Travel Bans vs. Social Distancing: A Mathematical Analysis

TL;DR

This work analyzes an $SIR$ epidemic on a two‑community dynamic network connected by travel, modeled with two sparse Erdős–Rényi graphs whose inter‑community edges form via a memoryful travel process. With travel rate $\rho_T=\Theta(n^{-\alpha})$, the authors derive precise temporal and size scaling: the outbreak probability converges to $\pi$, cross‑community infection occurs at time scales $\tau_{1\to 2}=\Theta(\tfrac{1}{\lambda}\ln n)$ and the first‑to‑second wave timings satisfy $\frac{\lambda}{\ln n}(\tau_{1\to 2},\tau_1(\epsilon),\tau_2(\epsilon))\to (\alpha,1,1+\alpha)$, while the final sizes scale as $R_1(\infty)/n\to r_\infty$ and $R_2(\infty)/n\to r_\infty$ in the absence of interventions. The core finding is that travel bans barely affect these asymptotics, whereas social distancing in the second community can drive $R_2(\infty)$ to be sublinear in $n$, effectively containing the outbreak there; this provides a rigorous explanation for why reducing within‑community transmission is more impactful than restricting travel in this setting. The paper further develops a rigorous analysis framework using one‑community lower bounds, a 16‑type branching process upper bound, and herd‑immunity arguments to establish timing and size results, with extensions to more general network structures discussed. Overall, the results highlight the limited utility of travel bans as a standalone policy and emphasize the effectiveness of local transmission reduction in controlling multi‑community outbreaks.

Abstract

As the world grows increasingly connected, infectious disease transmission and outbreaks become a pressing global concern for public health officials and policymakers. While policy interventions to contain and prevent the spread of disease have been proposed and implemented, there has been little rigorous quantitative analysis of the effectiveness of such interventions. In this paper, we study the susceptible-infected-recovered (SIR) infection process on a dynamic network model that models two communities with travel between them. In particular, we consider two Erdős--Rényi graphs where edges are dynamically changing based on node travel between the graphs. We characterize the time evolution of the outbreaks in both communities and pin down the time for when the infection first reaches the second community. Finally, we analyze two interventions, social distancing and travel bans, and show that while social distancing is effective at reducing the burden of the disease in the second community, travel bans are not.

Figures (2)

• Figure 1: Schematic illustration of epidemic trajectory in the two communities (not to scale). $\tau_1$ and $\tau_2$ mark the major "waves" in the two communities: most infections occur within the narrow bands around them. $\tau_{1\to 2}$ marks the initial infection of any type-2 individual. $\tau_{\text{end}}$ indicates the end of the epidemic, i.e., when the number of infectious individual first drops to zero.
• Figure 2: Simulation on two communities of $1$ million individuals each, with an underlying network generated from a configuration model with geometrically distributed degrees (parameter chosen such the basic reproduction number $R_0 = 2$). Other parameters are: $\beta=1.5$, $\gamma = 3$, $\rho_\mathsf{T} = 10^{-3}$, and $\rho_\mathsf{H} = 1$.

Theorems & Definitions (45)

• Definition 3.1
• Theorem 3.2: SIR epidemic with no interventions
• Remark 3.3
• Definition 3.4: Travel Ban
• Definition 3.5: Social Distancing
• Theorem 3.6: Effect of Travel Ban
• Theorem 3.7: Effect of Social Distancing
• Theorem A.1
• proof
• Lemma A.2: Outbreak in a single community with multiple seeds