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A tuberculosis model with seasonality. (English) Zbl 1191.92027
Summary: The statistical data of tuberculosis (TB) cases show seasonal fluctuations in many countries. A TB model incorporating seasonality is developed and the basic reproduction ratio $R_{0}$ is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the disease eventually disappears if $R_{0}<1$, and there exists at least one positive periodic solution and the disease is uniformly persistent if $R_{0}>1$. Numerical simulations indicate that there may be a unique positive periodic solution which is globally asymptotically stable if $R_{0}>1$. Parameter values of the model are estimated according to demographic and epidemiological data in China. The simulation results are in good accordance with the seasonal variation of the reported cases of active TB in China.

92C60Medical epidemiology
34C60Qualitative investigation and simulation of models (ODE)
34D23Global stability of ODE
65C60Computational problems in statistics
34D05Asymptotic stability of ODE
Full Text: DOI
[1] Akhtar, S., Mohammad, H.G., 2008. Seasonality in pulmonary tuberculosis among migrant workers entering Kuwait. BMC Infect. Dis. 8. doi: 10.1186/1471-2334-8-3 .
[2] Altizer, S., Dobson, A., Hosseini, P., Hudson, P., Pascual, M., Rohani, P., 2006. Seasonality and the dynamics of infectious diseases. Ecol. Lett. 9, 467--484. · doi:10.1111/j.1461-0248.2005.00879.x
[3] Aron, J.L., Schwartz, I.B., 1984. Seasonality and period-doubling bifurcations in an epidemic model. J. Theor. Biol. 110, 665--679. · doi:10.1016/S0022-5193(84)80150-2
[4] Bass, J.B. Jr., Farer, L.S., Hopewell, P.C., O’Brien, R., Jacobs, R.F., Ruben, F., Snider, D.E. Jr., Thornton, G., 1994. American thoracic society, treatment of tuberculosis and tuberculosis infection in adults and children. Am. J. Respir. Crit. Care Med. 149, 1359--1374.
[5] Bleed, D., Watt, C., Dye, C., 2001. World health report 2001: global tuberculosis control. Technical Report, World Health Organization, WHO/CDS/TB/2001.287. http://whqlibdoc.who.int/hq/2001/WHO_CDS_TB_2001.287.pdf .
[6] Blower, S.M., 1995. The intrinsic transmission dynamics of tuberculosis epidemics. Nat. Med. 1, 815--821. · doi:10.1038/nm0895-815
[7] Blower, S.M., Chou, T., 2004. Modeling the emergence of the ’hot zones’: tuberculosis and the amplification dynamics of drug resistance. Nat. Med. 10, 1111--1116. · doi:10.1038/nm1102
[8] Blower, S.M., Small, P.M., Hopewell, P.C., 1996. Control strategies for tuberculosis epidemics: new models for old problems. Science 273, 497--500. · doi:10.1126/science.273.5274.497
[9] Douglas, A.S., Strachan, D.P., Maxwell, J.D., 1996. Seasonality of tuberculosis: the reverse of other respiratory disease in the UK. Thorax 51, 944--946. · doi:10.1136/thx.51.9.944
[10] Dye, C., Floyd, K., Uplekar, M., 2008. World health report 2008: Global tuberculosis control: surveillance, planning, financing. World Health organization, WHO/HTM/TB/2008.393. http://www.who.int/entity/tb/publications/global_report/2008/pdf/-fullreport.pdf .
[11] Grassly, N.C., Fraser, C., 2006. Seasonality infectious disease epidemiology. Proc. R. Soc. B 273, 2541--2550. · doi:10.1098/rspb.2006.3604
[12] Greenman, J., Kamo, M., Boots, M., 2004. External forcing of ecological and epidemiological systems: a resonance approach. Physica D 190, 136--151. · Zbl 1040.92046 · doi:10.1016/j.physd.2003.08.008
[13] Hethcote, H.W., Yorke, J.A., 1984. Gonorrhea Transmission Dynamics and Control, Lecture Notes in Biomathematics, vol. 56, p. 105. Springer, Berlin. · Zbl 0542.92026
[14] Janmeja, A.K., Mohapatra, P.R., 2005. Seasonality of tuberculosis. Int. J. Tuberc. Lung Dis. 9, 704--705.
[15] Leung, C.C., Yew, W.W., Chan, T.Y.K., Tam, C.M., Chan, C.Y., Chan, C.K., Tang, N., Chang, K.C., Law, W.S., 2005. Seasonal pattern of tuberculosis in Hong Kong. Int. J. Epidemiol. 34, 924--930. · doi:10.1093/ije/dyi080
[16] Lietman, T., Blower, S.M., 2000. Potential impact of tuberculosis vaccines as epidemic control agents. Clin. Infect. Dis. 30, s316--s322. · doi:10.1086/313881
[17] Ma, Z., Zhou, Y., Wang, W., Jin, Z., 2004. Mathematical Modeling and Studying of Dynamic Models of Infectious Diseases. Science Press, London.
[18] Ministry of Health of the People’s Republic of China, 2002. Report on nationwide random survey for the epidemiology of tuberculosis in 2000, Beijing: The Ministry of Health of the People’s Republic of China.
[19] Ministry of Health of the People’s Republic of China, 2005--2009. The Ministry of Health Bulletin. .
[20] Ministry of Health, China, 2006. Notifiable communicable Disease in China, 2007, http://www.moh.gov.cn/newshtml/17829.htm .
[21] Nagayama, N., Ohmori, M., 2006. Seasonality in various forms of tuberculosis. Int. J. Tuberc. Lung Dis. 10, 1117--1122.
[22] National Bureau of Statistics of China, 2008. Statistical Data. http://www.stats.gov.cn/tjsj/ndsj/2007/indexch.htm . · Zbl 1269.62035
[23] Porco, T.C., Blower, S.M., 1998. Quantifying the intrinsic transmission dynamics of tuberculosis. Theor. Popul. Biol. 54, 117--132. · Zbl 0921.92018 · doi:10.1006/tpbi.1998.1366
[24] Rios, M., Garcia, J.M., Sanchez, J.A., Perez, D., 2000. A statistical analysis of the seasonality in pulmonary tuberculosis. Eur. J. Epidemiol. 16, 483--488. · doi:10.1023/A:1007653329972
[25] Rodrigues, P., Gomes, M.G., Rebelo, C., 2007. Drug resistance in tuberculosis-a reinfection model. Theor. Popul. Biol. 71, 196--212. · Zbl 1118.92036 · doi:10.1016/j.tpb.2006.10.004
[26] Saltelli, A., Chan, K., Scott, M. (Eds.), 2000. Sensitivity Analysis, Probability and Statistics Series. Wiley, New York. · Zbl 0961.62091
[27] Schaaf, H.S., Nel, E.D., Beyers, N., Gie, R.P., Scott, F., Donald, P.R., 1996. A decade of experience with Mycobacterium tuberculosis culture from children: a seasonal influence of children tuberculosis. Tuber. Lung Dis. 77, 43--46. · doi:10.1016/S0962-8479(96)90074-X
[28] Sharomi, O., Podder, C.N., Gumel, A.B., Song, B., 2008. Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment. Math. Biosci. Eng. 5, 145--174. · Zbl 1140.92016
[29] Smith, H.L., 1995. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, vol. 41. Am. Math. Soc., Providence. · Zbl 0821.34003
[30] Smith, H.L., Walman, P., 1995. The Theory of the Chemostat. Cambridge University Press, Cambridge.
[31] Thieme, H.R., 1992. Convergence results and a Poincaré--Bendison trichotomy for asymptotical autonomous differential equations. J. Math. Biol. 30, 755--763. · Zbl 0761.34039 · doi:10.1007/BF00173267
[32] Thorpe, L.E., Frieden, T.R., Laserson, K.F., Wells, C., Khatri, G.R., 2004. Seasonality of tuberculosis in India: is it real and what does it tell us? Lancet 364, 1613--1614. · doi:10.1016/S0140-6736(04)17316-9
[33] Van Den Driessche, P., Watmough, J., 2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29--48. · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6
[34] Wang, W., Zhao, X.-Q., 2008. Threshold dynamics for compartmental epidemic models in periodic environments. J. Dyn. Differ. Equ. 20, 699--717. · Zbl 1157.34041 · doi:10.1007/s10884-008-9111-8
[35] Wang, L., Liu, J., Chin, D.P., 2007. Progress in tuberculosis control and the evolving public health system in China. Lancet 369, 691--696. · doi:10.1016/S0140-6736(07)60316-X
[36] WHO, 2006. Global tuberculosis control. WHO report. WHO/HTM/TB/20-06.362. Geneva: World Health Organization.
[37] WHO, 2007. Tuberculosis Fact Sheet. http://www.who.int/features/factfiles/tb_facts/en/index1.html .
[38] Zhang, F., Zhao, X.-Q., 2007. A periodic epidemic model in a patchy environment. J. Math. Anal. Appl. 325, 496--516. · Zbl 1101.92046 · doi:10.1016/j.jmaa.2006.01.085
[39] Zhao, X.-Q., 2003. Dynamical Systems in Population Biology. Springer, New York. · Zbl 1023.37047
[40] Zhou, Y., Khan, K., Feng, Z., Wu, J., 2008. Projection of tuberculosis incidence with increasing immigration trends. J. Theor. Biol. 254, 215--228. · doi:10.1016/j.jtbi.2008.05.026
[41] Ziv, E., Daley, C.L., Blower, S.M., 2001. Early therapy for latent tuberculosis infection. Am. J. Epidemiol. 153, 381--385. · doi:10.1093/aje/153.4.381