Stochastic Mean-Field Dynamics and Applications to Life Sciences

Abstract

We study a dissipative version of the contact process, with mean-field interaction, which admits a simple epidemiological interpretation. The propagation of chaos and the corresponding normal fluctuations reveal that the noise present in the finite-size system induces oscillations with a nearly deterministic period and a randomly varying amplitude. This is reminiscent of the emergence of pandemic waves in real epidemics.

Derivation of mean-field equations for stochastic particle systems

“…In this case our main result further implies the propagation of chaos and asymptotic independence of single site processes (see e.g. [9]). Note that no time rescaling is required and the limiting dynamics are non-linear, i.e.…”

“…Existence of limits follows from standard tightness arguments, and the deterministic limit equation arises from a vanishing martingale part for the empirical processes. Previous results along these lines in the context of fluid limits include stochastic hybrid systems [11], interacting diffusions [9], and a particular zero-range process using a different technique [12]. Our proof also includes a simple uniqueness argument for solutions of the limit equation similar to [7] and recent work [13].…”

Rare Event Simulation for Stochastic Dynamics in Continuous Time

“…The factor 1/N results from a self-averaging property of the mean-field interaction through selection dynamics, which is expected from results on other mean-field particle systems (see e.g. [35,36] and references therein), and is fully analogous to the central-limit type scaling of the empirical variance for the sum of N independent random variables. While this scaling remains the same for more general particle approximations with more than one particle being affected by selection events, the simple identity (3.16) does not hold exactly for any N ≥ 1 as we see in the next subsection.…”

$K$-averaging agent-based model: propagation of chaos and convergence to equilibrium

“…Thanks to a classical result (see for instance Proposition 1 in [17] or Proposition 2.2 in [35]), to justify the propagation of chaos, it suffices to show that…”