Fractional-Order Dynamics and Memory Effects in Interbank Liquidity–Risk Contagion

Banks are increasingly exposed to liquidity–risk contagion through interbank refinancing links, confidence effects, and balance sheet interactions. This paper develops a fractional-order extension of a SIR-type liquidity contagion model previously introduced in Mourad et al. (2022). The main purpose is not merely to replace the classical derivative by a fractional one, but to provide a memory-dependent framework in which current banking vulnerability depends on past liquidity shortages, delayed refinancing reactions, and persistent confidence shocks. The model is formulated by means of the Caputo derivative, which preserves classical initial conditions and reduces to the integer-order model when the fractional order is equal to one. We correct the initial shock formulation by allowing a positive initial number of distressed banks, establish well-posedness, positivity, and boundedness, and analyze the equilibrium manifold through a basic contagion threshold. A Lyapunov comparison argument is used to justify the Mittag–Leffler decay of the distressed compartment under the subthreshold condition. An explicit Adams–Bashforth–Moulton predictor–corrector method for Caputo systems is presented and used to produce numerical simulations calibrated from European banking data. The simulations compare several bankruptcy rates, countries, and fractional orders, and they show that stronger memory slows the adjustment process and prolongs liquidity distress.