Abstract
As the COVID-19 pandemic progressed, research on mathematical modeling became imperative and very influential to understand the epidemiological dynamics of disease spreading. The momentary reproduction ratio r(t) of an epidemic is used as a public health guiding tool to evaluate the course of the epidemic, with the evolution of r(t) being the reasoning behind tightening and relaxing control measures over time. Here we investigate critical fluctuations around the epidemiological threshold, resembling new waves, even when the community disease transmission rate beta is not significantly changing. Without loss of generality, we use simple models that can be treated analytically and results are applied to more complex models describing COVID-19 epidemics. Our analysis shows that, rather than the supercritical regime (infectivity larger than a critical value, beta > beta _c) leading to new exponential growth of infection, the subcritical regime (infectivity smaller than a critical value, beta < beta _c) with small import is able to explain the dynamic behaviour of COVID-19 spreading after a lockdown lifting, with r(t) approx 1 hovering around its threshold value.
Highlights
As the COVID-19 pandemic progressed, research on mathematical modeling became imperative and very influential to understand the epidemiological dynamics of disease spreading
With real data supporting the theory, we show that the subcritical regime β < βc can explain the dynamic behaviour of COVID-19 epidemic in the Basque country and in many other European regions, after the lockdown was lifted in summer 2020
The qualitative behaviour of the stochastic SIR type model with import is shown in Fig. 1 where we investigate the transition between a subcritical epidemic threshold regime and a supercritical threshold regime
Summary
As the COVID-19 pandemic progressed, research on mathematical modeling became imperative and very influential to understand the epidemiological dynamics of disease spreading. In a basic Susceptible-Infected-Recovered (SIR) epidemiological models with infectivity β and recovery rate γ we observe in a susceptible population N the extinction or the exponential growth of infections when the community transmission β if respectively below or above a certain threshold, the so-called critical infectivity βc. These behaviours are explained by the fact that, during a given time period, infected individuals are recovering from infection respectively faster or slower than susceptible individuals are becoming infected
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