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.

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

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