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An epidemiological model for the evolution of COVID-19 in Germany and in
Brazil
Adriano A. Batista 1∗and Severino Horácio da Silva2†
1Unidade Acadêmica de Física, Universidade Federal de
Campina Grande, 58051-900 Campina Grande PB, Brazil.
2Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande,
58051-900 Campina Grande PB, Brazil.
(Dated: June 19, 2020)
In this work, we adapt the well-known and tested epidemiological SIR model developed
by Kermack and McKendrick (1927) to model the dissemination dynamics of COVID-19
in three levels: national (Germany and Brazil), state (Paraíba) and local (City of Campina
Grande). We validate our epidemiological model by comparing the official data of confirmed,
recovered, and deaths due to the pandemic. We obtained very good fits by allowing a time
variation in the rate of contagion, what reflects a lower or higher adherence to social distancing
by the involved populations. We also make epidemiological predictions for the next twenty
days. We hope this model can be adapted and applied mostly at the local level. It could help
pinpoint hot spots in the contagion of COVID-19 and better identify which strategies work
best in containing the spread of the disease.
∗adriano@df.ufcg.edu.br
†horacio@mat.ufcg.edu.br
2
I. INTRODUCTION
The COVID-19 pandemic already has reached practically the whole planet. According to
the World Health Organization (WHO, Situation Report-41 [1]), although around 80% of the
infected people present mild symptoms (equivalent to the common flu), older people and those
with a history of other diseases like diabetes, cardiovascular, and chronic respiratory syndrome can
develop serious problems after being infected by the SARS-CoV-2 virus. It was first identified in
December 2019 in the City of Wuhan, Province of Hubei, China. From there it spread across Asia,
Europe, and the other continents [2]. On March 11th, the WHO declared it to be a pandemic [3].
In this work, we investigate the spread of the pandemic in 4 different scenarios: Germany,
Brazil, the Brazilian state of Paraíba, and Paraíba’s second largest city: Campina Grande. We use
the data from Germany mostly as a benchmark test for our model, since very likely their COVID-19
data is one of the most accurate in the world, which is due to the widespread testing of its population.
Furthermore, the social distancing, isolation, and use of personal protection equipments (PPEs)
there has been far more efficient than the measures taken in Brazil in containing the spread of
COVID-19 infections. Also, the number of confirmed cases reached, so far almost 187,000 (ninth
largest in the world on 06/12/2020).
The first confirmed case in Brazil occurred in February 25th of 2020, when a 61-year-old
man who had returned from Italy tested positive. In Brazil and in the majority of countries around
the world social distancing policies have been adopted in order to decrease the rate of contagion
and thus allow that health systems do not collapse and have conditions to treat the gravest cases of
the disease.
In the State of Paraíba, according to the Paraíba Department of Health, the first confirmed
case of COVID-19 was registered on March 18th of 2020. It was a man who lived in the City of
João Pessoa that had returned from a trip to Europe on February 29th. On March 31st the first
death due to the pandemic was recorded in the State.
Campina Grande is the second largest city in the State of Paraíba. The social distancing
policy was implanted in this city on March 20th of 2020, with the closure of universities, schools,
and non-essential stores. The social distancing was implemented in a preventive form since the
first case of infection by the SARS-CoV-2 virus only came to be registered one week afterwards,
on March 27th. The first death due to COVID-19 in this city was only registered on April 19th.
In this work, we adapted the well-known epidemiological model SIR, developed by Kermack
and Mckendrick in 1927 [4] to study the evolution of the dissemination of Covid-19, with emphasis
3
on the number of deaths caused by this contagious respiratory disease in Brazil, the State of Paraíba,
and the City of Campina Grande. We validated the proposed theoretical model with comparisons of
official data of the numbers of confirmed, recovered and death cases in consequence of COVID-19
infections.
The SIR model is a well-known and tested epidemiological model that has been applied to
very diverse epidemics (see for example Refs. [5–8]). The model receives its name for dividing
the population in three groups: susceptible, infected, and recovered. Adapted SIR models with a
time-dependent contagion constant have been proposed before. For the COVID-19 pandemic see
Chen et al. [9] and Dehning et al. [10].
We hope that this study of epidemiological dynamics be useful in stressing the importance of
public health policies about the application, maintenance, and reinforcement of social distancing
measures with the objective of avoiding the collapse of the health system.
We point out that an earlier version of this work, written in Portuguese, was pre-published in
ResearchGate [11].
This work is organized in the following way: In Section II, we introduce our epidemiological
model and present estimates for various parameters used in the model. In Section III, we investigate
the spread of COVID-19 in Germany as a benchmark test for our model. Afterwards, we present
our results and discuss about the evolution of the virus in Brazil, in the State of Paraíba, and in the
City of Campina Grande. We validate the predictions of our theoretical model by fitting the official
data. Furthermore, based on the good fits obtained so far, we offer predictions for the evolution
of the pandemic for the next 20 days in the studied regions. Finally, in Section IV we draw our
conclusions.
II. EPIDEMIOLOGICAL MODEL
The original SIR model [4,12] is given by the following system of ordinary differential
equations (ODEs)
𝑑𝑆
𝑑𝑡
=−𝜅𝑆𝐼 ,
𝑑𝐼
𝑑𝑡
=𝜅𝑆𝐼 −𝐼
𝜏,
𝑑𝑅
𝑑𝑡
=𝐼
𝜏,
(1)
in which 𝑆(𝑡),𝐼(𝑡), and 𝑅(𝑡)represent, respectively, the number of susceptible, infected, removed
(recovered or dead) individuals, 𝜅represents the rate of contagion and 𝜏is the average time of
4
infection (from contagion until recovery or death).
Similarly, in the model we propose, the variables in time 𝑡,𝑆(𝑡)and 𝐼(𝑡), mean the same
thing as in the original SIR model, but 𝑅(𝑡)is now the number of recovered individuals and we
introduce 𝑀(𝑡)that is the number of accumulated deaths due to the epidemic. To simplify our
model, we suppose that the population is homogeneous such that all the susceptible individuals have
the same probability of being contaminated and the infected individuals have the same probability
of recovery or death due to the infection. We also suppose that the population evolves in such a way
that the newborn babies are all susceptible and the recovered are all immune. The evolution of the
epidemiological model we propose is determined by the following ordinary differential equations
(ODE) system
𝑑𝑆
𝑑𝑡
=𝜈(𝑆+𝐼+𝑅) − 𝜇𝑆 −𝜅(𝑡)𝑆 𝐼 ,
𝑑𝐼
𝑑𝑡
=−(𝜇+𝜌+𝜆)𝐼+𝜅(𝑡)𝑆 𝐼 ,
𝑑𝑅
𝑑𝑡
=𝜌𝐼 −𝜇𝑅,
𝑑𝑀
𝑑𝑡
=𝜆𝐼,
(2)
in which 𝜈is the population birth rate, 𝜇is the usual death rate (before the onset of the pandemic),
𝜅(𝑡)is the contagion rate function, 𝜌is the recovery rate, and 𝜆is the lethality rate due to the
epidemic.
It is important to point out further differences between the original model (2) and the pro-
posed SIR model (1). Here, we allow variations in the total population, taking into account the
contributions of birth and death rates to the population evolution. This could become relevant if
the pandemic lasts for over a year. This is relevant also as a source of comparison with the death
rates due to the COVID-19. One important difference we introduced is that we now allow for time
variation in the contagion rate 𝜅(𝑡), which reflects the change in time of the confinement, social
distancing, mask use, etc in the population.
Before we proceed, we divide all variables involved, 𝑆(𝑡),𝐼(𝑡),𝑅(𝑡), and 𝑀(𝑡)by 𝑃0, the
total initial population of the region considered, which can be a city, a state, a country or the whole
world. Hence, 𝜅(𝑡)has to be multiplied by 𝑃0. In this way, if all the other parameters of the model
are the same, 𝜅(𝑡)becomes independent of 𝑃0.
For a given time 𝑇 > 0and non-negative constants 𝜅1< 𝜅2, by Picard-Lindelöf theorem
[13], for each initial value (𝑆0, 𝐼0, 𝑅0, 𝑀0), with 𝜅(𝑡)varying continuously in the bounded interval
[𝜅1, 𝜅2]and with the other parameters fixed, one can show that the ODE system of Eq. (2) admits
existence and uniqueness of solution in the time interval [0, 𝑇 ].
5
A. Estimates of the parameters used in the model
We now make some estimates for the parameters 𝜈,𝜇,𝜌, and 𝜆present in the dynamical
system given in Eq. (2).
1. Birth and death rates
In order to make our model more precise, we obtained the annual birth rate (ABR) and the
annual mortality rate (AMR) from the most recently published data from Germany and from Brazil
before the spread of the pandemic.
We also converted these annual rates into daily rates using the geometrical progression
formulas
(1+𝜈)365 =1+𝐴𝐵 𝑅,
(1−𝜇)365 =1−𝐴 𝑀 𝑅.
Hence, we find the daily birth and mortality rates
𝜈=(1+𝐴𝐵 𝑅)1/365 −1=𝑒1
365 ln(1+𝐴𝐵𝑅)−1
𝜇=1− (1−𝐴 𝑀 𝑅)1/365 =1−𝑒1
365 ln(1−𝐴 𝑀 𝑅)(3)
The German data on birth and death rates were obtained from the Federal Statistical Office [14,15].
The Brazilian data was obtained from the Brazilian Institute of Geography and Statistics (IBGE),
which are from 2018. For the birth rate we divided the number of born alive infants by the population
estimate for 2018 and, likewise, the number of deaths by the 2018 population estimate. The data on
the born alive was collected from the site https://sidra.ibge.gov.br/tabela/2609,
the number of deaths from the site https://sidra.ibge.gov.br/tabela/2654, and
the population estimate from the site https://www.ibge.gov.br/estatisticas/soc
iais/populacao/9103-estimativas-de-populacao.html?edicao=22367&t
=resultados. The birth and death rates used are shown in the table Ibelow. We provide table
II with information on the average total daily deaths (without the born dead) before the pandemic
based on annual death rates from 2018. This should be compared with the average daily deaths due
to COVID-19 as another way of assessing the severity of this pandemic.
6
TABLE I. Birth and death rates
Population ABR (year−1)𝜈(day−1) AMR (year−1)𝜇(day −1)
Germany 0.0095 2.591 ×10−50.0115 3.169 ×10−5
Brazil 0.0139 3.7844 ×10−56.1560 ×10−31.6918 ×10−5
Paraíba 0.0148 4.0139 ×10−56.5325 ×10−31.7956 ×10−5
Campina Grande 0.0158 4.2943 ×10−57.1342 ×10−31.9616 ×10−5
TABLE II. Pre-pandemic average daily deaths
Location 𝑃0𝜇(day −1) average deaths (day−1)
Germany 83,149,360 3.169 ×10−52635
Brazil 211,000,000 1.6918 ×10−53570
Paraíba 4,018,127 1.7956 ×10−572
Campina Grande 409,731 1.9616 ×10−58
2. Lethality and recovery rates
It is fundamental that we have good estimates for the recovery and the lethality rates. In
order to obtain these estimates we will use a very simple statistical model. Suppose that a person
is infected at a given moment 𝑛(which may be a day, an hour, or a minute for example), then the
probability that the infected remains sick in the following moment 𝑛+1is 𝑞, the probability that
the infected recovers in the next instant is 𝑝, and the probability that the infected dies is 𝑠, in such
a way that 𝑝+𝑞+𝑠=1. Here, we suppose that 𝑞and 𝑝+𝑠remain constant during the course of
the disease. Hence, we have the following probability table
TABLE III. Probabilistic model
Situation \Instant 0 1 2 . . . 𝑛...
Recovered 𝑝 𝑞 𝑝 𝑞2𝑝... 𝑞𝑛𝑝...
Death 𝑠 𝑞𝑠 𝑞2𝑠... 𝑞𝑛𝑠...
Note that the normalization
∞
Õ
𝑛=0
𝑞𝑛𝑝+∞
Õ
𝑛=0
𝑞𝑛𝑠=𝑝+𝑠
1−𝑞
=1,
implies that this probabilistic model is well defined.
7
If 𝑛is sufficiently large, only two outcomes are possible: either the infected individual
recovers or dies. Hence, based on the Table III we find that the recovery probability is 𝑃𝜌=
Í∞
𝑛=0𝑝𝑞𝑛=𝑝/(1−𝑞)and the probability of death is 𝑃𝜆=Í∞
𝑛=0𝑠𝑞𝑛=𝑠/(1−𝑞).
Using these probabilities we find that the average number of instants (minutes, hours, days,
etc) of the infection is given by
¯𝑛=∞
Õ
𝑛=1(𝑝+𝑠)𝑛𝑞𝑛=(𝑝+𝑠)𝐹1(𝑞)=(1−𝑞)∞
Õ
𝑛=1
𝑛𝑞𝑛,(4)
in which the summation 𝐹1(𝑞)=Í∞
𝑛=1𝑛𝑞𝑛can be calculated in the following form
𝑞𝐹1(𝑞)=∞
Õ
𝑛=1
𝑛𝑞𝑛+1=∞
Õ
𝑛=2(𝑛−1)𝑞𝑛=∞
Õ
𝑛=2
𝑛𝑞𝑛−∞
Õ
𝑛=2
𝑞𝑛=𝐹1(𝑞) − ∞
Õ
𝑛=1
𝑞𝑛=𝐹1(𝑞) − 𝑞
1−𝑞.
Hence, we obtain
𝐹1(𝑞)=𝑞
(1−𝑞)2.(5)
Therefore, we find that the average number of instants of the infection is
¯𝑛=𝑞
1−𝑞,(6)
from where we obtain that 𝑞=¯𝑛/(1+¯𝑛). We can also find that the average number of time intervals
until recovery is given by
¯𝑛𝜌=𝑝𝐹1(𝑞)=𝑝𝑞
(1−𝑞)2=𝑃𝜌¯𝑛
and the average number of time intervals until death is
¯𝑛𝜆=𝑠𝐹1(𝑞)=𝑠𝑞
(1−𝑞)2=𝑃𝜆¯𝑛.
Note that ¯𝑛=¯𝑛𝜌+¯𝑛𝜆, that is, the average infection time is the sum of the average time for recovery
and the average time until death. If only these two processes were present, it would lead to the
following difference equation for the number of infected
𝐼(𝑛+1)=𝐼(𝑛)−(𝑝+𝑠)𝐼(𝑛)=𝐼(𝑛)−(1−𝑞)𝐼(𝑛)=𝐼(𝑛) − 1
1+¯𝑛𝐼(𝑛).
If we take 𝑛to indicate the 𝑛-th time interval, such as a minute, in which there is not much variation
in the quantities 𝑆,𝐼,𝑅, and 𝑀, hence, we can approximate
𝑑𝐼
𝑑𝑡 ≈Δ𝐼
Δ𝑡
=−1
(1+¯𝑛)Δ𝑡𝐼(𝑡𝑛),
8
in which 𝑡𝑛=𝑛Δ𝑡. In this work, we take Δ𝑡=1minute =1 day/(24 ×60). The average time of
infection can be obtained from the following equation
1
𝜏
=𝜆+𝜌=1
(1+¯𝑛)Δ𝑡.(7)
Hence, we obtain the following expressions for the rates of lethality and recovery
𝜆=¯𝑛𝜆
¯𝑛𝜏
=𝑃𝜆
𝜏,
𝜌=¯𝑛𝜌
¯𝑛𝜏
=𝑃𝜌
𝜏.
(8)
3. Standard deviation
Here, we calculate the standard deviation for this probabilistic process in the number of time
intervals 𝑛of infection, from the contamination until recovery or death. In order to achieve that,
we have first to calculate the sum
𝐹2(𝑞)=∞
Õ
𝑛=1
𝑛2𝑞𝑛.(9)
We can calculate this sum by noting that
𝑞2𝐹2(𝑞)=∞
Õ
𝑛=2(𝑛−2)2𝑞𝑛=𝐹2(𝑞) + 4∞
Õ
𝑛=1(1−𝑛)𝑞𝑛=𝐹2(𝑞) + 4𝑞
1−𝑞−𝑞−4𝐹1(𝑞).
Hence, we find
𝐹2(𝑞)=𝑞(1+𝑞)
(1−𝑞)3.(10)
We then obtain
𝑛2=(1−𝑞)𝐹2(𝑞)=𝑞(1+𝑞)
(1−𝑞)2
and
𝑛2=𝑞2
(1−𝑞)2.
We can now write the standard deviation of 𝑛as
𝜎=q𝑛2−𝑛2=√𝑞
1−𝑞.(11)
This shows that the statistical fluctuations in the time duration of the infection grows as 𝑞→1−.
By reducing 𝑞one not only decreases 𝑛but also 𝜎.
9
B. Basic reproduction rate
It is of paramount importance to know if a contagious disease will become epidemic or not
in a population. It is also important to know when it will be possible to control an epidemic, that
is, when it will be possible to block its growth. This will happen when 𝑑𝐼
𝑑𝑡 ≤0. From Eq. (2), we
verify that this condition is equivalent to
− (𝜇+𝜌+𝜆) + 𝜅𝑆(𝑡) ≤ 0=⇒𝜅𝑆 (𝑡)
𝜇+𝜌+𝜆≤1.(12)
In the beginning of the epidemic we obtain that the value of the following ratio
𝑅0=𝜅𝑆0
𝜇+𝜌+𝜆
=𝜅𝑆0
𝜇+1/𝜏,
known in the literature [12] as the basic reproduction rate, is what indicates whether we will have
an epidemic or not. When 𝑅0>1, the disease will spread, whereas when 𝑅0<1the contagion
loses strength and the dissemination of the virus will be controled. In our case 𝑆0=1and 𝑆(𝑡) ≤ 1,
thus at any time after the onset of the epidemic the disease will stop spreading when
𝑅0(𝑡)=𝜅(𝑡)𝑆(𝑡)
𝜇+1/𝜏≤1.(13)
We have that 𝑅0<1is a sufficient condition that the epidemic will enter remission, but in general
it is not a necessary condition. The necessary condition is that 𝑅0(𝑡)<1. Although, as we are
still in the earlier stages of the current epidemic in Brazil and 𝑀(𝑡)<< 1,𝑆(𝑡) ≈ 1, the critical
condition is still approximately 𝑅0=1and the critical value of 𝜅(𝑡)is 𝜅∗=𝜇+1/𝜏.
As there is no efficacious treatment against COVID-19 at the time of writing this paper as far
as the authors’ knowledge, it is not yet possible to easily alter the average time of infection 𝜏. As
1/𝜏 >> 𝜇, thus the only viable manner of decreasing 𝑅0(𝑡) ≈ 𝜅(𝑡)𝜏is by reducing the value of
𝜅(𝑡), which can be obtained with social distancing measures and the use of PPEs.
III. RESULTS AND DISCUSSION
We used the Odeint function of the Python’s scientific library package SciPy [16] to integrate
the ODE system of Eq. (2) with the integration time-step 𝑑𝑡 =1.0/1440, which corresponds to a
minute when the time unit is a day. In the cases investigated, we took 𝜏=14 days. We suppose this
parameter is uniform and does not change appreciably during the time scale of a few months. The
initial values used are: 𝑆(0)=1−1/𝑃0,𝐼(0)=1/𝑃0,𝑅(0)=0, and 𝑀(0)=0, with 𝑃0adapted
for the case of each population studied. We now apply our model to four cases of COVID-19
dissemination: in Germany, in Brazil, in the State of Paraíba, and in the City of Campina Grande.
10
A. Evolution of Covid-19 in Germany
We consider the case of Germany as a benchmark test for our epidemiological model. This
is so because it is widely believed that the cases from Germany are better accounted for with
widespread testing of the population https://www.labmate-online.com/news/labo
ratory-products/3/breaking-news/how-germany-has-led-the-way-on-c
ovid-19-testing/52141. The initial population considered is 𝑃0=83.14936 millions. The
first contagion was registered on 01/27/2020.
In Fig. 1, we show the official data on the confirmed cases of COVID-19 plotted alongside
the theoretical prediction. One can see a very good agreement with the theoretical prediction. The
parameters used are shown on top of the figure. The contagion function used is given in Fig. 5.
In Fig. 2, the lethality probability used was 𝑃𝜆=0.05, which is very close to the ratio of
deaths by removed cases recorded on 06/10/2020 of 0.0493.
In Fig. 3, we plot the recovered cases data along with the theoretical predictions based on our
model.
In Fig. 4, we plot the active (or infected) cases data along with the theoretical predictions
based on our model. The theoretical fit is not as good as in the previous figures, but it is still quite
reasonable. We predict a slow decline in the infected. Once the total number of confirmed cases
basically saturates, the evolution of the active cases could also be traced with a purely statistical
model as the one we developed above. From the statistical point of view this slow decay has to do
with the large value of the dispersion in 𝑛.
In Fig. 5, the initial and final basic reproduction rates are, respectively, 𝑅0=3.443 and
𝑅0=0.5388. The rapid decrease of the contagion rate starts approximately around 03/22/2020,
when strict social distancing rules were imposed by the German Government as stated in the site
https://www.dw.com/en/what-are-germanys-new-coronavirus-social-d
istancing-rules/a-52881742. This shows that these measures were very efficient in
containing the spread of the epidemic.
12
FIG. 2. The number of official deaths compared with the theoretical prediction. The theoretical fit was
obtained with the same parameters of Fig. 1and with the same contagion function 𝜅(𝑡)of Fig. 5. The best
fit was obtained with the lethality probability 𝑃𝜆=0.05 and the average time of infection 𝜏=14 days.
15
FIG. 5. Time variation of the rate of contagion. The used function when 𝑡 < 𝑡1=48 days was 𝜅(𝑡)=
𝑎+𝑏tanh(ℓ(𝑡−𝑡0)), where 𝑎=𝜅1+𝜅2
2,𝑏=𝜅1−𝜅2
2,𝑡0=64.5days and ℓ=0.125 day−1. The initial value of
the contagion rate was 𝜅1=0.246 and the final value was 𝜅2=0.0385.
16
B. Evolution of Covid-19 in Brazil
We consider the initial time the day of the first confirmed case in Brazil. We take Brazil’s
population to be approximately 𝑃0=211 ×106. The data of the number of confirmed, recovered
and death cases from Brazil were obtained from the site <https://data.humdata.org/dataset/novel-
coronavirus-2019-ncov-cases>, (accessed on 06/18/2020, with data collected until 06/17).
In Fig. 6, we make a comparison between the official data (blue dots) and the number
of confirmed cases predicted by the proposed model. Once we had estimates on the birth and
pre-pandemic death rates, and on the lethality and recovery rates, we tuned 𝜅(𝑡)to obtain the best
fit. The time variation of the contagion rate reflects the fact that the population slowly took heed
of the gravity of the pandemic and started adopting social distancing measures and using personal
protective equipments (PPEs). According to the model proposed, after about ninety five days from
the first contagion, there is no end in sight of the rise of the daily contagions and the increase of
the active cases. More severe measures of social distancing and mandatory use of PPEs in public,
such as masks, by the population should be enforced. Also, widespread testing and contact tracing
should be implemented.
In Figure 7, we show a comparison between the number of cases of death due to COVID-19
(blue line) and the number of deaths predicted by the theoretical model. For this result, we used
the same parameters as in Fig 6and the same function 𝜅(𝑡)from Fig. 10. The parameters 𝜆and 𝜌
vary according to Fig. 11. One sees that the rate of growth in this curve after 113 days from the
first confirmed death has not abated yet.
In Fig. 8, we show a comparison between the number of recovered cases(blue line) and the
predicted number of recovered cases predicted by the theoretical model. Although, we did not
obtain such a good fit as in Fig. 6the agreement is still fairly good. The discrepancies may have to
do with delays in the confirmation of the recovered cases, as we can see in the jump that occurred
from 06/07 to 06/08 . One possible source of systematic error, towards sub-notification of recovered
cases, could occur in milder cases. Recovered outpatients may fail to take another test to confirm
their recoveries.
In Fig. 9, we plot the active (or infected) cases data along with the theoretical predictions
based on our model. The theoretical fit is not as good as in the previous figures, but it is still quite
reasonable. Until now, we do not see any indication that we have come near the peak of the active
cases in Brazil.
In Fig. 10, we show the time evolution of the contagion rate 𝜅(𝑡). The drop in this rate is
17
likely due to the increase of isolation and social distancing that grew at the second half of March in
Brazil. In the first 20 days after the first official confirmed case of COVID-19 infection in Brazil,
this rate was very high, what meant that the level of necessary precautions to prevent contagion
by the population was very low. Between the 20th and 40th first days, there was a decrease in the
rate of contagion, what is certainly due to better precautions by the population (isolation, social
distancing, hand washing, and the increased use of PPEs). Although, this is not enough since the
number of active cases continued increasing. After 40 days the rate of contagion stabilized. The
variation of 𝜅(𝑡)leads to changes in the basic reproduction rate 𝑅0that decreased from around
𝑅01 =4.76 (at the beginning of the pandemic in Brazil) to 𝑅02 =2.03 (roughly from the middle of
April to the middle of May), and more recently to around 𝑅03 =1.57. This value is still very large,
instead it should be below 1.
In Fig. 11, we show the time evolution of the lethality and recovery rates, 𝜆and 𝜌, respectively.
The lethality probability 𝑃𝜆is decreasing. Initially it was around 0.14 and more recently it is around
0.06. This seems to be related with the increased amount of testing in Brazil, which is still far
below the necessary though. It could also be related with increased sub-notification of death
cases. Another possibility is that the medical treatment and procedures for the more severe cases
of COVID-19 are being better treated after 80 days from the first notified case of confirmation in
Brazil. Whatever the case, this behavior should be further investigated.
In Fig. 12, we show results of numerical simulation for a range of values of 𝜅. Unlike the
other results, 𝜅is held constant during each time integration of the equations of motion given in
Eq. (2). This result is important in conveying the message of the paramount importance of the
contagion rate on the possible outcomes of the pandemic. Not only we observe an increase in the
number of deaths when the contagion rates increase, but we also see a sharp transition. When there
is a growth in the contagion rate from 0.1to 0.15, this gives rise to an extremely sharp increase in
the number of deaths. This implies that there is a critical value of 𝜅, around which there is a rapid
increase in the total number of deaths due to the pandemic. Beyond the critical value, we see a
saturation in the total number of deaths. The value of 𝜅that corresponds to 𝑅0=1in our model
is 𝜅=𝜇+1/𝜏=0.0714. Such a low value indicates that the precautionary measures taken by the
population are far below what they should be. We believe, this reinforces the great relevance of
social distancing, since increasing the average distance between people, we will be decreasing the
rate of contagion and, consequently, decreasing also the number of deaths due to COVID-19.
19
FIG. 7. The number of official deaths compared with the theoretical prediction. The theoretical fit was
obtained with the same parameters of Fig. 6and with the same contagion function 𝜅(𝑡)of Fig. 10. The best
fit was obtained with the lethality probability 𝑃𝜆=0.0947 and the average time of infection 𝜏=14 days.
22
FIG. 10. Time variation of the rate of contagion. The used function was 𝜅(𝑡)=𝑎+𝑏tanh(ℓ(𝑡−𝑡0)), where
𝑎=𝜅1+𝜅2
2,𝑏=𝜅1−𝜅2
2,𝑡0=30 days and ℓ=0.125 day−1. The initial value of the contagion rate was 𝜅1=0.34,
the intermediate value was 𝜅2=0.145, and the final value was 𝜅=0.112.
FIG. 11. Time variation of 𝜆and 𝜌given in Eq. (8). The used function for 𝑃𝜆was 𝑎+𝑏tanh(ℓ(𝑡−𝑡0)), where
𝑎=𝑃𝜆1+𝑃𝜆2
2,𝑏=𝑃𝜆1−𝑃𝜆2
2, and 𝑡0=80 day. The initial lethality probability was 𝑃𝜆1=0.14 and 𝑃𝜆2=0.06.
23
FIG. 12. Total number of accumulated deaths as a consequence of the epidemic as a function of the contagion
rate 𝜅. The total time duration for each value of 𝜅is 365 days. The used parameters in this simulation are
indicated above the figure. The value of 𝜅that corresponds to 𝑅0=1is 𝜅∗=𝜇+1/𝜏=0.0714.
C. Evolution of COVID-19 in Paraíba
We obtained the time series of confirmed and death cases from the State of Paraíba in the
site: https://data.Brazil.io/dataset/covid19/_meta/list.html(accessed
on 06/11/2020, with data collected until 06/11). We also provide this data in the supplementary
material. The initial population of Paraíba is 𝑃0=4,018,127. The first case of COVID-19
contamination was registered on 03/18/2020.
In Fig. 13, we make a comparison between the official data (blue dots) and the number of
confirmed cases predicted by our model. Similarly as we did before for the cases of Germany and
Brazil, once we had estimates on the birth and pre-pandemic death rates, and on the lethality and
recovery rates, we tuned 𝜅(𝑡)as in Fig. 15 to obtain the best fit. According to the data available, after
about 80 days after the first contagion, there is no end in sight of the rise of the daily contagions.
More severe measures of social distancing and mandatory use of personal protective equipments
(PPEs), such as masks, face shields, and gloves, by the population should be enforced.
24
In Fig. 14, we show a comparison between the number of cases of death due to COVID-19
(blue line) and the number of deaths predicted by the theoretical model. For this result, we used
the same parameters and the same function 𝜅(𝑡)from Fig. 15. To obtain this fit we had to let the
lethality probability 𝑃𝜆vary (and consequently also 𝑃𝜌). Initially, 𝑃𝜆was around 0.14 and more
recently it is around 0.05. This seems to be related with the increased amount of testing in Paraíba.
It might also be related with the sub-notification of deaths.
In Fig. 15, we see a decrease in the rate of contagion. The time variation of the contagion rate
followed approximately what happened in Brazil, its decrease reflects social distancing measures
and the use of PPEs. We believe that this is due to the increase in isolation and social distancing
that started increasing in the end of March. Although, the initial value of 𝜅(𝑡)was about 0.24, after
about 50 days from the onset of the pandemic here, it was about 0.165, and more recently it is about
0.115. Initially, 𝜅(𝑡)was smaller in in Paraíba than in Brazil, but at the moment it is the other way
around. Also, initially, 𝑅0≈3.36 in Paraíba, more recently it is around 1.61. This means that it is
necessary to restrict even further the social interactions in order that 𝑅0becomes smaller than 1.
In Fig. 16, we show the time variation of the lethality and recovery rates. This qualitatively
follows what happens in Brazil. The explanation is likely similar as well.
26
FIG. 14. Comparison of the number of official deaths due to COVID-19 infection in Paraíba with the
theoretical prediction. The theoretical fit is obtained with the indicated parameters above the figure and with
𝜅(𝑡)varying in time as the function represented in Fig. 15. Here, we also allowed the lethality to vary from
𝑃𝜆=0.15 down to 𝑃𝜆=0.05.
27
FIG. 15. Time variation of the contagion rate in Paraíba. The used function was 𝜅(𝑡)=𝑎+𝑏tanh(ℓ(𝑡−𝑡0)),
where 𝑎=𝜅1+𝜅2
2,𝑏=𝜅1−𝜅2
2,𝑡0=30 days and ℓ=0.125 day−1. The initial value of the rate of contagion was
𝜅1=0.24, the intermediate value was 𝜅2=0.165, and the final value was 𝜅3=0.115.
FIG. 16. Time variation of 𝜆and 𝜌given in Eq. (8). The used function for 𝑃𝜆was 𝑎+𝑏tanh(ℓ(𝑡−𝑡0)), where
𝑎=𝑃𝜆1+𝑃𝜆2
2,𝑏=𝑃𝜆1−𝑃𝜆2
2, and 𝑡0=48 day. The initial lethality probability was 𝑃𝜆1=0.15 and 𝑃𝜆2=0.05.
28
D. Evolution of COVID-19 in Campina Grande
We obtained the time series of confirmed cases and death cases in Campina Grande in the
site: https://data.Brazil.io/dataset/covid19/_meta/list.html(accessed
on 06/11/2020, with data collected on 06/10/2020). We also present this data in the supplementary
material. We consider the initial time as the first day of the first confirmed case of contagion in
Campina Grande. We take 𝑃0=409.731 as the initial population.
According to Fig. 17, the fit of the available data for Campina Grande is quite reasonable.
We believe that the small fluctuations that occurred in the beginning are due to failures and delays
in the reporting of confirmed and death cases. In addition, as this is a smaller population the effect
of statistical fluctuations is also larger than in the other two cases. Hence, it is inherently more
difficult to fit the data with the predictions of a deterministic model. Nevertheless, if everything
else is held constant, we believe that with the increase in the number of cases the fit will become
better. Furthermore, we have a shorter time series here. In Campina Grande, we are still at the
early stages of the spread of the pandemic.
In Fig. 18, we compare the number of deaths due to COVID-19 (blue cross) and the number
of deaths predicted by the theoretical model. We used the same parameters and the same function 𝜅
used in Figure 19. Based on the evolution shown in this figure, provided no more restrictive measures
are taken in relation to anti-contagion precautions, by 90 days since the start of the pandemic here
the number of confirmed cases will reach almost 9000 and the number of COVID-19 related deaths
will reach 140.
In Figure 19, we plot the rate of contagion as a function of time in Campina Grande. We
observe that the behavior of the contagion rate curve is not only quantitatively different, but also
qualitatively different to the contagion rate curves from Brazil and Paraíba, which are shown in
Figs. 10 and 15. The peaks in 𝜅(𝑡)are due to a relaxation of anti-contagion preventive measures,
such as social distancing. In Campina Grande, the rate of contagion started at a lower value than in
Brazil, with early social distancing, closures of schools and universities, even before the first official
confirmed case. The peaks seem to coincide with emergency payments made by the government
in banks, where many people gathered to receive the financial support.
30
FIG. 18. The number of official deaths compared with the theoretical prediction. The theoretical fit was
obtained with the same parameters of Fig. 17 and with the same contagion function 𝜅(𝑡)of Fig. 19. The best
fit was obtained with the lethality probability 𝑃𝜆=0.04 and the average time of infection 𝜏=14 days.
31
FIG. 19. The contagion function used was 𝜅(𝑡)=𝑎+𝑏tanh (ℓ(𝑡−𝑡0)) + ℎ1𝑒−(𝑡−𝑡1)2/(2𝜎2)+ℎ2𝑒−(𝑡−𝑡2)2/(2𝜎2),
where 𝑎=𝜅1+𝜅2
2,𝑏=𝜅1−𝜅2
2,𝑡0=𝑡1=20 days, 𝑡2=50 days and ℓ=0.125 day−1. The initial value of the rate
of contagion was 𝜅1=0.130 day−1and the final value was 𝜅2=0.140 day−1. The extra parameters from the
Gaussian function are ℎ1=ℎ2=0.1and 𝜎=5.0day.
32
IV. CONCLUSION
In this work, we propose an epidemiological model based on the SIR model [4], but with
some notable differences. Here, we take into account the contribution from the pre-pandemic birth
and death rates to the evolution of the populations investigated. This could become relevant if
the pandemic lasts for over a year and also it is important as a comparison for the lethality of the
pandemic. We also allow that the contagion rate varies temporally so that it reflects the fact that
social distancing and isolation changes over time. According to the results exposed in the previous
section, with our model we could fit the official case data from Germany, Brazil, the State of
Paraíba, and the City of Campina Grande quite well.
The modeling of the spread of the pandemic in Germany is very emblematic, since it clearly
shows that the strict social distancing measures imposed by the government on 03/22/2020 were
very effective in containing the disease, reducing the 𝑅0from 3.44 down to 0.54, according to our
model.
In the case of Brazil, we conclude that, based on the fit of the proposed model, the contagion
rate 𝜅(𝑡)is decreasing in steps roughly from 𝜅1=0.34 to 𝜅2=0.145 until more recently 𝜅3=0.112
The corresponding basic reproduction rate decreased from 𝑅01 =4.76 through 𝑅02 =2.03 to
𝑅03 =1.57. This means the implanted social distancing measures are having an effect in thwarting
the spread of the disease, but it is not enough, since to control the pandemic 𝑅0should be less
than 1. The scenario seems bleak, since the social distancing measures in Brazil are becoming less
effective by the day, although the use of PPEs, mostly masks, is still very high.
The spread of COVID-19 in the State of Paraíba is similar to the national case. The rate of
contagion and, consequently, the basic reproduction rate also decreased. In our best fit with official
data, the value of the contagion rate decreased in steps from 𝜅1=0.24, to 𝜅2=0.165, and more
recently to 𝜅3=0.115. Consequently, 𝑅0varied from 𝑅01 ≈3.36 to 𝑅02 ≈2.31 and more recently
to 𝑅03 ≈1.61. In a similar way to what happens in Brazil, the State of Paraíba needs to intensify the
social distancing so that 𝑅0becomes less than 1. Unlike all the other populations we investigated,
we had to vary the lethality probability 𝑃𝜆in time so that we could fit the number of death cases
from 0.15, initially, down to 0.04, lately. This likely has to do with the increasing number of tests
of the population and sub-notification of deaths. We did not plot the number of recovered cases
because we have not been able, so far, to obtain this data for Paraíba.
As was commented in the Introduction, Campina Grande adopted a social distancing policy
one week earlier than the report of the first confirmed case. Despite of this, the rate of contagion
33
did not decrease monotonically as it did in Germany, Brazil or in Paraíba. It presents two peaks in
𝜅(𝑡)since the outbreak of the disease here. To obtain the best fit for our model, it was necessary
to consider variations in the rate of contagion. With this allowance for the variation of 𝜅(𝑡),
we succeeded in fitting the official data of confirmed cases. With these values of 𝜅the basic
reproduction rate 𝑅0was initially 𝑅01 =1.82 and more recently it is around 𝑅02 =1.96. Around
April 6th-16th the rate of contagion grew rapidly, reaching a maximum on April 16th, suggesting
that in this period there was a relaxation of social distancing by the population. According to
the predictions of our theoretical model, in case the social distancing measures are not tightened
around 20 days we will have around 140 accumulated deaths due to COVID-19 in Campina Grande.
Unlike the trend in Brazil and in Paraíba in the studied period, the social distancing measures are
getting worse in Campina Grande. We further comment that the lethality probability 𝑃𝜆of Campina
Grande used was 0.03, a number we consider very low, even lower than the one from Germany.
This very likely has to do with sub-notifications and should be further investigated.
Based on the results shown here, we conclude that the public health officials should look into
the local dynamics of the spread of the disease as they compare with the theoretical predictions of
models such as the one developed here. In this way, they will know where the social distancing
and isolation is being more efficiently implemented. The models will be more relevant and
accurate if there is more testing of the population. Even random testing should be considered, as
one gains statistical information on the spread of the disease and discovers where there is more
under-notification. Also, cellphone data of the motion of the population should be considered
as a means of predicting the contagion and of identifying hot spots of COVID-19. This could
help identify whether the contagion occurs more on supermarkets, offices, pharmacies, hospitals,
bakeries, restaurants, by delivery services, family visits, etc. Furthermore, by comparing different
local strategies one could gain insight on what works better for slowing the spread of the disease.
In a more refined work, one could couple several nearby cities into a network of populations.
34
V. ACKNOWLEDGMENTS
The authors thank professors Francisco A. Brito, Antônio A. Lisboa de Souza, Michelli
Barros, Joelson Campos, and Aldo Trajano for suggestions during the development of this work.
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