Dynamic properties and $\alpha$-path of an uncertain SIRS epidemic model

The dynamic analysis of epidemic models plays a crucial role in understanding disease transmission mechanisms and prevention strategies. Building upon the research of Tan et al. ^[1], this paper investigates novel dynamic properties of solutions and $\alpha$-paths of an uncertain SIRS model. First, we prove the existence, uniqueness, and stability of a solution for the SIRS model. Using the Yao-Chen formula, we derive $\alpha$-paths of the model and prove that both the disease-free and endemic equilibria are globally asymptotically stable, thereby refining existing research. When the threshold ${\mathcal{R}}_0^u\le 1$, the disease-free equilibrium is globally asymptotically stable, while the endemic equilibrium is globally asymptotically stable if ${\mathcal{R}}_0^u>1$. Finally, numerical simulations demonstrate that increasing the recovery rate and decreasing the disease-induced mortality rate can effectively reduce the disease spread. The results show that the disease transmission rate and the intensity of the Liu process are essential to prevent the disease spread.