The impact of recovery rate heterogeneity in achieving herd immunity

TL;DR

This work addresses how heterogeneity in recovery times affects herd immunity in $SIR$ and $SEIR$ dynamics. It develops a trait-structured framework with a heterogeneous $\gamma(x)$ and proves that herd immunity is achieved if and only if $\mathbb{E}[1/\gamma(x)]<\infty$, highlighting the inadequacy of using $\mathbb{E}[\gamma]$ as a predictor. The authors provide empirical examples showing that identical $\mathbb{E}[\gamma]$ can yield both immunity and its failure, and extend the results to the $SEIR$ model. The findings have practical implications for epidemic forecasting and intervention design by emphasizing finite mean recovery time as the key determinant of herd immunity in heterogeneous populations.

Abstract

Herd immunity is a critical concept in epidemiology, describing a threshold at which a sufficient proportion of a population is immune, either through infection or vaccination, thereby preventing sustained transmission of a pathogen. In the classic Susceptible-Infectious-Recovered (SIR) model, which has been widely used to study infectious disease dynamics, the achievement of herd immunity depends on key parameters, including the transmission rate ($β$) and the recovery rate ($γ$), where $γ$ represents the inverse of the mean infectious period. While the transmission rate has received substantial attention, recent studies have underscored the significant role of $γ$ in determining the timing and sustainability of herd immunity. Additionally, it is becoming increasingly evident that assuming $γ$ as a constant parameter might oversimplify the dynamics, as variations in recovery times can reflect diverse biological, social, and healthcare-related factors. In this paper, we investigate how heterogeneity in the recovery rate affects herd immunity. We show empirically that the mean of the recovery rate is not a reliable metric for determining the achievement of herd immunity. Furthermore, we provide a theoretical result demonstrating that it is instead the mean recovery time, which is the mean of the inverse $1/γ$ of the recovery rate that is critical in deciding whether herd immunity is achievable within the SIR framework. A similar result is proved for the SEIR model. These insights have significant implications for public health interventions and theoretical modeling of epidemic dynamics.

Figures (2)

• Figure 1: Left: Solution of the standard SIR model for $\beta=1/4$, $\gamma=1/6$. Here $S(\infty)\simeq 0.42$. Middle: solution of a two group SIR model for $\beta=1/4$, $\gamma=0$ or $1/3$ (each with probability $1/2$). Here $S(\infty)= 0.0$. Right: solution of a SIR model with $\beta=1/4$, $\Omega=[0,1]$, $\gamma(x)=x/3$. Again $S(\infty)= 0.0$.
• Figure 2: Solution of a SIR model with $\beta=1/4$, $\Omega=[0,1]$, $\gamma(x)= \frac{3}{2} \sqrt{x}$. Herd immunity ($S(\infty)>0$) is observed numerically, coherent with the theoretical results.

Theorems & Definitions (8)

• Definition 1
• Lemma 2
• proof
• Proposition 3
• Remark 4
• proof
• Proposition 5
• proof