August  2015, 20(6): 1685-1713. doi: 10.3934/dcdsb.2015.20.1685

Modeling of contact tracing in epidemic populations structured by disease age

1. 

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, United States

Received  July 2014 Revised  October 2014 Published  June 2015

We consider an age-structured epidemic model with two basic public health interventions: (i) identifying and isolating symptomatic cases, and (ii) tracing and quarantine of the contacts of identified infectives. The dynamics of the infected population are modeled by a nonlinear infection-age-dependent partial differential equation, which is coupled with an ordinary differential equation that describes the dynamics of the susceptible population. Theoretical results about global existence and uniqueness of positive solutions are proved. We also present two practical applications of our model: (1) we assess public health guidelines about emergency preparedness and response in the event of a smallpox bioterrorist attack; (2) we simulate the 2003 SARS outbreak in Taiwan and estimate the number of cases avoided by contact tracing. Our model can be applied as a rational basis for decision makers to guide interventions and deploy public health resources in future epidemics.
Citation: Xi Huo. Modeling of contact tracing in epidemic populations structured by disease age. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1685-1713. doi: 10.3934/dcdsb.2015.20.1685
References:
[1]

G. K. Aldis and M. G. Roberts, An integral equation model for the control of a smallpox outbreak,, Math. Biosci., 195 (2005), 1.  doi: 10.1016/j.mbs.2005.01.006.  Google Scholar

[2]

J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, Simple models for containment of a pandemic,, J. R. Soc. Interface, 3 (2006), 453.  doi: 10.1098/rsif.2006.0112.  Google Scholar

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C. T. Bauch, J. O. Lloyd-Smith, M. P. Coffee and A. P. Galvani, Dynamically modeling SARS and other newly emerging respiratory illnesses: Past, present, and future,, Epidemiology, 16 (2005), 791.  doi: 10.1097/01.ede.0000181633.80269.4c.  Google Scholar

[4]

F. Carrat, E. Vergu, N. M. Ferguson, M. Lemaitre, S. Cauchemez and S. Leach, et al., Time lines of infection and disease in human influenza: A review of volunteer challenge studies,, Am. J. Epidemiol., 167 (2008), 775.  doi: 10.1093/aje/kwm375.  Google Scholar

[5]

Centers for Disease Control and Prevention (CDC), Use of quarantine to prevent transmission of Severe Acute Respiratory Syndrome - Taiwan,, Morb Mortal Wkly Rep., 290 (2003), 1021.   Google Scholar

[6]

P. K. S. Chan, W. K. To, K. C. Ng, R. K. Y. Lam, T. K. Ng and R. C. W. Chan, et al., Laboratory Diagnosis of SARS,, Emerg. Infect. Dis., 10 (2004), 825.  doi: 10.3201/eid1005.030682.  Google Scholar

[7]

T. Day, A. Park, N. Madras, A. Gumel and J. Wu, When is quarantine a useful control strategy for emerging infectious diseases?,, Am. J. Epidemiol., 163 (2006), 479.  doi: 10.1093/aje/kwj056.  Google Scholar

[8]

M. Eichner, Case isolation and contact tracing can prevent the spread of smallpox,, Am. J. Epidemiol., 158 (2003), 118.  doi: 10.1093/aje/kwg104.  Google Scholar

[9]

Z. Feng, S. Towers and Y. Yang, Modeling the effects of vaccination and treatment on pandemic influenza,, AAPS J., 13 (2011), 427.  doi: 10.1208/s12248-011-9284-7.  Google Scholar

[10]

Z. Feng, D. Xu and H. Zhao, Epidemiological models with non-exponentially distributed disease stages and applications to disease control,, Bull. Math. Biol., 69 (2007), 1511.  doi: 10.1007/s11538-006-9174-9.  Google Scholar

[11]

Z. Feng, Y. Yang, D. Xu, P. Zhang, M. M. McCauley and J. W. Glasser, Timely identification of optimal control strategies for emerging infectious diseases,, J. Theor. Biol., 259 (2009), 165.  doi: 10.1016/j.jtbi.2009.03.006.  Google Scholar

[12]

C. Fraser, S. Riley, R. M. Anderson and N. M. Ferguson, Factors that make an infectious disease outbreak controllable,, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 6146.  doi: 10.1073/pnas.0307506101.  Google Scholar

[13]

J. W. Glasser, N. Hupert, M. M. McCauley and R. Hatchett, Modeling and public health emergency responses: Lessons from SARS,, Epidemics., 3 (2011), 32.  doi: 10.1016/j.epidem.2011.01.001.  Google Scholar

[14]

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, Modelling strategies for controlling SARS outbreaks,, Proc. Biol. Sci., 271 (2004), 2223.  doi: 10.1098/rspb.2004.2800.  Google Scholar

[15]

M. E. Halloran, I. M. Longini, A. Nizam and Y. Yang, Containing bioterrorist smallpox,, Science., 298 (2002), 1428.  doi: 10.1126/science.1074674.  Google Scholar

[16]

H. Hethcote, M. Zhien and L. Shengbing, Effects of quarantine in six endemic models for infectious diseases,, Math. Biosci., 180 (2002), 141.  doi: 10.1016/S0025-5564(02)00111-6.  Google Scholar

[17]

Y. H. Hsieh, C. W. S. Chen and S. B. Hsu, SARS outbreak, Taiwan, 2003,, Emerg. Infect. Dis., 10 (2004), 201.  doi: 10.3201/eid1002.030515.  Google Scholar

[18]

L. Y. Hsu, C. C. Lee, J. A. Green, B. Ang, N. I. Paton, L. Lee, J. S. Villacian, P. L. Lim, A. Earnest and Y. S. Leo, Severe acute respiratory syndrome (SARS) in Singapore: Clinical features of index patient and initial contacts,, Emerg. Infect. Dis., 9 (2003), 713.  doi: 10.3201/eid0906.030264.  Google Scholar

[19]

S. B. Hsu and Y. H. Hsieh, Modeling intervention measures and severity-dependent public response during Severe Acute Respiratory Syndrome outbreak,, SIAM J. Appl. Math., 66 (2006), 627.  doi: 10.1137/040615547.  Google Scholar

[20]

H. Inaba and H. Nishiura, The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model,, Math. Biosci., 216 (2008), 77.  doi: 10.1016/j.mbs.2008.08.005.  Google Scholar

[21]

E. H. Kaplan, D. L. Craft and L. M. Wein, Emergency response to a smallpox attack: The case for mass vaccination,, Proc. Natl. Acad. Sci. U.S.A., 99 (2002), 10935.  doi: 10.1073/pnas.162282799.  Google Scholar

[22]

E. H. Kaplan, D. L. Craft and L. M. Wein, Analyzing bioterror response logistics: The case of smallpox,, Math. Biosci., 185 (2003), 33.  doi: 10.1016/S0025-5564(03)00090-7.  Google Scholar

[23]

M. Kretzschmar, S. Van Den Hof, J. Wallinga and J. Van Wijngaarden, Ring vaccination and smallpox control,, Emerg. Infect. Dis., 10 (2004), 832.  doi: 10.3201/eid1005.030419.  Google Scholar

[24]

V. Lakshmikantham and S. Leela, Differential and Integral Inequalities,, Academic Press, (1969).   Google Scholar

[25]

M. I. Meltzer, Multiple contact dates and SARS incubation periods,, Emerg. Infect. Dis., 10 (2004), 207.  doi: 10.3201/eid1002.030426.  Google Scholar

[26]

M. I. Meltzer, I. Damon, J. W. LeDuc and J. D. Millar, Modeling potential responses to smallpox as a bioterrorist weapon,, Emerg. Infect. Dis., 7 (2001), 959.   Google Scholar

[27]

J. Müller, M. Kretzschmar and K. Dietz, Contact tracing in stochastic and deterministic epidemic models,, Math. Biosci., 164 (2000), 39.  doi: 10.1016/S0025-5564(99)00061-9.  Google Scholar

[28]

H. Nishiura, K. Patanarapelert, M. Sriprom, W. Sarakorn, S. Sriyab and I. M. Tang, Modelling potential responses to severe acute respiratory syndrome in Japan: The role of initial attack size, precaution, and quarantine,, J. Epidemiol. Community Health, 58 (2004), 186.  doi: 10.1136/jech.2003.014894.  Google Scholar

[29]

J. S. M. Peiris, C. M. Chu, V. C. C. Cheng, K. S. Chan, I. F. N. Hung and L. L. M. Poon, et al., Clinical progression and viral load in a community outbreak of coronavirus-associated SARS pneumonia: A prospective study,, Lancet, 361 (2003), 1767.  doi: 10.1016/S0140-6736(03)13412-5.  Google Scholar

[30]

E. Rash, Smallpox Overview,, 1977., ().   Google Scholar

[31]

S. Del Valle, H. Hethcote, J. M. Hyman and C. Castillo-Chavez, Effects of behavioral changes in a smallpox attack model,, Math. Biosci., 195 (2005), 228.  doi: 10.1016/j.mbs.2005.03.006.  Google Scholar

[32]

B. Vidondo, M. Schwehm, A. Bühlmann and M. Eichner, Finding and removing highly connected individuals using suboptimal vaccines,, BMC Infect. Dis., 12 (2012).  doi: 10.1186/1471-2334-12-51.  Google Scholar

[33]

W. Wang and S. Ruan, Simulating the SARS outbreak in Beijing with limited data,, J. Theor. Biol., 227 (2004), 369.  doi: 10.1016/j.jtbi.2003.11.014.  Google Scholar

[34]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics,, Monographs and Textbooks in Pure and Applied Mathematics, (1985).   Google Scholar

[35]

G. F. Webb, Y. H. Hsieh, J. Wu and M. J. Blaser, Pre-symptomatic influenza transmission, surveillance, and school closings: implications for Novel Influenza A (H1N1),, Math. Model. Nat. Phenom., 5 (2010), 191.  doi: 10.1051/mmnp/20105312.  Google Scholar

[36]

M. Wharton, R. Strikas, R. Harpaz, L. D. Rotz, B. Schwartz and C. G. Casey, et al., Recommendations for using smallpox vaccine in a pre-event vaccination program. Supplemental recommendations of the Advisory Committee on Immunization Practices (ACIP) and the Healthcare Infection Control Practices Advisory Committee (HICPAC),, MMWR. Recomm. Rep., 52 (2003), 1.   Google Scholar

show all references

References:
[1]

G. K. Aldis and M. G. Roberts, An integral equation model for the control of a smallpox outbreak,, Math. Biosci., 195 (2005), 1.  doi: 10.1016/j.mbs.2005.01.006.  Google Scholar

[2]

J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, Simple models for containment of a pandemic,, J. R. Soc. Interface, 3 (2006), 453.  doi: 10.1098/rsif.2006.0112.  Google Scholar

[3]

C. T. Bauch, J. O. Lloyd-Smith, M. P. Coffee and A. P. Galvani, Dynamically modeling SARS and other newly emerging respiratory illnesses: Past, present, and future,, Epidemiology, 16 (2005), 791.  doi: 10.1097/01.ede.0000181633.80269.4c.  Google Scholar

[4]

F. Carrat, E. Vergu, N. M. Ferguson, M. Lemaitre, S. Cauchemez and S. Leach, et al., Time lines of infection and disease in human influenza: A review of volunteer challenge studies,, Am. J. Epidemiol., 167 (2008), 775.  doi: 10.1093/aje/kwm375.  Google Scholar

[5]

Centers for Disease Control and Prevention (CDC), Use of quarantine to prevent transmission of Severe Acute Respiratory Syndrome - Taiwan,, Morb Mortal Wkly Rep., 290 (2003), 1021.   Google Scholar

[6]

P. K. S. Chan, W. K. To, K. C. Ng, R. K. Y. Lam, T. K. Ng and R. C. W. Chan, et al., Laboratory Diagnosis of SARS,, Emerg. Infect. Dis., 10 (2004), 825.  doi: 10.3201/eid1005.030682.  Google Scholar

[7]

T. Day, A. Park, N. Madras, A. Gumel and J. Wu, When is quarantine a useful control strategy for emerging infectious diseases?,, Am. J. Epidemiol., 163 (2006), 479.  doi: 10.1093/aje/kwj056.  Google Scholar

[8]

M. Eichner, Case isolation and contact tracing can prevent the spread of smallpox,, Am. J. Epidemiol., 158 (2003), 118.  doi: 10.1093/aje/kwg104.  Google Scholar

[9]

Z. Feng, S. Towers and Y. Yang, Modeling the effects of vaccination and treatment on pandemic influenza,, AAPS J., 13 (2011), 427.  doi: 10.1208/s12248-011-9284-7.  Google Scholar

[10]

Z. Feng, D. Xu and H. Zhao, Epidemiological models with non-exponentially distributed disease stages and applications to disease control,, Bull. Math. Biol., 69 (2007), 1511.  doi: 10.1007/s11538-006-9174-9.  Google Scholar

[11]

Z. Feng, Y. Yang, D. Xu, P. Zhang, M. M. McCauley and J. W. Glasser, Timely identification of optimal control strategies for emerging infectious diseases,, J. Theor. Biol., 259 (2009), 165.  doi: 10.1016/j.jtbi.2009.03.006.  Google Scholar

[12]

C. Fraser, S. Riley, R. M. Anderson and N. M. Ferguson, Factors that make an infectious disease outbreak controllable,, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 6146.  doi: 10.1073/pnas.0307506101.  Google Scholar

[13]

J. W. Glasser, N. Hupert, M. M. McCauley and R. Hatchett, Modeling and public health emergency responses: Lessons from SARS,, Epidemics., 3 (2011), 32.  doi: 10.1016/j.epidem.2011.01.001.  Google Scholar

[14]

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, Modelling strategies for controlling SARS outbreaks,, Proc. Biol. Sci., 271 (2004), 2223.  doi: 10.1098/rspb.2004.2800.  Google Scholar

[15]

M. E. Halloran, I. M. Longini, A. Nizam and Y. Yang, Containing bioterrorist smallpox,, Science., 298 (2002), 1428.  doi: 10.1126/science.1074674.  Google Scholar

[16]

H. Hethcote, M. Zhien and L. Shengbing, Effects of quarantine in six endemic models for infectious diseases,, Math. Biosci., 180 (2002), 141.  doi: 10.1016/S0025-5564(02)00111-6.  Google Scholar

[17]

Y. H. Hsieh, C. W. S. Chen and S. B. Hsu, SARS outbreak, Taiwan, 2003,, Emerg. Infect. Dis., 10 (2004), 201.  doi: 10.3201/eid1002.030515.  Google Scholar

[18]

L. Y. Hsu, C. C. Lee, J. A. Green, B. Ang, N. I. Paton, L. Lee, J. S. Villacian, P. L. Lim, A. Earnest and Y. S. Leo, Severe acute respiratory syndrome (SARS) in Singapore: Clinical features of index patient and initial contacts,, Emerg. Infect. Dis., 9 (2003), 713.  doi: 10.3201/eid0906.030264.  Google Scholar

[19]

S. B. Hsu and Y. H. Hsieh, Modeling intervention measures and severity-dependent public response during Severe Acute Respiratory Syndrome outbreak,, SIAM J. Appl. Math., 66 (2006), 627.  doi: 10.1137/040615547.  Google Scholar

[20]

H. Inaba and H. Nishiura, The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model,, Math. Biosci., 216 (2008), 77.  doi: 10.1016/j.mbs.2008.08.005.  Google Scholar

[21]

E. H. Kaplan, D. L. Craft and L. M. Wein, Emergency response to a smallpox attack: The case for mass vaccination,, Proc. Natl. Acad. Sci. U.S.A., 99 (2002), 10935.  doi: 10.1073/pnas.162282799.  Google Scholar

[22]

E. H. Kaplan, D. L. Craft and L. M. Wein, Analyzing bioterror response logistics: The case of smallpox,, Math. Biosci., 185 (2003), 33.  doi: 10.1016/S0025-5564(03)00090-7.  Google Scholar

[23]

M. Kretzschmar, S. Van Den Hof, J. Wallinga and J. Van Wijngaarden, Ring vaccination and smallpox control,, Emerg. Infect. Dis., 10 (2004), 832.  doi: 10.3201/eid1005.030419.  Google Scholar

[24]

V. Lakshmikantham and S. Leela, Differential and Integral Inequalities,, Academic Press, (1969).   Google Scholar

[25]

M. I. Meltzer, Multiple contact dates and SARS incubation periods,, Emerg. Infect. Dis., 10 (2004), 207.  doi: 10.3201/eid1002.030426.  Google Scholar

[26]

M. I. Meltzer, I. Damon, J. W. LeDuc and J. D. Millar, Modeling potential responses to smallpox as a bioterrorist weapon,, Emerg. Infect. Dis., 7 (2001), 959.   Google Scholar

[27]

J. Müller, M. Kretzschmar and K. Dietz, Contact tracing in stochastic and deterministic epidemic models,, Math. Biosci., 164 (2000), 39.  doi: 10.1016/S0025-5564(99)00061-9.  Google Scholar

[28]

H. Nishiura, K. Patanarapelert, M. Sriprom, W. Sarakorn, S. Sriyab and I. M. Tang, Modelling potential responses to severe acute respiratory syndrome in Japan: The role of initial attack size, precaution, and quarantine,, J. Epidemiol. Community Health, 58 (2004), 186.  doi: 10.1136/jech.2003.014894.  Google Scholar

[29]

J. S. M. Peiris, C. M. Chu, V. C. C. Cheng, K. S. Chan, I. F. N. Hung and L. L. M. Poon, et al., Clinical progression and viral load in a community outbreak of coronavirus-associated SARS pneumonia: A prospective study,, Lancet, 361 (2003), 1767.  doi: 10.1016/S0140-6736(03)13412-5.  Google Scholar

[30]

E. Rash, Smallpox Overview,, 1977., ().   Google Scholar

[31]

S. Del Valle, H. Hethcote, J. M. Hyman and C. Castillo-Chavez, Effects of behavioral changes in a smallpox attack model,, Math. Biosci., 195 (2005), 228.  doi: 10.1016/j.mbs.2005.03.006.  Google Scholar

[32]

B. Vidondo, M. Schwehm, A. Bühlmann and M. Eichner, Finding and removing highly connected individuals using suboptimal vaccines,, BMC Infect. Dis., 12 (2012).  doi: 10.1186/1471-2334-12-51.  Google Scholar

[33]

W. Wang and S. Ruan, Simulating the SARS outbreak in Beijing with limited data,, J. Theor. Biol., 227 (2004), 369.  doi: 10.1016/j.jtbi.2003.11.014.  Google Scholar

[34]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics,, Monographs and Textbooks in Pure and Applied Mathematics, (1985).   Google Scholar

[35]

G. F. Webb, Y. H. Hsieh, J. Wu and M. J. Blaser, Pre-symptomatic influenza transmission, surveillance, and school closings: implications for Novel Influenza A (H1N1),, Math. Model. Nat. Phenom., 5 (2010), 191.  doi: 10.1051/mmnp/20105312.  Google Scholar

[36]

M. Wharton, R. Strikas, R. Harpaz, L. D. Rotz, B. Schwartz and C. G. Casey, et al., Recommendations for using smallpox vaccine in a pre-event vaccination program. Supplemental recommendations of the Advisory Committee on Immunization Practices (ACIP) and the Healthcare Infection Control Practices Advisory Committee (HICPAC),, MMWR. Recomm. Rep., 52 (2003), 1.   Google Scholar

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