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

Shuang-Lin Jing; Hai-Feng Huo; Hong Xiang

doi:10.3934/dcdsb.2021113

Article Contents

2022, Volume 27, Issue 4: 1877-1911. Doi: 10.3934/dcdsb.2021113

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

Received: November 30, 2020

Revised: November 30, 2020

Early access: April 2021

Published: April 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 37N25, 92B05; Secondary: 92B15.

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Figure 1. Flow chart of the influenza system (2)

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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

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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

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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

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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} $

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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

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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)

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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

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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 $)

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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)

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Figure B.11. The global stability of the disease-free periodic solution $ P_{0} $ of the system (2)

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Figure B.12. Uniform Persistence of the system (2)

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Figure C.13. Daily ozone concentration data in Jiayuguan, Lanzhou, Jinchuan, Qingyang, Tianshui, Baiyin, Wuwei, Zhangye, Jiuquan, Pingliang, Dingxi, Longnan, Linxia and Gannan

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Figure D.14. Posterior distribution of unknown parameters of the system (2)

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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

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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)

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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

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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]

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