Analysing the effects of dual time delays and terror funding class in terrorism dynamics

In this paper, we develop a novel five-compartment terrorism dynamics model that explicitly incorporates a terror funding class, thereby capturing the critical role of financial resources in sustaining recruitment, logistics, and operational activities. To better reflect real-world processes, the model introduces two discrete time delays: ${\tau }_1$, representing the indoctrination period required for susceptible individuals to become terrorists, and ${\tau }_2$, denoting the lag associated with transferring terrorists to the recovered or quarantined classes. The main contributions of this work include: (i) the formulation of a funding-integrated terrorism model with dual delays; (ii) a complete mathematical analysis of positivity, boundedness, and equilibrium stability; (iii) derivation of the basic reproduction number ${ℛ}_0$ and a sensitivity analysis identifying the parameters that most strongly influence terrorism persistence; and (iv) a rigorous investigation of delay-induced destabilisation and Hopf bifurcation. For the non-delayed system, we establish conditions ensuring the existence and local stability of the terror-free equilibrium when ${ℛ}_0<1$ and the terror-persistent equilibrium when ${ℛ}_0>1$. For the delayed system, we demonstrate that increasing either ${\tau }_1$ or ${\tau }_2$ beyond their respective critical thresholds leads to Hopf bifurcations and sustained oscillations, representing recurrent waves of terrorist activity. Numerical simulations are provided to validate the analytical results. Overall, the study offers insight into how the speed of radicalisation, operational delays, and financial resources interact to shape terrorism dynamics, with potential implications for the design of more effective counter-terrorism policies.