Dynamics And Data Analysis Of Stochastic Models Of Respiratory Infectious Diseases Induced By Air Pollutio

Air pollution increases the risk of getting respiratory diseases,cardiovascular diseases,infectious diseases,etc.There is no doubt that air pollution has hindered the improvement of people’s living standards,caused a lot of inconvenience and,worst of all,is severely hazardous to people’s health.In this paper,we will comprehensively consider random factors in the environment,weather,and human activities,and integrate the air diffusion process with the process of respiratory disease transmission.We will construct a stochastic differential system to describe the impact of air pollution on respiratory disease infection.This system can provide a qualitative and quantitative method for establish the complex relationship between air pollution and respiratory diseases.The analysis results can predict and evaluate the effectiveness of various policies.Firstly,a stochastic Susceptible-Infected-Susceptible(SIS)model related to respiratory disease driven by random diffusion of air pollutants has been developed,in which the transmission coefficient is a function of air quality index.By applying the statistical properties of stochastic process,we derive a one-dimensional stochastic differential equation(SDE)model for the number of infected individuals.Then the critical conditions that guarantee the persistence and extinction have been obtained,meanwhile the results reveal that strong noise intensity will make the disease extinct instead.Uncertainty and sensitivity analyses reveal that the parameters related to air pollution have great influence on the critical condition and dynamics of the proposed model.In fact,we find that the random fluctuation of the original two-dimensional coupling model and the reduced model are different by comparing their sample paths.The corresponding images of power spectral densities related to real data and the two models further illustrate this phenomenon.Secondly,we consider that both the removal of air pollutants and the transmission of diseases can be influenced by random perturbations.We develop a SDE model,coupled with seasonal air pollution,to study the dynamics of infectious respiratory disease spread.The periodicity of disease outbreaks is assumed to be caused by seasonal air pollution.The SDE of the air quality index(AQI)is proved to be ultimately bounded.We prove that the system driven by the periodic clearance rate has at least one stochastic periodic solution under certain conditions.By computing the closed form of the Hermite expansion of transformation density,we construct an approximate likelihood function to fit the data of AQI and influenza-like illness cases,as a case study.Data fitting shows that the stochastic periodic model can well capture the dynamical behavior of air pollution and outbreaks of the infectious respiratory disease.We also study the impact of parameters on the reduction of air pollution and disease spread,and the parameter range of disease persistence and extinction.Thirdly,by using stochastic branching process and change points,variation in a 6year 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 parameters are estimated by using Markov Chain Monte Carlo(MCMC)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.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 verified that government interventions on vehicles are effective in controlling air pollution,thus providing numerical support for policy formulation to address the haze crisis.Finally,we explore the deterministic and stochastic optimal control problems of the two-dimensional coupling model considering the three control measures including personal protection,medical treatment and reducing the influx of air pollutants.By using the Pontryagin’s minimum principle,we obtain the optimal control solution of the deterministic and stochastic model,respectively.We do numerical simulations of optimal control solution and state evolution trajectories under different weight coefficient ratios and different control objectives,which show that the stochastic optimal control problem is more suitable for the actual situation.In addition,we develop the optimal control problem in the form of control variables related to the concentration of air pollutants.The optimal control solution of the developed model can reflect the periodic variation of control strategy of air pollution well.The comparison of cost values for different combinations of the three control measures illustrates that the treatment of air pollution is the most effective control measure.In this paper,we consider random factors in the process of air pollutant concentration changes and respiratory disease infections.We develop three models,including a coupled stochastic SIS model with constant coefficient,coupled stochastic periodic SIS model driven by periodic changes of pollutant concentration,and a stochastic Susceptible-Exposed-Infected-Susceptible(SEIS)model driven by the randomness of the disease transmission process.The existence and uniqueness of the positive solution,the threshold of disease extinction and persistence are systematically studied.Combined with the practical meaning,we found the main factors affecting air pollution status and respiratory diseases infection.The change points of AQI indicate the trend of air pollution becoming more serious.The parameter estimation methods used in this study including maximum likelihood,MCMC and Bayesian theory.Then the good fitting results of the AQI and ILI data are obtained.Our numerical results can evaluate the effectiveness of the relevant measures on the control of air pollution environment and respiratory disease infections.