April  2022, 27(4): 1877-1911. doi: 10.3934/dcdsb.2021113

Modelling the effects of ozone concentration and pulse vaccination on seasonal influenza outbreaks in Gansu Province, China

1. 

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, China

2. 

College of Mathematical Sciences, Harbin Engineering University, Harbin, Heilongjiang, 150001, China

* Corresponding author: Shuang-Lin Jing, Hai-Feng Huo

Received  December 2020 Revised  December 2020 Published  April 2022 Early access  April 2021

Fund Project: This work is supported by the National Natural Science Foundation of China (11861044 and 11661050), and the HongLiu first-class disciplines Development Program of Lanzhou University of Technology

Common air pollutants, such as ozone ($ \rm{O}_{3} $), sulfur dioxide ($ \rm{SO}_2 $) and nitrogen dioxide ($ \rm{NO}_2 $), can affect the spread of influenza. We propose a new non-autonomous impulsive differential equation model with the effects of ozone and vaccination in this paper. First, the basic reproduction number of the impulsive system is obtained, and the global asymptotic stability of the disease-free periodic solution is proved. Furthermore, the uniform persistence of the system is demonstrated. Second, the unknown parameters of the ozone dynamics model are obtained by fitting the ozone concentration data by the least square method and Bootstrap. The MCMC algorithm is used to fit influenza data in Gansu Province to identify the most suitable parameter values of the system. The basic reproduction number $ R_{0} $ is estimated to be $ 1.2486 $ ($ 95\%\rm{CI}:(1.2470, 1.2501) $). Then, a sensitivity analysis is performed on the system parameters. We find that the average annual incidence of seasonal influenza in Gansu Province is 31.3374 per 100,000 people. Influenza cases started to surge in 2016, rising by a factor of one and a half between 2014 and 2016, further increasing in 2019 (54.6909 per 100,000 population). The average incidence rate during the post-upsurge period (2017-2019) is one and a half times more than in the pre-upsurge period (2014-2016). In particular, we find that the peak ozone concentration appears 5–8 months in Gansu Province. A moderate negative correlation is seen between influenza cases and monthly ozone concentration (Pearson correlation coefficient: $ r $ = -0.4427). Finally, our results show that increasing the vaccination rate and appropriately increasing the ozone concentration can effectively prevent and control the spread of influenza.

Citation: Shuang-Lin Jing, Hai-Feng Huo, Hong Xiang. Modelling the effects of ozone concentration and pulse vaccination on seasonal influenza outbreaks in Gansu Province, China. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 1877-1911. doi: 10.3934/dcdsb.2021113
References:
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S. T. Ali, P. Wu, S. Cauchemez, D. He, V. J. Fang, B. J. Cowling and L. Tian, Ambient ozone and influenza transmissibility in Hong Kong, European Respiratory Journal, 51 (2018), 1800369. doi: 10.1183/13993003.00369-2018.

[2]

G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis, Mathematical Biosciences, 38 (1978), 113-122.  doi: 10.1016/0025-5564(78)90021-4.

[3]

D. Baınov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, Harlow, UK, 1993.

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F. CarratE. VerguN. M. FergusonM. LemaitreS. CauchemezS. Leach and A.-J. Valleron, Time lines of infection and disease in human influenza: A review of volunteer challenge studies, American Journal of Epidemiology, 167 (2008), 775-785.  doi: 10.1093/aje/kwm375.

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R. CasagrandiL. BolzoniS. A. Levin and V. Andreasen, The SIRC model and influenza A, Mathematical Biosciences, 200 (2006), 152-169.  doi: 10.1016/j.mbs.2005.12.029.

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D. Dwyer, I. Barr, A. Hurt, A. Kelso, P. Reading, S. Sullivan, P. Buchy, H. Yu, J. Zheng and Y. Shu, et al., Seasonal influenza vaccine policies, recommendations and use in the world health organization's western pacific region, Western Pacific Surveillance and Response Journal: WPSAR, 4 (2013), 51-59.

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A. d'Onofrio, Stability properties of pulse vaccination strategy in SEIR epidemic model, Mathematical Biosciences, 179 (2002), 57–72. doi: 10.1016/S0025-5564(02)00095-0.

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H. HaarioE. Saksman and J. Tamminen, An adaptive metropolis algorithm, Bernoulli, 7 (2001), 223-242.  doi: 10.2307/3318737.

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H. HaarioM. LaineA. Mira and E. Saksman, DRAM: Efficient adaptive MCMC, Statistics and computing, 16 (2006), 339-354.  doi: 10.1007/s11222-006-9438-0.

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M. W. Hirsch, Systems of differential equations that are competitive or cooperative Ⅱ: Convergence almost everywhere, SIAM Journal on Mathematical Analysis, 16 (1985), 423–439. doi: 10.1137/0516030.

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G. J. Jakab and R. R. Hmieleski, Reduction of influenza virus pathogenesis by exposure to 0.5 ppm ozone, Journal of Toxicology and Environmental Health, 23 (1988), 455-472.  doi: 10.1080/15287398809531128.

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S.-L. JingH.-F. Huo and H. Xiang, Modeling the effects of meteorological factors and unreported cases on seasonal influenza outbreaks in Gansu province, China, Bulletin of Mathematical Biology, 82 (2020), 1-36.  doi: 10.1007/s11538-020-00747-6.

[23]

M. Laine, Adaptive MCMC Methods with Applications in Environmental and Geophysical Models, Finnish meteorological institute contributions, 2008.

[24]

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

[25]

H. V. LoverenP. RomboutP. FischerE. Lebret and L. Van Bree, Modulation of host defenses by exposure to oxidant air pollutants, Inhalation toxicology, 7 (1995), 405-423.  doi: 10.3109/08958379509029711.

[26]

S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of Theoretical Biology, 254 (2008), 178–196. doi: 10.1016/j.jtbi.2008.04.011.

[27]

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[28]

M. D. McKayR. J. Beckman and W. J. Conover, A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, 21 (1979), 239-245.  doi: 10.2307/1268522.

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H. I. Nakaya, J. Wrammert, E. K. Lee, L. Racioppi, S. Marie-Kunze, W. N. Haining, A. R. Means, S. P. Kasturi, N. Khan and G.-M. Li, et al., Systems biology of vaccination for seasonal influenza in humans, Nature Immunology, 12 (2011), 786–795. doi: 10.1038/ni.2067.

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National Bureau of Statistics of China, Annual Statistics of Gansu Province, Available from: http://data.stats.gov.cn/, Accessed 5 January 2020.

[33]

J. B. PlotkinJ. Dushoff and S. A. Levin, Hemagglutinin sequence clusters and the antigenic evolution of influenza A virus, Proceedings of the National Academy of Sciences, 99 (2002), 6263-6268.  doi: 10.1073/pnas.082110799.

[34]

T. SardarS. K. Sasmal and J. Chattopadhyay, Estimating dengue type reproduction numbers for two provinces of Sri Lanka during the period 2013-14, Virulence, 7 (2016), 187-200.  doi: 10.1080/21505594.2015.1096470.

[35]

S. SasakiM. C. JaimesT. H. HolmesC. L. DekkerK. MahmoodG. W. KembleA. M. Arvin and H. B. Greenberg, Comparison of the influenza virus-specific effector and memory b-cell responses to immunization of children and adults with live attenuated or inactivated influenza virus vaccines, Journal of Virology, 81 (2007), 215-228.  doi: 10.1128/jvi.01957-06.

[36]

S. K. Sasmal, I. Ghosh, A. Huppert and J. Chattopadhyay, Modeling the spread of zika virus in a stage-structured population: Effect of sexual transmission, Bulletin of Mathematical Biology, 80 (2018), 3038–3067. doi: 10.1007/s11538-018-0510-7.

[37]

B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bulletin of Mathematical Biology, 60 (1998), 1123–1148.

[38]

D. J. Smith, Mapping the antigenic and genetic evolution of influenza virus, Science, 305 (2004), 371-376.  doi: 10.1126/science.1097211.

[39] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511530043.
[40]

H. Tanaka, M. Sakurai, K. Ishii and Y. Matsuzawa, Inactivation of influenza virus by ozone gas, Journal of IHI technologies, 49 (2009), 74–77. (in Japanese).

[41]

S. Tang, Q. Yan, W. Shi, X. Wang, X. Sun, P. Yu, J. Wu and Y. Xiao, Measuring the impact of air pollution on respiratory infection risk in China, Environmental Pollution, 232 (2018), 477–486. doi: 10.1016/j.envpol.2017.09.071.

[42]

The Lancet, The incubation period of influenza, The Lancet, 192 (1918), 635. doi: 10.1016/S0140-6736(01)02929-4.

[43]

J. Wang, Y. Xiao and R. 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 and Continuous Dynamical Systems-B, 24 (2019), 5849–5870. doi: 10.3934/dcdsb.2019109.

[44]

L. Wang, Z. Jin and H. Wang, A switching model for the impact of toxins on the spread of infectious diseases, Journal of Mathematical Biology, 77 (2018), 1093–1115. doi: 10.1007/s00285-018-1245-7.

[45]

J. A. WolcottY. Zee and J. W. Osebold, Exposure to ozone reduces influenza disease severity and alters distribution of influenza viral antigens in murine lungs, Applied and Environmental Microbiology, 44 (1982), 723-731.  doi: 10.1128/aem.44.3.723-731.1982.

[46]

World Health Organization, Seasonal Influenza, Available from: https://www.who.int/zh/news-room/fact-sheets/detail/influenza-(seasonal), Accessed 18 January 2020.

[47]

J. YangK. E. AtkinsL. FengM. PangY. ZhengX. LiuB. J. Cowling and H. Yu, Seasonal influenza vaccination in China: Landscape of diverse regional reimbursement policy, and budget impact analysis, Vaccine, 34 (2016), 5724-5735.  doi: 10.1016/j.vaccine.2016.10.013.

[48]

Y. Yang and Y. Xiao, The effects of population dispersal and pulse vaccination on disease control, Mathematical and Computer Modelling, 52 (2010), 1591–1604. doi: 10.1016/j.mcm.2010.06.024.

[49]

Y. Yang and Y. Xiao, Threshold dynamics for compartmental epidemic models with impulses, Nonlinear Analysis: Real World Applications, 13 (2012), 224-234.  doi: 10.1016/j.nonrwa.2011.07.028.

[50]

F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, Journal of Mathematical Analysis and Applications, 325 (2007), 496–516. doi: 10.1016/j.jmaa.2006.01.085.

[51]

Y. Zhu and J. Xu, Study on $ O _{3}$-${\mbox NO}_{x}$ concentrations in various seasons and their correlatively, Shanghai Environmental Science, 1 (1998), 36–38. (in Chinese).

show all references

References:
[1]

S. T. Ali, P. Wu, S. Cauchemez, D. He, V. J. Fang, B. J. Cowling and L. Tian, Ambient ozone and influenza transmissibility in Hong Kong, European Respiratory Journal, 51 (2018), 1800369. doi: 10.1183/13993003.00369-2018.

[2]

G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis, Mathematical Biosciences, 38 (1978), 113-122.  doi: 10.1016/0025-5564(78)90021-4.

[3]

D. Baınov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, Harlow, UK, 1993.

[4]

F. CarratE. VerguN. M. FergusonM. LemaitreS. CauchemezS. Leach and A.-J. Valleron, Time lines of infection and disease in human influenza: A review of volunteer challenge studies, American Journal of Epidemiology, 167 (2008), 775-785.  doi: 10.1093/aje/kwm375.

[5]

R. CasagrandiL. BolzoniS. A. Levin and V. Andreasen, The SIRC model and influenza A, Mathematical Biosciences, 200 (2006), 152-169.  doi: 10.1016/j.mbs.2005.12.029.

[6]

N. J. Cox and C. A. Bender, The molecular epidemiology of influenza viruses, Seminars in Virology, 6 (1995), 359-370.  doi: 10.1016/S1044-5773(05)80013-7.

[7]

D. Dwyer, I. Barr, A. Hurt, A. Kelso, P. Reading, S. Sullivan, P. Buchy, H. Yu, J. Zheng and Y. Shu, et al., Seasonal influenza vaccine policies, recommendations and use in the world health organization's western pacific region, Western Pacific Surveillance and Response Journal: WPSAR, 4 (2013), 51-59.

[8]

A. d'Onofrio, Stability properties of pulse vaccination strategy in SEIR epidemic model, Mathematical Biosciences, 179 (2002), 57–72. doi: 10.1016/S0025-5564(02)00095-0.

[9]

K. ED, The Influenza Viruses and Influenza, Academic Press Inc. (London) Ltd, 24/28 Oval Road, London, NW1 7DX, 1975.

[10]

Gansu Provincial Center for Disease Control and Prevention, Epidemic Notification, Available from: http://www.gscdc.net/, Accessed 28 January 2020.

[11]

Gansu Provincial Bureau of Statistics, Gansu Province Statistical Yearbook, Available from: http://www.gstj.gov.cn/, Accessed 12 January 2020.

[12]

A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari and D. B. Rubin, Bayesian Data Analysis, Third Edition, Texts in Statistical Science Series. CRC Press, Boca Raton, FL, 2014.

[13]

I. Ghosh, P. K. Tiwari, S. Samanta, I. M. Elmojtaba, N. Al-Salti and J. Chattopadhyay, A simple SI-type model for HIV/AIDS with media and self-imposed psychological fear, Mathematical Biosciences, 306 (2018), 160–169. doi: 10.1016/j.mbs.2018.09.014.

[14]

H. HaarioE. Saksman and J. Tamminen, An adaptive metropolis algorithm, Bernoulli, 7 (2001), 223-242.  doi: 10.2307/3318737.

[15]

H. HaarioM. LaineA. Mira and E. Saksman, DRAM: Efficient adaptive MCMC, Statistics and computing, 16 (2006), 339-354.  doi: 10.1007/s11222-006-9438-0.

[16]

A. J. HayV. GregoryA. R. Douglas and Y. P. Lin, The evolution of human influenza viruses, Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences, 356 (2001), 1861-1870.  doi: 10.1098/rstb.2001.0999.

[17]

S. He, S. Tang, Y. Xiao and R. A. Cheke, Stochastic modelling of air pollution impacts on respiratory infection risk, Bulletin of Mathematical Biology, 80 (2018), 3127–3153. doi: 10.1007/s11538-018-0512-5.

[18]

H. W. Hethcote, The mathematics of infectious diseases, SIAM review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.

[19]

M. W. Hirsch, Systems of differential equations that are competitive or cooperative Ⅱ: Convergence almost everywhere, SIAM Journal on Mathematical Analysis, 16 (1985), 423–439. doi: 10.1137/0516030.

[20]

G. J. Jakab and R. R. Hmieleski, Reduction of influenza virus pathogenesis by exposure to 0.5 ppm ozone, Journal of Toxicology and Environmental Health, 23 (1988), 455-472.  doi: 10.1080/15287398809531128.

[21]

Z. Jin, The Study for Ecological and Epidemical Models Influenced by Impulses, Ph.D. thesis, Xi'an Jiaotong University, 2001.

[22]

S.-L. JingH.-F. Huo and H. Xiang, Modeling the effects of meteorological factors and unreported cases on seasonal influenza outbreaks in Gansu province, China, Bulletin of Mathematical Biology, 82 (2020), 1-36.  doi: 10.1007/s11538-020-00747-6.

[23]

M. Laine, Adaptive MCMC Methods with Applications in Environmental and Geophysical Models, Finnish meteorological institute contributions, 2008.

[24]

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

[25]

H. V. LoverenP. RomboutP. FischerE. Lebret and L. Van Bree, Modulation of host defenses by exposure to oxidant air pollutants, Inhalation toxicology, 7 (1995), 405-423.  doi: 10.3109/08958379509029711.

[26]

S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of Theoretical Biology, 254 (2008), 178–196. doi: 10.1016/j.jtbi.2008.04.011.

[27]

E. MassadM. N. BurattiniF. A. B. Coutinho and L. F. Lopez, The 1918 influenza A epidemic in the city of S$\tilde{a}$o Paulo, Brazil, Medical Hypotheses, 68 (2007), 442-445.  doi: 10.1007/s11538-007-9210-4.

[28]

M. D. McKayR. J. Beckman and W. J. Conover, A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, 21 (1979), 239-245.  doi: 10.2307/1268522.

[29]

Ministry of Ecology and Environment of the People's Republic of China, Ambient Air Quality Standard, Available from: http://www.mee.gov.cn/, Accessed 22 January 2020.

[30]

H. I. Nakaya, J. Wrammert, E. K. Lee, L. Racioppi, S. Marie-Kunze, W. N. Haining, A. R. Means, S. P. Kasturi, N. Khan and G.-M. Li, et al., Systems biology of vaccination for seasonal influenza in humans, Nature Immunology, 12 (2011), 786–795. doi: 10.1038/ni.2067.

[31]

N}ational Immunization Program Technical Working Group of China CDC, China CDC publishes "Technical Guidelines for Influenza Vaccination in China (2018-2019)", Disease Surveillance, 40 (2019), 1333–1349. (in Chinese).

[32]

National Bureau of Statistics of China, Annual Statistics of Gansu Province, Available from: http://data.stats.gov.cn/, Accessed 5 January 2020.

[33]

J. B. PlotkinJ. Dushoff and S. A. Levin, Hemagglutinin sequence clusters and the antigenic evolution of influenza A virus, Proceedings of the National Academy of Sciences, 99 (2002), 6263-6268.  doi: 10.1073/pnas.082110799.

[34]

T. SardarS. K. Sasmal and J. Chattopadhyay, Estimating dengue type reproduction numbers for two provinces of Sri Lanka during the period 2013-14, Virulence, 7 (2016), 187-200.  doi: 10.1080/21505594.2015.1096470.

[35]

S. SasakiM. C. JaimesT. H. HolmesC. L. DekkerK. MahmoodG. W. KembleA. M. Arvin and H. B. Greenberg, Comparison of the influenza virus-specific effector and memory b-cell responses to immunization of children and adults with live attenuated or inactivated influenza virus vaccines, Journal of Virology, 81 (2007), 215-228.  doi: 10.1128/jvi.01957-06.

[36]

S. K. Sasmal, I. Ghosh, A. Huppert and J. Chattopadhyay, Modeling the spread of zika virus in a stage-structured population: Effect of sexual transmission, Bulletin of Mathematical Biology, 80 (2018), 3038–3067. doi: 10.1007/s11538-018-0510-7.

[37]

B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bulletin of Mathematical Biology, 60 (1998), 1123–1148.

[38]

D. J. Smith, Mapping the antigenic and genetic evolution of influenza virus, Science, 305 (2004), 371-376.  doi: 10.1126/science.1097211.

[39] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511530043.
[40]

H. Tanaka, M. Sakurai, K. Ishii and Y. Matsuzawa, Inactivation of influenza virus by ozone gas, Journal of IHI technologies, 49 (2009), 74–77. (in Japanese).

[41]

S. Tang, Q. Yan, W. Shi, X. Wang, X. Sun, P. Yu, J. Wu and Y. Xiao, Measuring the impact of air pollution on respiratory infection risk in China, Environmental Pollution, 232 (2018), 477–486. doi: 10.1016/j.envpol.2017.09.071.

[42]

The Lancet, The incubation period of influenza, The Lancet, 192 (1918), 635. doi: 10.1016/S0140-6736(01)02929-4.

[43]

J. Wang, Y. Xiao and R. 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 and Continuous Dynamical Systems-B, 24 (2019), 5849–5870. doi: 10.3934/dcdsb.2019109.

[44]

L. Wang, Z. Jin and H. Wang, A switching model for the impact of toxins on the spread of infectious diseases, Journal of Mathematical Biology, 77 (2018), 1093–1115. doi: 10.1007/s00285-018-1245-7.

[45]

J. A. WolcottY. Zee and J. W. Osebold, Exposure to ozone reduces influenza disease severity and alters distribution of influenza viral antigens in murine lungs, Applied and Environmental Microbiology, 44 (1982), 723-731.  doi: 10.1128/aem.44.3.723-731.1982.

[46]

World Health Organization, Seasonal Influenza, Available from: https://www.who.int/zh/news-room/fact-sheets/detail/influenza-(seasonal), Accessed 18 January 2020.

[47]

J. YangK. E. AtkinsL. FengM. PangY. ZhengX. LiuB. J. Cowling and H. Yu, Seasonal influenza vaccination in China: Landscape of diverse regional reimbursement policy, and budget impact analysis, Vaccine, 34 (2016), 5724-5735.  doi: 10.1016/j.vaccine.2016.10.013.

[48]

Y. Yang and Y. Xiao, The effects of population dispersal and pulse vaccination on disease control, Mathematical and Computer Modelling, 52 (2010), 1591–1604. doi: 10.1016/j.mcm.2010.06.024.

[49]

Y. Yang and Y. Xiao, Threshold dynamics for compartmental epidemic models with impulses, Nonlinear Analysis: Real World Applications, 13 (2012), 224-234.  doi: 10.1016/j.nonrwa.2011.07.028.

[50]

F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, Journal of Mathematical Analysis and Applications, 325 (2007), 496–516. doi: 10.1016/j.jmaa.2006.01.085.

[51]

Y. Zhu and J. Xu, Study on $ O _{3}$-${\mbox NO}_{x}$ concentrations in various seasons and their correlatively, Shanghai Environmental Science, 1 (1998), 36–38. (in Chinese).

Figure 1.  Flow chart of the influenza system (2)
Figure 2.  (a) The mean value of ozone concentration in Gansu Province from January 1, 2014, to December 31, 2019. (b) The influenza cases were reported by the Gansu Provincial Center for Disease Control and Prevention from January 2014 to December 2019. (c) Pearson's correlation between the Monthly average ozone concentration and the number of reported cases. (d) The annual incidence rate of seasonal influenza in Gansu, 2014-2019
Figure 3.  The black dots represent the mean value of daily ozone concentration in Gansu Province from January 1, 2014, to December 31, 2019, the red curve represents the mean value of 100 simulations, and the light red area represents the 95% credible intervals
Figure 4.  (a) The fitting results of the number of new cases reported from January 2014 to December 2019. The solid red line represents the simulated curve of the system (2) and the Green circles represent the actual data. (b) The fitting result of the number of new cases unreported from January 2014 to December 2019. The solid red line represents the simulated curve of the system (2). The light red area represents the 95% credible intervals. (c) Pearson's correlation between the number of estimated cases and the number of reported cases
Figure 5.  The Markov chain of the last 10000 samples of $ R_{0} $. (a) The blue dots indicate the value of $ R_{0} $ within the 95% credible intervals, the red pluses indicate the value of $ R_{0} $ outside the 95% credible intervals, and the black lines indicate the upper and lower credible limits. (b) The frequency distribution of $ R_{0} $. The red curve is the probability density function curve of $ R_{0} $
Figure 6.  The dependence of the solution of the number of reported cases $ P_{C}(t) $ as a function of time on vaccination rate $ p $. (a) The new cases reported varying with the vaccination rate. (b) The cumulative cases reported in 2020 are predicted to change with the vaccination rate. The grey area represents the 95% credible intervals
Figure 7.  Ozone concentration changes. The black dot represents the actual data, the red curve represents the mean value of 100 simulations, and the yellow area represents more than $ 160\rm{ug/m}^{3} $ (i.e., ozone light pollution)
Figure 8.  The dependence of the solution of the number of reported cases $ P_{C}(t) $ as a function of time on the basic input rate of ozone $ c_{0} $. (a) The number of new cases reported changes with the basic input rate of ozone $ c_{0} $. (b) The cumulative cases reported change with the basic input rate of ozone $ c_{0} $ in 2020. The grey area represents the 95% credible intervals
Figure 9.  (a) and (b) The sensitivity of the parameters changes as the dynamics of the system (2) progress. The light gray area represents PRCC values that are not statistically significant ($ 0\leq|\rm{PRCC}|<0.2 $). The dark gray areas represent PRCC values that are moderate correlation ($ 0.2\leq|\rm{PRCC}|<0.4 $)
Figure 10.  Plots of the basic reproduction number $ R_{0} $ in terms of (a) $ \theta $ (the modification factor in transmission coefficient of the reported infected individuals), (b) $ p $ (the proportion of those vaccinated successfully), (c) $ \kappa $ (the diagnosis rate of unreported infected individuals), and (d) $ \beta(t) $ (the contact transmission rate between susceptible individuals and infected individuals)
Figure B.11.  The global stability of the disease-free periodic solution $ P_{0} $ of the system (2)
Figure B.12.  Uniform Persistence of the system (2)
Figure C.13.  Daily ozone concentration data in Jiayuguan, Lanzhou, Jinchuan, Qingyang, Tianshui, Baiyin, Wuwei, Zhangye, Jiuquan, Pingliang, Dingxi, Longnan, Linxia and Gannan
Figure D.14.  Posterior distribution of unknown parameters of the system (2)
Figure D.15.  Traces of the unknown parameter values as obtained by the MCMC sampling for 100, 00 iteration numbers for the system (2). The blue dots indicate the parameter value within the 95% credible intervals, the red pluses indicate the parameter value outside the 95% credible intervals, and the black lines indicate the upper and lower credible limits
Table 1.  The parameters description of the system (2)
Parameters Description (Units)
$ \Lambda $ The recruitment rate of the susceptible individuals (person/month)
$ d $ The natural mortality rate of the population (month$ ^{-1} $)
$ \theta $ The modification factor in transmission coefficient of the reported infected individuals (none)
$ \delta $ The proportion of infected individuals notified by CDC in Gansu Province (none)
$ 1/\sigma $ The mean incubation period of the infected individuals (month)
$ q $ The progression rate of the recovered individuals (month$ ^{-1} $)
$ p $ The proportion of those vaccinated successfully (none)
$ \gamma_{1} $ The recovery rate of reported infected individuals (month$ ^{-1} $)
$ \gamma_{2} $ The recovery rate of unreported infected individuals (month$ ^{-1} $)
$ \kappa $ The diagnosis rate of unreported infected individuals (month$ ^{-1} $)
$ a_{1} $ The maximum protection rate due to ozone sterilization (none)
$ a_{2} $ Saturated constant (none)
$ \beta(t) $ The basic contact transmission rate (none)
$ T>0 $ The vaccination interval (month)
Parameters Description (Units)
$ \Lambda $ The recruitment rate of the susceptible individuals (person/month)
$ d $ The natural mortality rate of the population (month$ ^{-1} $)
$ \theta $ The modification factor in transmission coefficient of the reported infected individuals (none)
$ \delta $ The proportion of infected individuals notified by CDC in Gansu Province (none)
$ 1/\sigma $ The mean incubation period of the infected individuals (month)
$ q $ The progression rate of the recovered individuals (month$ ^{-1} $)
$ p $ The proportion of those vaccinated successfully (none)
$ \gamma_{1} $ The recovery rate of reported infected individuals (month$ ^{-1} $)
$ \gamma_{2} $ The recovery rate of unreported infected individuals (month$ ^{-1} $)
$ \kappa $ The diagnosis rate of unreported infected individuals (month$ ^{-1} $)
$ a_{1} $ The maximum protection rate due to ozone sterilization (none)
$ a_{2} $ Saturated constant (none)
$ \beta(t) $ The basic contact transmission rate (none)
$ T>0 $ The vaccination interval (month)
Table 2.  The parameters values of the system (1)
Parameters Mean value Std $ 95\% $ CI Reference
$ c_{0} $ $ 1.4150 $ $ 0.03489 $ [$ 1.3466 $, $ 1.4833 $] Bootstrap
$ c_{1} $ $ 0.1696 $ $ 0.04515 $ [$ 0.08112 $, $ 0.2581 $] Bootstrap
$ c_{2} $ $ 0.1172 $ $ 0.03579 $ [$ 0.04707 $, $ 0.1874 $] Bootstrap
$ \phi_{0} $ $ -9.6959 $ $ 0.2251 $ [$ -10.1370 $, $ -9.2547 $] Bootstrap
$ b_{0} $ $ 0.01733 $ $ 0.001310 $ [$ 0.01476 $, $ 0.01990 $] Bootstrap
$ b_{1} $ $ 0.005276 $ $ 0.001394 $ [$ 0.002545 $, $ 0.008007 $] Bootstrap
Parameters Mean value Std $ 95\% $ CI Reference
$ c_{0} $ $ 1.4150 $ $ 0.03489 $ [$ 1.3466 $, $ 1.4833 $] Bootstrap
$ c_{1} $ $ 0.1696 $ $ 0.04515 $ [$ 0.08112 $, $ 0.2581 $] Bootstrap
$ c_{2} $ $ 0.1172 $ $ 0.03579 $ [$ 0.04707 $, $ 0.1874 $] Bootstrap
$ \phi_{0} $ $ -9.6959 $ $ 0.2251 $ [$ -10.1370 $, $ -9.2547 $] Bootstrap
$ b_{0} $ $ 0.01733 $ $ 0.001310 $ [$ 0.01476 $, $ 0.01990 $] Bootstrap
$ b_{1} $ $ 0.005276 $ $ 0.001394 $ [$ 0.002545 $, $ 0.008007 $] Bootstrap
Table 3.  The parameters and initial values of the system (2)
Parameters Mean value Std $ 95\% $ CI Reference
$ \Lambda $ $ 26166 $ $ - $ $ - $ [11]
$ d $ $ 1/(73\times12) $ $ - $ $ - $ [32]
$ \gamma_{1} $ $ 30/7 $ $ - $ $ - $ [9,27,5,22]
$ \gamma_{2} $ $ 30/10 $ $ - $ $ - $ [9,27,22]
$ \sigma $ $ 30/4 $ $ - $ $ - $ [42,4]
$ q $ $ 30/365 $ $ - $ $ - $ [33,5,6,38,16]
$ p $ $ 2\% $ $ - $ $ - $ [47]
$ \theta $ $ 0.3184 $ $ - $ $ - $ [22]
$ \delta $ $ 0.04211 $ $ - $ $ - $ [22]
$ \kappa $ $ 0.09102 $ $ - $ $ - $ [22]
$ \beta_{0} $ $ 1.9765\times10^{-7} $ $ 2.3187\times10^{-8} $ [$ 1.7748\times10^{-7} $, $ 2.6870\times10^{-7} $] MCMC
$ \beta_{1} $ $ 0.2386 $ $ 0.02253 $ [$ 0.1899 $, $ 0.2834 $] MCMC
$ \phi_{1} $ $ 2.5287 $ $ 0.09460 $ [$ 2.3321 $, $ 2.7052 $] MCMC
$ a_{1} $ $ 0.1013 $ $ 0.09390 $ [$ 4.1789\times10^{-3} $, $ 0.3652 $] MCMC
$ a_{2} $ $ 4.4024 $ $ 3.1987 $ [$ 0.2137 $, $ 9.9282 $] MCMC
Initial values Mean value Std $ 95\% $ CI Reference
$ S(0) $ $ 18403500 $ $ - $ $ - $ [22]
$ E(0) $ $ 1484 $ $ - $ $ - $ [22]
$ I_{C}(0) $ $ 1091 $ $ - $ $ - $ [10]
$ I_{N}(0) $ $ 3232 $ $ - $ $ - $ [22]
$ R(0) $ $ 9001 $ $ - $ $ - $ [22]
Parameters Mean value Std $ 95\% $ CI Reference
$ \Lambda $ $ 26166 $ $ - $ $ - $ [11]
$ d $ $ 1/(73\times12) $ $ - $ $ - $ [32]
$ \gamma_{1} $ $ 30/7 $ $ - $ $ - $ [9,27,5,22]
$ \gamma_{2} $ $ 30/10 $ $ - $ $ - $ [9,27,22]
$ \sigma $ $ 30/4 $ $ - $ $ - $ [42,4]
$ q $ $ 30/365 $ $ - $ $ - $ [33,5,6,38,16]
$ p $ $ 2\% $ $ - $ $ - $ [47]
$ \theta $ $ 0.3184 $ $ - $ $ - $ [22]
$ \delta $ $ 0.04211 $ $ - $ $ - $ [22]
$ \kappa $ $ 0.09102 $ $ - $ $ - $ [22]
$ \beta_{0} $ $ 1.9765\times10^{-7} $ $ 2.3187\times10^{-8} $ [$ 1.7748\times10^{-7} $, $ 2.6870\times10^{-7} $] MCMC
$ \beta_{1} $ $ 0.2386 $ $ 0.02253 $ [$ 0.1899 $, $ 0.2834 $] MCMC
$ \phi_{1} $ $ 2.5287 $ $ 0.09460 $ [$ 2.3321 $, $ 2.7052 $] MCMC
$ a_{1} $ $ 0.1013 $ $ 0.09390 $ [$ 4.1789\times10^{-3} $, $ 0.3652 $] MCMC
$ a_{2} $ $ 4.4024 $ $ 3.1987 $ [$ 0.2137 $, $ 9.9282 $] MCMC
Initial values Mean value Std $ 95\% $ CI Reference
$ S(0) $ $ 18403500 $ $ - $ $ - $ [22]
$ E(0) $ $ 1484 $ $ - $ $ - $ [22]
$ I_{C}(0) $ $ 1091 $ $ - $ $ - $ [10]
$ I_{N}(0) $ $ 3232 $ $ - $ $ - $ [22]
$ R(0) $ $ 9001 $ $ - $ $ - $ [22]
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