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Stochastic Modelling of Air Pollution Impacts on Respiratory Infection Risk

Abstract

The impact of air pollution on people’s health and daily activities in China has recently aroused much attention. By using stochastic differential equations, variation in a 6 year long time series of air quality index (AQI) data, gathered from air quality monitoring sites in Xi’an from 15 November 2010 to 14 November 2016 was studied. Every year the extent of air pollution shifts from being serious to not so serious due to alterations in heat production systems. The distribution of such changes can be predicted by a Bayesian approach and the Gibbs sampler algorithm. The intervals between changes in a sequence indicate when the air pollution becomes increasingly serious. Also, the inflow rate of pollutants during the main pollution periods each year has an increasing trend. This study used a stochastic SEIS model associated with the AQI to explore the impact of air pollution on respiratory infections. Good fits to both the AQI data and the numbers of influenza-like illness cases were obtained by stochastic numerical simulation of the model. Based on the model’s dynamics, the AQI time series and the daily number of respiratory infection cases under various government intervention measures and human protection strategies were forecasted. The AQI data in the last 15 months verified that government interventions on vehicles are effective in controlling air pollution, thus providing numerical support for policy formulation to address the haze crisis.

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References

  • Aguilera I, Pedersen M, Garcia-Esteban R, Ballester F, Basterrechea M, Esplugues A, Fernández-Somoano A, Lertxundi A, Tardón A, Sunyer J (2013) Early-life exposure to outdoor air pollution and respiratory health, ear infections, and eczema in infants from the INMA study. Environ Health Perspect 121(3):387–392

    Article  Google Scholar 

  • Allen LJS, Allen EJ, Jonsson CB (2006) The impact of environmental variation on hantavirus infection in rodents. In: Gumel AB, Castillo-Chavez C, Mickens RE, Clemence DP (eds) Contemporary mathematics series. Proceedings of the joint summer research conference on modeling the dynamics of human diseases: emerging paradigms and challenges, vol 410. AMS, Providence, pp 1–15

  • Becker S, Dailey LA, Soukup JM, Grambow S, Devlin RB, Huang YT (2005) Seasonal variations in air pollution particle-induced inflammatory mediator release and oxidative stress. Environ Health Perspect 113(8):1032–1038

    Article  Google Scholar 

  • Bernstein JA, Alexis N, Barnes C, Bernstein IL, Nel A, Peden D, Diaz-Sanchez D, Tarlo SM, Williams PB (2004) Health effects of air pollution. J Allergy Asthma Clin Immunol 114:116–1123

    Google Scholar 

  • Bing Z, Wilson E, Jun B (2011) Controlling air pollution from coal power plants in China: incremental change or a great leap forward. Environ Sci Technol 45(24):10294–5

    Article  Google Scholar 

  • Botchev MA, Verwer JG (2003) A new approximate matrix factorization for implicit time integration in air pollution modeling. J Comput Appl Math 157:309–327

    MathSciNet  Article  Google Scholar 

  • Chan-Yeung MNW (2000) Air pollution and health. Hong Kong Med J 6:390–8

    Google Scholar 

  • Chauhan AJ, Johnston SL (2003) Air pollution and infection in respiratory illness. Br Med Bull 68:95–112

    Article  Google Scholar 

  • Chen BH, Kan HD, Chen RJ, Jiang SH, Hong CJ (2011) Air pollution and health studies in China-policy implications. J Air Waste Manag Assoc 61(11):1292–1299

    Article  Google Scholar 

  • Chen RJ, Zhang YH, Yang CX, Zhao ZH, Xu XH, Kan HD (2013) Acute effect of ambient air pollution on stroke mortality in the China air pollution and health effects study. Stroke 44:954–960

    Article  Google Scholar 

  • Chin JC, Broemeling LD (1980) Some Bayesian inferences for a changing linear model. Technometrics 22:71–78

    MathSciNet  Article  Google Scholar 

  • Chow JC, Watson JG, Mauderly JL, Costa DL, Wyzga RE, Vedal S, Hidy GM, Altshuler Sam L, Marrack D, Heuss JM, Wolff GT, Pope CA, Dockery DW (2006) Health effects of fine particulate air pollution: lines that connect. J Air Waste Manag 56(10):1368–1378

    Article  Google Scholar 

  • Cropper M (2010) What are the health effects of air pollution in China? In: Heal G (ed) Is economic growth sustainable?. Palgrave-Macmillan, London

    Google Scholar 

  • Genc DD, Yesilyurt C, Tuncel G (2010) Air pollution forecasting in Ankara, Turkey using air pollution index and its relation to assimilative capacity of the atmosphere. Environ Monit Assess 166:11–27

    Article  Google Scholar 

  • HEI-International Scientific Oversight Committee (2010) Outdoor air pollution and health in the developing countries of Asia: a comprehensive review. Special Report 18. Health Effects Institute, Boston

  • Huang L, Zhou L, Chen J, Chen K, Liu Y, Chen XD, Tang FY (2016) Acute effects of air pollution on influenza-like illness in Nanjing, China: a population-based study. Chemosphere 147:180–187

    Article  Google Scholar 

  • Ji Y (2014) Economic, growth, urbanization and air pollution in China: an empirical research based on panel data. Energies 7(7):4202–4220

    Article  Google Scholar 

  • Kampa M, Castanas E (2008) Human health effects of air pollution. Environ Pollut 151:362–367

    Article  Google Scholar 

  • Katsouyanni K (2003) Ambient air pollution and health. Br Med Bull 68:143–156

    Article  Google Scholar 

  • Kou SC, Kou SG (2004) A diffusion model for growth stocks. Math Method Oper Res 29:191–212

    MathSciNet  Article  Google Scholar 

  • Li FZ, Liu Y, Lü JJ, Liang LC, Harmer P (2015) Ambient air pollution in China poses a multifaceted health threat to outdoor physical activity. J Epidemiol Commun 69(3):201–204

    Article  Google Scholar 

  • Low RB, Bielory L, Qureshi AI, Dunn V, Stuhlmiller DFE, Dickey DA (2006) The relation of stroke admissions to recent weather, airborne allergens, air pollution, seasons, upper respiratory infections, and asthma incidence, September 11, 2001, and day of the week. Stroke 37:951–957

    Article  Google Scholar 

  • Lyons TJ, Zheng WA (1990) On conditional diffusion processes. Proc R Soc Edinb Sect A 115:243–255

    MathSciNet  Article  Google Scholar 

  • Mehta S, Shin H, Burnett R, North T, Cohen AJ (2013) Ambient particulate air pollution and acute lower respiratory infections: a systematic review and implications for estimating the global burden of disease. Air Qual Atmos Health 6:69–83

    Article  Google Scholar 

  • Mostofsky E, Schwartz J, Coull BA, Koutrakis P, Wellenius GA, Suh HH, Gold DR, Mittleman MA (2012) Modeling the association between particle constituents of air pollution and health outcomes. Am J Epidemiol 176(4):317–C326

    Article  Google Scholar 

  • Nielsen CP, Ho MS (2007) Air pollution and health damages in China: an introduction and review. In: Ho MS, Nielsen CP (eds) Clearing the air: the health and economic damages of air pollution in China. MIT Press, Cambridge

    Google Scholar 

  • Peng RD, Dominici F, Pastor-Barriuso R, Zeger SL, Samet JM (2005) Seasonal analyses of air pollution and mortality in 100 US cities. Am J Epidemiol 161:585–594

    Article  Google Scholar 

  • Poon JPH, Casas I, He CF (2006) The impact of energy, transport, and trade on air pollution in China Eurasian geography and economics. Eurasian Geogr Econ 47(5):568–584

    Article  Google Scholar 

  • Raaschou-Nielsen O, Andersen ZJ, Beelen R, Samoli E, Stafoggia M (2013) Air pollution and lung cancer incidence in 17 European cohorts: prospective analyses from the European Study of Cohorts for Air Pollution Effects (ESCAPE). Lancet Oncol 14(9):813

    Article  Google Scholar 

  • Rohde RA, Muller RA (2015) Air pollution in China: mapping of concentrations and sources. PLoS ONE 10(8):e0135749

    Article  Google Scholar 

  • Samoli E, Schwartz J, Wojtyniak B, Touloumi G, Spix C, Balducci F, Medina S, Rossi G, Sunyer J, Bacharova L, Anderson HR, Katsouyanni K (2001) Investigating regional differences in short-term effects of air pollution on daily mortality in the APHEA Project: a sensitivity analysis for controlling long-term trends and seasonality. Stoch Environ Res Risk Assess 109(4):349–353

    Google Scholar 

  • Shostya A (2016) Ambient air pollution in China: predicting a turning point. Int Adv Econ Res 22:295–307

    Article  Google Scholar 

  • Song CB, Wu L, Xie YC, He JJ, Chen X, Wang T, Lin YC, Jin TS, Wang AX, Liu Y, Dai QL, Liu BS, Wang YN, Mao HJ (2017) Air pollution in China: status and spatiotemporal variations. Environ Pollut 227:334–347

    Article  Google Scholar 

  • Stern G, Latzin P, Röösli M, Fuchs O, Proietti E, Kuehni C, Frey U (2013) A prospective study of the impact of air pollution on respiratory symptoms and infections in infants. Am J Respir Crit Care Med 187(12):1341–1348

    Article  Google Scholar 

  • Sun CW, Yuan X, Yao X (2016) Social acceptance towards the air pollution in China: evidence from public’s willingness to pay for smog mitigation. Energy Policy 92:313–324

    Article  Google Scholar 

  • Tang SY, Heron EA (2008) Bayesian inference for a stochastic logistic model with switching points. Ecol Model 219:153–169

    Article  Google Scholar 

  • Tang SY, Xiao YN, Yang YP, Zhou YC, Wu JH, Ma ZE (2010) Community-based measures for mitigating the 2009 H1N1 pandemic in China. PLoS ONE 5:e10911

    Article  Google Scholar 

  • Tang SY, Yan QL, Shi W, Wang X, Sun XD, Yu PB, Wu JH, Xiao YN (2018) Measuring the impact of air pollution on respiratory infection risk in China. Environ Pollut 232:477–486

    Article  Google Scholar 

  • Touloumi G, Samoli E, Pipikou M, Tertre AL, Atkinson R, Katsouyanni K (2006) Seasonal confounding in air pollution and health time-series studies: effect on air pollution effect estimates. Stat Med 25:4164–4178

    MathSciNet  Article  Google Scholar 

  • Wang XK, Lu WZ (2006) Seasonal variation of air pollution index: Hong Kong case study. Chemosphere 63:1261–1272

    Article  Google Scholar 

  • Xia XL, Zhang A, Liang S, Qi QW, Jiang L, Ye YJ (2017) The association between air pollution and population health risk for respiratory infection: a case study of Shenzhen, China. Int J Environ Res Public Health 14(9):950

    Article  Google Scholar 

  • Xiao QY, Ma ZW, Li SS, Liu Y (2015) The impact of winter heating on air pollution in China. PLoS ONE 10(1):e0117311

    Article  Google Scholar 

  • Yu B, Huang CM, Liu ZH, Wang HP, Wang LL (2011) A chaotic analysis on air pollution index change over past 10 years in Lanzhou, northwest China. Stoch Environ Res Risk Assess 25:643–653

    Article  Google Scholar 

  • Zhao Y, Wang SX, Duan L, Lei Y, Cao PF, Hao JM (2008) Primary air pollutant emissions of coal-fired power plants in China: current status and future prediction. Atmos Environ 42:8442–8452

    Article  Google Scholar 

  • Zhang JF, Smith KR (2007) Household air pollution from coal and biomass fuels in China: measurements, health impacts, and interventions. Environ Health Perspect 115(6):848–855

    Article  Google Scholar 

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Correspondence to Sanyi Tang.

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This work is supported by the National Natural Science Foundation of China [NSFCs 61772017, 11471201 (ST) and 11631012 (YX)], and by the Fundamental Research Funds for the Central Universities GK201701001.

Appendix A: The Prior Distribution of \(\beta \) and \(\sigma ^2\) in Bayes’ Linear Model

Appendix A: The Prior Distribution of \(\beta \) and \(\sigma ^2\) in Bayes’ Linear Model

The probability density function of the prior distribution of \(\beta \) and \(\sigma ^2\) is given by

$$\begin{aligned} \begin{array}{rcl} p(\beta , \sigma ^2)&{}=&{}p(\beta |\sigma ^2)p(\sigma ^2)= {N}(m, \sigma ^2 V)\text{ IG }(n_0,s_0)\\ &{}=&{}\displaystyle \frac{s_0^{n_0}}{(2\pi )|V|^{1/2}\varGamma (n_0)}(\sigma ^2)^{-(n_0+2)}\\ &{}&{}\quad \times \exp \Big [-\Big ((\beta -m)'V^{-1}(\beta -m)+2s_0\Big )/(2\sigma ^2)\Big ], \end{array} \end{aligned}$$
(18)

where \(\varGamma (\cdot )\) represents the standard gamma function and \(\text{ IG }\) is the inverse-gamma distribution. With m as the prior mean of the coefficient vector \(\beta \), its prior variance is given by \(\sigma ^2V\).

Using the Bayes rule, we can then form the posterior distribution of the parameters, the posterior is determined by multiplying together the expressions in (9) and (18), the constant term p(Y) is a regularization factor that does not depend on either \(\beta \) or \(\sigma ^2\). The posterior distribution of \(\beta \) and \(\sigma ^2\) is

$$\begin{aligned} \begin{array}{rcl} p(\beta , \sigma ^2| Y)&{}\propto &{}(\sigma ^2)^{-(n^*+2)}\exp \Big [- \frac{(\beta -m^*)'(V^*)^{-1}(\beta -m^*)+2s^*}{2\sigma ^2}\Big ]\\ &{}=&{}(\sigma ^2)^{-1}\exp \Big [-\frac{(\beta -m^*) '(V^*)^{-1}(\beta -m^*)}{2\sigma ^2}\Big ]\times (\sigma ^2)^{-(n^*+1)} \exp \left( -\frac{s^*}{\sigma ^2}\right) \\ &{}\propto &{}{N}(m^*, \sigma ^2 V^*)\text{ IG }(n^*,s^*), \end{array} \end{aligned}$$
(19)

where

$$\begin{aligned} \begin{array}{rcl} m^*&{}=&{}(V^{-1}+B'B)^{-1}(V^{-1}m+B'Y),\\ V^*&{}=&{}(V^{-1}+B'B)^{-1},\\ n^*&{}=&{}n_0+(N_1-1)/2, \\ s^*&{}=&{}s_0+[m'V^{-1}m+Y'Y-(m^*)'(V^*)^{-1}m^*]/2. \end{array} \end{aligned}$$

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He, S., Tang, S., Xiao, Y. et al. Stochastic Modelling of Air Pollution Impacts on Respiratory Infection Risk. Bull Math Biol 80, 3127–3153 (2018). https://doi.org/10.1007/s11538-018-0512-5

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  • DOI: https://doi.org/10.1007/s11538-018-0512-5

Keywords

  • Air pollution
  • Stochastic differential equation
  • Change point
  • Respiratory infection
  • Intervention measures