [2] M. E. Alexander, C. Bowman, Z. Feng, M. Gardam, S. Moghadas, G. Rost, J. Wu and P. Yan, Emergence of drug-resistance: Dynamical implications for pandemic influenza, Proc. R. Soc. B., 274 (2007), 1675-1684. doi: 10.1098/rspb.2007.0422. [3] J. Arino and P. van den Driessche, A multi-city epidemic model, Math. Popul. Studies, 10 (2003), 175-193. doi: 10.1080/08898480306720. [4] J. Arino and P. van den Driessche, The basic reproduction number in a multi-city compartmental epidemic model, in "Positive Systems" (Rome, 2003), Lect. Notes Contr. Inf. Sci., 294, Springer, Berlin, (2003), 135-142. [5] J. Arino, R. Jordan and P. van den Driessche, Quarantine in a multi-species epidemics model with spatial dynamics, Math. Biosci., 206 (2007), 46-60. doi: 10.1016/j.mbs.2005.09.002. [6] F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001. [7] C. Castillo-Chavez and A. Yakubu, Dispersal disease and life-history evolution, Math. Biosci., 173 (2001), 35-53. doi: 10.1016/S0025-5564(01)00065-7. [8] G. Chowell, P. W. Fenimore, M. A. Castillo-Garsow and C. Castillo-Chavez, SARS outbreaks in Ontario, Hong Kong and Singapore: The role of diagnosis and isolation as a control mechanism, J. Theor. Biol., 224 (2003), 1-8. doi: 10.1016/S0022-5193(03)00228-5. [9] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases Model Building, Analysis and Interpretation," Wiley Series in Mathematical and Computational Biology, Wiley, Chichester, 2000. [10] Z. Feng, Final and peak epidemic sizes for SEIR models with quarantine and isolation, Mathematical Biosciences and Engineering, 4 (2007), 675-686. doi: 10.3934/mbe.2007.4.675. [11] Z. Feng, Y. Yang, D. Xu, P. Zhang, M. McCauley and J. Glasser, Timely identification of optimal control strategies for emerging infectious diseases, J. Theor. Biol., 259 (2009), 165-171. doi: 10.1016/j.jtbi.2009.03.006. [12] J. Glasser, D. Taneri, Z. Feng, J. H. Chuang, P. Tull, W. Thompson, M. M. McCauley and J. Alexander, Evaluation of targeted influenza vaccination strategies via population modeling, PLoS One, 5 (2010), e12777. doi: 10.1371/journal.pone.0012777. [13] A. B. Gumel, S. Ruan, T. Day, J. Watmough, F. Brauer, P. van den Driessche, D. Gabrielson, C. Bowman, M. E. Alexander, S. Ardal, J. Wu and B. M. Sahai, Modeling strategies for controlling SARS outbreaks, Proc. R. Soc. B., 271 (2004), 2223-2232. doi: 10.1098/rspb.2004.2800. [14] H. W. Hethcote, A thousand and one epidemic models, in "Frontiers in Mathematical Biology" (ed. S. A. Levin), Lecture Notes in Biomathematics, 100, Springer-Verlag, Berlin-Heidelberg-New York, (1994), 504-515. [15] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [16] M. Lipsitch, T. Cohen, M. Muray and B. R. Levin, Antiviral resistance and the control of pandemic influenza, PLoS Med., 4 (2007), 111-120. [17] J. O. Lloyd-Smith, A. P. Galvani and W. M. Getz, Curtailing transmission of severe acute respiratory syndrome within a community and its hospital, Proc. R. Soc. B., 170 (2003), 1979-1989. doi: 10.1098/rspb.2003.2481. [18] I. M. Longini, M. E. Halloran, A. Nizam and Y. Yang, Containing pandemic influenza with antiviral agents, Am. J. Epidemiol., 159 (2004), 623-633. doi: 10.1093/aje/kwh092. [19] J. M. McCaw and J. McVernon, Prophylaxis or treatment? Optimal use of an antiviral stockpile during an influenza pandemic, Math. Biosci., 209 (2007), 336-360. doi: 10.1016/j.mbs.2007.02.003. [20] C. N. Podder, A. B. Gumel, C. Bowman and R. G. McLeod, Mathematical study of the impact of quarantine, isolation and vaccination in curtailing an epidemic, Journal of Biological Sciences, 15 (2007), 1-18. [21] Z. Qiu and Z. Feng, Transmission dynamics of an influenza model with vaccination and antiviral treatment, Bull. Math. Biol., 72 (2010), 1-33. doi: 10.1007/s11538-009-9435-5. [22] R. R. Regoes and S. Bonhoeffer, Emergence of drug-resistant influenza virus: Population dynamical considerations, Science, 312 (2006), 389-391. doi: 10.1126/science.1122947. [23] S. Ruan, W. Wang and S. A. Levin, The effect of global travel on the spread of SARS, Math. Biosci. Eng., 3 (2006), 205-218. doi: 10.3934/mbe.2006.3.205. [24] A. R. Tuite, D. N. Fisman, J. C. Kwong and A. L. Greer, Optimal pandemic influenza vaccine allocation strategies for the Canadian population, PLoS One, 5 (2010), e10520. doi: 10.1371/journal.pone.0010520. [25] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [26] W. Wang and X. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001. [27] P. Yan and Z. Feng, Variability order of the latent and the infectious periods in a deterministic SEIR epidemic model and evaluation of control effectiveness, Math. Biosci., 224 (2010), 43-52.
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