Stochastic fluctuations of the transmission rate in the susceptible-infected-susceptible epidemic model

We analyze the dynamics of the susceptible-infected-susceptible epidemic model when the transmission rate displays Gaussian white noise fluctuations around its mean value. We obtain analytic expressions for the final size distribution of infectives and the mean infected number of individuals. The model displays a variety of noise-induced transitions as a function of the basic reproductive rate and the noise intensity. We derive a threshold criterion for epidemic invasion in the presence of external noise and determine the mean epidemic duration and the mean epidemic extinction time

Stationary energy probability density of oscillators driven by a random external force

We derive rigorous analytical results for the stationary energy probability density function of linear and nonlinear oscillators driven by additive Gaussian noise. Our study focuses on two cases: (i) a harmonic oscillator subjected to Gaussian colored noise with an arbitrary correlation function and (ii) nonlinear oscillators with a general potential driven by Gaussian white noise. We also derive analytical expressions for the stationary moments of the energy and investigate the partition of the mean energy between kinetic and potential energy. To illustrate our general results, we consider specifically the case of exponentially correlated noise for (i) and power-law and bistable potentials for (ii). Our theoretical results are substantiated by Langevin simulations

Generalized Fokker-Planck equation: Derivation and exact solutions

We derive the generalized Fokker-Planck equation associated with the Langevin equation (in the Ito sense) for an overdamped particle in an external potential driven by multiplicative noise with an arbitrary distribution of the increments of the noise generating process. We explicitly consider this equation for various specific types of noises, including Poisson white noise and L\'{e}vy stable noise, and show that it reproduces all Fokker-Planck equations that are known for these noises. Exact analytical, time-dependent and stationary solutions of the generalized Fokker-Planck equation are derived and analyzed in detail for the cases of a linear, a quadratic, and a tailored potential.Comment: 9 page

Biased random walks and propagation failure

The critical value of the reaction rate able to sustain the propagation of an invasive front is obtained for general non-Markovian biased random walks with reactions. From the Hamilton-Jacobi equation corresponding to the mean field equation we find that the critical reaction rate depends only on the mean waiting time and on the statistical properties of the jump length probability distribution function and is always underestimated by the diffusion approximation. If the reaction rate is larger than the jump frequency, invasion always succeeds, even in the case of maximal bias. Numerical simulations support our analytical predictions

Instabilities of the harmonic oscillator with fluctuating damping

We investigate the instabilities of a linear damped oscillator due to fluctuations of the damping parameter. The fluctuations are driven either by Gaussian white noise or Poisson white noise (white shot noise). We consider three notions of stability. The first two are the well-known notions of stability in the mean and stability in the mean square. We introduce the concept of thermodynamic stability, corresponding to a nonpositive rate of energy dissipation at all times. We derive analytical results for the various instability thresholds, confirm the validity of our approach for white shot noise by numerical simulations, and obtain the unexpected result that mean-square and thermodynamic stability coincide for the two types of white noise

Segregation and pursuit waves in activator-inhibitor systems

We investigate the effects of cross-diffusion on propagating waves in an activator-inhibitor system. The model consists of a piecewise linear approximation of FitzHugh-Nagumo kinetics and a cross-diffusion term for either the activator or the inhibitor. We obtain exact analytic solutions for traveling fronts and solitary pulses and discuss the corresponding speed diagrams. A detailed comparison with the corresponding Rinzel-Keller model for the usually studied case of self-diffusion is performed