Statistical inference and neural network training based on stochastic difference model for air pollution and associated disease transmission

A polluted air environment can potentially provoke infections of diverse respiratory diseases. The development of mathematical models can study the mechanism of air pollution and its effect on the spread of diseases. The key is to characterize the intrinsic correlation between the disease infection and the change in air pollutant concentration. In this paper, we establish a coupled discrete susceptible–exposed–infectious–susceptible (SEIS) model with demography to characterize the transmission of disease, and the change in the concentration of air pollutants is described in the form of the Beverton–Holt (BH) model with a time-varying inflow rate of air pollutants. Considering the periodic variation characteristics of data, time-varying parameters are defined as specific functional forms. We estimate the change point at which the parameters switch and the parameter values within the switching interval based on Bayesian statistical theory. The data fitting of the model can reflect the seasonal peaks and annual growth trends of values of air quality index (AQI) and the number of influenza-like illnesses (ILI) cases. However, the bias in data fitting indicates a more complex correlation pattern between disease and pollutant concentration changes. To explore unknown mechanisms, we propose the extended transmission-dynamics-informed neural network (TDINN) algorithm by combining deep learning with difference equations and obtain the curves of the transmission rate and inflow rate functions over time. The results show that neural network models can help us determine time-varying parameters in the model, thereby better reflecting the trend of data changes.