The analysis of mode-locking topology in an SIR epidemic dynamics model with impulsive vaccination control: Infinite cascade of Stern-Brocot sum trees

•Based on GPU parallel computing technology, the topology of mode-locking in a hybrid discrete/continuous non-smooth system is investigated.•A new cloned dynamical approach is employed to calculate the Lyapunov exponent of the hybrid system.•The mode-locking topology is organized regularly according to the elusive Stern-Brocot sum tree.•The mode-locking order is unfolded in a way that never encountered before to emerge in perfect agreement with the reciprocal of the Stern-Brocot sum tree. We report the topology of stable periodic solutions of an SIR epidemic dynamics model with impulsive vaccination control, a hybrid discrete/continuous non-smooth system, in the parameter plane spanned by the pulse period T and the quantity of pulse vaccination b. This mode-locking topology is governed by an invariant torus (a pair of frequencies) initiated from Hopf bifurcations rather than the fast-slow time scales, and its periodicity is in perfect agreement with the so-called Stern-Brocot sum tree, a derived tree from “Farey algorithm”. More surprisingly, the mode-locking order is unfolded in a way that never encountered before to emerge organized according to the reciprocal of the Stern-Brocot sum tree. Furthermore, the global organization of mode-locking is not isolated structure but an infinite Stern-Brocot sum tree cascade when tuning the control parameter T. The results obtained contribute a new framework to classify mode-locking oscillations observed in the hybrid system.