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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

Zi Sang 1, , Zhipeng Qiu 2, , Xiefei Yan 1, and  Yun Zou 1,

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.
Keywords: isolation, metapopualtion epidemic model., Quarantine, entry screening.
Mathematics Subject Classification: Primary: 92B05, 92D3.
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

[15]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

J. Arino and P. van den Driessche, A multi-city epidemic model, Math. Popul. Studies, 10 (2003), 175-193. doi: 10.1080/08898480306720.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[15]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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