Bulletin of Mathematical Biology

, Volume 74, Issue 5, pp 1226–1251 | Cite as

Modeling Seasonal Rabies Epidemics in China

  • Juan Zhang
  • Zhen Jin
  • Gui-Quan Sun
  • Xiang-Dong Sun
  • Shigui Ruan
Original Article


Human rabies, an infection of the nervous system, is a major public-health problem in China. In the last 60 years (1950–2010) there had been 124,255 reported human rabies cases, an average of 2,037 cases per year. However, the factors and mechanisms behind the persistence and prevalence of human rabies have not become well understood. The monthly data of human rabies cases reported by the Chinese Ministry of Health exhibits a periodic pattern on an annual base. The cases in the summer and autumn are significantly higher than in the spring and winter. Based on this observation, we propose a susceptible, exposed, infectious, and recovered (SEIRS) model with periodic transmission rates to investigate the seasonal rabies epidemics. We evaluate the basic reproduction number R 0, analyze the dynamical behavior of the model, and use the model to simulate the monthly data of human rabies cases reported by the Chinese Ministry of Health. We also carry out some sensitivity analysis of the basic reproduction number R 0 in terms of various model parameters. Moreover, we demonstrate that it is more reasonable to regard R 0 rather than the average basic reproduction number \(\bar{R}_{0}\) or the basic reproduction number \(\hat{R}_{0}\) of the corresponding autonomous system as a threshold for the disease. Finally, our studies show that human rabies in China can be controlled by reducing the birth rate of dogs, increasing the immunization rate of dogs, enhancing public education and awareness about rabies, and strengthening supervision of pupils and children in the summer and autumn.


Rabies SEIRS model Basic reproduction number Periodic solution Vaccination 



The research was partially supported by the National Natural Science Foundation of China (11171314, 11147015 and 10901145), Program for Basic Research (2010011007), International and Technical Cooperation Project (2010081005) and Bairen Project of Shan’xi Province, and the National Science Foundation of USA (DMS-1022728). The authors would like to thank Dr. Luju Liu for her help on using the Matlab program. The authors are also grateful to the referees for their helpful comments and suggestions.


  1. AnshanCDC (2011). Rabies knowledge for 20 questions. http://www.ascdc.com.cn/newscontent.asp?lsh=5.
  2. Bacaer, N., & Guernaoui, S. (2006). The epidemic threshold of vector-borne diseases with seasonality. J. Math. Biol., 53, 421–436. MathSciNetMATHCrossRefGoogle Scholar
  3. Bai, Z., & Zhou, Y. (2011). Threshold dynamics of a Bacillary Dysentery model with seasonal fluctuation. Discrete Contin. Dyn. Syst., Ser. B, 15(1), 1–14. MathSciNetMATHCrossRefGoogle Scholar
  4. Bjornstad, O. N., Finkenstadt, B. F., & Grenfell, B. T. (2002). Dynamics of measles epidemics: Estimating scaling of transmission rates using a time series SIR model. Ecol. Monogr., 72(2), 169–184. Google Scholar
  5. CDC (2010a). Rabies—How is rabies transmitted? http://www.cdc.gov/rabies/transmission/index.html.
  6. CDC (2010b). Rabies—What are the signs and symptoms of rabies? http://www.cdc.gov/rabies/symptoms/index.html.
  7. ChinaCDC (2011). Rabies answer of knowledge and hot question. http://www.chinacdc.cn/jkzt/crb/kqb/kqbzstd/201109/t20110922_52966.htm.
  8. Chowell, G., Ammon, C., Hengartner, N., & Hyman, J. (2006). Transmission dynamics of the great influenza pandemic of 1918 in Geneva, Switzerland: Assessing the effects of hypothetical interventions. J. Theor. Biol., 241, 193–204. MathSciNetCrossRefGoogle Scholar
  9. Diekmann, O., Heesterbeek, J. A. P., & Roberts, M. G. (1990). On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. J. Math. Biol., 28, 365–382. MathSciNetMATHCrossRefGoogle Scholar
  10. Diekmann, O., Heesterbeek, J. A. P., & Roberts, M. G. (2010). The construction of next-generation matrices for compartmental epidemic models. J. R. Soc. Interface, 7, 873–885. CrossRefGoogle Scholar
  11. Dowell, S. F. (2001). Seasonal variation in host susceptibility and cycles of certain infectious diseases. Emerg. Infect. Dis., 7(3), 369–374. Google Scholar
  12. Dushoff, J., Plotkin, J. B., Levin, S. A., & Earn, D. J. D. (2004). Dynamical resonance can account for seasonality of influenza epidemics. Proc. Natl. Acad. Sci. USA, 101, 16915–16916. CrossRefGoogle Scholar
  13. Earn, D., Rohani, P., Bolker, B., & Grenfell, B. (2000). A simple model for complex dynamical transitions in epidemics. Science, 287, 667–670. CrossRefGoogle Scholar
  14. Greenhalgh, D., & Moneim, I. A. (2003). SIRS epidemic model and simulations using different types of seasonal contact rate. Syst. Anal. Model. Simul., 43(5), 573–600. MathSciNetMATHCrossRefGoogle Scholar
  15. Hampson, K., Dushoff, J., Bingham, J., Bruckner, G., Ali, Y., & Dobson, A. (2007). Synchronous cycles of domestic dog rabies in Sub-Saharan Africa and the impact of control effort. Proc. Natl. Acad. Sci. USA, 104, 7717–7722. CrossRefGoogle Scholar
  16. Hou, Q., Jin, Z., & Ruan, S. (2012). Dynamics of rabies epidemics and the impact of control efforts in Guangdong Province, China. J. Theor. Biol., 300, 39–47. MathSciNetCrossRefGoogle Scholar
  17. Liu, J. (2010). Threshold dynamics for a HFMD epidemic model with periodic transmission rate. Nonlinear Dyn., 64(1–2), 89–95. Google Scholar
  18. Liu, L., Zhao, X., & Zhou, Y. (2010). A Tuberculosis model with seasonality. Bull. Math. Biol., 72, 931–952. MathSciNetMATHCrossRefGoogle Scholar
  19. London, W., & Yorke, J. A. (1973). Recurrent outbreaks of measles, chickenpox and mumps. i. Seasonal variation in contact rates. Am. J. Epidemiol., 98(6), 453–468. Google Scholar
  20. Ma, J., & Ma, Z. (2006). Epidemic threshold conditions for seasonally forced SEIR models. Math. Biosci. Eng., 3(1), 161–172. MathSciNetMATHCrossRefGoogle Scholar
  21. MOHC (2009). Ministry of health of the People’s Republic of China, the status of prevention and control of rabies in China (Zhongguo Kuangquanbing Fangzhi Xiankuang), 27 September 2009. http://www.moh.gov.cn/publicfiles/business/htmlfiles/mohbgt/s9513/200909/42937.htm.
  22. MOHC (2011). Ministry of health of the People’s Republic of China, bulletins. http://www.moh.gov.cn/publicfiles/business/htmlfiles/mohbgt/pwsbgb/index.htm.
  23. Moneim, I. (2007). The effect of using different types of periodic contact rate on the behaviour of infectious diseases: A simulation study. Comput. Biol. Med., 37, 1582–1590. CrossRefGoogle Scholar
  24. Nakata, Y., & Kuniya, T. (2010). Global dynamics of a class of SEIRS epidemic models in a periodic environment. J. Math. Anal. Appl., 363, 230–237. MathSciNetMATHCrossRefGoogle Scholar
  25. NBSC (2009). National Bureau of Statistics of China, China Demographic Yearbook of 2009. http://www.stats.gov.cn/tjsj/ndsj/2009/indexch.htm.
  26. Perko, L. (2000). Differential equations and dynamical systems. New York: Springer. Google Scholar
  27. Ruan, S., & Wu, J. (2009). Modeling spatial spread of communicable diseases involving animal hosts. In S. Cantrell, C. Cosner, & S. Ruan (Eds.), Spatial ecology (pp. 293–316). Boca Raton: Chapman Hall/CRC. Google Scholar
  28. Schenzle, D. (1984). An age-structured model of pre- and pose-vaccination measles transmission. Math. Med. Biol., 1, 169–191. MathSciNetMATHCrossRefGoogle Scholar
  29. Schwartz, I. (1992). Small amplitude, long periodic out breaks in seasonally driven epidemics. J. Math. Biol., 30, 473–491. MathSciNetMATHCrossRefGoogle Scholar
  30. Schwartz, I., & Smith, H. (1983). Infinite subharmonic bifurcation in an SIER epidemic model. J. Math. Biol., 18, 233–253. MathSciNetMATHCrossRefGoogle Scholar
  31. Smith, H. (1983). Multiple stable subharmonics for a periodic epidemic model. J. Math. Biol., 17, 179–190. MathSciNetMATHCrossRefGoogle Scholar
  32. Smith, H., & Waltman, P. (1995). The theory of the chemostat. Cambridge: Cambridge University Press. MATHCrossRefGoogle Scholar
  33. Song, M., Tang, Q., Wang, D.-M., Mo, Z.-J., Guo, S.-H., Li, H., Tao, X.-Y., Rupprecht, C. E., Feng, Z.-J., & Liang, G.-D. (2009). Epidemiological investigations of human rabies in China. BMC Infect. Dis., 9(1), 210. CrossRefGoogle Scholar
  34. Stafford, M., Corey, L., Cao, Y., Daar, E., Ho, D., & Perelson, A. (2000). Modeling plasma virus concentration during primary HIV infection. J. Theor. Biol., 203, 285–301. CrossRefGoogle Scholar
  35. Thieme, H. (1992). Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations. J. Math. Biol., 30, 755–763. MathSciNetMATHCrossRefGoogle Scholar
  36. van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci., 18, 29–48. CrossRefGoogle Scholar
  37. Wang, W., & Zhao, X. (2008). Threshold dynamics for compartmental epidemic models in periodic environments. J. Dyn. Differ. Equ., 20, 699–717. MATHCrossRefGoogle Scholar
  38. Wesley, C., & Allen, L. (2009). The basic reproduction number in epidemic models with periodic demographics. J. Biol. Dyn., 3(2–3), 116–129. MathSciNetCrossRefGoogle Scholar
  39. WHO (2010a). Human rabies. http://www.who.int/rabies/human/en/.
  40. WHO (2010b). Rabies. http://www.who.int/rabies/en/.
  41. Williams, B. (1997). Infectious disease persistence when transmission varies seasonally. Math. Biosci., 145, 77–88. MathSciNetMATHCrossRefGoogle Scholar
  42. Zhang, F., & Zhao, X. (2007). A periodic epidemic model in a patchy environment. J. Math. Anal. Appl., 325, 496–516. MathSciNetMATHCrossRefGoogle Scholar
  43. Zhang, J., Jin, Z., Sun, G.-Q., Zhou, T., & Ruan, S. (2011). Analysis of rabies in China: Tranmission dynamics and control. PLoS ONE, 6(7), e20891. doi: 10.1371/journal.pone.0020891. CrossRefGoogle Scholar
  44. Zhang, J., Jin, Z., Sun, G.-Q., Sun, X.-D., & Ruan, S. (2012). Spatial spread of rabies in China. J. Appl. Anal. Comput., 2 (to appear). Google Scholar
  45. Zhao, X.-Q. (2003). Dynamical systems in population biology. New York: Springer. MATHGoogle Scholar
  46. Zinsstag, J., Durr, S., Penny, M., Mindekem, R., Roth, F., Gonzalez, S., Naissengar, S., & Hattendorf, J. (2009). Transmission dynamic and economics of rabies control in dogs and humans in an African city. Proc. Natl. Acad. Sci. USA, 106, 14996–15001. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2012

Authors and Affiliations

  • Juan Zhang
    • 1
  • Zhen Jin
    • 1
  • Gui-Quan Sun
    • 1
  • Xiang-Dong Sun
    • 2
  • Shigui Ruan
    • 3
  1. 1.Department of MathematicsNorth University of ChinaTaiyuanPeople’s Republic of China
  2. 2.The Laboratory of Animal Epidemiological SurveillanceChina Animal Health and Epidemiology CenterQingdaoPeople’s Republic of China
  3. 3.Department of MathematicsUniversity of MiamiCoral GablesUSA

Personalised recommendations