Global dynamics of a fractional order SIRS epidemic model by the way of generalized continuous time random walk.

In this paper, we propose a novel fractional-order SIRS (frSIRS) model incorporating infection forces under intervention strategies, developed through the framework of generalized continuous-time random walks. The model is first transformed into a system of Volterra integral equations to identify the disease-free equilibrium (DFE) state and the endemic equilibrium (EE) state. Additionally, we introduce a new method for calculating the basic reproduction number . Through several examples, we demonstrate the broad applicability of this method in determining for fractional-order epidemic models. Next, we establish that serves as a critical threshold governing the model's dynamics: if , the unique DFE is globally asymptotically stable; while if , the unique EE is globally asymptotically stable. Furthermore, we apply our findings to two fractional-order SIRS (frSIRS) models incorporating infection forces under various intervention strategies, thereby substantiating our results. From an epidemiological perspective, our analysis reveals several key insights for controlling disease spread: (i) when the death rate is high, it is essential to increase the memory index; (ii) when the recovery rate is high, decreasing the memory index is advisable; and (iii) enhancing psychological or inhibitory effects-factors independent of the death rate, recovery rate, or memory index-can also play a critical role in mitigating disease transmission. These findings offer valuable insights into how the memory index influences disease outbreaks and the overall severity of epidemics.