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2013, 10(4): 1159-1171. doi: 10.3934/mbe.2013.10.1159

## Modelling seasonal HFMD with the recessive infection in Shandong, China

 1 Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China, China, China 2 Department of Mathematics, North University of China, School of Mechatronic Engineering, North University of China, Taiyuan, Shanxi 030051, China

Received  March 2012 Revised  February 2013 Published  June 2013

Hand, foot and mouth disease (HFMD) is one of the major public-health problems in China. Based on the HFMD data of the Department of Health of Shandong Province, we propose a dynamic model with periodic transmission rates to investigate the seasonal HFMD. After evaluating the basic reproduction number, we analyze the dynamical behaviors of the model and simulate the HFMD data of Shandong Province. By carrying out the sensitivity analysis of some key parameters, we conclude that the recessive subpopulation plays an important role in the spread of HFMD, and only quarantining the infected is not an effective measure in controlling the disease.
Citation: Yangjun Ma, Maoxing Liu, Qiang Hou, Jinqing Zhao. Modelling seasonal HFMD with the recessive infection in Shandong, China. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1159-1171. doi: 10.3934/mbe.2013.10.1159
##### References:
 [1] O. N. Bjornstad, B. F. Finkenstadt and B. T. Grenfell, Dynamics of measles epidemics: Estimating scaling of transmission rates using a time series SIR model, Ecol. Monogr., 72 (2002), 169-184. [2] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0. [3] N. Bacaër, Approximation of the basic reproduction number $R_{0}$ for vector-borne diseases with a periodic vector population, Bull. Math. Biol., 69 (2007), 1067-1091. doi: 10.1007/s11538-006-9166-9. [4] CDC, "Hand, Foot, and Mouth Disease (HFMD)$-$About Hand, Foot, and Mouth (HFMD)," http://www.cdc.gov/hand-foot-mouth/about/index.html. [5] CDC, Notes from the Field: Severe Hand, Foot, and Mouth Disease Associated with Coxsackievirus A6-Alabama, Connecticut, California, and Nevada, November 2011-February 2012, http://www.cdc.gov/mmwr/preview/mmwrhtml/mm6112a5.htm. [6] S. F. Dowell, Seasonal variation in host susceptibility and cycles of certain infectious diseases, Emerg. Infect. Dis., 7 (2001), 369-374. [7] J. Dushoff, J. B. Poltkin, S. A. Levin and D. J. D. Earn, Dynamical resonance can account for seasonality of influenza epidemics, Proc. Natl. Acad. Sci., 101 (2004), 16915-16916. doi: 10.1073/pnas.0407293101. [8] Z. Grossman, Oscillatory phenomena in a model of infectious diseases, Theory. Pop. Biol., 18 (1980), 204-243. doi: 10.1016/0040-5809(80)90050-7. [9] J. L. Liu, Threshold dynamics for a HFMD epidemic model with periodic transmission rate, Nonlinear. Dyn., 64 (2011), 89-95. doi: 10.1007/s11071-010-9848-6. [10] M. Y. Liu, W. Liu, J. Luo, Y. Liu, Y. Zhu, H. Berman and J. Wu, Characterization of an Outbreak of Hand, Foot, and Mouth Disease in Nanchang, China in 2010, PLoS ONE., 6 (2011), e25287. doi: 10.1371/journal.pone.0025287. [11] W. London and J. A. Yorke, Recurrent outbreaks of measles, chickenpox and mumps.i.seasonal variation in contact rates, Am. J. Epidemiol., 98 (1973), 453-468. [12] J. Ma and Z. Ma, Epidemic threshold conditions for seasonally forced SEIR models, Math. Biosci. Eng., 3 (2006), 161-172. doi: 10.3934/mbe.2006.3.161. [13] I. A. Moneim and D. Greenhalgh, Use of a periodic vaccination strategy to control the spread of epidemics with seasonally varying contact rate, Math. Biosci. Eng., 2 (2005), 591-611. doi: 10.3934/mbe.2005.2.591. [14] Z. Ma, Y. Zhou, W. Wang and Z. Jin, "Mathematical Modeling and Studying of Dynamic Models of Infectious Disease," Science Press, London, 2004. [15] L. Perko, "Differential Equations and Dynamical System," Springer-Verlag, New York, 2000. [16] I. Schwartz, Small amplitude, long periodic out breaks in seasonally driven epidemics, J. Math. Biol., 30 (1992), 473-491. doi: 10.1007/BF00160532. [17] I. Schwartz and H. Smith, Infinite subharmonic bifurcation in an SIER epidemic model, J. Math. Biol., 18 (1983), 233-253. doi: 10.1007/BF00276090. [18] , Shandong Statistical Information, http://www.stats-sd.gov.cn/2007/tjsj/tjsj.asp?lbbm=1. [19] F. C. S. Tiing and J. Labadin, A simple deterministic model for the spread of hand, foot and mouth disease (HFMD) in Sarawak, in "Second Asia International Conference on Modelling and Simulation," Conference Publications, (2008), 947-952. doi: 10.1109/AMS.2008.139. [20] M. Urashima, N. Shindo and N. Okable, Seasonal model of herpangina and hand-foot-mouth disease to simulate annual fluctuations in urban warming in Tokyo, Jpn. J. Infect. Dis., 56 (2003), 48-53. [21] WHO, Emerging disease surveillance and response, http://www.wpro.who.int/emerging_diseases/HFMD/en/index.html. [22] D. Wu, C. Ke, W. Li, M. Corina, J. Yan, C. Ma, H. Zen and J.Su, A large outbreak of hand, foot, and mouth disease caused by EV71 and CAV16 in Guangdong, China, 2009, Arch. Virol., 156 (2011), 945-953. [23] A. Weber, M. Weber and P. Milligan, Modeling epidemics caused by respiratory syncytial virus (RSV), Math. Biosci., 172 (2001), 95-113. doi: 10.1016/S0025-5564(01)00066-9. [24] L. J.White, J. N.Mandl, M. G. Gomes, A. T. Bodley-Tickell, P. A.Cane, P. Perez-Brena, J. C. Aguilar, M. M. Siqueira, S. A. Portes, S. M. Straliotto, M. Waris, D. J. Nokes and G. F. Medley, Understanding the transmissiondynamics of respiratorysyncytialvirus using multiple time series and nested models, Math. Biosci., 209 (2007), 222-239. doi: 10.1016/j.mbs.2006.08.018. [25] W. Wang and X. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Biol. Dyn., 3 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. [26] Q. Zhu, Y. T. Hao, J. Q. Ma , S. C. Yu and Y. Wang, Surveillance of Hand, Foot, and Mouth Disease in Mainland China (2008-2009), Biomed. Environ. Sci., 4 (2011), 349-356. [27] Y. Zhang, X. J. Tan, H. Y. Wang, D. M. Yan, S. L. Zhu, D. Y. Wang, F. Ji, X. J. Wang, Y. J. Gao, L. Chen, H. Q. An, D. X. Li, S. W. Wang, A. Q. Xu, Z. J. Wang and W. B. Xu, An outbreak of hand, foot, and mouth disease associated with subgenotype C4 of human enterovirus 71 in Shandong, China, J. Clin. Virol., 44 (2009), 262-267. [28] J. Zhang, Z. Jin, G.-Q. Sun, X.-D. Sun and S. Ruan, Modeling seasonal rabies epidemics in China, Bull. Math. Biol., 74 (2012), 1226-1251. doi: 10.1007/s11538-012-9720-6. [29] F. Zhang and X. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085.

show all references

##### References:
 [1] O. N. Bjornstad, B. F. Finkenstadt and B. T. Grenfell, Dynamics of measles epidemics: Estimating scaling of transmission rates using a time series SIR model, Ecol. Monogr., 72 (2002), 169-184. [2] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0. [3] N. Bacaër, Approximation of the basic reproduction number $R_{0}$ for vector-borne diseases with a periodic vector population, Bull. Math. Biol., 69 (2007), 1067-1091. doi: 10.1007/s11538-006-9166-9. [4] CDC, "Hand, Foot, and Mouth Disease (HFMD)$-$About Hand, Foot, and Mouth (HFMD)," http://www.cdc.gov/hand-foot-mouth/about/index.html. [5] CDC, Notes from the Field: Severe Hand, Foot, and Mouth Disease Associated with Coxsackievirus A6-Alabama, Connecticut, California, and Nevada, November 2011-February 2012, http://www.cdc.gov/mmwr/preview/mmwrhtml/mm6112a5.htm. [6] S. F. Dowell, Seasonal variation in host susceptibility and cycles of certain infectious diseases, Emerg. Infect. Dis., 7 (2001), 369-374. [7] J. Dushoff, J. B. Poltkin, S. A. Levin and D. J. D. Earn, Dynamical resonance can account for seasonality of influenza epidemics, Proc. Natl. Acad. Sci., 101 (2004), 16915-16916. doi: 10.1073/pnas.0407293101. [8] Z. Grossman, Oscillatory phenomena in a model of infectious diseases, Theory. Pop. Biol., 18 (1980), 204-243. doi: 10.1016/0040-5809(80)90050-7. [9] J. L. Liu, Threshold dynamics for a HFMD epidemic model with periodic transmission rate, Nonlinear. Dyn., 64 (2011), 89-95. doi: 10.1007/s11071-010-9848-6. [10] M. Y. Liu, W. Liu, J. Luo, Y. Liu, Y. Zhu, H. Berman and J. Wu, Characterization of an Outbreak of Hand, Foot, and Mouth Disease in Nanchang, China in 2010, PLoS ONE., 6 (2011), e25287. doi: 10.1371/journal.pone.0025287. [11] W. London and J. A. Yorke, Recurrent outbreaks of measles, chickenpox and mumps.i.seasonal variation in contact rates, Am. J. Epidemiol., 98 (1973), 453-468. [12] J. Ma and Z. Ma, Epidemic threshold conditions for seasonally forced SEIR models, Math. Biosci. Eng., 3 (2006), 161-172. doi: 10.3934/mbe.2006.3.161. [13] I. A. Moneim and D. Greenhalgh, Use of a periodic vaccination strategy to control the spread of epidemics with seasonally varying contact rate, Math. Biosci. Eng., 2 (2005), 591-611. doi: 10.3934/mbe.2005.2.591. [14] Z. Ma, Y. Zhou, W. Wang and Z. Jin, "Mathematical Modeling and Studying of Dynamic Models of Infectious Disease," Science Press, London, 2004. [15] L. Perko, "Differential Equations and Dynamical System," Springer-Verlag, New York, 2000. [16] I. Schwartz, Small amplitude, long periodic out breaks in seasonally driven epidemics, J. Math. Biol., 30 (1992), 473-491. doi: 10.1007/BF00160532. [17] I. Schwartz and H. Smith, Infinite subharmonic bifurcation in an SIER epidemic model, J. Math. Biol., 18 (1983), 233-253. doi: 10.1007/BF00276090. [18] , Shandong Statistical Information, http://www.stats-sd.gov.cn/2007/tjsj/tjsj.asp?lbbm=1. [19] F. C. S. Tiing and J. Labadin, A simple deterministic model for the spread of hand, foot and mouth disease (HFMD) in Sarawak, in "Second Asia International Conference on Modelling and Simulation," Conference Publications, (2008), 947-952. doi: 10.1109/AMS.2008.139. [20] M. Urashima, N. Shindo and N. Okable, Seasonal model of herpangina and hand-foot-mouth disease to simulate annual fluctuations in urban warming in Tokyo, Jpn. J. Infect. Dis., 56 (2003), 48-53. [21] WHO, Emerging disease surveillance and response, http://www.wpro.who.int/emerging_diseases/HFMD/en/index.html. [22] D. Wu, C. Ke, W. Li, M. Corina, J. Yan, C. Ma, H. Zen and J.Su, A large outbreak of hand, foot, and mouth disease caused by EV71 and CAV16 in Guangdong, China, 2009, Arch. Virol., 156 (2011), 945-953. [23] A. Weber, M. Weber and P. Milligan, Modeling epidemics caused by respiratory syncytial virus (RSV), Math. Biosci., 172 (2001), 95-113. doi: 10.1016/S0025-5564(01)00066-9. [24] L. J.White, J. N.Mandl, M. G. Gomes, A. T. Bodley-Tickell, P. A.Cane, P. Perez-Brena, J. C. Aguilar, M. M. Siqueira, S. A. Portes, S. M. Straliotto, M. Waris, D. J. Nokes and G. F. Medley, Understanding the transmissiondynamics of respiratorysyncytialvirus using multiple time series and nested models, Math. Biosci., 209 (2007), 222-239. doi: 10.1016/j.mbs.2006.08.018. [25] W. Wang and X. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Biol. Dyn., 3 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. [26] Q. Zhu, Y. T. Hao, J. Q. Ma , S. C. Yu and Y. Wang, Surveillance of Hand, Foot, and Mouth Disease in Mainland China (2008-2009), Biomed. Environ. Sci., 4 (2011), 349-356. [27] Y. Zhang, X. J. Tan, H. Y. Wang, D. M. Yan, S. L. Zhu, D. Y. Wang, F. Ji, X. J. Wang, Y. J. Gao, L. Chen, H. Q. An, D. X. Li, S. W. Wang, A. Q. Xu, Z. J. Wang and W. B. Xu, An outbreak of hand, foot, and mouth disease associated with subgenotype C4 of human enterovirus 71 in Shandong, China, J. Clin. Virol., 44 (2009), 262-267. [28] J. Zhang, Z. Jin, G.-Q. Sun, X.-D. Sun and S. Ruan, Modeling seasonal rabies epidemics in China, Bull. Math. Biol., 74 (2012), 1226-1251. doi: 10.1007/s11538-012-9720-6. [29] F. Zhang and X. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085.
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