How macroscopic laws describe complex dynamics: asymptomatic population and CoviD-19 spreading

Macroscopic growth laws, solutions of mean field equations, describe in an effective way an underlying complex dynamics. They are applied to study the spreading of infections, as in the case of CoviD-19, where the counting of the cumulated number $N(t)$ of detected infected individuals is a generally accepted, coarse-grain, variable to understand the epidemic phase. However $N(t)$ does not take into account the unknown number of asymptomatic, not detected, cases $A(t)$. Therefore, the question arises if the observed time series of data of $N(t)$ is a reliable tool for monitoring the evolution of the infectious disease. We study a system of coupled differential equations which includes the dynamics of the spreading among symptomatic and asymptomatic individuals and the strong containment effects due to the social isolation. The solution is therefore compared with a macroscopic law for the population $N(t)$ coming from a single, non-linear, differential equation with no explicit reference to $A(t)$, showing the equivalence of the two methods. Indeed, $N(t)$ takes into account a more complex and detailed population dynamics which permits the evaluation of the number of asymptomatic individuals also. The model is then 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. Then the information contained in the data analysis of $N(t)$ is 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.