Quantifying Inaccuracies in Modeling COVID-19 Pandemic within a Continuous Time Picture

Typically, mathematical simulation studies on COVID-19 pandemic forecasting are based on deterministic differential equations which assume that both the number ($n$) of individuals in various epidemiological classes and the time ($t$) on which they depend are quantities that vary continuous. This picture contrasts with the discrete representation of $n$ and $t$ underlying the real epidemiological data reported in terms daily numbers of infection cases, for which a description based on finite difference equations would be more adequate. Adopting a logistic growth framework, in this paper we present a quantitative analysis of the errors introduced by the continuous time description. This analysis reveals that, although the height of the epidemiological curve maximum is essentially unaffected, the position $T_{1/2}^{c}$ obtained within the continuous time representation is systematically shifted backwards in time with respect to the position $T_{1/2}^{d}$ predicted within the discrete time representation. Rather counterintuitively, the magnitude of this temporal shift $\tau \equiv T_{1/2}^{c} - T_{1/2}^{d} < 0$ is basically insensitive to changes in infection rate $\kappa$. For a broad range of $\kappa$ values deduced from COVID-19 data at extreme situations (exponential growth in time and complete lockdown), we found a rather robust estimate $\tau \simeq -2.65\,\mbox{day}^{-1}$. Being obtained without any particular assumption, the present mathematical results apply to logistic growth in general without any limitation to a specific real system.