SIS epidemics on open networks: A replacement-based approximation

TL;DR

This work tackles SIS epidemics on open networks where agents enter and leave according to Poisson processes, causing the system size to vary. To enable analysis, it introduces a replacement-based approximation that preserves the average population and replaces arrivals/departures with random replacements on a fixed Erdős–Rényi topology, tracking the aggregate infection level $V(x)$. The authors derive upper bounds for the first and second moments of $V$—showing dependence on epidemic parameters ($β$, $δ$) and on replacement-tuning moments ($m$, $σ^2$)—and demonstrate that, as the replacement rate grows, the asymptotic behavior of $V$ converges to $σ^2+m^2$. Numerical simulations confirm that the replacement model closely mirrors the original open-SIS dynamics, supporting the approach as a tractable tool for analyzing real open epidemic systems.

Abstract

In this paper we analyze continuous-time SIS epidemics subject to arrivals and departures of agents, by using an approximated process based on replacements. In defining the SIS dynamics in an open network, we consider a stochastic setting in which arrivals and departures take place according to Poisson processes with similar rates, and the new value of the infection probability of an arriving agent is drawn from a continuous distribution. Since the system size changes with time, we define an approximated process, in which replacements take place instead of arrivals and departures, and we focus on the evolution of an aggregate measure of the level of infection. So long as the reproduction number is less than one, the long-term behavior of this function measures the impact of the changes of the set of agents in the epidemic. We derive upper bounds for the expectation and variance of this function and we include a numerical example to show that the approximated process is close to the original SIS process.

Figures (2)

• Figure 1: Sample realization of the aggregate function $V(x(t))$ and the number of agents $n(t)$ for a graph with $n_0=50$ agents and probability $p=0.5$. The SIS epidemic is considered with $\beta_n=0.1/n$ and $\delta_n=0.15p$, and the rate of the Poisson processes is $\mu_a=\mu_d=\mu=7$. The arrival and departure instants correspond to the green and red lines respectively.
• Figure 2: Evolution of the moments of the aggregate function $V(x(t))$ for a graph with $n_0=50$ agents and probability $p=0.5$. The SIS epidemic is considered with $\beta=0.1/n$ and $\delta=0.15p$, and the rate of the Poisson processes is $\mu_a=\mu_d=\mu=7$. In the top plot, the solid blue line and dashed red line correspond to the estimation of $\mathbb{E}\left[V(x(t))\right]$ for the original and approximate process respectively, while the dash-dotted yellow line is the upper bound \ref{['eq:SIS_limit']}. In the middle plot, the solid blue line and dashed red line correspond to the estimation of $\mathbb{E}\left[V^2(x(t))\right]$ for the original and approximate process respectively, while the dash-dotted yellow line is the upper bound \ref{['eq:second_moment']}. In the bottom plot, the solid blue line and dashed red line correspond to the estimation of $\, \mathrm{Var} (V(x(t)))$ for the original and approximate process respectively, while the dash-dotted yellow line is the upper bound \ref{['eq:second_moment']}. The simulated values were obtained considering 1000 realizations of the process.

Theorems & Definitions (16)

• Proposition 1: Stability lajmanovich1976deterministic
• Definition 1: Departure
• Definition 2: Arrival
• Definition 3: Replacement
• Proposition 2: Replacement
• proof
• Remark 1: Other systems
• Proposition 3: Pure replacement process
• proof
• Theorem 1: Expectation