Entrainment in the master equation

R Soc Open Sci. 2018 Apr 25;5(4):172157. doi: 10.1098/rsos.172157. eCollection 2018 Apr.


The master equation plays an important role in many scientific fields including physics, chemistry, systems biology, physical finance and sociodynamics. We consider the master equation with periodic transition rates. This may represent an external periodic excitation like the 24 h solar day in biological systems or periodic traffic lights in a model of vehicular traffic. Using tools from systems and control theory, we prove that under mild technical conditions every solution of the master equation converges to a periodic solution with the same period as the rates. In other words, the master equation entrains (or phase locks) to periodic excitations. We describe two applications of our theoretical results to important models from statistical mechanics and epidemiology.

Keywords: Metzler matrix; asymmetric simple exclusion process; contractive systems; cooperative dynamical systems; first integral; stability.