key: cord-0443808-hkmg9lsc
authors: D.Lanteri,; D.Carco',; P.Castorina,
title: How macroscopic laws describe complex dynamics: asymptomatic population and CoviD-19 spreading
date: 2020-03-27
journal: nan
DOI: nan
sha: 2f2760eb495a1868f3dfd82fa58e5528a61556fd
doc_id: 443808
cord_uid: hkmg9lsc

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.

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 cumulative 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.

The time evolution of N (t) follows a macroscopic growth law, solution of a single differential equation which is apparently independent on the asymptomatic population, A(t). 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.

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.

There is an impressive number of experimental verifications, in many different scientific sectors, that coarse-grain properties of systems, with simple laws with respect to the fundamental microscopic algorithms, emerge at different levels of magnification providing important tools for explaining and predicting new phenomena.

For example, the Gompertz law (GL) [1] , initially applied to human mortality tables (i.e. aging), describes tumor growth [2, 3] , kinetics of enzymatic reactions, oxygenation of hemoglobin, intensity of photosynthesis as a function of CO2 concentration, drug dose-response curve, dynamics of growth, (e.g., bacteria, normal eukaryotic organisms). Analogously, the Logistic Law (LL) [4] has been applied in population dynamics, in economics, in material science and in many other sectors.

The ability of macroscopic growth laws in describing the underlying complex dynamics is sometime surprising. A clear and timely example is given by the infection spreading (Coronavirus [5, 6] in particular), where different Governments impose strong containment efforts on the basis of the mortality rate and of the growth rate of the cumulative total number of detected infected people N (t) at time t, which has a strong impact on the national health systems.

However, there is a large number of infected people without any symptoms who contribute to the disease spreading but are not explicitly taken into account in N (t), since not detected. More precisely, a macroscopic growth law for N (t) is solution of the general differential equation

where α ef f (t) is the specific growth rate at time t. For example, if α ef f (t) =constant one obtains the exponential behavior. Various growth patterns have been very recently applied to the time evolution of the CoviD-19 infection [7] [8] [9] [10] [11] [12] [13] . On the other hand, by using the previous equation to describe the epidemic phase, the cumulative number of asymptomatic infected individuals, A(t), seems completely neglected. Indeed, the second term of the equation depends on N (t) only. For the CoviD-19 infection, since A(t) is unknown, many Governments have correctly applied strong constraints to slow down the spreading: social isolation, information on the localization of infected individuals and the use of a very large number of medical swabs.

However, the question arises: is the information obtained by monitoring N (t) reliable in understanding the epidemic phase? By a simple model of the interaction between the symptomatic cumulative detected population N (t) and the asymptomatic one, A(t), we discuss how the effective coarse-grain equation in eq.(1) takes into account the asymptomatic population in the transient phase of the spreading. Moreover one gets useful indications on the number of asymptomatic individuals.

In the model, we call T (t) the cumulative total number of infected people, which is the sum of the number of the cumulative detected infected individuals N (t) and of the asymptomatic ones A(t):

In the data of N (t) the number of dead and of healed people is included, since they have been previously infected. In the fast growing phase, their total number is much smaller than N (t) and therefore they are not included in the dynamics (the re-infection is considered very unlikely). The previous condition defines the transient phase.

Therefore one has to take into account the rates of the following processes (n = detected infected person, a = asymptomatic infected person): n → n + n, a → n + a, n → n + a, a → a + a with rates

Accordingly, the corresponding mean field equations are

The functions c i (t), i = 1, 4, describe the damping effects due to the containment effort and we assume they have an exponential decreasing behavior:

i.e. the rates of the dynamical processes decrease in time due to social isolation and other external constraints. This effect should be independent on the status of the infected individual (n or a). On the other hand, in the processes involving a symptomatic individual, he/she is (or should be) rapidly detected, together with those who belong to his/her chain of infection transmission, independently on their symptomatic or asymptomatic condition. Therefore the rate of the processes involving infected persons should be suppressed with respect to the transmission rate among asymptomatic persons, which is "invisible".

In other terms, one expects that, in the transient phase, the rate of the process among asymptomatic individuals only, a → a + a, decreases slowly than the other ones. As a first step, one assumes that the functions c 1 (t), c 2 (t), c 3 (t), c 4 (t) are given by 

The data of the cumulative number of detected infected individuals of the Covid-19 in Italy are depicted in Fig.1 and clearly show a transition from an exponential growth to the Gompertz macroscopic law, which is solution of the particular case of eq.(1) given by

where k and N ∞ (the so called carrying capacity) are constants. Eq.(6) seems to neglect the role of the asymptomatic population, but, as we shall see, it is an effective way to take into account the dynamics in eqs. (2, 3, 5) .

To clarify this aspect, the dynamical system in eqs. (2, 3, 5) is analytically solved (see supplementary material) for λ 1 = λ 2 . For λ 2 < λ 1 there is no analytical solution and the numerical one has to be compared with the Italian data and the previous Gompertz fit.

As discussed, the fits of the Italian data of N (t) from February 22nd to March 22nd are plotted in Fig.1 , showing that the Gompertz growth (from eq.(6)) is in very good agreement with data.

The Gompertz fit is then compared with the numerical solution of the dynamical system (eqs. (2, 3, 5) with λ 2 < λ 1 ) in Fig.2 . The two curves are well compatible, indicating that the interaction between symptomatic and asymptomatic populations is contained in eq.(6) in an effective way.

The dynamical approach is more rich than the macroscopic description and it permits to evaluate the cumulative number of asymptomatic individuals as a function of time. Let us recall that our analysis applies to the transient region of fast growth and not to the saturation phase of the spreading. As shown in Fig. 3 , in the transient regime the cumulative number of asymptomatic people turns out to be larger than N (t), but smaller than other proposed models [18] .

The previous analysis clearly suggests that: +++++ ++ + ++ + + + + + + + + + + + + + + + + + + + + b) in the fast growing phase there is a large asymptomatic population: to stop the infection spreading the strong social isolation is the best method.

On the other hand, one expects that in the saturation, final, phase of the spreading, the asymptomatic individuals reduce their viral load, damping the transmission of the infection.

Let us go back to the initial point: the time evolution of N (t), which indeed contains the underlying spreading coming from the asymptomatic individuals, is an effective macroscopic growth laws solution of eq.(1) and, in particular, of eq.(6) for Italy, as shown in Fig.2 .

In this respect the choice of monitoring the time evolution of N (t) to understand the epidemic phase is reliable, since the underling dynamics is included in the time dependence of α ef f in eq.(1). This is probably related to the result that a general classification of the growth laws depends on the time derivative of α ef f [14, 15] .

However, it should be clear that the previous analysis implicitly assumes that the protocol concerning the detection of infected individuals, as for example the number of medical swabs (per day), does not change in time. Indeed, a modification in the detecting protocol (as in the Chinese case) or the sudden, strong, variation in the number of swabs could mimic an increase or a decrease in N (t) of not dynamical origin.

The model in eqs. (2, 3, 5) is a simplified version of a more complex dynamics [16, 17] and can be improved including more specific populations [18] . However it is useful to understand that the macroscopic laws in eq.(1) are an important tool to describe the evolution of the epidemic spreading, with the advantage of a strong reduction of the free parameters: the system in eqs. (2, 3, 5) has in principle 5 parameters plus 2 initial conditions, but the Gompertz curve has 2 parameters and 1 initial condition.

In general, in the spreading of infections there is a large asymptomatic population which drives the time dependence of N (t) in the transient phase. Both N (t) and A(t) reach a steady-state condition after some time, difficult to evaluate by mathematical models, where asymptomatic individuals are still present, but with a low viral load. From this point of view, the asymptomatic population should decline in the saturation phase.

Different analyses of the CoviD-19 spreading predict a large asymptomatic population but the results are strongly model dependent. In ref. 

Finally, for the specific case c 0 3 = c 0 1 and c 0 2 = c 0 4 = k c 0 1 , with k some constant,

and R(t) = 1 + (R 0 − 1)(1 + k)

The typical time dependence of the previous solutions , N (t) and A(t), is plotted in figure. 

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Let us consider the system of differential equationswith.It can be solved by defining the ratiowhich satisfies the equationThe analytical solution of the previous eq. (10) iswhereand