Math Biology Seminar: May Ann Mata

We present an approximate description of sustained oscillations produced by a linear stochastic differential equation (SDE) of the form: dx(t)=A(t) x(t) dt + C(t) dW(t), a linear diffusion equation in two dimensions with a time-dependent periodic parameter, i.e. periodic forcing. Our work uses Floquet theory and a stochastic approximation by Baxendale and Greenwood (2011). Here we show that x(t), in an approximate sense, follows a cyclic path whose periodicity is related to the frequency of A(t) and the frequency predicted by the Floquet exponents. The radius of this approximate process is modulated by a slowly-varying bi-variate standard Ornstein-Uhlenbeck process. Moreover, we find that the typical amplitude of the approximate process is directly proportional to the square-root of the variance of the noise. We demonstrate the theory using a simulated stochastic model for a driven harmonic oscillator with noise. We discuss the applicability of our approximation in the context of stochastic epidemic model with seasonal forcing (e.g. avian flu).