Stochastic dynamics of an SIR model for respiratory diseases coupled air pollutant concentration changes

Industrial development has made air pollution increasingly severe, and many respiratory diseases are closely related to air quality in terms of infection and transmission. In this work, we used the classic stochastic susceptible–infectious–recovered (SIR) model to reflect the spread of respiratory disease, coupled with the diffusion process of air pollutants to the infectious disease model, and we investigated the impact of various environmental noises on the process of disease transmission and air pollutant diffusion. The value of this study lies in two aspects. Mathematically, we define threshold \(\mathcal{R}_{1}^{s}\) for extinction and threshold \(\mathcal{R}_{2}^{s}\) for persistence of the disease in the stochastic model (\(\mathcal{R}_{2}^{s}<\mathcal{R}_{1}^{s}\)) when the parameters are constant, and we show that (i) when \(\mathcal{R}_{1}^{s}\) is less than 1, the disease will go to stochastic extinction; (ii) when \(\mathcal{R}_{2}^{s}\) is larger than 1, the disease will persist almost surely and the model has a unique ergodic stationary distribution; (iii) when \(\mathcal{R}_{1}^{s}\) is larger than 1 and \(\mathcal{R}_{2}^{s}\) is less than 1, the extinction of the disease has randomness, which is demonstrated through numerical experiments. In addition, we derive the exact expression of the probability density function of the stationary distribution by solving the corresponding Fokker–Planck equation under the condition of disease persistence and analyze the effects of random noises on stationary distribution characteristics and the disease extinction. Epidemiologically, the change of the concentration of air pollutants affects the conditions for disease extinction and persistence. The increase in the inflow of pollutants and the increase in the clearance rate have negative and positive impacts on the spread of diseases, respectively. We found that an increase in random noise intensity will increase the variance, reduce the kurtosis of distribution, which is not conducive to predicting and controlling the development status of the disease; however, large random noise intensity can also increase the probability of disease extinction and accelerates disease extinction. We further investigate the dynamic of the stochastic model, assuming that the inflow rate switches between two levels by numerical experiments. The results show that the random noise has a significant impact on disease extinction. The data fitting of the switching model shows that the model can effectively depict the relationship and changes in trends between air pollution and diseases.