Super-Spreaders Out, Super-Spreading In: The Effects of Infectiousness Heterogeneity and Lockdowns on Herd Immunity

TL;DR

This paper analyzes how heterogeneity in infectiousness and susceptibility, decomposed into Personal-Trait and Event-Based spreading, reshapes the Herd Immunity Threshold (HIT) and the long-term impact of lockdowns. Using a mixed model with $S^i(a)=p\,S_p(a)+(1-p)S_e^i(a)$ and $I^i(a)=p\,I_p(a)+(1-p)I_e^i(a)$, it derives an expression for the effective reproduction number $R(n)$ and shows HIT depends sensitively on the mix parameter $p$, ranging from about $5\%$ (personal-trait only) to $67\%$ (event-based only) under $R_0\approx 3$. A gamma-distributed spreading analysis and COVID-19 data illustrate how the personal-trait component wanes early while event-based spreading persists, pushing HIT up as $p$ decreases. The study also demonstrates that lockdowns have opposite effects: targeting personal-trait spreading can raise HIT, whereas targeting event-based spreading can lower HIT, with important implications for policy design in epidemic management.

Abstract

Recently, [8] has proposed that heterogeneity of infectiousness (and susceptibility) across individuals in infectious diseases, plays a major role in affecting the Herd Immunity Threshold (HIT). Such heterogeneity has been observed in COVID-19 and is recognized as overdispersion (or "super-spreading"). The model of [8] suggests that super-spreaders contribute significantly to the effective reproduction factor, R, and that they are likely to get infected and immune early in the process. Consequently, under R_0 = 3 (attributed to COVID-19), the Herd Immunity Threshold (HIT) is as low as 5%, in contrast to 67% according to the traditional models [1, 2, 4, 10]. This work follows up on [8] and proposes that heterogeneity of infectiousness (susceptibility) has two "faces" whose mix affects dramatically the HIT: (1) Personal-Trait-, and (2) Event-Based- Infectiousness (Susceptibility). The former is a personal trait of specific individuals (super-spreaders) and is nullified once those individuals are immune (as in [8]). The latter is event-based (e.g cultural super-spreading events) and remains effective throughout the process, even after the super-spreaders immune. We extend [8]'s model to account for these two factors, analyze it and conclude that the HIT is very sensitive to the mix between (1) and (2), and under R_0 = 3 it can vary between 5% and 67%. Preliminary data from COVID-19 suggests that herd immunity is not reached at 5%. We address operational aspects and analyze the effects of lockdown strategies on the spread of a disease. We find that herd immunity (and HIT) is very sensitive to the lockdown type. While some lockdowns affect positively the disease blocking and increase herd immunity, others have adverse effects and reduce the herd immunity.

Figures (6)

• Figure 1: The over-time reduction in the effective reproduction number, $R(n)$, and its contributing factors as a function of $n$, assuming $p=0.5$, $k=0.1$ and $R_0=3$. Note that $n$ (horizontal-axis) is normalized to percentage. In red - $R(n)$ (Eq. (\ref{['eqnarr1']})); In blue - $R_p(n)$ (Eq. (\ref{['eqnarr2']})); In Green - $R_e(n)$ (Eq. (\ref{['eqnarr3']})); In yellow - $R_{mix}(n)$ (Eq. (\ref{['eqnarr4']}));
• Figure 2: The Herd-Immunity Threshold (HIT) as a function of $p$ assuming Gamma distribution with shape parameter $k=0.1$. In green - $R_0=3$. In red - $R_0 = 9$.
• Figure 3: The effective reproduction number $R(n)$ as a function of $n$ throughout the disease spread for different $p$ values where $R_0=3$ and $k=0.1$. Note that $n$ (horizontal-axis) is normalized to percentage.
• Figure 4: The expected value of $R_L(n)$ under a personal-trait spreading targeted lockdown (vs. $R(n)$ in a natural evolution) assuming $p=0.5$, $R_0=3$ and $k=0.1$. The lockdown begins at $n_b = 5\%$ and ends at $n_e = 30\%$.
• Figure 5: The expected value of $R_L(n)$ under a personal-trait spreading targeted lockdown (vs. $R(n)$ in a natural evolution) assuming $p=0.5$, $R_0=3$ and $k=0.1$. The lockdown begins at $n_b = 5\%$ and ends at $n_e = 30\%$. We use $x = 10\%$. The dashed purple line demonstrates a linear reduction in the value of $R(n')$. As proven, the purple and red curves intersect at $n$.

Theorems & Definitions (28)

• Theorem 3.1: General Case Herd Immunity Threshold
• Claim 1: The likelihood of an individual to be infected
• proof : Proof of Claim \ref{['lem32']}
• Lemma 1: Heterogeneity of the population during the process
• proof : Proof of Lemma \ref{['lem33']}
• Lemma 2: The size of the susceptible population
• proof : Proof of Lemma \ref{['lem34']}
• proof : Proof of Theorem \ref{['thm1']}
• Definition 1: Personal-Trait Spreading Targeted Lockdown
• Definition 2: Event-Based Spreading Targeted Lockdown