For anyone interested in understanding the dynamics behind bank runs, the Diamond–Dybvig model [1] is worth reading. It describes a very simplified situation where due to banks short-term liabilities and long-term assets, a bank run is a valid Nash equilibrium. I think they won a Nobel prize for this model.
There’s also some very interesting discussion at the end about preventing runs: first if banks can suspend withdrawals, and second through central bank backing. I’ll avoid summarizing it because I’m too dumb — but to quote: “Deposit insurance provided by the government allows bank contracts that can dominate the best that can be offered without insurance and never do worse.”