Data-driven physics-informed neural networks for predicting terrorism dynamics under time delay and economic stress

In this study, we develop a delay differential equation model for terrorism dynamics by extending the framework of Kumar et al. (2023). The population is divided into susceptible individuals, active terrorists, and those who have deserted or resisted recruitment, with an additional component incorporating economic stress to capture the influence of socioeconomic conditions such as unemployment and inequality on radicalization. A time delay is introduced to represent the ideological incubation period associated with the transition to terrorist activity. We investigate the qualitative dynamics of the model, including the existence and stability of equilibrium points. The analysis shows that the terror-free equilibrium is locally asymptotically stable when the basic reproduction number R0<1, while a terrorism-persistent equilibrium exists and remains stable when R0>1. Notably, numerical simulations indicate that time delay has a limited influence on the system dynamics for the selected parameter set, with no significant qualitative changes or sustained oscillations observed in the long-term behavior. Furthermore, a physics-informed neural networks (PINNs) framework is employed to solve the model and estimate key parameters from world data. In particular, terrorism data from Pakistan over the period 2014–2020 are used to validate the model, and the PINN approach demonstrates high accuracy in capturing the observed dynamics. The results highlight the role of economic stress in influencing terrorism persistence within the modeling framework. However, these findings should be interpreted as qualitative insights rather than direct policy recommendations. Overall, the proposed framework provides a data-informed and theoretically grounded approach for analyzing terrorism dynamics under socio-economic influences.