Modelling financial contagion and optimal policy design for bank runs and systemic risk

Bank runs can destabilize individual institutions and, through financial networks, spread into general economic crises. The study explores the interconnection of systemic risk in the banking system, emphasizing interbank networks as the primary means of propagating financial contagion. We propose a compartmental system through which contagion is propagated. The system classifies the banks in the network into six compartments (undistressed, exposed, distressed, liquid, run, and failed states). We capture the dynamics of distress transmission through interbank interactions and depositor behaviours. We derive the basic reproduction number R0 to characterize the threshold conditions for systemic stability and identify both risk-free and risk-persistent equilibrium points. Through sensitivity experiments, we identify the parameters that exert the strongest influence on contagion dynamics—the contact rate between banks, the level of behavioural compliance, and transition intensities. Building on these insights, we formulate an optimal-control framework that incorporates three forms of intervention: deposit-insurance protection, policies aimed at calming depositors, and targeted liquidity intervention. Using Pontryagin’s Maximum Principle, we derive the time paths of these interventions that jointly reduce the spread of distress while keeping regulatory costs manageable. The numerical results highlight the importance of acting early: even a moderate level of deposit-insurance coverage, when implemented at the right moment, substantially dampens the transmission of shocks across the network. The study offers practical guidance for the design of policy tools intended to contain systemic risk in interconnected banking systems.