Reshaping transmission and recovery: How memory effects govern global dynamics thresholds in the SIR model

Biological processes inherently operate with memory: past infections modulate present transmissibility, and cumulative immune experience shapes future recovery. Conventional epidemic models, however, largely neglect such history-dependent mechanisms. To address this gap, we develop a fractional SIR model with dual memory kernels-for infectivity and recovery-extending the framework introduced by Angstmann et al. This formulation captures the non-Markovian dynamics of transmission and immunity. We prove that the basic reproduction number R₀ serves as a sharp threshold: the disease-free equilibrium is globally stable when R₀<1, and a unique endemic equilibrium emerges and remains globally stable when R₀>1, extending classical threshold theory to coupled memory effects. Moreover, we identify context-dependent intervention strategies by modulating memory indices: reducing recovery memory or enhancing infectivity memory suppresses spread in low-mortality settings, while the reverse approach is effective under high mortality. This work transforms memory effects from mathematical concepts into actionable targets for epidemic control, bridging mechanistic modeling with public health strategy.