November  2019, 24(11): 5849-5870. doi: 10.3934/dcdsb.2019109

Modelling the effects of contaminated environments in mainland China on seasonal HFMD infections and the potential benefit of a pulse vaccination strategy

1. 

School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China

2. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

3. 

Natural Resources Institute, University of Greenwich, Central Avenue, Chatham Maritime, Chatham, Kent, ME4 4TB, UK

* Corresponding author: Yanni Xiao

Received  August 2018 Revised  December 2018 Published  June 2019

Substantial and increasing outbreaks of EV71-related hand, foot and mouth disease (HFMD) have occurred recently in mainland China with serious consequences for child health. The HFMD pathogens can survive for long periods outside the host in suitable conditions, and hence indirect transmission via free-living pathogens in the environment cannot be ignored. We propose a novel mathematical model of both periodic direct transmission and indirect transmission followed by incorporation of an impulsive vaccination strategy. By applying Floquet theory and the comparison theorem of impulsive differential equations, we obtained a threshold parameter which governs the extinction or the uniform persistence of the disease. The rate, frequency and timing of pulse vaccination were found to affect the basic reproduction number and the number of infected individuals significantly. In particular, frequent vaccination with a high coverage rate leads to declines in the basic reproduction number. Moreover, for a given rate of vaccination or frequency, numerical studies suggested that there was an optimal time (September, just before the start of new school terms) when the basic reproduction number and hence new HFMD infections could be minimised. Frequent high intensity vaccinations at a suitable time (e.g. September) and regular cleaning of the environment are effective measures for controlling HFMD infections.

Citation: Jinyan Wang, Yanni Xiao, Robert A. Cheke. Modelling the effects of contaminated environments in mainland China on seasonal HFMD infections and the potential benefit of a pulse vaccination strategy. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5849-5870. doi: 10.3934/dcdsb.2019109
References:
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D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Asymptotic Properties if the Solution, World Scientific, Singapore, 1995. doi: 10.1142/9789812831804. Google Scholar

[2]

J. M. BibleM. Iturriza-GomaraB. Megson and D. Brown, Molecular epidemiology of human enterovirus 71 in the United Kingdom from 1998 to 2006, J. Clin Microbiol., 46 (2008), 3192-3200. doi: 10.1128/JCM.00628-08. Google Scholar

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L. BourouibaS. L. Teslya and J. Wu, Highly pathogenic Avian influenza outbreak mitigated by seasonal low pathogenic strains: insights from dynamic modeling, J. Theor. Biol., 271 (2011), 181-201. doi: 10.1016/j.jtbi.2010.11.013. Google Scholar

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C. J. BrowneR. J. Smith? and L. Bourouiba, From regional pulse vaccination to global disease eradication: Insights from a mathematical model of poliomyelitis, J. Math. Biol., 71 (2015), 215-253. doi: 10.1007/s00285-014-0810-y. Google Scholar

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WPRO. Hand, Foot and Mouth Disease (HFMD). WPRO, Available from: http://www.wpro.who.int/emerging_diseases/HFMD/en/, 2016.Google Scholar

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L. Y. ChangK. C. Tsao and H. H. Shao, Transmission and Clinical Features of Enterovirus 71 Infections in Household Contacts in Taiwan, J. America Medical Assocation, 291 (2004), 222-227. doi: 10.1001/jama.291.2.222. Google Scholar

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Chinese Center for Disease Control and Prevention (China CDC):, Statistics of HFMD confirmed cases in Beijing, China: China CDC; 2010.Google Scholar

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K. B. Chua and A. R. Kasri, Hand foot and mouth disease due to enterovirus 71 in Malaysia, Virol. Sin., 26 (2011), 221-228. doi: 10.1007/s12250-011-3195-8. Google Scholar

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P. W. ChungY. C. HuangL. Y. ChangT. Y. Lin and H. C. Ning, Duration of enterovirus shedding in stool, J. Microbiol Immunol Infect., 34 (2001), 167-170. Google Scholar

[11]

F. Chuo and S. Ting, A simple deterministic model for the spread of hand, foot and mouth disease (HFMD) in Sarawak, in 2008 Second Asia International Conference on Modelling and Simulation, 2008,947–952. doi: 10.1109/AMS.2008.139. Google Scholar

[12]

M. F. Duff, Hand-foot-and-mouth syndrome in humans: Coxackie A10 infections in New Zealand, B.M.J., 2 (1968), 661-664. doi: 10.1136/bmj.2.5606.661. Google Scholar

[13]

A. D'Onofrio, Pulse vaccination strategy in the SIR epidemic model: Global asymptotic stable eradication in presence of vaccine failures, Math. Comput. Model., 36 (2002), 473-489. doi: 10.1016/S0895-7177(02)00177-2. Google Scholar

[14]

T. FujimotoM. ChikahiraS. YoshidaH. EbiraA. Hasegawa and A. Totsuka, Outbreak of central nervous system disease associated with hand, foot, and mouth disease in Japan during the summer of 2000: detection and molecular epidemiology of enterovirus 71, Microbiol. Immunol., 46 (2002), 621-627. doi: 10.1111/j.1348-0421.2002.tb02743.x. Google Scholar

[15]

H. Gomez-AcevedoM. Y. Li and S. Jacobson, Multistability in a model for CTL response to HTLV-I infection and Its implications to HAM/TSP decelopment and prevention, Bulletin of Mathematical Biology, 72 (2010), 681-696. doi: 10.1007/s11538-009-9465-z. Google Scholar

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T. HamaguchiH. FujisawaK. SakaiS. Okino and N. Kurosaki, Acute encephalitis caused by intrafamilial transmission of enterovirus 71 in adults, Emerg. Infect. Dis., 14 (2008), 828-830. doi: 10.3201/eid1405.071121. Google Scholar

[17]

J. HanX. J. Ma and J. F. Wan, Long persistence of EV71 specific nucleotides in respiratory and feces samples of the patients with Hand-Foot-Mouth Disease after recovery, BMC Infect. Dis., 10 (2010), 178-182. doi: 10.1186/1471-2334-10-178. Google Scholar

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H. W. Hethcote, The Mathematics of Infectious Diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. Google Scholar

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D. M. Knipe and P. M. Howley, Enteroviruses: Polioviruses, coxsackie-viruses, echoviruses, and newer enteroviruses, in Fields Virology, 5th edition, Lippincott/The Williams Wilkins Co., Philadelphia, 2007,840–892. doi: 10.1002/0470857285.ch6. Google Scholar

[20]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. doi: 10.1142/0906. Google Scholar

[21]

R. C. LiL. D. LiuZ. J. Mo and X. Y. Wang, An Inactivated Enterovirus 71 Vaccine in Healthy Children, N. Engl. J. Med., 370 (2014), 829-837. doi: 10.1056/NEJMoa1303224. Google Scholar

[22]

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. Google Scholar

[23]

Y. J. MaM. X. Liu and Q. Hou, Modelling seasonal HFMD with recessive infection in Shangdong, China, Math. Biosci. Eng., 10 (2013), 1159-1171. doi: 10.3934/mbe.2013.10.1159. Google Scholar

[24]

S. MarinoI. B. HogueC. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196. doi: 10.1016/j.jtbi.2008.04.011. Google Scholar

[25]

P. McMinn, K. Lindsay and D. Perera, et al., Phylogenetic analysis of enterovirus 71 Strains Isolated during Linked Epidemics in Malaysia, Singapore, and Western Australia, Journal of Virology, (2001), 7732–7738. doi: 10.1128/JVI.75.16.7732-7738.2001. Google Scholar

[26]

The National Bureau of Statistics of China (China NBS):, Available from: http://data.stats.gov.cn/workspace/index?m=hgnd.Google Scholar

[27]

N. O. Onyango and J. M$\ddot{u}$ller, Determination of optimal vaccination strategies using an orbital stability threshold from periodically driven systems, J. Math. Biol., 68 (2014), 763-784. doi: 10.1007/s00285-013-0648-8. Google Scholar

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[29]

L. Stone and Z. Shulgin, Theoretical examination of the pulse vaccination policy in the SIR epidemic model, Math. Comput. Model., 31 (2000), 207-215. doi: 10.1016/S0895-7177(00)00040-6. Google Scholar

[30]

J. R. WangY. C. Tuan and H. P. Tsai, Change of major genotype of enterovirus 71 in outbreaks of hand-foot-and-mouth disease in Taiwan between 1998 and 2000, J. Clin. Micro. Biol., 40 (2002), 10-15. doi: 10.1128/JCM.40.1.10-15.2002. Google Scholar

[31]

A. L. Wang and Y. N. Xiao, A Filippov system describing media effects on the spread of infectious diseases, Nonl. Anal. Hybrid Systems, 11 (2014), 84-97. doi: 10.1016/j.nahs.2013.06.005. Google Scholar

[32]

W. D. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equ., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. Google Scholar

[33]

J. Y. WangY. N. Xiao and R. A. Cheke, Modelling the effects of contaminated environments on HFMD infections in mainland China, BioSystems, 140 (2016), 1-7. doi: 10.1016/j.biosystems.2015.12.001. Google Scholar

[34]

J. Y. WangY. N. Xiao and Z. H. Peng, Modelling seasonal HFMD infections with the effects of contaminated environments in mainland China, Appl. Math. Comput., 274 (2016), 615-627. doi: 10.1016/j.amc.2015.11.035. Google Scholar

[35]

Y. N. XiaoS. Y. TangY. C. ZhouR. J. SmithJ. Wu and N. Wang, Predicting an HIV/AIDS epidemic and measuring the effect on it of population mobility in mainland China, J. Theor. Bio., 317 (2013), 271-285. doi: 10.1016/j.jtbi.2012.09.037. Google Scholar

[36]

Y. N. Xiao and G. R. D. Clancy, Dynamics of infection with multiple transmission mechanisms in unmanaged/managed animal populations, Theor. Popul. Biol., 71 (2007), 408-423. doi: 10.1016/j.tpb.2007.02.003. Google Scholar

[37]

Y. N. XiaoT. T. Zhao and S. Y. Tang, Dynamics of an infectious disease with media/ psychology induced non-smooth incidence, Math. Biosci. Eng., 10 (2013), 445-461. doi: 10.3934/mbe.2013.10.445. Google Scholar

[38]

J. Y. YangY. M. Chen and F. Q. Zhang, Stability analysis and optimal control of a hand-foot-mouth disease (HFMD) model, J. Appl. Math. Comput., 41 (2013), 99-117. doi: 10.1007/s12190-012-0597-1. Google Scholar

[39]

Y. P. Yang and Y. N. Xiao, The effects of population dispersal and pulse vaccination on disease control, Math. Comput. Model., 52 (2010), 1591-1604. doi: 10.1016/j.mcm.2010.06.024. Google Scholar

[40]

Y. ZhangX. TanH. WangZ. Wang and W. Xua, 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. doi: 10.1016/j.jcv.2009.02.002. Google Scholar

[41]

F. Zhang and X. Q. 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. Google Scholar

[42]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1. Google Scholar

[43]

F. C. ZhuW. B. XuJ. L. Xia and Z. L. Liang, Efficacy, safety, and immunogenicity of an enterovirus 71 vaccine in China, N. Engl. J. Med., 370 (2014), 818-828. doi: 10.1056/NEJMoa1304923. Google Scholar

[44]

Y. T. Zhu and B. Y. Xu, et al., A hand-foot-and-mouth disease model with periodic transmission Rate in Wenzhou, China, Abstract and Applied Analysis, (2014), Article ID 234509, 11pp. doi: 10.1155/2014/234509. Google Scholar

show all references

References:
[1]

D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Asymptotic Properties if the Solution, World Scientific, Singapore, 1995. doi: 10.1142/9789812831804. Google Scholar

[2]

J. M. BibleM. Iturriza-GomaraB. Megson and D. Brown, Molecular epidemiology of human enterovirus 71 in the United Kingdom from 1998 to 2006, J. Clin Microbiol., 46 (2008), 3192-3200. doi: 10.1128/JCM.00628-08. Google Scholar

[3]

L. BourouibaS. L. Teslya and J. Wu, Highly pathogenic Avian influenza outbreak mitigated by seasonal low pathogenic strains: insights from dynamic modeling, J. Theor. Biol., 271 (2011), 181-201. doi: 10.1016/j.jtbi.2010.11.013. Google Scholar

[4]

C. J. BrowneR. J. Smith? and L. Bourouiba, From regional pulse vaccination to global disease eradication: Insights from a mathematical model of poliomyelitis, J. Math. Biol., 71 (2015), 215-253. doi: 10.1007/s00285-014-0810-y. Google Scholar

[5]

CDC. CDC Hand, Foot and Mouth Disease (HFMD). Atlanta, GA: US Department of Health and Human Services, CDC, Health Alert Network; Available from: http://www.cdc.gov/hand-foot-mouth/index.html, 2016.Google Scholar

[6]

WPRO. Hand, Foot and Mouth Disease (HFMD). WPRO, Available from: http://www.wpro.who.int/emerging_diseases/HFMD/en/, 2016.Google Scholar

[7]

L. Y. ChangK. C. Tsao and H. H. Shao, Transmission and Clinical Features of Enterovirus 71 Infections in Household Contacts in Taiwan, J. America Medical Assocation, 291 (2004), 222-227. doi: 10.1001/jama.291.2.222. Google Scholar

[8]

Chinese Center for Disease Control and Prevention (China CDC):, Statistics of HFMD confirmed cases in Beijing, China: China CDC; 2010.Google Scholar

[9]

K. B. Chua and A. R. Kasri, Hand foot and mouth disease due to enterovirus 71 in Malaysia, Virol. Sin., 26 (2011), 221-228. doi: 10.1007/s12250-011-3195-8. Google Scholar

[10]

P. W. ChungY. C. HuangL. Y. ChangT. Y. Lin and H. C. Ning, Duration of enterovirus shedding in stool, J. Microbiol Immunol Infect., 34 (2001), 167-170. Google Scholar

[11]

F. Chuo and S. Ting, A simple deterministic model for the spread of hand, foot and mouth disease (HFMD) in Sarawak, in 2008 Second Asia International Conference on Modelling and Simulation, 2008,947–952. doi: 10.1109/AMS.2008.139. Google Scholar

[12]

M. F. Duff, Hand-foot-and-mouth syndrome in humans: Coxackie A10 infections in New Zealand, B.M.J., 2 (1968), 661-664. doi: 10.1136/bmj.2.5606.661. Google Scholar

[13]

A. D'Onofrio, Pulse vaccination strategy in the SIR epidemic model: Global asymptotic stable eradication in presence of vaccine failures, Math. Comput. Model., 36 (2002), 473-489. doi: 10.1016/S0895-7177(02)00177-2. Google Scholar

[14]

T. FujimotoM. ChikahiraS. YoshidaH. EbiraA. Hasegawa and A. Totsuka, Outbreak of central nervous system disease associated with hand, foot, and mouth disease in Japan during the summer of 2000: detection and molecular epidemiology of enterovirus 71, Microbiol. Immunol., 46 (2002), 621-627. doi: 10.1111/j.1348-0421.2002.tb02743.x. Google Scholar

[15]

H. Gomez-AcevedoM. Y. Li and S. Jacobson, Multistability in a model for CTL response to HTLV-I infection and Its implications to HAM/TSP decelopment and prevention, Bulletin of Mathematical Biology, 72 (2010), 681-696. doi: 10.1007/s11538-009-9465-z. Google Scholar

[16]

T. HamaguchiH. FujisawaK. SakaiS. Okino and N. Kurosaki, Acute encephalitis caused by intrafamilial transmission of enterovirus 71 in adults, Emerg. Infect. Dis., 14 (2008), 828-830. doi: 10.3201/eid1405.071121. Google Scholar

[17]

J. HanX. J. Ma and J. F. Wan, Long persistence of EV71 specific nucleotides in respiratory and feces samples of the patients with Hand-Foot-Mouth Disease after recovery, BMC Infect. Dis., 10 (2010), 178-182. doi: 10.1186/1471-2334-10-178. Google Scholar

[18]

H. W. Hethcote, The Mathematics of Infectious Diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. Google Scholar

[19]

D. M. Knipe and P. M. Howley, Enteroviruses: Polioviruses, coxsackie-viruses, echoviruses, and newer enteroviruses, in Fields Virology, 5th edition, Lippincott/The Williams Wilkins Co., Philadelphia, 2007,840–892. doi: 10.1002/0470857285.ch6. Google Scholar

[20]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. doi: 10.1142/0906. Google Scholar

[21]

R. C. LiL. D. LiuZ. J. Mo and X. Y. Wang, An Inactivated Enterovirus 71 Vaccine in Healthy Children, N. Engl. J. Med., 370 (2014), 829-837. doi: 10.1056/NEJMoa1303224. Google Scholar

[22]

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. Google Scholar

[23]

Y. J. MaM. X. Liu and Q. Hou, Modelling seasonal HFMD with recessive infection in Shangdong, China, Math. Biosci. Eng., 10 (2013), 1159-1171. doi: 10.3934/mbe.2013.10.1159. Google Scholar

[24]

S. MarinoI. B. HogueC. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196. doi: 10.1016/j.jtbi.2008.04.011. Google Scholar

[25]

P. McMinn, K. Lindsay and D. Perera, et al., Phylogenetic analysis of enterovirus 71 Strains Isolated during Linked Epidemics in Malaysia, Singapore, and Western Australia, Journal of Virology, (2001), 7732–7738. doi: 10.1128/JVI.75.16.7732-7738.2001. Google Scholar

[26]

The National Bureau of Statistics of China (China NBS):, Available from: http://data.stats.gov.cn/workspace/index?m=hgnd.Google Scholar

[27]

N. O. Onyango and J. M$\ddot{u}$ller, Determination of optimal vaccination strategies using an orbital stability threshold from periodically driven systems, J. Math. Biol., 68 (2014), 763-784. doi: 10.1007/s00285-013-0648-8. Google Scholar

[28] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge Univ. Press, 1995. doi: 10.1017/CBO9780511530043. Google Scholar
[29]

L. Stone and Z. Shulgin, Theoretical examination of the pulse vaccination policy in the SIR epidemic model, Math. Comput. Model., 31 (2000), 207-215. doi: 10.1016/S0895-7177(00)00040-6. Google Scholar

[30]

J. R. WangY. C. Tuan and H. P. Tsai, Change of major genotype of enterovirus 71 in outbreaks of hand-foot-and-mouth disease in Taiwan between 1998 and 2000, J. Clin. Micro. Biol., 40 (2002), 10-15. doi: 10.1128/JCM.40.1.10-15.2002. Google Scholar

[31]

A. L. Wang and Y. N. Xiao, A Filippov system describing media effects on the spread of infectious diseases, Nonl. Anal. Hybrid Systems, 11 (2014), 84-97. doi: 10.1016/j.nahs.2013.06.005. Google Scholar

[32]

W. D. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equ., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. Google Scholar

[33]

J. Y. WangY. N. Xiao and R. A. Cheke, Modelling the effects of contaminated environments on HFMD infections in mainland China, BioSystems, 140 (2016), 1-7. doi: 10.1016/j.biosystems.2015.12.001. Google Scholar

[34]

J. Y. WangY. N. Xiao and Z. H. Peng, Modelling seasonal HFMD infections with the effects of contaminated environments in mainland China, Appl. Math. Comput., 274 (2016), 615-627. doi: 10.1016/j.amc.2015.11.035. Google Scholar

[35]

Y. N. XiaoS. Y. TangY. C. ZhouR. J. SmithJ. Wu and N. Wang, Predicting an HIV/AIDS epidemic and measuring the effect on it of population mobility in mainland China, J. Theor. Bio., 317 (2013), 271-285. doi: 10.1016/j.jtbi.2012.09.037. Google Scholar

[36]

Y. N. Xiao and G. R. D. Clancy, Dynamics of infection with multiple transmission mechanisms in unmanaged/managed animal populations, Theor. Popul. Biol., 71 (2007), 408-423. doi: 10.1016/j.tpb.2007.02.003. Google Scholar

[37]

Y. N. XiaoT. T. Zhao and S. Y. Tang, Dynamics of an infectious disease with media/ psychology induced non-smooth incidence, Math. Biosci. Eng., 10 (2013), 445-461. doi: 10.3934/mbe.2013.10.445. Google Scholar

[38]

J. Y. YangY. M. Chen and F. Q. Zhang, Stability analysis and optimal control of a hand-foot-mouth disease (HFMD) model, J. Appl. Math. Comput., 41 (2013), 99-117. doi: 10.1007/s12190-012-0597-1. Google Scholar

[39]

Y. P. Yang and Y. N. Xiao, The effects of population dispersal and pulse vaccination on disease control, Math. Comput. Model., 52 (2010), 1591-1604. doi: 10.1016/j.mcm.2010.06.024. Google Scholar

[40]

Y. ZhangX. TanH. WangZ. Wang and W. Xua, 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. doi: 10.1016/j.jcv.2009.02.002. Google Scholar

[41]

F. Zhang and X. Q. 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. Google Scholar

[42]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1. Google Scholar

[43]

F. C. ZhuW. B. XuJ. L. Xia and Z. L. Liang, Efficacy, safety, and immunogenicity of an enterovirus 71 vaccine in China, N. Engl. J. Med., 370 (2014), 818-828. doi: 10.1056/NEJMoa1304923. Google Scholar

[44]

Y. T. Zhu and B. Y. Xu, et al., A hand-foot-and-mouth disease model with periodic transmission Rate in Wenzhou, China, Abstract and Applied Analysis, (2014), Article ID 234509, 11pp. doi: 10.1155/2014/234509. Google Scholar

Figure 1.  Goodness-of-fit to the real data from 2010-2015 in mainland China without pulse vaccination
Figure 2.  The influence of the introduction of pulse vaccination in 2015 on (a) the number of infected individuals and (b) the number of recovered individuals
Figure 3.  The effect of the seasonal phase $ \phi $ on the monthly average number of symptomatic infected individuals
Figure 4.  The effect of different phase $ \phi $ and proportion of pulse vaccination $ \tau $ on $ R_0 $. The seasonal transmissions are of the forms $ \beta_1(t) = a_1(1+\sin (\frac{2\pi t}{12}+\phi)) $, $ \beta_2(t) = a_2(1+\sin (\frac{2\pi t}{12}+\phi)) $ and $ \nu(t) = a_3(1+\sin (\frac{2\pi t}{24}+\phi)) $, where $ \phi $ is the seasonal phase shift; (a) the pulse vaccination period $ T = 12 $ months, (b) the pulse vaccination period $ T = 24 $ months
Figure 5.  The effect of pulse vaccination on the monthly average number of symptomatic infected individuals with a different vaccination period, where $ T = 12 $ is depicted by a blue line and $ T = 24 $ is depicted by a green line; (a) proportion of pulse vaccination $ \tau = 0.25 $, and (b) proportion of pulse vaccination $ \tau = 0.75 $
Figure 6.  The effect of phase differences between pulse vaccinations on the monthly average number of symptomatic infected individuals (pulse vaccinations occur at times $ t = nT+\psi $), where the pulse vaccination proportion $ \tau = 0.75 $, the vaccination period $ T = 12 $ months. (a) $ \psi = 0 $, (b) $ \psi = 3 $, (c) $ \psi = 6 $, (d) $ \psi = 9 $
Figure 7.  The effect of phase differences between pulse vaccinations on the monthly average number of symptomatic infected individuals (pulse vaccinations occur at times $ t = nT+\psi $), where the pulse vaccination proportion $ \tau = 0.75 $, the vaccination period $ T = 24 $ months. (a) $ \psi = 0 $, (b) $ \psi = 6 $, (c) $ \psi = 12 $, (d) $ \psi = 18 $
Figure 8.  The effect of different phases $ \psi $ and proportions of pulse vaccinations $ \tau $ on $ R_0 $. Pulse vaccinations occur at times $ t = nT+\psi $, where $ 0\leq \psi < T $, $ n \in \mathbb{N} $. (a) pulse vaccination period $ T = 12 $ months, (b) pulse vaccination period $ T = 24 $ months
Figure 9.  PRCCs value for the outcome of the monthly average number $ I $. All the parameters are listed in Table 1
Table 1.  Definitions of the parameters used in the model
Para. Definition(Units) Value References
$ \Lambda $ Recruitment rate (/month) 1,328,556 [26]
$ \mu $ Natural death rate (/month) $ 1.126\times 10^{-3} $ [26]
$ p $ Proportion of HFMD symptomatic infected individuals 0.025 [34]
$ \eta $ Rate from recovered to susceptible (/month) 0.115 [34]
$ \beta_1(t) $ Periodic transmission rate between $ S(t) $ and $ I(t) $ $ a_1(1+\sin (\frac{2\pi t}{12}+\phi)) $ [23]
$ \beta_2(t) $ Periodic transmission rate between $ S(t) $ and $ I_e(t) $ $ a_2(1+\sin (\frac{2\pi t}{12}+\phi)) $ [23]
$ \nu(t) $ Periodic indirect transmission rate $ a_3(1+\sin (\frac{2\pi t}{24}+\phi)) $ [34]
$ 1/\sigma $ Average incubation period (month) 1/6 [23]
$ \gamma_1 $ Recovery rate of the symptomatic infected individuals (/month) 0.1922 [11]
$ \gamma_2 $ Recovery rate of the asymptomatic infected individuals (/month) 0.1922 [11]
$ \delta_1 $ Disease-related death for symptomatic HFMD individuals (/month) $ 6.86\times 10^{-4} $ [23]
$ \delta_2 $ Disease-related death for asymptomatic HFMD individuals (/month) $ 6.86\times 10^{-4} $ [23]
$ \lambda_1 $ Virus shedding rate from symptomatic infected individuals (/month) $ 9.38\times 10^2 $ [34]
$ \lambda_2 $ Virus shedding rate from asymptomatic infected individuals (/month) $ 7.89\times 10^2 $ [34]
$ \zeta $ Clearance rate of the virus (/month) 27 [34]
$ a_1 $ Coefficient of transmission rate between $ S(t) $ and $ I(t) $ (none) $ 1.5\times 5^{-8} $ [34]
$ a_2 $ Coefficient of transmission rate between $ S(t) $ and $ I_e(t) $ (none) $ 2.25\times 10^{-9} $ [34]
$ a_3 $ Coefficient of indirect transmission rate (none) $ 1.8\times 10^{-11} $ [34]
$ \tau $ Proportion of susceptible who are vaccinated successfully varied
Para. Definition(Units) Value References
$ \Lambda $ Recruitment rate (/month) 1,328,556 [26]
$ \mu $ Natural death rate (/month) $ 1.126\times 10^{-3} $ [26]
$ p $ Proportion of HFMD symptomatic infected individuals 0.025 [34]
$ \eta $ Rate from recovered to susceptible (/month) 0.115 [34]
$ \beta_1(t) $ Periodic transmission rate between $ S(t) $ and $ I(t) $ $ a_1(1+\sin (\frac{2\pi t}{12}+\phi)) $ [23]
$ \beta_2(t) $ Periodic transmission rate between $ S(t) $ and $ I_e(t) $ $ a_2(1+\sin (\frac{2\pi t}{12}+\phi)) $ [23]
$ \nu(t) $ Periodic indirect transmission rate $ a_3(1+\sin (\frac{2\pi t}{24}+\phi)) $ [34]
$ 1/\sigma $ Average incubation period (month) 1/6 [23]
$ \gamma_1 $ Recovery rate of the symptomatic infected individuals (/month) 0.1922 [11]
$ \gamma_2 $ Recovery rate of the asymptomatic infected individuals (/month) 0.1922 [11]
$ \delta_1 $ Disease-related death for symptomatic HFMD individuals (/month) $ 6.86\times 10^{-4} $ [23]
$ \delta_2 $ Disease-related death for asymptomatic HFMD individuals (/month) $ 6.86\times 10^{-4} $ [23]
$ \lambda_1 $ Virus shedding rate from symptomatic infected individuals (/month) $ 9.38\times 10^2 $ [34]
$ \lambda_2 $ Virus shedding rate from asymptomatic infected individuals (/month) $ 7.89\times 10^2 $ [34]
$ \zeta $ Clearance rate of the virus (/month) 27 [34]
$ a_1 $ Coefficient of transmission rate between $ S(t) $ and $ I(t) $ (none) $ 1.5\times 5^{-8} $ [34]
$ a_2 $ Coefficient of transmission rate between $ S(t) $ and $ I_e(t) $ (none) $ 2.25\times 10^{-9} $ [34]
$ a_3 $ Coefficient of indirect transmission rate (none) $ 1.8\times 10^{-11} $ [34]
$ \tau $ Proportion of susceptible who are vaccinated successfully varied
Table 2.  PRCC values for monthly average number of $ I $
Parameters Distribution PRCC p-Value
$ \Lambda $ U(1328000, 1329000) 0.0336 0.1346
$ \eta $ U(0.1, 0.1) 0.0816 0.0328
$ p $ U(0.022, 0.028) 0.4598 0
$ a_1 $ U(0.000000012, 0.000000018) 0.7430 0
$ a_2 $ U(0.000000002, 0.0000000025) 0.9579 0
$ \sigma $ U(5.9, 6.1) 0.0968 0.1147
$ \gamma_1 $ U(0.19, 0.196) -0.0679 0.0243
$ \gamma_2 $ U(0.19, 0.196) -0.2475 0.0165
$ a_3 $ U(0.000000000015, 0.000000000021) 0.9635 0
$ \lambda_1 $ U(800, 30000) 0.2611 0
$ \lambda_2 $ U(600,800) 0.5977 0
$ \tau $ U(0.1, 1) -0.7813 0
$ \zeta $ U(25, 35) -0.3812 0.0043
Parameters Distribution PRCC p-Value
$ \Lambda $ U(1328000, 1329000) 0.0336 0.1346
$ \eta $ U(0.1, 0.1) 0.0816 0.0328
$ p $ U(0.022, 0.028) 0.4598 0
$ a_1 $ U(0.000000012, 0.000000018) 0.7430 0
$ a_2 $ U(0.000000002, 0.0000000025) 0.9579 0
$ \sigma $ U(5.9, 6.1) 0.0968 0.1147
$ \gamma_1 $ U(0.19, 0.196) -0.0679 0.0243
$ \gamma_2 $ U(0.19, 0.196) -0.2475 0.0165
$ a_3 $ U(0.000000000015, 0.000000000021) 0.9635 0
$ \lambda_1 $ U(800, 30000) 0.2611 0
$ \lambda_2 $ U(600,800) 0.5977 0
$ \tau $ U(0.1, 1) -0.7813 0
$ \zeta $ U(25, 35) -0.3812 0.0043
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