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Noise-induced oscillations for the mean-field Dissipative Contact Process

Paolo Dai Pra, Elisa Marini

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.

Noise-induced oscillations for the mean-field Dissipative Contact Process

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.
Paper Structure (14 sections, 7 theorems, 133 equations, 3 figures)

This paper contains 14 sections, 7 theorems, 133 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Microscopic dynamics and propagation of chaos
  3. Macroscopic dynamics of aggregated variables
  4. Evolution of the mean values
  5. The distribution of $\bar{x}$
  6. The fluctuation process and its Gaussian limit
  7. Power spectral density of the fluctuation process in the supercritical regime
  8. Rescaling the parameters
  9. Proofs
  10. Proof of Theorem \ref{['thm:prop:chaos']}
  11. Proof of Corollary \ref{['cor:LLN']}
  12. Proof of Theorem \ref{['thm:limit:rho:lambda']}
  13. Proof of Theorem \ref{['thm:fluct:convergence']}
  14. Proof of Theorem \ref{['th:rescal']}

Key Result

Theorem 2.1

Suppose $\left(x_i(0)\right)_{i = 1}^N$ are independent identically distributed random variables with values in $[0,1]$ and law $\mu_0$. Denote by $(\left(x_i(t)\right)_{t\geq 0})_{i=1}^N$ the corresponding solution to system eq:micro:dyn. Also, consider $N$ independent copies of the solution to Eq. where

Figures (3)

  • Figure 1: An individual $i$ who is initially susceptible ($x_i=0$), can become infectious with rate $\frac{\lambda}{N}\sum_{j=1}^{N}x_j$ (mean-field interaction). When this happens, her viral load $x_i$ jumps to the value $1$. When infectious, individual $i$ can recover with rate $r$ and, in this case, she becomes immediately susceptible again. In the meantime, the viral load of the individual decreases deterministically in time with rate $\alpha$. Notice that this dynamics is invariant under permutations of the particles. This model is a piecewise deterministic Markov process, as the $x_i$s have a deterministic continuous dynamics with discontinuities determined by random jump events.
  • Figure 2: The spectral density of the process $\xi(t)$ in \ref{['eq:fluct:limit']} (in orange) compared to the estimated spectrum of $\xi_N$ in \ref{['eq:def:fluctuations']} (blue points) for the parameters $N=10000$, $r=5$, $\lambda=100$, $\rho= 0.7$. To obtain the blue curve, $100$ simulations of system \ref{['eq:micro:dyn']} were performed employing an Euler scheme with time-step $0.001$.
  • Figure 3: Sample trajectory of $\hat{\xi}(t)$ (system \ref{['eq:rescal']}) at stationarity.

Theorems & Definitions (9)

  • Theorem 2.1: Propagation of chaos
  • Corollary 2.2
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 4.1: Diffusion approximation of the fluctuation process
  • Proposition 4.2
  • Theorem 5.1
  • Theorem 6.1: Diffusion approximation (Theorem VII, 4.1 EtKu2005)