# A Stochastic Compartmental Model for COVID-19

Giorgio SONNINO1, Fernando MORA2, and Pasquale NARDONE1
1Université Libre de Bruxelles (ULB), Faculté de Sciences
Bvd du Triomphe, Campus de la Plaine, CP 231, 1050 Brussels, Belgium
2Centro de Física No Lineal y Sistemas Complejos de Santiago (CFNL)
Vitacura 2902, Las Condes, Santiago, Chile
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We propose two stochastic models for the Coronavirus pandemic. The statistical
properties of the models, in particular the correlation functions and the
probability density function, have duly been computed. Our models, which
generalises a model previously proposed and published in a specialised
journal, take into account the adoption of the lockdown measures as well as
the crucial role of the hospitals and Health Care Institutes. To accomplish
this work we have analysed two scenarios: the $SIS$-model (Susceptible
$\Rightarrow$ Infectious $\Rightarrow$ Susceptible) in presence of the
lockdown measures and the $SIS$-model integrated with the action of the
hospitals (always in presence of the lockdown measures). We show that in the
case of the pure $SIS$-model, once the lockdown measures are removed, the
Coronavirus will start growing again. However, in the second scenario, beyond
a certain threshold of the hospital capacities, the Coronavirus is not only
kept under control, but its capacity to spread tends to diminish in time.
Therefore, the combined effect of the lockdown measures with the action of the
hospitals and health Institutes is able to contain and dampen the spread of
the SARS-CoV-2 epidemic. This result can be used during a period of time when
the massive distribution of delivery of a limited number of vaccines in a
given population is not yet feasible. By way of example, we analysed the data
for USA and France where the intensities of the noise have been estimated by
Statistical Mechanics. In particular, for USA we have analysed two possible
hypotheses: USA is still subject to the first wave of infection by and USA is
in the second (or third) wave of SARS-CoV-2 infection.The agreement between
theoretical predictions and real data confirms the validity of our approach.
Key words: Mathematical model; COVID-19; Dynamics of population; Pneumonia.

##  1 Introduction

The current number of infectious people by SARS-CoV-2 brought us back to the
lockdown phase, but the tracking and swabs give a clearer vision of the
situation. By reading the figures on new cases of positivity, it seems that we
are returning back to mid-April 2020. Hospitalisations are also growing up
including new hospitalisations in intensive care. In this respect, the
situation appears deeply different compared to the first months of the
epidemic, when the prolonged lockdown was necessary, which allowed to reduce
the spread of the Coronavirus significantly. The percentage ratio between new
cases and swabs has now stood at around 5$\%$ [1]. I is therefore far from
almost 30 per cent of the ”darkest” periods of the pandemic in certain
European Countries. However, these data induce to increase further preventive
strategies, focusing on tracing and beyond, taking into account that in some
countries the ”weight” of the pandemic is starting to be felt again on
hospitals. So, we are in the middle of the second wave, but the virus
circulates among humans and therefore the lockdown period has limited its
spread, so much so that we have assisted to a sort of ”stagnation” in the
months of June and July for the subsequent resumption with August. However,
the virus did not changed or that the clinical manifestations that it induces
have not changed [cauda]. In this second period of COVID-19, we may trust much
more of statistical data thanks to the increased number of swabs done and the
tracing and monitoring of people who have contracted the SARS-CoV-2 infection.
Of course, there are still some cases, especially of asymptomatic subjects who
unfortunately can transmit the infection: about 20$\%$ of cases of contagion
would in fact be referred to subjects who do not show any health problems.
This situation also explains why the percentage of positive swabs in the first
weeks of the lockdown was around 30 percent while today it is much lower.
Anyway, we are still far from the figures, of several thousand, of last March-
April, but the trend is still on the rise. For this reason, it is essential to
contain the infections, also through ”ad hoc” measures (lockdown, quarantine,
isolation). The carelessness during the summer leave has created a clear
epidemiological link, given that the cases resumed in August and now we have
to remedy, also because there are signs coming from nearby France that warn
under this aspect.

$\bullet$ USA situation. After sustained declines in the number of COVID-19
cases over recent months, restrictions are starting to ease across the United
States. Numbers of new cases are falling or stable at low numbers in some
states, but they are surging in many others. Overall, the USA. is experiencing
a sharp increase in the number of new cases a day, and by late June, had
surpassed the peak rate of spread in early April. In the USA, cases are rising
quickly right now. With more than 11 million confirmed cases, USA has the
highest number of infections in the world and the spread of the Coronavirus
shows no sign of slowing down. During the spring wave, swabs were mostly
limited to confirming cases in people who were already in hospital, meaning
the true scale of that outbreak wasn’t fully captured. But the latest data
compiled by the COVID Tracking Project shows the current surge is not just
down to increased testing, the number of tests carried out in USA was up by
12.5$\%$ week on week, while the number of cases increased by more than 40$\%$
[3]. One likely cause is the change of season and colder weather driving
people indoors to socialise, where the risk of spread is heightened due to
less social distancing and poor ventilation. The strain on hospitals is
growing. Because of the change in the level of testing, a better way to
compare waves is to look at the number of people being admitted to hospital
because of Covid-19. This data shows that roughly the same number of people
across the USA were in hospital during the first and second waves of the
outbreak. But there are already more people in hospital during the current
wave - more than 70000 at the moment. New deaths per day have not begun to
climb, but some hospitals’ intensive care units have recently reached full
capacity [4]. This wave is hitting every USA region. We can say that the USA
as a whole is not in a second wave because the first wave never really
stopped. The virus is simply spreading into new populations or resurgent in
places that let down their guard too soon. This time around, every region is
seeing a spike in new cases [3].

$\bullet$ France situation. Coronavirus cases in France rose dramatically, but
the number of daily deaths declined. The total number of infections stands at
around of 1829659, with the death toll standing at 42207, according to the
Ministry of Health. The overall downward trend is a positive sign after
Coronavirus numbers increased drastically by reaching a peak of 86852 cases.
Some 99 departments throughout France remain in a vulnerable position with
3042 clusters of infection considered serious [5]. France now has the fourth
highest number of infections in the world behind the USA, India and Brazil.
Despite the French government’s efforts to slow down the spread of the virus
with targeted measures such as lockdown and isolation, the rate of infections
has continued to rise across the Country. Talk of a second wave, and how to
avoid it, has been around since the end of the spring lockdown, but the
current situation seems to suggest it was inevitable. One month after
instituting the lockdown measures were applied in major cities starting from
Paris and three weeks after extending it to almost the whole Country, France
sees the light at the end of the dark tunnel of the second wave of
Coronavirus. French Health Minister announced that he believes that the peak
of the pandemic has passed, and the control has been regained. Indeed, looking
at the data, after the peak of 86000 cases recorded on 7 November, the Country
has moved on figures of around 30000 daily infections with a minimum level of
230000 recorded on 13 November. Figures that are returning in line with those
of a month ago when the lockdown began [6]. However, the Country will remain
in a second nationwide lockdown that will continue until Dec. 1 unless the
number of infections and deaths decrease significantly.

$\bullet$ Modelling the spread of SARS-CoV-2 infection. Until vaccine arrives,
the tried-and-true public health measures of the last months (social
distancing, universal mask wearing, frequent hand-washing and avoiding crowded
indoor spaces) are the ways to stop the first wave and thwart a second one.
And when there are surges like what is happening now in the word, further
reopening plans need to be put on hold. Meanwhile, in order to understand the
situation, theoretic studies, mainly based on mathematical model with the use
of up-to-date data, must be carried in order to understand the spread of this
pandemic in future and provide predictions. In this study we propose a simple
stochastic model with the objective to show how public health interventions
can influence the outcome of the epidemic, by limiting and even damping the
spread of the SARS-CoV-2 epidemic, waiting for the production and delivering
of an effective vaccine. The analysis is carried by assigning population in
different compartments with labels, such as $S$, $I$, $R$ and $D$
(Susceptible, Infectious, Recovered, and Deaths). People can progress between
the compartments. The order of the labels shows the flow patterns between the
compartments; for example $SIS$ means susceptible, infectious and, therefore,
Susceptible again. The origin of these models is at the beginning of the
twentieth century [7]. The goal of our model is to predict how a disease
spreads, or the dynamics of the number of infectious people, and the duration
of the COVID-19 pandemic. The dynamics of this kind of models is governed by
ordinary differential equations (O.D.E.s). Generally, these O.D.E.s are
deterministic, but in this work the dynamics is described by a stochastic
differential equation, which is more realistic.

The work is organised as follows. In Section (2), we study the deterministic
$SIS$-model in presence of the lockdown measures. We shall see that as soon as
the lockdown measures are stopped, the spread of the Coronavirus begins to
grow back vigorously once a certain threshold is exceeded. In the Subsection
(1.2) we ask ourselves the following question: Let us suppose that a
government needs a number $x$ of weeks (e.g. 6 weeks) before being able to
deliver an effective vaccine. What is the minimum value of infectious people
that should be expected, after the adoption of severe lockdown measures
imposed on the population, to be sure that after $x$-weeks (e.g. after 6
weeks) the total number of the infectious individuals by Coronavirus remains
below a threshold pre-set in advance, and acceptable from a medical and
political point of view ?. Subsection (2.1) answers the above-mentioned
question for the $SIS$-model subject to the lockdown measures. In Section (3)
we introduce, and study, the Stochastic $SIS$-model subject to the lockdown
measures (the $(SIS)_{L}$-model). We compute the relevant correlation
functions as well as the probability density function. Hospitals and health
institutions play a crucial role in hindering the spread of the Coronavirus.
In Section (4), we propose a model that accounts for people who are only
traced back to hospitalised infectious individuals. This model is referred to
as the $(SISI_{h}r_{h}d_{h})_{L}$-model. We shall see that beyond a certain
threshold of the hospital capacities, the coronavirus not only is kept under
control, but its spread even tends to diminish in time. Therefore, the
combined effect of the lockdown measures with efficient Hospitals and health
Institutes is able to contain and dampen the spread of the SARS-CoV-2
epidemic, waiting for the delivery of a limited number of vaccines in a given
population. In Section (5), we study the simplified $(SISI_{h})_{L}$
stochastic model by computing the relevant correlation functions. Comparison
between the theoretical predictions and the real data for USA and France can
be found in Section (6). Concluding remarks are reported in Section (7). An
estimation of the intensity of the noise by Statistical Mechanics can be found
in Annex 1. The derivation of the exact solution of Richards equation is
reported in Appendix 2. Tables of real data for infectious individuals in USA
and France are shown in Annex 3.

##  2 The Deterministic Model in Presence of the Lockdown Measures - The
$(SIS)_{L}$-Model

According to ref. [8], Respiratory viruses remain quiet for months, inactive
but viable, within living cells. Then suddenly they activate and become
virulent as they say, the infectious capacity grows to a maximum, after which
it decreases. So, there is no valid reason to think that this Coronavirus
behaves differently from others infections by SARS. [8]. The present work
starts from the following hypothesis: the SARS-CoV-2 behaves like the other
viruses that cause respiratory diseases. The common cold and influenza, do not
confer any long-lasting immunity. Such infections do not give immunity upon
recovery from infection, and individuals become susceptible again. Hence,
according to the above-cited hypothesis we propose the following simplest
compartmental model:

$S+I\xrightarrow{\sigma}2I$ (1)

In our model the SARS-CoV-2 infection does not leave any immunity, thus
individuals return back into the $S$ compartment. Hence, infectious people,
after recovery, return back to the compartmental $S$. This added detail can be
shown by including an $R$ class in the middle of the model

$\displaystyle S+I\xrightarrow{\sigma}2I$ (2) $\displaystyle
I\xrightarrow{\gamma}R\xrightarrow{\gamma_{1}}S$

From scheme (2), we get the O.D.E.s for $S$, $R$, and $I$:

$\displaystyle\frac{dS}{dt}=-\sigma\frac{S}{N_{Tot.}}I+\gamma_{1}R$ (3)
$\displaystyle\frac{dI}{dt}=\sigma\frac{S}{N_{Tot.}}I-\gamma I$
$\displaystyle\frac{dR}{dt}=\gamma I-\gamma_{1}R$

with $N_{Tot.}$ denoting the total population. By assuming that the dynamics
of $R$ is much faster that those of $S$ and $I$, we may set $dR/dt\simeq 0$
and system (3) reduces to

$\displaystyle\frac{dS}{dt}\simeq-\sigma\frac{S}{N_{Tot.}}I+\gamma I$ (4)
$\displaystyle\frac{dI}{dt}=\sigma\frac{S}{N_{Tot.}}I-\gamma I$

which corresponds to the model

$\displaystyle S+I\xrightarrow{\sigma}2I$ (5) $\displaystyle
I\xrightarrow{\gamma}S$

In literature, the model (5) is referred to as the $SIS$-model (see, for
example, [9]). From Eq. (4) we get the conservation relation

$\frac{dS}{dt}+\frac{dI}{dt}=0\qquad{\rm or}\quad S+I=N_{Tot.}$ (6)

Hence, the dynamics of infectious is governing by the logistic model

$\frac{dI}{dt}=(\sigma-\gamma)I\left(1-\frac{\sigma}{N_{Tot.}(\sigma-\gamma)}I\right)$
(7)

or

$\displaystyle\frac{dI}{dt}={\tilde{\alpha}}I\left(1-\frac{I}{K}\right)\qquad{\rm
with}$ (8)
$\displaystyle{\tilde{\alpha}}\equiv\sigma\left(1-\frac{\gamma}{\sigma}\right)\quad;\quad
K\equiv N_{Tot.}\left(1-\frac{\gamma}{\sigma}\right)$

where $\tilde{\alpha}$ and $K$ denote the linear growing rate of the COVID-19
and the carrying capacity, respectively. The lockdown measures are mainly
based on the isolation of the susceptible individuals, eventually with the
removal of infected people by hospitalisation. In our model, the effect of the
lockdown measures are taken into account by introducing in the $SIS$-model the
lockdown-induced decrease rate $c(t)$

$\displaystyle S+I\xrightarrow{\sigma}2I$ (9) $\displaystyle
I\xrightarrow{\gamma+c(t)}S$

where

$\displaystyle c(t_{L})=0\qquad{\rm for}\quad t=t_{L}$ (10) $\displaystyle
c(t)>0\qquad\ \ {\rm for}\quad t>t_{L}$

with $t_{L}$ denotes the time when the lockdown measures are applied. The
corresponding deterministic differential equations for the COVID-19 model in
presence of the lockdown measures reads then:

$\displaystyle\frac{dS}{dt}=-\sigma\frac{S}{N_{Tot.}}I+\gamma I+c(t)I$ (11)
$\displaystyle\frac{dI}{dt}=\sigma\frac{S}{N_{Tot.}}I-\gamma I-c(t)I$

Scheme (9) may be referred to as the $(SIS)_{L}$-model where $L$ stands for
$Lockdown$. The general expression for the lockdown-contribution, $c(t)$, may
be cast into the form111Here, we shall not discuss on the completeness of the
basis functions $\\{{\hat{t}}^{i+1}-{\hat{t}}^{-j}\\}_{i,j=0,1,\dots}$.

$c({\hat{t}})=\sum_{i,j=0}^{\infty}\beta_{ij}\left({\hat{t}}^{i+1}-{\hat{t}}^{-j}\right)\qquad{\rm
with}\quad{\hat{t}}\geq 1$ (12)

with ${\hat{t}}$ denoting the normalised time ${\hat{t}}\equiv t/t_{L}$ and
$\beta_{ij}$ are real numbers, subject to the condition $c({\hat{t}})>0$ for
${\hat{t}}>1$, respectively. Notice that expression (12) satisfies condition
(10) and is able to dampen the growing trend of the number of infectious. We
are looking for an expression for the lockdown, which grows at most linearly
in time. So, the relevant terms in expression (12) are

$c({\hat{t}})\simeq\beta_{00}({\hat{t}}-1)+\beta_{01}\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)\qquad{\rm
with}\quad{\hat{t}}\geq 1$ (13)

However, fittings with the real data correspond to $\beta_{00}\simeq 0$ (ref.
to [11], [12], and [13]). Finally, we get

$c({\hat{t}})\simeq\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)\qquad{\rm
with}\quad\beta>0\quad{\rm and}\quad{\hat{t}}\geq 1$ (14)

with $\beta\equiv\beta_{01}=const.$ denoting the intensity of the lockdown
measures. Eq. (14) is the expression for the lockdown measures that will be
considered in this work. Notice that Eq. (14) generalises the lockdown term
introduced in ref. [11]. By taking into account Eq. (6), the deterministic
differential equation for the infectious people, in presence of the lockdown
measures, reads

$\frac{dI}{d{\hat{t}}}=\alpha
I\left(1-\frac{I}{K}\right)-\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)I$
(15)

with $\alpha\equiv{\tilde{\alpha}}t_{L}$. As shown in the Appendix, the exact
solution of Eq. (15) reads

$I=\frac{A{\hat{t}}^{\beta}\exp(-({\hat{t}}-\alpha/\beta)^{2}/\sigma)}{1+A\alpha/K\int_{1}^{\hat{t}}x^{\beta}\exp(-(x-\alpha/\beta)^{2}/\sigma)dx}$
(16)

where $\sigma\equiv 2/\beta$. Expression (16) can be brought into the form

$I=\frac{I_{0}{\hat{t}}^{\beta}\exp((1-\alpha/\beta)^{2}/\sigma)\exp(-({\hat{t}}-\alpha/\beta)^{2}/\sigma)}{1+(I_{0}\alpha/K)\exp((1-\alpha/\beta)^{2}/\sigma)\int_{1}^{\hat{t}}x^{\beta}\exp(-(x-\alpha/\beta)^{2}/\sigma)dx}$
(17)

with $I_{0}$ denoting the value of the total cases at the time when the
lockdown measures are applied, i.e. $I_{{\hat{t}}=1}=I_{0}$. Eq. (17) can be
cast into the form

$I\simeq
I_{0}{\hat{t}}^{\beta}\exp((1-\alpha/\beta)^{2}/\sigma)\exp(-({\hat{t}}-\alpha/\beta)^{2}/\sigma)$
(18)

Fig.(1) shows two solutions of the model (15) for Italy, during the first wave
of infection by SARS-CoV-2. The red dotted line refers to the solution without
the lockdown measures and the dark dotted line to the solution when the
lockdown measures were applied.

Figure 1:  Solution of Eq. (15) for Italy, first wave of infection by SARS-
CoV-2. The black and the red dotted lines refer to the solutions of Eq. (15)
with and without the application of the lockdown measures, respectively. The
values of the parameters are $I_{0}$=5000, $K=150000$, $\alpha=1.5$, and
$\beta=0.1$, respectively.

###  2.1 The Second Wave of the COVID-19 Pandemic in the $(SIS)_{L}$-model

The basic reproduction number, $R_{0}$, is defined as the expected number of
secondary cases produced by a single (typical) infection in a completely
susceptible population. An epidemic occurs if the number of infected
individuals increases, i.e., $dI/dt>0$

$\sigma\frac{S}{N_{Tot.}}I-\gamma I>0\qquad{\rm
or}\quad\frac{\sigma}{\gamma}\frac{S}{N_{Tot.}}>1$ (19)

In this case, $R_{0}=\sigma/\gamma$. When the lockdown measures are
interrupted, the dynamics of the process are governed again by Eq. (8). For
Italy, in the first wave we had $R_{0}>1$, so $K>0$. From Eq. (8) we get

$\frac{\sigma}{\gamma}>1\qquad{\rm hence}\quad\alpha>0\qquad{\rm
and}\quad\lim_{{\hat{t}}\rightarrow\infty}I=K$ (20)

The above relations biologically mean that the disease will remain permanently
endemic in the population. Notice that the inflection point of the logistic
curve (8) occurs at time $t=t_{f}$ with

${\hat{t}}_{f}=\frac{1}{\alpha}\log\left(\frac{K}{I_{L}}-1\right)$ (21)

with $I_{L}$ denoting the number of the infectious people when the lockdown
measures start to be applied. It is easily checked that the inflection point,
$P_{f}$, has coordinates $P_{f}=({\hat{t}}_{f},I_{f}=K/2)$. In our forthcoming
reasoning, we shall see that the inflection point plays an important role in
our analysis. To summarize, after removing the lockdown measurements, say at
time ${\hat{t}}_{1}$, the number of infectious people will start to rise again
from the value $I_{1}$, with $I_{1}\equiv I_{t={\hat{t}}_{1}}$, following
again a growth rate given by the logistic curve (8). This new growth of
outbreak is referred to as the second wave of COVID-19. Now, our goal is to
answer the following question: ”Given a fixed value of the infectious people,
say ${\tilde{I}}$, determine the minimum value that $I_{1}$ should have so
that the number of infectious people is less than ${\hat{I}}$, during the
entire period from ${\hat{t}}_{1}$ to ${\hat{t}}_{f}$”. We note that within
the interval ${\hat{t}}_{1}\leq{\hat{t}}\leq{\hat{t}}_{f}$ the number of
infectious people grows quite slowly. Let us indicate with ${\hat{t}}_{2}$ the
time when it is decided to interrupt the lockdown measures. Hence, during the
time interval $\Delta{\hat{t}}={\hat{t}}_{2}-{\hat{t}}_{1}$, population is no
longer subject to the lockdown measures. To answer the above-mentioned
question we need firstly to find the range of values that can be assumed by
$I_{1}$ so that in the interval $\Delta{\hat{t}}<{\hat{t}}_{f}-{\hat{t}}_{1}$,
the number of infectious individuals is lower than the ordinate of the
inflection point, i.e. $I_{1}<I_{f}=K/2$ for ${\hat{t}}\in\Delta{\hat{t}}$. It
is easily checked that $I_{1}$ should be within the range

$0\leq I_{1}<\frac{K}{1+\exp(\alpha\Delta{\hat{t}})}$ (22)

Now, we have to require that after a lapse of time $\Delta{\hat{t}}$, the
number of the infectious individuals must be lower than a value ${\tilde{I}}$
fixed a priori, with ${\tilde{I}}<K/2$. The result is easily obtained by
writing the initial condition as

$I_{1}=\frac{K}{1+\exp(\alpha\Delta{\hat{t}})}-Kx\qquad{\rm with}\quad
0<x\leq\frac{1}{1+\exp(\alpha\Delta{\hat{t}})}$ (23)

and by determining the range of values of $x$ such that the inequality

$\frac{K}{1+(K/I_{1}-1)\exp(-\alpha\Delta{\hat{t}})}\leq{\tilde{I}}<\frac{K}{2}$
(24)

is satisfied. After a little algebra, we get

$0\leq
I_{1}\leq{\tilde{I}}\frac{2+\exp{\alpha\Delta{\hat{t}}}+\exp(-\alpha\Delta{\hat{t}})}{(1+\exp(\alpha\Delta{\hat{t}})({\tilde{I}}/K(\exp(\alpha\Delta{\hat{t}})+\exp(-\alpha\Delta{\hat{t}})+1+\exp(\alpha\Delta{\hat{t}}))}$
(25)

To summarise, if the lockdown measures are interrupted at the time
${\hat{t}}_{1}$, during the successive period of time
$\Delta{\hat{t}}={\hat{t}}_{2}-{\hat{t}}_{1}$ the number of infectious remains
inferior to a preset number ${\tilde{I}}$ if the number of the infectious at
the end of the lockdown measures is inferior to $I_{1}$, with $I_{1}$ given by
the r.h.s. of Eq. (25). A Schematic representation of the second wave of SARS-
CoV-2 infection is shown in Fig. 2.

As an example, let us consider the case of the Italian situation. Fig. 4.
reports $I_{1}$ against ${\tilde{I}}$ and $\Delta{\hat{t}}$ in the range
$50000\leq{\tilde{I}}\leq K_{IT}$. As shown in Fig. 4., if we set the value of
${\tilde{I}}=50000$, the maximum value of $I_{1}$ is $I_{1}\simeq 9500$ for
$t\simeq t_{L}$, and $I_{1}\simeq 700$ for $t\simeq 2t_{L}$, respectively.

Figure 2:  Schematic representation of the second wave of SARS-CoV-2
infection. The Lockdown measures are applied during the period
${{\hat{t}}_{0}}-{{\hat{t}}_{1}}$. At the end of the confinement measures, the
number of infectious dropped to $I_{1}$. During the time interval
$\Delta{\hat{t}}={{\hat{t}}_{2}}-{{\hat{t}}_{1}}$, the lockdown measures are
not applied and the infection by SARS-CoV-2 (starts growing again (second wave
of Coronavirus infection) until the preset value ${\tilde{I}}$ is reached,
which corresponds to $I({\hat{t}}_{2})$. During the whole time interval
$\Delta{\hat{t}}$, the number of infectious individuals still stay below a set
threshold ${\tilde{I}}$ if $I_{1}$ satisfies condition (25).

Figure 3: $I_{1}$ against $\tilde{I}$ and $\Delta{\hat{t}}$ in the range
$50000\leq{\tilde{I}}\leq K_{IT}$.

Figure 4: As shown, if we set the value of ${\tilde{I}}=50000$, the maximum
value of $I_{1}$ is $I_{1}=9500$ for $t=t_{L}$, and $I_{1}=700$ for
$t=2t_{L}$, respectively.

##  3 The Stochastic $(SIS)_{L}$-Model

The Stochastic version of Eq. (15) reads

$\frac{dI}{d{\hat{t}}}=\alpha
I\left(1-\frac{I}{K}\right)-\beta\left(\frac{{\hat{t}}^{2}-1}{{\hat{t}}}\right)I+\
\xi({\hat{t}})$ (26)

with $\xi({\hat{t}})$ denoting a white noise

$\displaystyle<\xi({\hat{t}})>=0$ (27)
$\displaystyle<\xi({\hat{t}})\xi({\hat{t}}^{\prime})>=\eta\delta({\hat{t}}-{\hat{t}}^{\prime})$

and $\sqrt{\eta}$ is the intensity of the noise. $\delta$ denotes the Dirac
delta function (distribution). Let us now consider the system at the reference
state $I_{RS}$, which is the solution of the deterministic equation

$\frac{dI_{RS}}{d{\hat{t}}}=\alpha
I_{RS}\left(1-\frac{I_{RS}}{K}\right)-\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)I_{RS}$
(28)

subject to a perturbation of small amplitude $\delta I$, i.e.

$I(t)=I_{RS}(t)+\delta I(t)$ (29)

Our goal is to compute the relevant statistical correlation functions of this
processes i.e. $<\delta I({\hat{t}})\xi({\hat{t}})>$ and $<\delta I(t)\delta
I(t^{\prime})>$. Our task can easily be accomplished by recalling the
following theorem (see, for example, [14], [15], [16], [17]):

For a system of Langevin equations describing the temporal evolution of the
processes $(x_{1},\dots,x_{n})$:

$\displaystyle\frac{dx_{i}}{dt}=f_{i}(x_{1},\dots,x_{n})+\xi_{i}(t)\qquad
i=1,\dots,n$ (30)
$\displaystyle<\xi_{i}(t)>=0\quad;\quad<\xi_{i}(t)\xi_{j}(t^{\prime})>=\eta_{ij}\delta(t-t^{\prime})$

the following equations are valid:

$\displaystyle<x_{i}(t)\xi_{j}(t^{\prime})>=0\qquad{\rm if}\quad t^{\prime}>t$
(31) $\displaystyle<x_{i}(t)\xi_{j}(t)>=\frac{1}{2}\eta_{ij}$

Now, we assume that the perturbation $\delta I$ is of order $\sqrt{\eta}$
(considered as a small parameter) and we develop Eq. (26) up to the first
order in $\sqrt{\eta}$. We get

$\displaystyle\frac{dI_{RS}({\hat{t}})}{d{\hat{t}}}=\alpha
I_{RS}({\hat{t}})\left(1-\frac{I_{RS}({\hat{t}})}{K}\right)-\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)I_{RS}({\hat{t}})\
{\rm(at\ the\ zero\ order)}$ $\displaystyle\frac{d}{d{\hat{t}}}\delta
I({\hat{t}})=\left(\alpha-\frac{2\alpha}{K}I_{RS}({\hat{t}})-\beta\frac{{\hat{t}}^{2}-1}{{\hat{t}}}\right)\delta
I({\hat{t}})+\xi({\hat{t}})\ \ {\rm(at\ the\ first\ order)}$ (32)

Solution $I_{RS}({\hat{t}})$ is given by Eq. (17) and the exact solution for
$\delta I({\hat{t}})$ is easily obtained:

$\displaystyle I_{RS}({\hat{t}})=$
$\displaystyle\frac{I_{0RS}{\hat{t}}^{\beta}\exp((1-\alpha/\beta)^{2}/\sigma)\exp(-({\hat{t}}-\alpha/\beta)^{2}/\sigma)}{1+(I_{0RS}\alpha/K)\exp((1-\alpha/\beta)^{2}/\sigma)\int_{1}^{\hat{t}}x^{\beta}\exp(-(x-\alpha/\beta)^{2}/\sigma)dx}$
$\displaystyle\delta I({\hat{t}})=$ $\displaystyle
t^{\beta}\exp\left(\alpha({\hat{t}}-1)-1/2\beta({\hat{t}}^{2}-1)-2\alpha
G({\hat{t}})/K\right)\Big{(}\delta I(1)$ $\displaystyle+\int_{1}^{\hat{t}}dx\
\xi(x)\left(x^{-\beta}\exp(-\alpha(x-1)+1/2\beta(x^{2}-1)+2\alpha
G(x)/K\right)$ $\displaystyle{\rm with}\qquad
G({\hat{t}})\equiv\int_{1}^{\hat{t}}I_{RS}(x)dx$ (33)

Solution (3) allows to check the validity of Eq. (31). Indeed, the correlation
function $\delta I(t)$-noise at the same time reads

$<\delta I({\hat{t}})\xi({\hat{t}})>=\frac{\eta}{2}$ (34)

where Eqs (30) and the properties of the Dirac’s delta have been taken into
account. The equation for the second moment $<\delta I^{2}({\hat{t}})>$ is
easily obtained by multiplying both sides of the second equation of the system
(3) by $2\delta I(t)$ and by taking into account Eq (31). We get

$\frac{d}{d{\hat{t}}}<\left(\delta
I({\hat{t}}\right))^{2}>=2\left(\alpha-\frac{2\alpha}{K}I_{RS}({\hat{t}})-\beta\frac{{\hat{t}}^{2}-1}{{\hat{t}}}\right)<\left(\delta
I({\hat{t}})\right)^{2}>+\eta$ (35)

The solution of Eq. (35) provides the expression for the second moment

$\displaystyle<\left(\delta I({\hat{t}}\right))^{2}>=$
$\displaystyle\left({\hat{t}}^{2\beta}\exp(2\alpha({\hat{t}}-1)-\beta({\hat{t}}^{2}-1)-4\alpha
G({\hat{t}})/K\right)\\!\\!\Big{(}<\\!\left(\delta I(1)\right)^{2}\\!>$
$\displaystyle\\!+\eta\int_{1}^{\hat{t}}x^{-2\beta}\exp(-2\alpha(x-1)+\beta(x^{2}-1)+4\alpha
G(x)/K)\\!dx\Bigl{)}$ (36)

From solution (3) we also get the correlation function for $\delta I$

$<\delta I(1)\delta
I({\hat{t}})>={\hat{t}}^{\beta}\exp(\alpha({\hat{t}}\\!-\\!1)\\!-\\!\beta/2({\hat{t}}^{2}\\!-\\!1)\\!-\\!2\alpha
G({\hat{t}})/K)\\!<(\delta I(1))^{2}>$ (37)

Fig. 5. shows the simulation of 200 trajectories of Eq. (26) for Italy using
the Order-2 Stochastic Runge-Kutta integration method. The thick black curve
is the numerical solution of the deterministic equation (15). The intensity of
the noise is $\eta=0.05$ and the values of other parameters are reported in
the figure caption of Fig. 5.

Figure 5:  Solution of the stochastic equation for Italy. Solutions of Eqs
(26) and (27) for Italy - first wave of infection by SARS-CoV-2 - with
$\eta=0.05$ and for 200 realisations. The values of the other parameters are
$I_{0}$=5000, $K=150000$, $\alpha=1.5$, and $\beta=0.1$, respectively. The
black thick curve is the numerical solution of the deterministic equation
(15). The black curve is the solution of the deterministic equation.

Figs. (7) and (7) show the correlation functions $<(\delta I({\hat{t}}))^{2}>$
and $<\delta I(1)\delta({\hat{t}})>$ for Italy, first wave of infection by
SARS-CoV-2. The values of the parameters are reported in the figure captions.

Figure 6:  Correlation Function $<(\delta I({\hat{t}}))^{2}>$ for Italy, first
wave of infection by SARS-CoV-2. This correlation function corresponds to the
solution of Eq. (35). The values of the parameters are $I_{0}$=5000,
$K=150000$, $\alpha=1.5$, and $\beta=0.1$, respectively.

Figure 7:  Correlation Function for the Infectious people $<\delta
I(1)\delta({\hat{t}})>$. Plot of Eq. (37) with $I_{0}$=5000, $K=150000$,
$\alpha=1.5$, and $\beta=0.1$, respectively.

The Fokker-Planck equation, associated to Eq. (26), describing the time
evolution of the probability density function (PDF), $p(I,t)$, to find the
number of infectious people at time $\hat{t}$ reads [19]

$\displaystyle\frac{\partial}{\partial{\hat{t}}}p(I,{\hat{t}})=-\frac{\partial}{\partial
I}[\mu({\hat{t}})p(I,{\hat{t}})]+D\frac{\partial^{2}}{\partial
I^{2}}p(I,{\hat{t})}$ (38) $\displaystyle\mu({\hat{t}})=\alpha
I\left(1-\frac{I}{K}\right)-\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)\quad;\quad
D=\frac{\eta^{2}}{2}$

with $\mu$ denoting the drift coefficient and $D$ the diffusion coefficient,
respectively. Eq. (38) is subject to the initial condition

$p(I,0)=\delta(I-I_{0})$ (39)

To get the PDF, we proceed as follows. First, we simulate paths or
realisations of Eq. (26) with the indicated parameters. The employed algorithm
is an Order-2 Stochastic Runge-Kutta like scheme. Regarding the discretisation
in the numerical solution of the Langevin Eq. (26) we use 500 time steps to
build each stochastic path. The PDF is computed numerically by constructing an
histogram at each instant of time with a sufficiently number of realisations.
The standard number used here is 22000 paths. The values of the parameters are
$\alpha=1.5$, $\beta=0.1$, respectively, and for the PDF algorithm we use 800
bins in the construction of the corresponding histograms at the indicated
times. The intensity of the noise, $\eta=0.05$, has been estimated according
to Eq. (64) (see Appendix 1). Figures (9) to (15) represent a set of
screenshots at the indicated times ${\hat{t}}=t/t_{L}$.

Figure 8:  Probability density function (PDF), solution of Eq. (3) for Italy,
first wave of infection by SARS-CoV-2. Screenshoot at time ${\hat{t}}=1.129$.
The values of the parameters are $I_{0}$=5000, $K=150000$, $\alpha=1.5$, and
$\beta=0.1$, respectively.

Figure 9:  Probability density function (PDF), solution of Eq. (3).
Screenshoot at time ${\hat{t}}=2.0304$.

Figure 10:  Probability density function (PDF), solution of Eq. (3) for Italy,
first wave of infection by SARS-CoV-2. Screenshoot at time ${\hat{t}}=3.0647$.

Figure 11:  Probability density function (PDF), solution of Eq. (3).
Screenshoot at time ${\hat{t}}=4.0397$.

Figure 12:  Probability density function (PDF), solution of Eq. (3) for Italy,
first wave of infection by SARS-CoV-2. Screenshoot at time ${\hat{t}}=5.0131$.

Figure 13:  Probability density function (PDF), solution of Eq. (3).
Screenshoot at time ${\hat{t}}=6.0375$.

Figure 14:  Probability density function (PDF), solution of Eq. (3) for Italy,
first wave of infection by SARS-CoV-2. Screenshoot at time ${\hat{t}}=7.0656$.

Figure 15:  Probability density function (PDF), solution of Eq. (3).
Screenshoot at time ${\hat{t}}=7.9942$.

##  4 The $(SISI_{h}r_{h}d_{h})_{L}$-Model

As seen, according to the $SIS$-model, after the lockdown measures the number
of infectious people starts to increase again. Hospitals and health
institutions play a crucial role in hindering the spread of the Coronavirus.
In this Section, we propose a model that accounts for people who are only
traced back to hospitalised infectious individuals. However, we anticipate
that the model for the hospitalised people should be considered as an purely
idealistic, rather than a realistic. model. Finally, we propose the following
model222In refs. [20] and [21] it is proposed to adopt the Michaelis-Menten’s
enzyme-substrate reaction model (the MM reaction) for describing the dynamics
of the recovered people previously hospitalised. In our case, the enzyme is
compared to the available hospital beds, the substrate to the infected people,
and the product to the recovered people, respectively. Of course, the reverse
MM reaction has no sense in our case and, consequently, the kinetic constant
is equal to zero.:

$\displaystyle S+I\xrightarrow{\sigma}2I$ (40) $\displaystyle
I\xrightarrow{\gamma+c(t)}S$ $\displaystyle
I+b\xrightarrow{k_{1}}I_{h}\xrightarrow{k_{r}}r_{h}+b$
$\displaystyle\qquad\qquad\ \\!I_{h}\xrightarrow{k_{d}}d_{h}+b$
$\displaystyle\qquad\qquad\ \\!r_{h}\xrightarrow{\gamma_{1}}S$

where the hypothesis that an individual acquires immunity, after having
contracted the Coronavirus and being recovered, is not adopted. In the scheme
(17), $b$ denotes the number of available hospital beds, $I_{h}$ the number of
infected people blocking an hospital bed, $r_{h}$ the number of recovered
people previously hospitalised, and $d_{h}$ the number of people deceased in
the hospital, respectively. In the scheme (17), we have into account that
people, once recovered, are subject to the same existing lockdown measures as
any other people.. Of course, we have

$I_{h}+b=C_{h}=const.\qquad{\rm where}\quad{C_{h}={\rm Total\
hospital^{\prime}s\ capacity}}$ (41)

The dynamic equations for the entire process are then:

$\displaystyle\frac{d}{d{\hat{t}}}S=-\sigma\frac{S}{N_{Tot.}}I+\gamma
I+\gamma_{1}r_{h}+\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)(I+r_{h})$
(42) $\displaystyle\frac{d}{d{\hat{t}}}I=\sigma\frac{S}{N_{Tot.}}I-\gamma
I-k_{1}I(C_{h}-I_{h})-\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)I$
$\displaystyle\frac{d}{d{\hat{t}}}I_{h}=k_{1}I(C_{h}-I_{h})-k_{r}{I_{h}}-k_{d}{I_{h}}$
$\displaystyle\frac{d}{d{\hat{t}}}r_{h}=k_{r}{I_{h}}-\gamma_{1}r_{h}$
$\displaystyle\frac{d}{d{\hat{t}}}d_{h}=k_{d}{I_{h}}$

where, for simplicity, the average recovery time delay and the average death
time delay have been neglected [20]. In this case $I$ stands for the
infectious individuals not hospitalised. From system (42) we get the following
conservation law

$S+I+I_{h}+r_{h}+d_{h}=N_{Tot.}=const.$ (43)

###  4.1 The Deterministic $(SISI_{h})_{L}$-Model

To simplify as much as possible the set of O.D.E.s (42), we adopt several
hypotheses that will not compromise the validity of our model. First, we
assume that that $S+I+I_{h}\simeq N_{Tot.}=const.$. Secondly, let
$\gamma\simeq\gamma_{1}$. Finally, we take into account the current Belgian
hospital-protocol: ”Only the seriously sick people are hospitalised, the
remaining infectious individuals have to be sent home and they must be subject
to quarantine measures”. Hence, $I\gg I_{h}$ and the total number of recovered
people, $R$, is much larger than the total number of the recovered people,
previously hospitalised (i.e. $R\gg r_{h}$). Under these assumptions, the
model simplifies to

$\displaystyle S+I\xrightarrow{\sigma}2I$ (44) $\displaystyle
I\xrightarrow{\gamma+c(t)}S$ $\displaystyle I+b\xrightarrow{k_{1}}I_{h}$ (45)

with $I_{h}+b=C_{h}$. Hence, under these assumptions, after hospitalisation,
individuals will be removed from the disease, either due to immunisation (e.g.
due to vaccination or special health care received) or due to death. The
governing O.D.E.s, associated to the model (44), read

$\displaystyle\frac{d}{d{\hat{t}}}S\simeq-\sigma\frac{S}{N_{Tot.}}I+\gamma
I+\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)I$ (46)
$\displaystyle\frac{d}{d{\hat{t}}}I=\sigma\frac{S}{N_{Tot.}}I-\gamma
I-k_{1}I(C_{h}-I_{h})-\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)I$
$\displaystyle\frac{d}{d{\hat{t}}}I_{h}\simeq k_{1}I(C_{h}-I_{h})$

The recovered people $r_{h}$ and the deceased people $d_{h}$ may be obtained
by solving, respectively, the following O.D.E.s

$\frac{d}{d{\hat{t}}}r_{h}\simeq
k_{r}{I_{h}}\qquad;\qquad\frac{d}{d{\hat{t}}}d_{h}=k_{d}{I_{h}}$ (47)

From system (46), we obtain

$\displaystyle\frac{d}{d{\hat{t}}}I=\alpha_{1}I\left(1-\frac{I}{K_{1}}\right)-\rho
II_{h}-\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)I$ (48)
$\displaystyle\frac{d}{d{\hat{t}}}I_{h}=k_{1}C_{h}I-k_{1}II_{h}\qquad{\rm
with}$
$\displaystyle\alpha_{1}=\sigma\\!\left(\\!1-\frac{\gamma}{\sigma}-\frac{k_{1}}{\sigma}C_{h}\\!\right)\
;\
K_{1}=N_{Tot.}\left(\\!1-\frac{\gamma}{\sigma}-\frac{k_{1}}{\sigma}C_{h}\\!\right)\
;\ \rho=\frac{\sigma}{N_{Tot.}}\\!-\\!k_{1}$

In absence of the lockdown measures ($\beta=0$), we have the following
scenarios: $\forall I_{0}>0$

$\displaystyle{\bf i)}\ \ {\rm if}\quad
C_{h}<C_{hCrit.}\equiv\frac{\sigma}{k_{1}}\left(1-\frac{\gamma}{\sigma}\right)\
{\rm the\ equilibrium\ with}\ I=K_{1}\ {\rm is\ stable}$ (49)
$\displaystyle{\bf ii)}\ \ \ {\rm if}\quad
C_{h}>C_{hCrit.}\equiv\frac{\sigma}{k_{1}}\left(1-\frac{\gamma}{\sigma}\right)\
{\rm the\ equilibrium\ with}\ I=0\ {\rm is\ stable}$

In words:

$\bullet$ for the case ${\bf i}$), there will be a proper epidemic outbreak
with an increase of the number of the infectious people;

$\bullet$ for the case ${\bf ii}$), independently of the initial size of the
susceptible population, the disease can never cause a proper epidemic
outbreak.

This result highlights the crucial role of the Hospitals and the Health Care
Institutes:

If the threshold of the hospital capacities exceeds a minimum limit, the
spread of the Coronavirus tends to decrease over time, and the stable solution
corresponds to zero infectious individuals.

Fig. 17. and 17. illustrate the situation. Notice that ,with the values of
parameters reported in the figure captions, $C_{hCrit.}=18434$.

Figure 16:  Infectious people and Infectious hospitalised people vs time -
$C_{h}>C_{HCrit.}$. In this case, there is an epidemic outbreak with an
increase of the number of the infectious people. The values of the parameters
are: $\beta=0$, $k_{1}=0.00001$, $\rho=0.0001$, $K_{1}=150000$,
$\gamma=0.0001$, $\sigma=2.7651$, and $C_{h}=5100$ .

Figure 17:  Infectious people and Infectious hospitalised people vs time -
$C_{h}>C_{HCrit.}$. Independently of the initial size of the susceptible
population, the disease can never cause a proper epidemic outbreak. The values
of the parameters are: $\beta=0$, $k_{1}=0.00001$, $\rho=0.0001$,
$K_{1}=150000$, $\gamma=0.0001$, $\sigma=2.7651$, and $C_{h}=18436$.

##  5 The Stochastic $(SISI_{h})_{L}$ Model

If the dynamics is subject to a white noise, the related stochastic equations
read

$\displaystyle\frac{d}{d{\hat{t}}}I=\alpha_{1}I\left(1-\frac{I}{K_{1}}\right)-\rho
II_{h}-\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)I+\xi_{1}({\hat{t}})$
(50)
$\displaystyle\frac{d}{d{\hat{t}}}I_{h}=k_{1}C_{h}I-k_{1}II_{h}+\xi_{2}({\hat{t}})$

where

$\displaystyle<\xi_{i}(t)>=0\qquad(i=1,2)$ (51)
$\displaystyle<\xi_{i}(t)\xi_{j}(t^{\prime})>=\eta_{ij}\delta_{ij}\delta(t-t^{\prime})\qquad{\rm
with}\quad\eta_{12}=\eta_{21}$

with $\delta_{ij}$ denoting Kronecker’s delta. The statistical properties of
this processes, i.e. $<\delta I({\hat{t}})\delta I({\hat{t}})>$, $<\delta
I({\hat{t}})\delta I_{h}({\hat{t}})>$ and $<\delta I_{h}({\hat{t}})\delta
I_{h}({\hat{t}})>$, may be obtained, firstly, by determining the reference
state. This state satisfies the following O.D.E.s

$\displaystyle\frac{d}{d{\hat{t}}}I_{RS}=\alpha_{1}I_{RS}\left(1-\frac{I_{RS}}{K_{1}}\right)-\rho
I_{RS}I_{hRS}-\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)I_{RS}$ (52)
$\displaystyle\frac{d}{d{\hat{t}}}I_{hRS}=k_{1}C_{h}I_{RS}-k_{1}I_{RS}I_{hRS}$

Let us choose, for example, $I_{hRS}=C_{h}$, then

$\displaystyle
I_{RS}({\hat{t}})=\frac{I_{0RS}\exp((1-\alpha_{2}/\beta)^{2}/\sigma){\hat{t}}^{\beta}\exp(-({\hat{t}}-\alpha_{2}/\beta)^{2}/\sigma)}{1+(I_{0RS}\alpha_{2}/K_{2})\exp((1-\alpha_{2}/\beta)^{2}/\sigma)\int_{1}^{\hat{t}}x^{\beta}\exp(-(x-\alpha_{2}/\beta)^{2}/\sigma)dx}$
$\displaystyle\alpha_{2}\equiv\sigma\left(1-\frac{\gamma}{\sigma}-\frac{C_{h}}{N_{Tot.}}\right)\quad;\quad
K_{2}=N_{Tot.}\left(1-\frac{\gamma}{\sigma}-\frac{C_{h}}{N_{Tot.}}\right)$
(53)

Successively, we have to find the solution of the Fokker-Planck equation,
$p(I,I_{h},{\hat{t}})$, associated to the system (50), which in this case
reads

$\frac{d}{d{\hat{t}}}p=-\frac{\partial}{\partial
I}(\mu_{1}p)-\frac{\partial}{\partial
I_{h}}(\mu_{2}p)+D_{11}\frac{\partial}{\partial
I^{2}}p+D_{12}\frac{\partial}{\partial I\partial
I_{h}}p+D_{22}\frac{\partial}{\partial I_{h}^{2}}p$ (54)

with

$\displaystyle\mu_{1}=\alpha_{1}I\left(1-\frac{I}{K_{1}}\right)-\rho
II_{h}-\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)I\quad;\quad\mu_{2}=k_{1}C_{h}I-k_{1}II_{h}$
(55) $\displaystyle D_{11}=\frac{1}{2}(\eta_{11}+\eta_{12})\ ;\
D_{12}=\sqrt{\eta_{12}}(\sqrt{\eta_{11}}+\sqrt{\eta_{22}})\ ;\
D_{22}=\frac{1}{2}(\eta_{22}+\eta_{12})$

Finally, with the expression of $p(I,I_{h},t)$ we may compute the correlation
functions $<\delta I({\hat{t}})\delta I({\hat{t}})>$, $<\delta
I({\hat{t}})\delta I_{h}({\hat{t}})>$ and $<\delta I_{h}({\hat{t}})\delta
I_{h}({\hat{t}})>$ with

$\delta I=I-I_{RS}\qquad;\qquad\delta I_{h}=I_{h}-I_{hRS}$ (56)

However, this is a quite long and tedious procedure. We may circumvent the
obstacle by recalling that, as we did in Section (3), it is possible to obtain
such correlation functions in a direct way, by using the theorem (30-31). For
this, we have to determine the dynamic O.D.E.s for the fluctuations $\delta
I({\hat{t}})$ and $\delta I_{h}({\hat{t}})$ around the reference state
($I_{RS},I_{hRS}$). We get

$\displaystyle\frac{d}{d{\hat{t}}}\delta I=\left(\alpha_{1}-\rho
C_{h}-2\frac{\alpha_{1}}{K_{1}}I_{RS}-\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)\right)\delta
I-\rho I_{RS}\delta I_{h}+\zeta_{1}({\hat{t}})$
$\displaystyle\frac{d}{d{\hat{t}}}\delta I_{h}=-k_{1}I_{RS}(t)\delta
I_{h}+\zeta_{2}({\hat{t}})$ (57)

The correlation functions $<\delta I({\hat{t}}_{0})\delta I({\hat{t}})>$ and
$<\delta I_{h}({\hat{t}}_{0})\delta I_{h}({\hat{t}})>$ are easily obtained

$\displaystyle<\\!\delta I({\hat{t}}_{0})\delta
I({\hat{t}})\\!>\\!=\\!\left(\frac{{\hat{t}}}{{\hat{t}}_{0}}\right)^{\beta}\exp(H({\hat{t}}))\exp\left(-2\frac{\alpha_{1}}{K_{1}}G({\hat{t}}))\right)\\!\\!\Bigg{(}\\!\\!<\\!\delta
I({\hat{t}}_{0})^{2}\\!>$ (58) $\displaystyle-\rho<\delta
I({\hat{t}}_{0})\delta
I_{h}({\hat{t}}_{0})>\int_{{\hat{t}}_{0}}^{\hat{t}}\left(\frac{{\hat{t}}_{0}}{s}\right)^{\beta}\\!I_{RS}(s)\exp(-H(s))\exp\left((2\frac{\alpha_{1}}{K_{1}}-k_{1})G(s)\right)\\!ds\Bigg{)}$
$\displaystyle<\delta I_{h}({\hat{t}}_{0})\delta I_{h}({\hat{t}})>=<\delta
I_{h}({\hat{t}}_{0})^{2}>\exp(-k_{1}G({\hat{t}}))\qquad{\rm with}$
$\displaystyle
G({\hat{t}})\equiv\int_{{\hat{t}}_{0}}^{\hat{t}}I_{RS}(s)ds\quad;\quad
H({\hat{t}})\equiv(\alpha_{1}-\rho
C_{h})({\hat{t}}-{\hat{t}}_{0})-1/2\beta({\hat{t}}^{2}-{\hat{t}}_{0}^{2})$

In addition, by using again Eqs. (5), we obtain the O.D.E.s for the
correlation functions for $<(\delta I({\hat{t}}))^{2}>$, $<(\delta
I_{h}({\hat{t}}))^{2}>$, and $<\delta I({\hat{t}})\delta I_{h}({\hat{t}})>$

$\displaystyle\frac{d}{d{\hat{t}}}<(\delta
I({\hat{t}}))^{2}>=2\left(\alpha_{1}-\rho
C_{h}-2\frac{\alpha_{1}}{K_{1}}I_{RS}-\beta\left(\frac{{\hat{t}}^{2}-1}{\hat{t}}\right)\right)<(\delta
I({\hat{t}}))^{2}>$ (59) $\displaystyle\qquad\qquad\qquad\qquad\ \ -2\rho
I_{RS}<\delta I({\hat{t}})\delta I_{h}({\hat{t}})>+\eta_{11}({\hat{t}})$
$\displaystyle\frac{d}{d{\hat{t}}}<(\delta
I_{h}({\hat{t}}))^{2}>=-2k_{1}I_{RS}<(\delta I_{h}({\hat{t}}))^{2}>+\eta_{22}$
$\displaystyle\frac{d}{d{\hat{t}}}\\!<\\!\delta I({\hat{t}})\delta
I_{h}({\hat{t}})\\!>=\\!\left(\\!\alpha_{1}\\!-\\!\rho
C_{h}\\!-\left(\\!2\frac{\alpha_{1}}{K_{1}}+k_{1}\\!\right)I_{RS}\\!-\\!\beta\left(\\!\frac{{\hat{t}}^{2}\\!-\\!1}{\hat{t}}\right)\\!\right)\\!\\!<\\!\delta
I({\hat{t}})\delta I_{h}({\hat{t}})\\!>$
$\displaystyle\qquad\qquad\qquad\qquad\ \ -\rho I_{RS}<(\delta
I_{h}({\hat{t}}))^{2}>+\eta_{12}({\hat{t}})$

where Eqs (31) have been taken into account. The solutions of system (59) read

$\displaystyle<\\!(\delta
I_{h}({\hat{t}}))^{2}\\!>\\!=\\!-\exp(-2k_{1}G({\hat{t}}))\\!\left(\\!<\\!(\delta
I_{h}({\hat{t}}_{0}))^{2}\\!>\\!+\eta_{22}\\!\int_{{\hat{t}}_{0}}^{\hat{t}}\\!\\!\exp(2k1G(s)\\!ds\right)$
(60) $\displaystyle<\\!\delta I({\hat{t}})\delta
I_{h}({\hat{t}})\\!>=\\!\left(\frac{{\hat{t}}}{{\hat{t}}_{0}}\right)^{\beta}\\!\\!\\!\exp(H({\hat{t}}))\\!\exp\left(-(2\alpha_{1}/K_{1}+k_{1})\right)\\!G({\hat{t}}))\\!\Bigg{(}\\!\\!\\!<\delta
I({\hat{t}}_{0})\delta I_{h}({\hat{t}}_{0})\\!>$
$\displaystyle-\int_{{\hat{t}}_{0}}^{\hat{t}}\\!\\!\left(\frac{{\hat{t}}_{0}}{s}\right)^{\beta}\\!\\!\\!\\!\exp(\\!-\\!H(s))\exp\left((2\alpha_{1}/K1\\!+\\!k_{1})G(s)\right)\\!\\!\left(\eta_{12}\\!+\\!\rho
I_{RS}(s)<\\!(\delta I_{h}(s))^{2}\\!>\right)\\!ds\Bigg{)}$
$\displaystyle<(\delta
I({\hat{t}}))^{2}>=\left(\frac{{\hat{t}}}{{\hat{t}}_{0}}\right)^{2\beta}\exp(2H({\hat{t}}))\exp\left(-4\frac{\alpha_{1}}{K_{1}}\right)G({\hat{t}}))\Bigg{(}<(\delta
I({\hat{t}}_{0}))^{2}>$
$\displaystyle-\int_{{\hat{t}}_{0}}^{\hat{t}}\\!\\!\left(\frac{{\hat{t}}_{0}}{s}\right)^{2\beta}\\!\\!\\!\\!\\!\\!\exp(\\!-\\!2H(s))\\!\exp\left((4\alpha_{1}/K1\\!)G(s)\right)\\!\left(\eta_{11}\\!\\!-2\\!\rho
I_{RS}(s)<\\!\delta I(s)\delta I_{h}(s)\\!>\right)\\!ds\Bigg{)}\ $

The value of $C_{h}$ (the total Italian hospitals’ capacity) may be obtained
by making reference to the data published in [22]. More specifically, in 2017,
when there were $518$ public hospitals and $482$ accredited private ones, in
Italy there were $151646$ beds for ordinary hospitalisation in public
hospitals ($2.5$ per $1000$ inhabitants) and $40458$ in private ones ($0.7$
per $1000$ inhabitants), for a total of over $192$ thousand beds ($3.2$ per
$1000$ inhabitants).

The number of public and private beds destined for intensive care was 5.090 (a
number very close to the $5100$ cited by the newspapers these days), about
$8.42$ per $100000$ inhabitants [22].

Fig. 18. shows the correlation function $<\delta I_{h}(t)\delta I_{h}(t)>$,
solution of the second O.D.E. of system (59). As se can see, this is a typical
correlation function that random fluctuations at equilibrium look like. This
is not surprising as our reference state correspond to the maximum capacity of
the hospitals (i.e. $I_{hRS}=C_{h}$) so, fluctuations at equilibrium are the
only possible ones. The three correlation functions $<\delta I(t)\delta
I(t)>$, $<\delta I_{h}(t)\delta I_{h}(t)>$, and $<\delta I(t)\delta
I_{h}(t)>$, solution of the system (59), are reported in Fig. 19.

Figure 18:  Correlation function $<(\delta I_{h}(t))^{2}>$ for Italy, first
wave of infection by SARS-CoV-2. This correlation function is the solution of
second equation of the system (59). Having chosen our reference state
$I_{hRS}=C_{h}$, we get a typical correlation function for random fluctuations
at equilibrium. Figure 19:  Correlation functions $<\delta I(t)\delta I(t)>$,
$<\delta I_{h}(t)\delta I_{h}(t)>$, and $<\delta I(t)\delta I_{h}(t)>$. These
three correlation functions are obtained by solving numerically system (59).

##  6 Comparison between the Theoretical Predictions and Real Data for USA and
France

To perform the comparison with real data, it is convenient to rewrite Eq. (26)
in dimensional form

$\frac{dI}{dt}={\tilde{\alpha}}I\left(1-\frac{I}{K}\right)-\beta\left(\frac{t^{2}-t_{L}^{2}}{t_{L}^{2}t}\right)I+\
\sqrt{\tilde{\eta}}\ {\tilde{\xi}}(t)$ (61)

where ${\tilde{\alpha}}\equiv{\alpha}/t_{L}$ and the white noise
$\tilde{\xi}(t)$ satisfies the relations

$\displaystyle<{\tilde{\xi}}(t)>=0$ (62)
$\displaystyle<{\tilde{\xi}}(t){\tilde{\xi}}(t^{\prime})>=\delta(t-t^{\prime})$

The order of magnitude of the intensity of the noise $\tilde{\eta}$ may be
derived by using Eq. (64) reported in Appendix 1:

${\tilde{\eta}}\sim K^{-1/2}$ (63)

Appendix 2 reports the tables of infectious people by SARS-CoV-2 for USA and
France according to refs [23], [24] and [25], [26], respectively. As the data
show, France is currently subject to the second wave of Coronavirus while the
USA data may induce to think that they are in full second (or third) wave.
However, looking at the behaviour of infectious curve we may also argue that,
in agreement with ref. [27], USA as a whole is not in a second (or third) wave
because the first wave never really stopped. The virus is simply spreading
into new populations or resurgent in places that let down their guard too
soon. So, we are interested in analysing both of these scenarios.

Fig. 23. and Fig. 23. show the comparison between the theoretical predictions
(blue line) and real data (black dots) for USA. Fig. 23. refers to the
hypothesis that US is still in the first wave of Coronavirus, whereas Fig. 23.
has been obtained by assuming that US population is currently subject to the
third wave of Coronavirus, respectively. Fig. 25. and Fig. 25. show the
comparison between the theoretical predictions (blue line) and real data
(black dots) for France. The values of the parameters for both cases, USA and
France, are reported in the figure captions.

Figure 20:  Number of Infectious individuals by SARS-CoV-2 in USA.

Figure 21:  Number of Infectious individuals by SARS-CoV-2 in France.

Figure 22:  Comparison between the theoretical predictions (blue line) and
real data (black dots) for USA by assuming that US is still in the first wave
of Coronavirus. The values of the parameters are; ${\tilde{\alpha}}=0.021\
{\rm day}^{-1}$, $\beta=0.095\ {\rm day}^{-2}$, $K=25000000$, and $t_{L}=54\
{\rm days}$, respectively.

Figure 23:  Comparison between the theoretical predictions (blue lines) and
real data (black dots) for USA by assuming that US is in the third wave of
Coronavirus. The three series of values of the parameters, corresponding to
the three waves, are, respectively: (${\tilde{\alpha}}=0.043\ {\rm day}^{-1}$,
$\beta=0.45\ {\rm day}^{-2}$) ; (${\tilde{\alpha}}=0.021\ {\rm day}^{-1}$,
$\beta=0.008\ {\rm day}^{-2}$), and (${\tilde{\alpha}}=0.01\ {\rm day}^{-1}$,
$\beta=0.02\ {\rm day}^{-2}$). $K=25000000$ and $t_{L}=54\ {\rm days}$.

Figure 24:  Comparison between the theoretical predictions (blue line) and
real data (black dots) for France during the first wave of Coronavirus. The
values of the parameters are; ${\tilde{\alpha}}=0.145\ {\rm day}^{-1}$,
$\beta=0.5\ {\rm day}^{-2}$, $K=110000$, and $t_{L}=53\ {\rm days}$,
respectively.

Figure 25:  Comparison between the theoretical predictions (blue line) and
real data (black dots) for France during the second wave of Coronavirus. The
values of the parameters are; ${\tilde{\alpha}}=0.035\ {\rm day}^{-1}$,
$\beta=0.5\ {\rm day}^{-2}$, $K=5000000$, and $t_{L}=278\ {\rm days}$,
respectively.

Now, let us deal with the stochastic processes. Fig. 27. and Fig. 27.
illustrate the comparison between the theoretical predictions (blue lines) and
real data (black dots) for France concerning the first and the second waves of
SARS-CoV-2, respectively. The intensity of the noise has been estimated by
using Eq. (63). As we can see, the predictions of our model are in a fairly
good agreement with real data.

Figure 26: Solutions (200 realisations) of the stochastic differential
equation for France - First wave. The values of the parameters are those
reported in Fig. 25. ${\tilde{\eta}}=0.002$, which has been estimated
according to Eq. (63). The black curve is the solution of the deterministic
equation.

Figure 27: Solutions (200 realisations) of the stochastic differential
equation for France - Second wave. The values of the parameters are those
reported in Fig. 25. ${\tilde{\eta}}=0.001$, which has been estimated
according to Eq. (63).

As for US, we have analised two possible hypotheses:

1) USA is still subject to the first wave of Coronavirus infection;

2) USA is in the second or third wave of SARS-CoV-2 infection.

Figs. 29. and 29. illustrate these two scenarios. More specifically, Fig. 29.
refers to scenario 1). In order to obtain the best possible agreement with
real data, we have increased, arbitrarily, the intensity of the noise up to a
value that is $45$ times higher than that estimated by Eq. (63) (we set
${\tilde{\eta}}=0.009$). However, we did not reach this objective is a
satisfactory way.

Fig. 29. refers to scenario 2). In this case the intensity of the noise
corresponds to the value estimated by Eq. (63) (we have
${\tilde{\eta}}=0.0002$). Here, we get a fairly good agreement with real data.
Finally, our results lead to two possible interpretations:

a) USA is experiencing a second (or even third) wave of Coronavirus infection;

b) USA is still subject to the first wave of infection. However, given the
heterogeneity and vastness of the territory, the variables of the system (i.e.
$S$, $I$, $R$, and $D$) must depend on time as well as on space. In this case,
the correct equations governing the evolution of the compartments must be of
Stochastic Partial Differential Equations (S.P.D.E.s)-type.

In our opinion, scenario b) is the correct interpretation.

Figure 28: Solutions (200 realisations) of the stochastic differential
equation for USA - First-Wave Model. The values of the parameters are those
reported in Fig. 23. The intensity of the noise, has been set arbitrarily at
the value ${\tilde{\eta}}=0.009$, i.e. $45$ times greater than the value
estimated by Eq. (63).

Figure 29: Solutions (200 realisations) of the stochastic differential
equation for USA - Three-Waves Model. The values of the parameters are those
reported in Fig. 23. The intensity of the noise is ${\tilde{\eta}}=0.0002$,
which has been obtained by using Eq. (63). The black curve is the solution of
the deterministic equation.

##  7 Conclusions

Waiting for the mass production of the vaccine, we studied two scenarios by
means of two mathematical models: the $(SIS)_{L}$-model and the
$(SISI_{h}r_{h}d_{h})_{L}$-model.

The first scenario, described by the $(SIS)_{L}$-model, refers to the
situation in which the role of Hospitals is not taken into account. The second
model (the $(SISI_{h}r_{h}d_{h})_{L}$-model) integrates the contribution
provided by the Hospitals and Health Care Institutes. The dynamics are
governed by stochastic differential equations, which are suitable for
describing realistic situations. We compute the relevant correlation functions
as well as the probability density functions.

In the ${SIS}_{L}$ model, we determined the minimum value of infectious
individuals that must be reached after the lockdown measures so that, once
these measures are removed, during the time interval $\Delta{\hat{t}}$, the
number of infectious still stay below a set threshold.

The $(SISI_{h}r_{h}d_{h})_{L}$-model shows the crucial role played by the
Hospitals. More specifically, we showed that the Health Care Institutions can
influence the outcome of the outbreak, limiting, and even dampening, the
spread of the Coronavirus. By way of example, we dealt with the cases of the
United States and France. The comparison between the theoretical predictions
with real data confirmed the validity of our models. In particular, for US, we
have examined two possible scenarios: USA is still subject to the first wave
of infection by Coronavirus and USA is in the second (or third) wave of SARS-
CoV-2 infection. We concluded that, for a so vast and heterogeneous Country,
spatial dependance of the variables can no longer be ignored and the correct
equations governing the evolution of the different compartments, $S$, $I$,
$R$, and $D$, should be of S.P.D.E.s-type.

##  8 Acknowledgments

GS is grateful to Prof. M. Malek Mansour, of the Université Libre de Bruxelles
(ULB), for his useful suggestions.

##  9 APPENDIX 1: A Rough Estimate of the Intensity of the Noise

The intensity of the noise for the entire process may be estimated by adopting
the concept of Grand Ensemble in statistical mechanics. More specifically, we
may establish the following correspondences: Quantity of SARS-CoV-2
corresponds to the Energy of the system, the Quantity of SARS-CoV-2 that can
be absorbed or released due to a change of the Infectious individuals models
the Chemical Potential ($\mu^{ch.}$), and the Surface of a Country corresponds
to the Volume of the system, respectively. Temperature of a system (T) is
mimicked by the surrounding environment, which is more or less favourable to
the spread of Coronavirus depending on whether particular prevention measures
are adopted (e.g. use of masks, maintaining a safe distance, tracking of the
infection, adoption of lockdown, quarantine and isolation measures, etc.).
Hence, in this scenario:

i) Temperature equilibrium (”$T_{Country}=T_{Env.}$”) stands for Environment
equilibrium;

ii) Identical chemical potentials ($\mu^{ch.}_{Country}=\mu^{ch.}_{Env.}$)
stands for equal variation of the quantity of Coronavirus with respect to a
change of the Infectious people;

iii) Of course the Surface of a Country (corresponding to the Volume of the
system) is kept constant;

iv) The total number of Infectious individuals (corresponding to the ”number
of particles”) and the quantity of the SARS-CoV-2 in a Country (corresponding
to the ”Energy of the system”) are not fixed.

Under the above conditions, we may consider that the system is governed by the
Grand Ensemble statistical mechanics. So, according to this statistics, the
relative fluctuations of the number of the infectious people behaves as
$K_{Country}^{-1/2}$ [28], with $K_{Country}$ denoting the capacity of the
Country’s population. It is easily checked that this implies that the
intensity of the noise in Eq. (26) is of the order of

$\eta\sim t_{L}K_{Country}^{-1/2}$ (64)

with $t_{L}$ denoting the time when the lockdown measures have been applied.
We may object that this reasoning is based on equilibrium conditions. However,
we would like to point out that the aim of this Section is only to provide a
(rough) estimate of the order of magnitude of the noise.

##  10 APPENDIX 2: Exact Solution of Richards’ differential Equation with the
Lockdown Term

The Richards curve is an extension of the logistic function, allowing for more
flexible $S$-shaped curves. This curve satisfies the O.D.E. [29]

$\frac{dI}{dt}=\alpha I\left(1-\left(\frac{I}{K(t)}\right)^{\nu}\right)$ (65)

with $\nu>0$ (with $\nu>0$) denoting a parameter affecting the point where the
asymptote maximum growth occurs. In presence of the therapy-induced death
rate, the Richards’ equation takes the form

$\frac{dI}{dt}=\alpha I\left(1-\left(\frac{I}{K(t)}\right)^{\nu}\right)-c(t)I$
(66)

The goal is to determine the analytic solution of Eq. (11)333We report the
procedure to find the exact solution of Eq. (11) since it is not easy to find
it in literature.. Let us first rewrite Eq. (11) as

$\frac{1}{I}\frac{dI}{dt}=\alpha\left(1-\left(\frac{I}{K(t)}\right)^{\nu}\right)-c(t)$
(67)

Successively, we introduce the new functions $n(t)$ and $C(t)$ defined as

$n(t)\equiv\ln(I(t))+C(t)\qquad{\rm where}\qquad
C(t)\equiv\int_{t_{0}}^{t}c(s)ds$ (68)

with $t_{0}$ denoting the initial time. We get

$\displaystyle\frac{dn}{dt}=\frac{1}{I}\frac{dI}{dt}+\frac{dC}{dt}=\frac{1}{I}\frac{dI}{dt}+c(t)\qquad{\rm
and}$ (69) $\displaystyle\frac{d}{dt}(n(t)-\alpha
t)=-\alpha\exp(\nu(n(t)-\alpha t))\frac{\exp(-\nu(C(t)-\alpha
t))}{K(t)^{\nu}}$

By setting

$m(t)\equiv n(t)-\alpha t$ (70)

we obtain

$\int_{m_{0}}^{m}\frac{dm}{\exp(\nu
m)}=-\alpha\int_{t_{0}}^{t}\frac{\exp(-\nu(C(s)-\alpha s)}{K(s)^{\nu}}ds$ (71)

which, after integration, reads

$\exp(m(t))=\left(c_{0}+\nu\alpha\int_{t_{0}}^{t}\frac{\exp(-\nu(C(s)-\alpha
s)}{K(s)^{\nu}}ds\right)^{-1/\nu}$ (72)

where $c_{0}$ is a constant independent of time. Finally, we get

$I(t)=\exp(\alpha
t-C(t))\left(c_{0}+\nu\alpha\int_{t_{0}}^{t}\frac{\exp(-\nu(C(s)-\alpha
s)}{K(s)^{\nu}}ds\right)^{-1/\nu}$ (73)

Solution (73) may be cast in the following, more convenient, form

$I=I_{0}\frac{\exp(\alpha(t-t_{0})-C(t))}{\\!\left(1+\nu\alpha
I_{0}^{\nu}\exp(-\alpha\nu t_{0})\int_{t_{0}}^{t}(\exp(-\nu(C(s)-\alpha
s))/K(s)^{\nu}ds\right)^{1/\nu}}$ (74)

with $I_{0}$ denoting the number of population $I$ at time $t_{0}$
($I_{0}\equiv I(t_{0})$).

##  11 APPENDIX 3: Tables of Infectious individuals by SARS-CoV-2 - USA and
France

Date | USA | France
---|---|---
22-Jan | 1 | 0
23-Jan | 1 | 0
24-Jan | 2 | 2
25-Jan | 2 | 3
26-Jan | 5 | 3
27-Jan | 5 | 3
28-Jan | 5 | 4
29-Jan | 6 | 5
30-Jan | 6 | 5
31-Jan | 8 | 5
1-Feb | 8 | 6
2-Feb | 8 | 6
3-Feb | 11 | 6
4-Feb | 11 | 6
5-Feb | 11 | 6
6-Feb | 12 | 6
7-Feb | 12 | 6
8-Feb | 12 | 11
9-Feb | 9 | 11
10-Feb | 9 | 11
11-Feb | 10 | 11
12-Feb | 10 | 9
13-Feb | 11 | 9
14-Feb | 11 | 9
15-Feb | 11 | 7
16-Feb | 11 | 7
17-Feb | 11 | 7
18-Feb | 11 | 7
19-Feb | 11 | 7
20-Feb | 11 | 7
21-Feb | 11 | 7
22-Feb | 11 | 7
23-Feb | 11 | 7
24-Feb | 11 | 7
25-Feb | 10 | 2
26-Feb | 10 | 5
27-Feb | 11 | 25
28-Feb | 10 | 44
29-Feb | 17 | 86
1-Mar | 24 | 116
2-Mar | 42 | 176
3-Mar | 60 | 188
4-Mar | 89 | 272
5-Mar | 165 | 362
6-Mar | 216 | 635
7-Mar | 379 | 936
8-Mar | 491 | 1105
9-Mar | 565 | 1188
10-Mar | 746 | 1749
11-Mar | 1106 | 2233
12-Mar | 1531 | 2233
13-Mar | 2156 | 3590
14-Mar | 2908 | 4393
15-Mar | 3130 | 4429
Table 1: Number of infectious individuals by SARS-CoV2 in USA and France. Date | USA | France
---|---|---
16-Mar | 4565 | 6522
17-Mar | 6363 | 7554
18-Mar | 8873 | 8963
19-Mar | 13278 | 10714
20-Mar | 19518 | 12295
21-Mar | 25385 | 13888
22-Mar | 34103 | 13361
23-Mar | 45146 | 17055
24-Mar | 55347 | 18270
25-Mar | 67083 | 20360
26-Mar | 84191 | 22898
27-Mar | 102165 | 25698
28-Mar | 120937 | 30064
29-Mar | 137630 | 30871
30-Mar | 156037 | 34176
31-Mar | 179881 | 39782
1-Apr | 203134 | 42282
2-Apr | 231492 | 41983
3-Apr | 261362 | 44547
4-Apr | 287888 | 46354
5-Apr | 311723 | 46970
6-Apr | 337987 | 48989
7-Apr | 364651 | 49297
8-Apr | 392895 | 50718
9-Apr | 423395 | 51725
10-Apr | 451289 | 53328
11-Apr | 475982 | 54349
12-Apr | 499386 | 79831
13-Apr | 512755 | 82407
14-Apr | 534593 | 85829
15-Apr | 557816 | 85461
16-Apr | 584439 | 95341
17-Apr | 611175 | 94981
18-Apr | 630837 | 92778
19-Apr | 649097 | 96713
20-Apr | 673345 | 97717
21-Apr | 693988 | 98189
22-Apr | 718702 | 93981
23-Apr | 746583 | 94357
24-Apr | 759298 | 94211
25-Apr | 787874 | 94433
26-Apr | 806768 | 94458
27-Apr | 824054 | 97154
28-Apr | 841962 | 98504
29-Apr | 862462 | 94225
30-Apr | 856306 | 93444
1-May | 878601 | 92496
2-May | 893311 | 93092
3-May | 911895 | 93140
4-May | 926931 | 92903
5-May | 946531 | 92308
6-May | 969157 | 94333
7-May | 989843 | 93737
8-May | 1010883 | 94077
9-May | 1020748 | 94321
10-May | 1035084 | 94384
Table 2: Number of infectious individuals by SARS-CoV2 in USA and France. Date | USA | France
---|---|---
11-May | 1036945 | 94066
12-May | 1060502 | 93457
13-May | 1066573 | 92321
14-May | 1088973 | 91847
15-May | 1108231 | 91536
16-May | 1113346 | 91536
17-May | 1127017 | 90255
18-May | 1136924 | 89966
19-May | 1150239 | 90230
20-May | 1166863 | 90093
21-May | 1187411 | 89757
22-May | 1157787 | 89735
23-May | 1166628 | 89694
24-May | 1180460 | 89604
25-May | 1185875 | 89290
26-May | 1198869 | 88317
27-May | 1209288 | 87737
28-May | 1222490 | 90390
29-May | 1239114 | 90285
30-May | 1252000 | 91592
31-May | 1242033 | 91731
1-Jun | 1244711 | 91954
2-Jun | 1259414 | 90709
3-Jun | 1263066 | 90337
4-Jun | 1277666 | 90539
5-Jun | 1295466 | 90576
6-Jun | 1307001 | 90822
7-Jun | 1318704 | 91115
8-Jun | 1323590 | 91051
9-Jun | 1334677 | 90924
10-Jun | 1346078 | 91120
11-Jun | 1361476 | 91201
12-Jun | 1378478 | 91474
13-Jun | 1393707 | 91740
14-Jun | 1407202 | 92087
15-Jun | 1411814 | 92024
16-Jun | 1427516 | 91966
17-Jun | 1444596 | 92064
18-Jun | 1464830 | 92283
19-Jun | 1487551 | 92850
20-Jun | 1508109 | 93280
21-Jun | 1529107 | 93497
22-Jun | 1541333 | 93607
23-Jun | 1569358 | 93808
24-Jun | 1594422 | 93622
25-Jun | 1626842 | 93377
26-Jun | 1664318 | 94641
27-Jun | 1696670 | 94641
28-Jun | 1730945 | 93921
29-Jun | 1750571 | 95582
30-Jun | 1780503 | 95818
1-Jul | 1821928 | 96443
2-Jul | 1824839 | 96835
3-Jul | 1867592 | 97141
4-Jul | 1808946 | 97141
5-Jul | 1845794 | 97141
Table 3: Number of infectious individuals by SARS-CoV2 in USA and France. Date | USA | France
---|---|---
6-Jul | 1872210 | 98230
7-Jul | 1919330 | 98356
8-Jul | 1961331 | 98356
9-Jul | 2007257 | 99079
10-Jul | 2060216 | 99495
11-Jul | 2107164 | 99495
12-Jul | 2155023 | 99495
13-Jul | 2187840 | 100886
14-Jul | 2237159 | 100886
15-Jul | 2277035 | 101500
16-Jul | 2338454 | 101800
17-Jul | 2392640 | 102417
18-Jul | 2438729 | 102417
19-Jul | 2490585 | 102417
20-Jul | 2522582 | 104175
21-Jul | 2563770 | 104578
22-Jul | 2605612 | 105346
23-Jul | 2650612 | 105882
24-Jul | 2694315 | 106659
25-Jul | 2740987 | 106661
26-Jul | 2777024 | 106661
27-Jul | 2803934 | 108928
28-Jul | 2838789 | 109410
29-Jul | 2875126 | 109408
30-Jul | 2916772 | 110596
31-Jul | 2959434 | 113165
1-Aug | 2992425 | 113166
2-Aug | 3031480 | 113166
3-Aug | 3030750 | 113166
4-Aug | 3071148 | 116515
5-Aug | 3075303 | 115981
6-Aug | 3112757 | 118410
7-Aug | 3144584 | 121912
8-Aug | 3178688 | 121913
9-Aug | 3210796 | 121939
10-Aug | 3245109 | 126051
11-Aug | 3246746 | 126056
12-Aug | 3263231 | 130104
13-Aug | 3293106 | 130092
14-Aug | 3334854 | 135252
15-Aug | 3358391 | 138562
16-Aug | 3384449 | 138562
17-Aug | 3388006 | 141885
18-Aug | 3399126 | 141886
19-Aug | 3418273 | 141886
20-Aug | 3439319 | 141886
21-Aug | 3468440 | 156471
22-Aug | 3490679 | 159943
23-Aug | 3512398 | 164816
24-Aug | 3525433 | 166529
25-Aug | 3531255 | 169596
Table 4: Number of infectious individuals by SARS-CoV2 in USA and France. Date | USA | France
---|---|---
26-Aug | 3544757 | 175014
27-Aug | 3571899 | 180633
28-Aug | 3600855 | 187851
29-Aug | 3622993 | 187851
30-Aug | 3644136 | 198707
31-Aug | 3647867 | 201556
1-Sep | 3670637 | 206266
2-Sep | 3681431 | 212950
3-Sep | 3688945 | 219847
4-Sep | 3721623 | 228610
5-Sep | 3745253 | 228610
6-Sep | 3762544 | 228611
7-Sep | 3768258 | 247958
8-Sep | 3769148 | 254072
9-Sep | 3773487 | 263245
10-Sep | 3792628 | 271956
11-Sep | 3824476 | 281099
12-Sep | 3847957 | 281464
13-Sep | 3865128 | 281545
14-Sep | 3875410 | 304480
15-Sep | 3892945 | 312082
16-Sep | 3899974 | 321520
17-Sep | 3929266 | 331398
18-Sep | 3960964 | 343464
19-Sep | 3981720 | 342771
20-Sep | 4006695 | 342771
21-Sep | 4032529 | 371216
22-Sep | 4039967 | 380763
23-Sep | 4054155 | 380511
24-Sep | 4059018 | 408785
25-Sep | 4089001 | 423837
26-Sep | 4109430 | 423842
27-Sep | 4130607 | 423861
28-Sep | 4134248 | 449109
29-Sep | 4157434 | 460025
30-Sep | 4170312 | 473413
1-Oct | 4194775 | 485798
2-Oct | 4235533 | 497007
3-Oct | 4259573 | 497057
4-Oct | 4280939 | 497057
5-Oct | 4295944 | 531099
6-Oct | 4321937 | 541910
7-Oct | 4324142 | 559079
8-Oct | 4360138 | 576623
9-Oct | 4397371 | 596601
10-Oct | 4426857 | 596601
11-Oct | 4460175 | 596601
12-Oct | 4468893 | 639359
13-Oct | 4503100 | 660072
14-Oct | 4530389 | 680944
15-Oct | 4572309 | 710808
Table 5: Number of infectious individuals by SARS-CoV2 in USA and France. Date | USA | France
---|---|---
16-Oct | 4619704 | 735003
17-Oct | 4652793 | 735003
18-Oct | 4688096 | 735003
19-Oct | 4716172 | 809342
20-Oct | 4753794 | 828832
21-Oct | 4787492 | 854579
22-Oct | 4832242 | 895092
23-Oct | 4889906 | 936487
24-Oct | 4940522 | 936512
25-Oct | 4985683 | 936512
26-Oct | 5014178 | 1058635
27-Oct | 5061982 | 1091260
28-Oct | 5109189 | 1124979
29-Oct | 5162232 | 1171071
30-Oct | 5235613 | 1218518
31-Oct | 5289578 | 1252788
1-Nov | 5346628 | 1298278
2-Nov | 5385178 | 1351392
3-Nov | 5478077 | 1387595
4-Nov | 5541476 | 1426229
5-Nov | 5628278 | 1480142
6-Nov | 5723823 | 1537990
7-Nov | 5810136 | 1536134
8-Nov | 5889532 | 1660602
9-Nov | 5962848 | 1680027
10-Nov | 6066108 | 1681031
11-Nov | 6172962 | 1733142
12-Nov | 6278908 | 1733471
13-Nov | 6411436 | 1785521
14-Nov | 6523581 | 1815277
15-Nov | 6629769 | 1841533
16-Nov | 6719435 | 1849775
17-Nov | 6829398 | 1892168
18-Nov | 6940785 | 1917979
19-Nov | 7066805 | 1936750
20-Nov | 7213710 | 1956173
21-Nov | 7318556 | 1972675
22-Nov | 7463554 | 1985114
23-Nov | 7524768 | 1988458
Table 6: Number of infectious individuals by SARS-CoV2 in USA and France.

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