How macroscopic laws describe complex dynamics: Asymptomatic population and Covid-19 spreading

Macroscopic growth laws describe in an effective way the underlying complex dynamics of the spreading of infections, as in the case of Covid-19, where the counting of the cumulative number N(t) of detected infected individuals is a generally accepted variable to understand the epidemic phase. However, N(t) does not take into account the unknown number of asymptomatic cases A(t). The considered model of Covid-19 spreading is based on a system of coupled differential equations, which include the dynamics of the spreading among symptomatic and asymptomatic individuals and the strong containment effects due to the social isolation. The solution has been compared with N(t), determined by a single differential equation with no explicit reference to A(t), showing the equivalence of the two methods. The model is applied to Covid-19 spreading in Italy where a transition from an exponential behavior to a Gompertz growth for N(t) has been observed in more recent data. The information contained in the time series N(t) turns out to be reliable to understand the epidemic phase, although it does not describe the total infected population. The asymptomatic population is larger than the symptomatic one in the fast growth phase of the spreading.