Dynamic analysis and optimal control of a class of SISP respiratory diseases.

In this paper, the actual background of the susceptible population being directly patients after inhaling a certain amount of PM${}_{2.5}$ is taken into account. The concentration response function of PM${}_{2.5}$ is introduced, and the SISP respiratory disease model is proposed. Qualitative theoretical analysis proves that the existence, local stability and global stability of the equilibria are all related to the daily emission $P_0$ of PM${}_{2.5}$ and PM${}_{2.5}$ pathogenic threshold K. Based on the sensitivity factor analysis and time-varying sensitivity analysis of parameters on the number of patients, it is found that the conversion rate β and the inhalation rate η has the largest positive correlation. The cure rate γ of infected persons has the greatest negative correlation on the number of patients. The control strategy formulated by the analysis results of optimal control theory is as follows: The first step is to improve the clearance rate of PM${}_{2.5}$ by reducing the PM${}_{2.5}$ emissions and increasing the intensity of dust removal. Moreover, such removal work must be maintained for a long time. The second step is to improve the cure rate of patients by being treated in time. After that, people should be reminded to wear masks and go out less so as to reduce the conversion rate of susceptible people becoming patients.