| In the process of human urbanization and industrialization,the burning of fossil fuels has caused air pollution.Epidemiologists have established a number of models of infectious diseases with air pollution,and a series of research results have been obtained.However,many of the existing models do not reflect diffusion of individuals and pollutants and difference in survival environments.In epidemiology,mobility and heterogeneity have a serious impact on the transmission of respiratory diseases.Therefore,it is important to study respiratory infectious disease models that reflect these two properties.Considering the diffusion of individuals and pollutants and the difference in survival environments,two kinds of respiratory infectious disease models with air pollution are proposed,and the details of the study are as follows:1.In order to research the influences of the difference in survival environments and the diffusion of individuals and pollutants on respiratory infectious diseases,a reaction-diffusion model is proposed,where the infection rate is associated with the concentration of pollutants,and the recovery rate of infected individuals and the inflow and clearance rates of pollutants are spatially heterogeneous.By using the maximum principle and the comparison principle,we investigate the dynamical behaviors of the model and obtain that both spatial heterogeneity and limiting the movement of infected individuals would promote the spread of respiratory infectious diseases.On the other hand,in order to control the disease pandemic,the intensity of treatment of infected individuals and the intensity of implementation of artificial rainfall are introduced into the model as strategies.Necessary condition for optimal control is obtained by applying Pontryagin’s maximum principle.Numerical simulations are performed to verify the corresponding theoretical results.2.In order to reflect the dispersal of pollutants in non-adjacent areas and the large-scale movement of individuals,a nonlocal dispersal epidemic model with air pollution is proposed,where the infection rate is also related to the concentration of pollutants in the air.The existence uniqueness of global positive solution is discussed with the help of semigroup theory,Frechet derivative and constant variational method.By using the spectral theory,the basic reproduction number R0 is defined.The dynamics behavior of this model is investigated by the Lyapunov function.In addition,the numerical algorithm for R0 is given.Numerical simulations have corroborated the theoretical results and revealed that increasing the nonlocal dispersion rate of infected individuals or air pollutants can decrease R0. |
