# American Institute of Mathematical Sciences

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] M. E. Halloran, I. M. Longini, A. Nizam and Y. Yang, Containing bioterrorist smallpox,, Science., 298 (2002), 1428. doi: 10.1126/science.1074674. [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. [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. [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. [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. [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. [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. [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. [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. [24] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities,, Academic Press, (1969). [25] M. I. Meltzer, Multiple contact dates and SARS incubation periods,, Emerg. Infect. Dis., 10 (2004), 207. doi: 10.3201/eid1002.030426. [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. [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. [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. [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. [30] E. Rash, Smallpox Overview,, 1977., (). [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. [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. [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. [34] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics,, Monographs and Textbooks in Pure and Applied Mathematics, (1985). [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] M. E. Halloran, I. M. Longini, A. Nizam and Y. Yang, Containing bioterrorist smallpox,, Science., 298 (2002), 1428. doi: 10.1126/science.1074674. [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. [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. [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. [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. [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. [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. [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. [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. [24] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities,, Academic Press, (1969). [25] M. I. Meltzer, Multiple contact dates and SARS incubation periods,, Emerg. Infect. Dis., 10 (2004), 207. doi: 10.3201/eid1002.030426. [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. [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. [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. [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. [30] E. Rash, Smallpox Overview,, 1977., (). [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. [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. [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. [34] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics,, Monographs and Textbooks in Pure and Applied Mathematics, (1985). [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. [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.
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