A Dynamical Model with Time Delay for Risk Contagion

Abstract

The explanation of risk contagion among economic players—not only in financial crises—and how they spread across the world has fascinated scholars and scientists in the last few decades. Inspired by the literature dealing with the analogy between financial systems and ecosystems, we model risk contagion by revisiting the mathematical approach of epidemiological models for infectious disease spread in a new paradigm. We propose a time delay differential system describing risk diffusion among companies inside an economic sector by means of a SIR dynamics. Contagion is modelled in terms of credit and financial risks with low and high levels. A complete theoretical analysis of the problem is carried out: well-posedness and solution positivity are proven. The existence of a risk-free steady state together with an endemic equilibrium is verified. Global asymptotic stability is investigated for both equilibria by the classical Lyapunov functional theory. The model is tested on a case study of some companies operating in the food economic sector in a specific Italian region. The analysis allows for understanding the crucial role of both incubation time and financial immunity period in the asymptotic behaviour of any solution in terms of endemic permanence of risk rather than its disappearance.

Keywords

• financial distress
• financial immunity
• bankruptcy
• outbreak
• SIR model
• delay differential equation