Renormalisation group approach to pandemics: The COVID-19 case

Abstract

We investigate the spreading dynamics of infected cases for SARS-2013 and COVID-19 epidemics for different regions of the world, in terms of the renormalisation group language. The latter provides an alternative way to describe the underlying dynamics of disease spread. This allows us to introduce important quantities, for a given disease, such as the slope of the beta function at fixed points and the time scale of the epidemic spread inflection point. We discover that for COVID-19 the epidemic slope is of order one inverse week and the inflection point occurs roughly four weeks after the outbreak. We use these results to attempt long term estimates for the epidemic evolution in several regions of the world. The accuracy of the results vary depending on the epidemic stage for each region. We also provide a webpage where we daily update our analyses.

Full text

Renormalisation group approach to pandemics: The COVID-19 case
Michele Della Morte∗
IMADA & CP3-Origins. University of Southern Denmark. Campusvej 55, DK-5230 Odense, Denmark
Domenico Orlando†
INFN sezione di Torino — Arnold–Regge Center via Pietro Giuria 1, 10125 Turin, Italy
Albert Einstein Center for Fundamental Physics — Institute for Theoretical Physics,
University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
Francesco Sannino‡
CP3-Origins & the Danish Institute for Advanced Study. University of Southern Denmark. Campusvej 55,
DK-5230 Odense, Denmark
Dipartimento di Fisica “E. Pancini”, Universit`a di Napoli Federico II — INFN sezione di Napoli
Complesso Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italy.
We investigate the spreading dynamics of infected cases for SARS-2013 and COVID-19 epidemics
for different regions of the world, in terms of the renormalisation group language. The latter
provides an alternative way to describe the underlying dynamics of disease spread. This allows us
to introduce important quantities, for a given disease, such as the slope of the beta function at fixed
points and the time scale of the epidemic spread inflection point. We discover that for COVID-19
the epidemic slope is of order one inverse week and the inflection point occurs roughly four weeks
after the outbreak. We use these results to attempt long term estimates for the epidemic evolution
in several regions of the world. The accuracy of the results vary depending on the epidemic stage
for each region. We also provide a webpage where we daily update our analyses.
INTRODUCING THE FRAMEWORK
Prompted by the COVID-19 pandemic outbreak we
investigate its spreading dynamics using the language of
the renormalisation group approach that is extremely ef-
fective in statistical and high energy physics [1, 2]. Our
approach is complementary to other traditional methods
nicely summarised in [3, 4] for complex network methods
and in [5–7] when taking into account also spatial effects.
As for the widely adopted choice to represent the data
by fit them to a logistic function we refer to [8–13].
We find convenient to discuss rather than the number
of cases its logarithm which, being a slowly varying func-
tion, is more suited for modelling. We define through it
an epidemic strength function whose derivative with re-
spect to time provides a new quantity that we interpret
as the beta-function of an underlying microscopic model.
In statistical and high energy physics the latter governs
the time (inverse energy) dependence of the interaction
strength among fundamental particles. Here it regulates
social interactions.
To establish and test the framework we use the epi-
demic data from China for COVID-19 and from Hong-
Kong (HK) for SARS-2003 since they represent statis-
tically significative and precise ensembles with, to date,
over 80k infected cases and 3k deaths reported for China
COVID-19 and around 2k infected cases for (HK) SARS-
2003 data. We start empirically by collecting data1
for the cumulative confirmed infected cases2from the
World Health Organization (WHO) -China COVID-19
and for HK-SARS 2003 from World Health Organization
(WHO)-HK SARS-2003. For the remaining countries we
also cross-check the data with Worldometers.
As simple characterisation of the logarithmic value of
the number of infected data points we use the following
function:
αES[t] = aexp[γ t]
b+ exp[γ t].(1)
By construction the function grows quickly at small t
and then it approaches rapidly the large tplateau of the
data. Here tis time measured in weeks. The parameter a
determines the height of the plateau, ln(b) the offsetting
time of the spread of the disease whereas γcontrols the
slope and it is measured in inverse time units. Since we
will be using weeks as time-unit, this means that γhas
to be understood as given in inverse weeks. Naturally a
1Although we are aware of possible inaccuracies and dishomo-
geneities in the data provided by each country we believe that
the overall results of our analyses are robust when it comes to
discuss the dynamics and temporal evolution of the epidemic.
One check to ensure stability of our results has been to thin the
data by either considering them weekly or daily, obtaining in the
end results consistent within 90% confidence level.
2There is only one exception and it deals with the Danish estimate
for infected cases as detailed in the corresponding subsection.

2
encodes information about the number of next-neighbour
infectious transmissions. This is the parameter that is di-
rectly affected, in a logarithmic manner, by containment
measures and the size of the population. The analytic
function in Eq. (1) can be further extended at the cost of
introducing new parameters. For example acan be itself
made a function of time to model changes in containment
measures.
Additionally, the number of infected cases, which is the
exponential of Eq. (1), has an inflection point at the time
tInfl =γ−1ln b
2a+p4 + a2,(2)
which corresponds to the point where the second deriva-
tive of the number of the infected cases vanishes. Phys-
ically it is associated to the time when the number of
reported new cases starts decreasing. This means that
the first derivative of the fitted function at the inflection
point displays a maximum (second derivative vanishes).
It is convenient to measure the inflection point relative
to an initial arbitrary value of the time t0corresponding,
for example, to the point when the number of infected
cases is 10. We arrive at:
∆tInfl := tInfl −t0=γ−1ln a−ln [10]
2 ln [10] a+p4 + a2.
(3)
which has the advantage of being bindependent
and that for sufficiently large ais approximately
∆tInfl ≈γ−1ln a2−aln [10]
ln [10] . Since the dependence on
ais only logarithmic we will see that this time scale is
fairly constant across different regions of the world with
respect to the COVID-19 epidemic. This, in turn, al-
lows for a certain degree of predictive power to the model
parameterisation when the data approach the inflection
point.
To better elucidate how our parameterisation encodes
the epidemic dynamics occurring in each region of the
globe we better clarify the role played by the parameter γ
in what is commonly known as flattening the curve. The
latter has come a rallying cry in the COVID-19 battle.
From Fig. 1 we learn that the smaller γthe longer it takes
to reach the peak of the number of new cases (right panel)
which also decreases with decreasing γ. As a consequence
the epidemic will last longer. This is the price to pay for
trying to reduce the number of new cases per day required
to minimise the impact on the health system.
Hong Kong SARS-2003
It is instructive to analyse first the Hong Kong (HK)
SARS-2003 data to test the robustness of our approach
before applying it to the evolving COVID-19 epidemic.
The number of infected cases as function of time as well
as the epidemic strength (the logarithm of the number of
infected cases) is depicted in Fig. 2 along with the best
fit that yields:
aHK= 7.47 , bHK= 1.11 , γHK= 0.60 .(4)
This is the final picture for the HK SARS-2003 epi-
demic, however it is interesting to learn about an evolv-
ing epidemic. This can be simulated by replaying the HK
SARS-2003 data as function of time and for each given
time obtain the time-dependent parameters a,band γ.
The results are reported in Fig. 3. We obtain excellent
fits for each specific time. Nevertheless, as it is clear
from the figures, unless the inflection point has occurred
the time-dependent fitted parameters cannot be used to
predict the entire evolution of the epidemic. The dashed-
blue line in the plot corresponds to the time when the
inflection occurs.
Another observation is that while band γare corre-
lated they are anti-correlated with respect to a. Ad-
ditionally, γand b(a) decrease (increase) till near the
inflection point where they overshoot (undershoot) the
asymptotic value which is however quickly reached after
the inflection point. We also discover that the inflection
point occurs around 3.5 weeks after the first report.
Remarkably our analysis is able to reproduce the data
with excellent accuracy.
China COVID-19
We now move to the China COVID-19 data to corrob-
orate the findings above.
The outcome is reported in Fig. 4 for the number of
infected cases as function of time as well as the epidemic
strength along with the best fit that yields
aChina = 11.35 , bChina = 2.55 , γChina = 0.97 .(5)
We report in Fig. 5 the time-evolution of the fit parame-
ters a,band γ. Although the fit does not have the same
quality as in the HK SARS-2003 case, the main features
remain unchanged. Namely, the results stabilises only
when the inflection point has occurred and the param-
eters are still correlated as observed for the HK SARS-
2003 case. The inflection point occurs after roughly 3.5
weeks from the outbreak. Another interesting observa-
tion is that the parameter aseems to stabilise earlier than
the other parameters. This is a welcome news given that
it gives us the log of the asymptotic number of infected
cases. It will be interesting to test whether these trends
persist for the epidemic spread in other regions of the
world including whether the peak in the number of new
cases also occurs between three and four weeks from the
outbreak.

3
0 2 4 6 8 10 12
0
5000
10000
15000
20000
week number
N. Infected cases example
0 2 4 6 8 10 12
0
2000
4000
6000
week number
New cases example
FIG. 1. How to flatten the curve. Left panel: Hypothetical number of infected cases for two values of γ, i.e. γ= 1 (blue) and
0.5 (orange) as function of time keeping fixed the total number of infected cases. Right panel: The corresponding number of
new cases.
0 5 10 15
0
500
1000
1500
week number
N. infected Hong Kong, SARS
0 5 10 15
4.5
5.0
5.5
6.0
6.5
7.0
7.5
week number
Log N. infected Hong Kong, SARS
FIG. 2. Left panel: SARS-2003 infected cases in Hong Kong as function of the week number starting from March 17 2003. In
red, we report the exponential of the epidemic strength model result. Right panel: Epidemic strength (the logarithm of the
number of infected cases) and the best fit result as in Eq. (4).
0 5 10 15
5.5
6.0
6.5
7.0
7.5
8.0
8.5
t in weeks
aHK(t)
0 5 10 15
0.5
1.0
1.5
2.0
t in weeks
bHK(t)
0 5 10 15
0.5
1.0
1.5
2.0
t in weeks
γHK(t)
FIG. 3. Fit parameters to HK SARS-2003 data as function of time. Left panel: a(t) with the red band corresponding to a
change of 0.1 in the asymptotic value of a. This translates into a 10% change in the asymptotic number of infected cases.
Center and Right panel: b(t) and γ(t) with the red band corresponding to a 10% change in their asymptotic value. The vertical
dashed blue line marks the occurrence of the inflection point in the infected cases.
RENORMALISATION GROUP DICTIONARY
Behaviour such as the one in Fig. 4 are common in
physics from Fermi distributions to out-of-equilibrium
thermodynamics to energy dependence of the interaction
strengths in physical systems. We focus on the latter
similarity with the goal to obtain an anatomic descrip-
tion of the epidemic strength as function of time and
possibly identify universal quantities underlying disease
spread mechanisms.
We now introduce the following dictionary: The time
is naturally identified with t=−ln µ/µ0with µan en-

4
02468
0
20 000
40 000
60 000
80 000
week number
N. infected China, COVID-19
02468
0
2
4
6
8
10
week number
Log N. infected China, COVID-19
FIG. 4. Left panel: COVID-19 infected cases in China as function of the week number starting from January 21st 2019. In
red, we report the exponential of the epidemic strength model result. Right panel: Epidemic strength (the logarithm of the
number of infected cases) and the best fit result as in Eq. (5).
02468
10
12
14
16
t in weeks
aChina(t)
02468
2.0
2.5
3.0
3.5
4.0
t in weeks
bChina(t)
02468
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
t in weeks
γChina(t)
FIG. 5. Fit parameters to China COVID-19 data as function of time. Left panel: a(t) with the red band corresponding to
a change of 0.1 in the asymptotic value of a. This translates into a 10% change in the asymptotic number of infected cases.
Center and Right panel: b(t) and γ(t) with the red band corresponding to a 10% change in their asymptotic value. The vertical
dashed blue line marks the occurrence of the inflection point in the infected cases.
ergy scale and µ0a reference energy scale; The epidemic
coupling strength is identified with αES in Eq. (1) and
therefore we can now introduce the βEfunction of the
epidemic:
βE=dαES
dln (µ/µ0)=−dαES
dt .(6)
αES captures the essential information of the China data
and it is shown in Fig. 6. We now differentiate αES w.r.t.
-6-4-20246
0
2
4
6
8
10
t
αES
FIG. 6. αES[t] as function of time.
time and obtain βEwhich we plot it in Fig. 7 as a func-
tion of αES. The epidemic beta function has two zeros,
2 4 6 8 10 αES
-2.5
-2.0
-1.5
-1.0
-0.5
βE
FIG. 7. βEas function of αES .
one at αES = 0 corresponding to no disease and one for a
finite value α∗
ES corresponding to the log of the plateau of
the total infected cases. This is mapped into an underly-
ing epidemic dynamics for which the system features two
fixed points one of them being non-interacting and the
other still interacting but time dilation invariant. This
observation has important consequences in characteris-
ing the universal properties of the underlying dynamics
governing the epidemics as the critical theory of phase
transitions teaches us [1, 2]. For example, the slope of
the beta function evaluated in the infrared (i.e. at low
energy/large times) is
∂βE
∂αES α∗
ES
≡θE=−γ , (7)
with α∗
ES = 11.35 (see Eq. 5) the interacting fixed point

5
which is approached at infinite time. The negative sign
of the scaling epidemic exponent θEtells us that the fixed
point is attractive, meaning that any small external per-
turbation will not destabilise the system. This is different
from the fixed point at zero for which the scaling expo-
nent θE0=γis positive meaning that any external per-
turbation drives the system away from the non-disease
limit. Intuitively this means that one has to work hard
to stop the spreading of the disease.
Additionally, the analytic time form of the epidemic
strength encodes its history allowing us, for example, to
estimate (taking the time for which αES= ln 10) when
it actually started. From Fig. 5 this can be estimated to
be around end of November to beginning of December in
agreement with the general expectations [14].
The above goes under the name of the Renormalisa-
tion Group (RG) analysis in condensed matter and high
energy physics.
One can actually construct an ab initio RG analysis
for the spread of the pandemic to show that the model in
Eq. (1) reproduces the general qualitative behavior that
we expect for the pandemic. The simplest model for the
epidemic spread is
dP
dt =βP(t) = r0P(t) (8)
where P(t) is the number of infections at time tand r0
is a characteristic constant. This predicts an exponential
growth modelling the first phase of the spread. The flow
has a fixed point at P= 0, with critical exponent r0: in
RG parlance this is the ultraviolet (UV) repulsive fixed
point.
Clearly this description is too naive. We can improve if
we think of the RHS as of the first term in a polynomial
expansion in P. Adding the next term we have
βP(t) = r0P(t)1−P(t)
K(9)
where Kis a new constant. There are now two fixed
points. Together with the initial P= 0, we have a
new one at P=K. The latter is an infrared (IR) at-
tractive fixed point. It models the later stages of the
epidemy, when the number of infections reaches its final
value P=K. The flow equation can be solved explicitly
and reproduces a logistic growth:
P(t) = K
1 + e−r0(t−T0),(10)
where T0is a constant indicating the inflection point in
time. Expanding around the two fixed points, we find
that the corresponding critical exponents are the same,
up the sign marking the fact that one is repulsive P= 0
and the other attractive P=K:
(β(P)∼r0Pfor P∼0,
β(P)∼ −r0(P−K) for P∼K.(11)
The strength αES satisfies Eq. 9 upon identifying a
with K,γwith r0and exp (r0T0) with b. We have already
noticed that our parameterisation works fairly well and
it will continue to do so for the other regions of the globe.
Nevertheless, it is possible to generalise the approach
to allow for the exponents in the UV and the IR to differ.
The UV describes the beginning of the epidemic, when
no measures are taken, while the IR describes the end of
it, which strongly depends on the social behaviour of the
population. To take this into account we add another
term in the RG equation:
βP(t) = r0P(t)1−P(t)
K1−r0−rf
r0
P(t)
K.(12)
without loss of generality we will assume rf≤r0. This
model has three fixed points. The UV repulsive point at
P= 0, the IR attractive at P=Kand a new unphysical
repulsive one at P=r0K/(r0−rf) = K2> K. A nice
feature of this model is that now the critical exponents
of the UV and IR are independent:
(β(P)∼r0Pfor P∼0,
β(P)∼ −rf(P−K) for P∼K.(13)
(the spurious point is repulsive with β(P)∼r0rf/(r0−
rf)(P−K2)). This allows for a more flexible description
of the data at the price of adding a new parameter for
the fit. Adding extra terms to the βfunction will not
change this description qualitatively.
In fact, this RG-flow picture is useful in the sense that
it clearly separates UV quantities, related to the begin-
ning of the epidemic curve, such as r0, from IR quanti-
ties, related to its end, such as rfand the total number of
infected K. We expect UV quantities to be universal, be-
cause they typically depend on the virus properties. On
the other hand IR quantities, such as the total number
of infected cases as well as the slope will depend on con-
tainment measures. The crossing point between the two
behaviours is the inflection point tInfl, where the deriva-
tive of the β-function changes sign. We are in the UV
region if dβ/dP > 0 and in the IR region if dβ/dP < 0.
It follows that before the inflection point, where the UV
fixed point dominates, it is difficult to estimate IR quan-
tities such as the total number of infected K.
As mentioned above, we will consider the simpler case
with a single scaling exponent.
We have therefore provided a useful map between mod-
els of infectious diseases that are typically introduced
directly for modelling P(t) and the epidemic strength
αES (t).
COVID-19 IN OTHER REGIONS OF THE
WORLD
We now employ the formalism above to analyse the epi-
demic spread in other regions of the world, namely South

6
Korea, Italy, Denmark, the United States and the United
Kingdom. We chose these countries because they are at
different stages of the epidemic evolution. However, at
the end of the paper, we provide a link to a webpage in
which other countries will be analysed and all the data
daily updated.
South Korea
We report the outcome for the fit adjourned to March
12th to the South Korea data in Fig. 8 for the number of
infected cases as function of time as well as the epidemic
strength along with the best fit that yields
aKor = 9.18 , bKor = 727 , γKor = 1.29 .(14)
We report in Fig. 9 the time-evolution of the fit param-
eters a,band γ. The time evolution of aand γindicate
that an almost stable value has been reached whereas the
situation for bis still uncertain. However, the sensitivity
of the data to bis very flat and therefore we can trust
the observed stabilisation of the other parameters and
especially of a. Additionally, the latest-time value of a
translates in the following number of asymptotic infected
cases of about 9.7k±1k.
Taking the starting point of the epidemic around the
order of 10 reported cases (here occurring at week 2.5)
we find that the inflection point is about 4 weeks after
that in reasonable agreement with the China COVID-19
and HK SARS-2003 observations. Additionally, we can
expect a stabilisation in week 10-11 which corresponds
to end of March beginning of April.
The data indicates that South Korea has implemented
efficient containment measures when, as we shall see, we
compare to other countries such as Italy, Denmark, US
and UK. We find convenient to introduce, for each coun-
try, denoted by Xthe parameter
aKor
X=aKor + ln popX
popKor ,(15)
where ”pop” stands for population. We believe that when
fixing the parameter ato this value aXit will allow us
to measure the overall effects of the containment mea-
sures compared to the South Korean case. The following
quantity
EX= exp aX−aKor
X,(16)
measures the relative (in)efficiency of the containment
policies enforced by a given county w.r.t. another coun-
try, such as South Korea. The measures adopted by a
country are more efficient than the ones taken by South
Korea if EXis less than unity.
Italy
We report the outcome for the fit for the Italian data
in Fig. 10 for the number of infected cases as function
of time. To see if we have reached the inflection point
we provide in Fig. 11 the evolution of a,band γ. It
seems that if the current trends continues we have just
approached the inflection point.
The curve and the band (90% confidence level) in
Fig. 10 have been obtained by performing a fit with re-
spect to a,band γto the red data points. For the over-
all fit we have aIta = 12.29 ±0.12 , bIta = 214 ±80
and γIta = 0.58 ±0.04 with the errors estimated at 90%
confidence level.
Currently, we therefore expect a minimal and maximal
number of infected cases to be respectively 187k and 255k
corresponding a 90% confidence level.
We also find the efficiency of the containment measures
taken by Italy compared to Korea to be
EIta = 16 −22 .(17)
Denmark
We report the outcome for Denmark in Fig. 12 for
the number of infected cases as function of time. The
curve and the band, corresponding to 90% confidence
level, have been obtained by performing a fit with respect
to a,band γto the red data points. The best values are:
a= 9.30 ±0.23, b= (1.8±2.8) ×104and γ= 0.92 ±
0.14. This corresponds to a variation in the number of
asymptotic infected cases spanning from 8.7k to 14k. The
inflection point is expected in between weeks 13 and 14
corresponding to end of March while stabilisation around
end of April.
With these very preliminary data we find the efficiency
of the containment measures taken by Denmark to be
EDenmark = 8 −13 ,(18)
at the 90% confidence level.
United States
The data and the fits for the United States are reported
in Fig. 13. The United States are still at the beginning of
the exponential growth. The curve and the band, corre-
sponding to 90% confidence level, have been obtained by
performing a fit with respect to a,band γto the red data
points. The best values from the fit are: a= 13.3±0.5,
b= (1.3±1.6)×104and γ= 0.84±0.11. This corresponds
to a variation in the number of asymptotic infected cases
spanning from 359k to 927k at the 90% confidence level.
We are still far away from the inflection point.

7
0246810
0
2000
4000
6000
8000
10 000
week number
N. infected S. Korea, COVID-19
02468
0
2
4
6
8
week number
Log N. infected S. Korea, COVID-19
FIG. 8. Left panel: COVID-19 infected cases in South Korea as function of the week number starting from January 21st 2019.
In red, we report the exponential of the epidemic strength model result. Right panel: Epidemic strength (the logarithm of the
number of infected cases) and the best fit result.
45678
3
4
5
6
7
8
9
10
t in weeks
aKor(t)
45678
100
200
300
400
500
600
700
t in weeks
bKor(t)
45678
0.5
1.0
1.5
2.0
t in weeks
γKor(t)
FIG. 9. Fit parameters to South Korea COVID-19 data as function of time. Left panel: a(t) with the red band corresponding
to a change of 0.1 in the last value of aas function of time. This translates into a 10% change in the number of infected cases.
Center and Right panel: b(t) and γ(t) with the red band for γcorresponding to a 10% change w.r.t. its last value. The vertical
dashed blue line marks the occurrence of the inflection point in the infected cases according to the overall fit.
10 11 12 13 14 15
0
50 000
100 000
150 000
calendar week
Italy N. infected COVID-19
Italy
FIG. 10. COVID-19 infected cases in Italy as function of
the calendar week number. The curve and the band (90%
confidence level) have been obtained by performing a fit w.r.t.
a,band γto the red data points.
With these very preliminary data we find the efficiency
of the containment measures taken by the United States
to be
EUS = 6 −15 ,(19)
at the 90% confidence level.
United Kingdom
Following the previous analyses the fit for the United
Kingdom is given in Fig. 14. We are still at the be-
ginning of the exponential growth. The curve and the
band, corresponding to 90% confidence level, have been
obtained by performing a fit yielding a= 11.2±0.5,
b= (1.1±1.1) ×103and γ= 0.64 ±0.09 to the red data
points. This corresponds to a variation in the number of
asymptotic infected cases ranging from 43.5k to 124k at
the 90% confidence level. We are still far away from the
inflection point.
With these very preliminary data we find the efficiency
of the containment measures taken by the United King-
dom to be
EUK = 3.5−9.7,(20)
at the 90% confidence level.

8
10.0 10.5 11.0 11.5 12.0 12.5 13.0
7
8
9
10
11
12
13
calendar week
a(t)
11.5 12.0 12.5 13.0
50
100
200
500
calendar week
b(t)
10.0 10.5 11.0 11.5 12.0 12.5 13.0
0.0
0.5
1.0
1.5
2.0
calendar week
γ(t)
FIG. 11. Fit parameters to Italy COVID-19 data as function of time. Left panel: a(t). Center panel: b(t). Right panel: γ(t).
The errors correspond to 90% confidence level.
10 11 12 13 14 15 16
0
2000
4000
6000
8000
10 000
12 000
calendar week
Denmark N. infected COVID-19
Denmark
FIG. 12. COVID-19 infected cases in Denmark as function
of the calendar week number. The curve and the band (90%
confidence level) have been obtained by performing a fit w.r.t.
a,band γto the red data points. Nota bene that after the
13th of March we multiplied the number of reported new cases
by five to take into account the policy change in the way
potential cases were tested.
A preliminary global analysis
We now attempt to provide a very preliminary analysis
for the world pandemic which must be taken cum grano
salis. One of the advantages of performing such an anal-
ysis is that, given the large numbers involved, the results
are less sensitive to individual countries ways of repre-
senting the data. As it is clear from Fig. 15 the world
is, overall, still at the beginning of the pandemic growth
and far away from the inflection point. For this reason
the predictions are only reasonable on a short time scale.
The curve and the band, correspond to 90% confidence
level. We stress that this band is for the current fit and
it can very will be that when new data arrive the band
moves up in the number of tested infected cases.
10 11 12 13 14 15 16
0
100 000
200 000
300 000
400 000
500 000
600 000
calendar week
N. infected COVID-19 USA
USA
FIG. 13. COVID-19 infected cases in the United States as
function of the calendar week number. The curve and the
band (90% confidence level) have been obtained by perform-
ing a fit w.r.t. a,band γto the red data points.
10 11 12 13 14 15 16
0
20 000
40 000
60 000
calendar week
N. infected COVID-19 UK
UK
FIG. 14. COVID-19 infected cases in the United Kingdom
as function of the calendar week number. The curve and the
band (90% confidence level) have been obtained by perform-
ing a fit w.r.t. a,band γto the red data points.

9
FIG. 15. COVID-19 infected cases for the World as function
of the calendar week number. The curve and the band (90%
confidence level) have been obtained by performing a fit w.r.t.
a,band γto the red data points.
CONCLUSIONS AND ONLINE UPDATES
Summarising, we interpreted the epidemic data in
terms of a renormalisation group approach which pro-
vides an alternative way to investigate the underlying
dynamics of disease spread. We noticed, for example,
that universal quantities, for a given disease, can be de-
fined such as the slope of the beta function at fixed points
and the time scale of the epidemic spread inflection point.
The slope characterises the speed with which the asymp-
totic number of infected cases is approached while the
inflection point marks the deceleration in the number
of new infected cases. Our results are corroborated by
the experimental findings that show, indeed, that in the
cases considered these quantities are of order one for the
slope γfor COVID-19 and the inflection point typically
occurs between three and four weeks after the outbreak.
Encouraged by our findings we attempted long term esti-
mates for Italy, Denmark, the United States and United
Kingdom. These countries are at different stages of the
epidemic with Italy and to some extent Denmark being
at or close to the inflection point and the United States
and United Kingdom at the initial exponential growth
phase. Consequently, the estimates for the asymptotic
values of the corresponding number of infected cases are
more accurate for Italy and for Denmark.
Additionally, we have shown that the parameter a
which determines the final number of infected cases sta-
bilises earlier than the other two parameters (see Figs. 3
5 9, 11). We used this parameter to compare the effects
of the containment measures on the overall number of
infected cases among different countries by defining an
efficiency factor. The latter can also be employed to de-
vise better control strategies [15]. For example, it is clear
that the measures taken by South Korea have been highly
impactful at the beginning of the outbreak. Other coun-
tries, such as Italy that didn’t employ the South Korean
model impose social distancing that affect the γand apa-
rameters. Additionally we have also seen that reducing γ
flattens the curve of new infected cases. Therefore control
strategies are further naturally monitored by the value of
this parameter. Being able to predict with a certain de-
gree of confidence the inflection point and, once this is
reached, when we expect the overall number of infected
cases to be reached is key to devise control strategies such
as either reducing or increasing social distancing.
Our work should be envisioned as a first step to-
wards establishing a connection between epidemology
and quantum field theory.
Online updates
To keep up with the evolving situation you will find
the updated analyses for several countries including
the examples reported here on the following webpage:
https://www.cp3-origins.dk/COVID-19.
∗dellamor@cp3.sdu.dk
†domenico.orlando@to.infn.it
‡sannino@cp3.sdu.dk
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