$\mathcal{R}_{0}$ fails to predict the outbreak potential in the presence of natural-boosting immunity

TL;DR

This paper introduces a mathematical model incorporating natural-boosting immunity to analyze short-term epidemic dynamics, revealing delayed epidemic onset and limitations of the basic reproduction number $_{0}$ in predicting outbreak potential.

Contribution

The study develops an explicit solution for a new model that accounts for natural immunity boosting, highlighting phenomena like delayed epidemics and challenging traditional $_{0}$ predictions.

Findings

Delayed epidemic phenomenon identified

Explicit conditions for epidemic classification derived

$_{0}$ not sufficient to predict short-term outbreak potential

Abstract

Time varying susceptibility of host at individual level due to waning and boosting immunity is known to induce rich long-term behavior of disease transmission dynamics. Meanwhile, the impact of the time varying heterogeneity of host susceptibility on the shot-term behavior of epidemics is not well-studied, even though the large amount of the available epidemiological data are the short-term epidemics. Here we constructed a parsimonious mathematical model describing the short-term transmission dynamics taking into account natural-boosting immunity by reinfection, and obtained the explicit solution for our model. We found that our system show "the delayed epidemic", the epidemic takes off after negative slope of the epidemic curve at the initial phase of epidemic, in addition to the common classification in the standard SIR model, i.e., "no epidemic" as $\mathcal{R}_{0}\leq1$ or normal epidemic as $\mathcal{R}_{0}>1$. Employing the explicit solution we derived the condition for each classification.