In recent times, all world banks have been threatened by the liquidity risk problem. This phenomenon represents a devastating financial threat to banks and may lead to irrecoverable consequences in case of negligence or underestimation. In this article, we study a mathematical model that describes the contagion of liquidity risk in the banking system based on the SIR epidemic model simulation. The model consists of three ordinary differential equations illustrating the interaction between banks susceptible or affected by liquidity risk and tending towards bankruptcy. We have demonstrated the bornness and positivity of the solutions, and we have mathematically analyzed this system to demonstrate how to control the banking system's stability. Numerical simulations have been illustrated by using real data to support the analytical results and prove the effects of different system parameters studied on the contagion of liquidity risk.

The study of contagion dynamics is a well-established domain within epidemiology, where the spread of infectious diseases is modeled and analyzed. In recent years, similar methodologies have been applied to the financial sector to understand and predict the propagation of risks within banking systems better. This paper examines the application of contagion models to assessing liquidity risk in the banking sector, leveraging optimal control theory to evaluate potential interventions by central banks. Using data from the largest European banks, we simulate the impact of central bank measures on liquidity risk. By employing optimal control techniques, we construct a model capable of simulating various scenarios to evaluate the effectiveness of policy interventions in mitigating financial contagion. Our approach provides a robust framework for analyzing the systemic risk propagation within the banking network, offering qualitative insights into the contagion mechanisms and their implications for the financial and macroeconomic landscape. The model simulates three distinct scenarios, with each representing varying levels of intervention and market conditions. The results demonstrate the model’s ability to capture the intricate interactions among major European banks, reflecting the complex realities of the financial system. These findings emphasize the critical role of central bank policies in maintaining financial stability and underscore the necessity of coordinated international efforts to manage systemic risks. This analysis contributes to a broader understanding of financial contagion, offering valuable insights for policymakers and financial institutions aiming to strengthen their resilience against future crises. The data used for the parameters are historical, which may not reflect recent changes in the banking system. The model could also be improved by incorporating non-financial factors, such as the behaviors of market actors. For future research, several improvements are possible. One improvement would be to make the bank interactions more dynamic to reflect rapid market changes better. It would also be interesting to add financial crisis scenarios to test the system’s resilience. Using more up-to-date data and incorporating new regulations would help refine the model. Finally, it would be relevant to examine the impact of external events, such as geopolitical crises, on the propagation of systemic risk. In conclusion, while the model is useful, there are several avenues for improving it and making it more suitable for our current realities.

The application of contagion risk spread modeling within the banking sector is a relatively recent development, emerging as a response to the persistent threat of liquidity risk that has affected financial institutions globally. Liquidity risk is recognized as one of the most destructive financial threats to banks, capable of causing severe and irreparable damage if overlooked or underestimated. This study aims to identify the most effective control strategy for managing financial contagion using a Susceptible–Infected–Recovered (SIR) epidemic model, incorporating time delays in both state and control variables. The proposed strategy seeks to maximize the number of resilient (vulnerable) banks while minimizing the number of infected institutions at risk of bankruptcy. Our goal is to formulate intervention policies that can curtail the propagation of financial contagion and mitigate associated systemic risks. Our model remains a simplification of reality. It does not account for strategic interactions between banks (e.g., panic reactions, network coordination), nor for adaptive regulatory mechanisms. The integration of these aspects will be the subject of future work. We establish the existence of an optimal control strategy and apply Pontryagin’s Maximum Principle to characterize and analyze the control dynamics. To numerically solve the control system, we employ a discretization approach based on forward and backward finite difference approximations. Despite the model’s simplifications, it captures key dynamics relevant to major European banks. Simulations performed using Python 3.12 yield significant results across three distinct scenarios. Notably, in the most severe case (${\alpha }_3=1.0$), the optimal control strategy reduces bankruptcies from 25% to nearly 0% in Spain, and from 12.5% to 0% in France and Germany, demonstrating the effectiveness of timely intervention in containing financial 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.