Global stability for a respiratory disease model with distributed or discrete delay on complex network

Distributed delay is a core concept in time-delay systems and it has been incorporated into the networked respiratory disease model that elucidates the occurrence of respiratory diseases induced by air pollution. Next we studied a respiratory disease model with distributed delay and discrete delay. By analyzing the linearized system, we showed that if the disease-free equilibrium $E_0$ exists, exhibits global asymptotic stability without any constraint on the variable space. In addition, we proved the global stability of the endemic equilibrium $E_{\ast }$ by constructing the Liapunov functional. Our findings contributed networked reaction-diffusion models with distributed delay and discrete delay to the existing body of knowledge. Our research found that distributed delay altered the transmission rhythm of respiratory diseases, which weakened local stability and disrupted global stability, which leads to disease recurrence. Discrete delay could disrupt the "synchrony" of respiratory disease transmission, thereby inducing Hopf branches that lead to periodic disease outbreaks and undermined global stability, making it impossible to completely eradicate the disease.