Epidemiological Model with Fluctuations
Abstract
We study a three-compartment system based on the SIR model [1]. From the view of dynamical systems, the inherent property of SIR and SIRS models is the change of stability. A stable fixed point (healthy state) turns into an unstable saddle if the infection rate reaches a critical value and the trajectories end in a newborn (stable) fixed point, the endemic equilibrium. In our model, noise effects from the environment that allow for a fluctuating infection rate play a crucial role. This results in multiplicative noise terms and our model turns into a set of stochastic nonlinear differential equations (SDEs). A periodic behavior of the infection number is observed that becomes more and more irregular if the noise is increasing. For rather large fluctuations repeated disease outbreaks in the form of sharp pulses with a variety of frequencies and amplitudes are found. The results based on SDEs are confirmed by numerical and approximate analytical solutions of the corresponding Fokker-Planck equation.
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