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