Abstract
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.
Highlights
Consider a physical system that can be in one of exactly N possible configurations and let xi(t) denote the probability that the system is in configuration i at time t
The appendix includes all the proofs. These are based on known tools, yet we are able to use the special structure of the master equation to derive stronger results than those available in the literature on monotone dynamical systems
Soper raised the question of whether the observed periodicity in epidemic outbreaks is the result of a ‘seasonal change in perturbing influences, such as might be brought about by school break-up and reassembling, or other annual recurrences?’ In modern terms, this amounts to asking whether the solutions of the system describing the dynamics of the epidemics entrain to periodic variations in the transmission parameters
Summary
Consider a physical system that can be in one of exactly N possible configurations and let xi(t) denote the probability that the system is in configuration i at time t. A mean-field approximation of this master equation yields a model in the form (1.2), where xi represents the average density at settlement i, and pij = exp((xj − xi)kij), with kij > 0 This models the fact that the rate of transition from settlement i to settlement j increases when the population in settlement j is larger than in i, i.e. the tendency of individuals to migrate to larger cities. The tendency to migrate to colder cities may decrease (increase) in the winter (summer) This can be modelled by adding time dependence, say, changing the scaling parameters kij to functions kij(t) that are periodic with a period of 1 year. These are based on known tools, yet we are able to use the special structure of the master equation to derive stronger results than those available in the literature on monotone dynamical systems
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