# American Institute of Mathematical Sciences

2012, 9(4): 785-807. doi: 10.3934/mbe.2012.9.785

## The impact of migrant workers on the tuberculosis transmission: General models and a case study for China

 1 School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023, China 2 Centre for Disease Modeling, York University, 4700 Keele Street, Toronto, ON M3J1P3, Canada 3 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7

Received  October 2011 Revised  July 2012 Published  October 2012

A tuberculosis (TB) transmission model involving migrant workers is proposed and investigated. The basic reproduction number $\mathcal{R}_{0}$ is calculated, and is shown to be a threshold parameter for the disease to persist or become extinct in the population. The existence and global attractivity of an endemic equilibrium, if $\mathcal{R}_{0}>1$, is also established under some technical conditions. A case study, based on the TB epidemiological and other statistical data in China, indicates that the disease spread can be controlled if effective measures are taken to reduce the reactivation rate of exposed/latent migrant workers. Impact of the migration rate and direction, as well as the duration of home visit stay, on the control of disease spread is also examined numerically.
Citation: Luju Liu, Jianhong Wu, Xiao-Qiang Zhao. The impact of migrant workers on the tuberculosis transmission: General models and a case study for China. Mathematical Biosciences & Engineering, 2012, 9 (4) : 785-807. doi: 10.3934/mbe.2012.9.785
##### References:
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Winston, et al., Tuberculosis among foreign-born persons in the United States, JAMA., 300 (2008), 405-412. doi: 10.1001/jama.300.4.405. [8] C. Castillo-Chavez and Z. Feng, To treat or not to treat: the case of tuberculosis, J. Math. Biol., 35 (1997), 629-656. doi: 10.1007/s002850050069. [9] C. Castillo-Chavez and Z. Feng, Global stability of an age-structure model for TB and its application to optimal vaccination strategies, Math. Biosci., 151 (1998), 135-154. doi: 10.1016/S0025-5564(98)10016-0. [10] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404. [11] [12] [13] [14] T. Cohen and M. Murry, Modeling epidemics of multidrug-resistant M.tuberculosis of heterogeneous fitness, Nature Med., 10 (2004), 1117-1121. doi: 10.1038/nm1110. [15] P. D. O. Davies, Tuberculosis and migration, J. R. Coll. Physicians Lond., 29 (1995), 113-118. [16] [17] M. G. Farah, H. E. meyer, R. 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Voss, M. Baker, et al., Tuberculosis in New Zealand, 1992-2001: a resurgence, Arch. Dis. Child., 90 (2005), 1157-1161. doi: 10.1136/adc.2004.066415. [24] Z. Jia, X. Jia, Y. Liu, et al., Spatial analysis of tuberculosis cases in migrants and permanent residents, Beijing, 2000-2006, Emerg. Infec. Dis., 14 (2008), 1413-1419. doi: 10.3201/1409.071543. [25] D. Kelly and X. Luo, SARS and China's rural migrant labour: roots of a government crisis, in "Population Dynamics and Infectious diseases in Asia" (eds. A. C. Sleigh, H. L. Chee, B. S. A Yeoh, K. H. Phua and R. Safman), Singapore: World Scientific, (2006), 389-408. [26] [27] C. C. McCluskey and P. van den Driessche, Global analysis of two tuberculosis models, J. Dyn. Diff. Equations, 16 (2004), 139-166. doi: 10.1023/B:JODY.0000041283.66784.3e. [28] M. T. McKenna, E. McCray and I. Onorato, The epidemiology of tuberculosis among foreign-born persons in the United States, 1986 to 1993, N. Engl. J. Med., 332 (1995), 1071-1076. doi: 10.1056/NEJM199504203321606. [29] B. M. Murphy, B. H. Singer, S. Anderson, et al., Comparing epidemic tuberculosis in demographically distinct heterogeneous populations, Math. Biosci., 180 (2002), 161-185. doi: 10.1016/S0025-5564(02)00133-5. [30] The Ministry of Health of the People's Republic of China, Report on nationwide random survey for the epidemiology of tuberculosis in 2000, Beijing: The Ministry of Health of The People's Republic of China, 2002. [31] [32] , http://www.stats.gov.cn/, ., (). [33] C. Parry and P. D. O. Davies, The resurgence of tuberculosis, J. Applied. Bacteriol., 81 (1996), 23S-26S. doi: 10.1111/j.1365-2672.1996.tb04829.x. [34] T. C. Porco and S. M. Blower, Quantifying the intrinsic transmission dynamics of tuberculosis, Theor. Popul. Biol., 54 (1998), 117-132. doi: 10.1006/tpbi.1998.1366. [35] A. Saltelli, K. Chan and M. Scott, "Sensitivity Analysis," Probability and Statistics series. John Wiley & Sons: New York, 2000. [36] E. Schneider, M. Moore and K. G. Castro, Epidemiology of tuberculosis in the United States, Clin. Chest. Med., 26 (2005), 183-195. doi: 10.1016/j.ccm.2005.02.007. [37] O. Sharomi, C. N. Podder, A. B. Gumel, et al., Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment, Math. Biosci. Eng., 5 (2008), 145-174. [38] A. C. Sleigh, Health-system reforms to control tuberculosis in China, Lancet., 369 (2007), 626-627. doi: 10.1016/S0140-6736(07)60292-X. [39] H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Cystems," Mathematical Surveys and Monographs, Vol. 41, Amer. Math. Soc., Providence, 1995. [40] H. L. Smith and P. Walman, "The Theory of the Chemostat," Cambridge Univ. Press, 1995. [41] H. R. Thieme, Convergence results and a Poincaré-Bendison trichotomy for asymptotical autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267. [42] 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. [43] L. Wang, J. Liu, and D. P. Chin, Progress in tuberculosis control and the evolving public-health system in China, Lancet., 369 (2007), 691-696. doi: 10.1016/S0140-6736(07)60316-X. [44] World health report 1998: WHO report on the global tuberculosis epidemic 1998., World Health Organization., , (). [45] , http://www.who.int/tb/en/, ., (). [46] WHO, Tuberculosis Fact Sheet. 2007., (, (). [47] L. Zhang, D. Tu, Y. An, et al., The impact of migrants on the epidemiology of tuberculosis in Beijing, China, Int. J. Tuberc. Dis., 10 (2006), 959-962. [48] X.-Q. Zhao, "Dynamical Systems in Population Biology," Springer-Verlag, New York, 2003. [49] Y. Zhou, K. Khan, Z. Feng, et al., Projection of tuberculosis incidence with increasing immigration trends, J. Theor. Biol., 254 (2008), 215-228. doi: 10.1016/j.jtbi.2008.05.026. [50] E. Ziv, C. L. Daley and S. M. Blower, Early therapy for latent tuberculosis infection, Am. J. Epidemiol., 153 (2001), 381-385. doi: 10.1093/aje/153.4.381.

show all references

##### References:
 [1] L. Aggarwal, Tuberculosis-diagnosis and investigation, Hospital Pharmacist, 13 (2006), 73-78. [2] , http://www.agri.gov.cn/, ., (). [3] S. Akhtar and H. G. Mohammad, Seasonality in pulmonary tuberculosis among migrant workers entering Kuwait, BMC Infect Dis., 8 (2008), 8:3. doi: 10.1186/1471-2334-8-3. [4] S. M. Blower, P. M. Small and P. C. Hopewell, Control strategies for tuberculosis epidemics: New models for old problems, Science, 273 (1996), 497-500. doi: 10.1126/science.273.5274.497. [5] S. M. Blower and T. Chou, Modeling the emergence of the 'hot zones': tuberculosis and the amplification dynamics of drug resistance, Nat. Med., 10 (2004), 1111-1116. doi: 10.1038/nm1102. [6] F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci., 171 (2001), 143-154. doi: 10.1016/S0025-5564(01)00057-8. [7] K. P. Cain, S. R. Benoit, C. A. Winston, et al., Tuberculosis among foreign-born persons in the United States, JAMA., 300 (2008), 405-412. doi: 10.1001/jama.300.4.405. [8] C. Castillo-Chavez and Z. Feng, To treat or not to treat: the case of tuberculosis, J. Math. Biol., 35 (1997), 629-656. doi: 10.1007/s002850050069. [9] C. Castillo-Chavez and Z. Feng, Global stability of an age-structure model for TB and its application to optimal vaccination strategies, Math. Biosci., 151 (1998), 135-154. doi: 10.1016/S0025-5564(98)10016-0. [10] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404. [11] [12] [13] [14] T. Cohen and M. Murry, Modeling epidemics of multidrug-resistant M.tuberculosis of heterogeneous fitness, Nature Med., 10 (2004), 1117-1121. doi: 10.1038/nm1110. [15] P. D. O. Davies, Tuberculosis and migration, J. R. Coll. Physicians Lond., 29 (1995), 113-118. [16] [17] M. G. Farah, H. E. meyer, R. Selmer, et al., long-term risk of tuberculosis among immigrants in Norway, Int. J. Epidemiol., 34 (2005), 1005-1011. doi: 10.1093/ije/dyi058. [18] Z. Feng, C. Castillo-Chavez and A. F. Capurro, A model for tuberculosis with exogenous reinfection, Theor. Popul. Biol., 57 (2000), 235-247. doi: 10.1006/tpbi.2000.1451. [19] Z. Feng, W. Huang and C. Castillo-Chavez, On the role of variable latent periods in mathematical models for tuberculosis, J. Dyn. Diff. Equations, 13 (2001), 425-452. doi: 10.1023/A:1016688209771. [20] Z. Feng, M. Iannelli and F. A. Milner, A two-strain tuberculosis model with age of infection, SIAM J. Appl. Math., 62 (2002), 1634-1656. doi: 10.1137/S003613990038205X. [21] J. R. Glynn, Resurgence of tuberculosis and the impact of HIV infection, Br. Med. Bull., 54 (1998), 579-593. doi: 10.1093/oxfordjournals.bmb.a011712. [22] H. Guo and M. Y. Li, Global dynamics to a three-population TB model with immigration and cross-infectin,, preprint., (). [23] S. Howie, L. Voss, M. Baker, et al., Tuberculosis in New Zealand, 1992-2001: a resurgence, Arch. Dis. Child., 90 (2005), 1157-1161. doi: 10.1136/adc.2004.066415. [24] Z. Jia, X. Jia, Y. Liu, et al., Spatial analysis of tuberculosis cases in migrants and permanent residents, Beijing, 2000-2006, Emerg. Infec. Dis., 14 (2008), 1413-1419. doi: 10.3201/1409.071543. [25] D. Kelly and X. Luo, SARS and China's rural migrant labour: roots of a government crisis, in "Population Dynamics and Infectious diseases in Asia" (eds. A. C. Sleigh, H. L. Chee, B. S. A Yeoh, K. H. Phua and R. Safman), Singapore: World Scientific, (2006), 389-408. [26] [27] C. C. McCluskey and P. van den Driessche, Global analysis of two tuberculosis models, J. Dyn. Diff. Equations, 16 (2004), 139-166. doi: 10.1023/B:JODY.0000041283.66784.3e. [28] M. T. McKenna, E. McCray and I. Onorato, The epidemiology of tuberculosis among foreign-born persons in the United States, 1986 to 1993, N. Engl. J. Med., 332 (1995), 1071-1076. doi: 10.1056/NEJM199504203321606. [29] B. M. Murphy, B. H. Singer, S. Anderson, et al., Comparing epidemic tuberculosis in demographically distinct heterogeneous populations, Math. Biosci., 180 (2002), 161-185. doi: 10.1016/S0025-5564(02)00133-5. [30] The Ministry of Health of the People's Republic of China, Report on nationwide random survey for the epidemiology of tuberculosis in 2000, Beijing: The Ministry of Health of The People's Republic of China, 2002. [31] [32] , http://www.stats.gov.cn/, ., (). [33] C. Parry and P. D. O. Davies, The resurgence of tuberculosis, J. Applied. Bacteriol., 81 (1996), 23S-26S. doi: 10.1111/j.1365-2672.1996.tb04829.x. [34] T. C. Porco and S. M. Blower, Quantifying the intrinsic transmission dynamics of tuberculosis, Theor. Popul. Biol., 54 (1998), 117-132. doi: 10.1006/tpbi.1998.1366. [35] A. Saltelli, K. Chan and M. Scott, "Sensitivity Analysis," Probability and Statistics series. John Wiley & Sons: New York, 2000. [36] E. Schneider, M. Moore and K. G. Castro, Epidemiology of tuberculosis in the United States, Clin. Chest. Med., 26 (2005), 183-195. doi: 10.1016/j.ccm.2005.02.007. [37] O. Sharomi, C. N. Podder, A. B. Gumel, et al., Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment, Math. Biosci. Eng., 5 (2008), 145-174. [38] A. C. Sleigh, Health-system reforms to control tuberculosis in China, Lancet., 369 (2007), 626-627. doi: 10.1016/S0140-6736(07)60292-X. [39] H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Cystems," Mathematical Surveys and Monographs, Vol. 41, Amer. Math. Soc., Providence, 1995. [40] H. L. Smith and P. Walman, "The Theory of the Chemostat," Cambridge Univ. Press, 1995. [41] H. R. Thieme, Convergence results and a Poincaré-Bendison trichotomy for asymptotical autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267. [42] 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. [43] L. Wang, J. Liu, and D. P. Chin, Progress in tuberculosis control and the evolving public-health system in China, Lancet., 369 (2007), 691-696. doi: 10.1016/S0140-6736(07)60316-X. [44] World health report 1998: WHO report on the global tuberculosis epidemic 1998., World Health Organization., , (). [45] , http://www.who.int/tb/en/, ., (). [46] WHO, Tuberculosis Fact Sheet. 2007., (, (). [47] L. Zhang, D. Tu, Y. An, et al., The impact of migrants on the epidemiology of tuberculosis in Beijing, China, Int. J. Tuberc. Dis., 10 (2006), 959-962. [48] X.-Q. Zhao, "Dynamical Systems in Population Biology," Springer-Verlag, New York, 2003. [49] Y. Zhou, K. Khan, Z. Feng, et al., Projection of tuberculosis incidence with increasing immigration trends, J. Theor. Biol., 254 (2008), 215-228. doi: 10.1016/j.jtbi.2008.05.026. [50] E. Ziv, C. L. Daley and S. M. Blower, Early therapy for latent tuberculosis infection, Am. J. Epidemiol., 153 (2001), 381-385. doi: 10.1093/aje/153.4.381.
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