Deterministic Analysis of Extrinsic and Intrinsic Noise in an Epidemiological Model
We couple a stochastic collocation method with an analytical expansion of the canonical epidemiological master equation to analyze the effects of both extrinsic and intrinsic noise. It is shown that depending on the distribution of the extrinsic noise, the master equation yields quantitatively different results compared to using the expectation of the distribution for the stochastic parameter. This difference is incident to the nonlinear terms in the master equation, and we show that the deviation away from the expectation of the extrinsic noise scales nonlinearly with the variance of the distribution. The method presented here converges linearly with respect to the number of particles in the system and exponentially with respect to the order of the polynomials used in the stochastic collocation calculation. This makes the method presented here more accurate than standard Monte Carlo methods, which suffer from slow, nonmonotonic convergence. In epidemiological terms, the results show that extrinsic fluctuations should be taken into account since they effect the speed of disease breakouts and that the gamma distribution should be used to model the basic reproductive number.
FIG 4 Monte Carlo simulation. Shown by the orange noisy lines are 100 Monte Carlo simulations along with the mean of 2 x 104 trajectories (green circles) for the infectious species. The Ο ( Ω 3/2) expansion with extrinsic gamma-distributed noise (blue dashed line, intrinsic and extrinsic noise) coincides with the full Monte Carlo simulation. The continuum equation (black[solid line, no noise) is also shown for reference.
The deterministic method presented here is able to capture the expectation over time for a process which displays relatively large fluctuations.