# Assessing the Time Evolution of COVID-19 Effective Reproduction Number in
Brazil

Edson Porto da Silva, and Antonio M.N. Lima Edson Porto da Silva and Antonio
M.N. Lima are with the Department of Electrical and Computer Engineering,
Center for Electrical Engineering and Informatics, Universidade Federal de
Campina Grande).

###### Abstract

In this paper, we use a Bayesian method to estimate the effective reproduction
number ($R(t)$), in the context of monitoring the time evolution of the
COVID-19 pandemic in Brazil at different geographic levels. The focus of this
study is to investigate the similarities between the trends in the evolution
of such indicators at different subnational levels with the trends observed
nationally. The underlying question addressed is whether national surveillance
of such variables is enough to provide a picture of the epidemic evolution in
the country or if it may hide important localized trends. This is particularly
relevant in the scenario where health authorities use information obtained
from such indicators in the design of non-pharmaceutical intervention policies
to control the epidemic. A comparison between $R(t)$ estimates and the moving
average (MA) of daily reported infections is also presented, which is another
commonly monitored variable. The analysis carried out in this paper is based
on the data of confirmed infected cases provided by a public repository. The
correlations between the time series of $R(t)$ and MA in different geographic
levels are assessed. Comparing national with subnational trends, higher
degrees of correlation are found for the time series of $R(t)$ estimates,
compared to the MA time series. Nevertheless, differences between national and
subnational trends are observed for both indicators, suggesting that local
epidemiological surveillance would be more suitable as an input to the design
of non-pharmaceutical intervention policies in Brazil, particularly for the
least populated states.

###### Index Terms:

COVID-19, reproduction numbers, epidemiological models

##  I Introduction

THE spread of the SARS-CoV-2 which led to the COVID-19 pandemic has brought
many challenges to policymakers around the globe. To prevent explosive growth
in the number of infections that may result in the collapse of the health
systems, non-pharmaceutical interventions (NPI) had to be put in place to
contain the virus spread. NPI is any non-pharmaceutical measure aimed to
reduce the number of contagious contacts in the population, from promoting the
use of hand sanitizers to lockdowns. However, NPIs such as closing schools and
non-essential services, or implementing full lockdowns have significant
negative economic side-effects, making them difficult to be sustained over
long periods. Hence, the health authorities faced the hard task of defining
epidemiological policies, trying to find the best balance in adopting the
necessary NPIs to contain the virus spread, while minimizing the undesired
consequences to the population. Defining the appropriate policy requires a
careful assessment of the pandemic status and its evolution. In this context,
the use of mathematical epidemiological models is paramount (Kermack and
Mckendrick, 1927; Fred Brauer, 2008; Britton and Pardoux, 2019).
Unsurprisingly, since the beginning of the COVID-19 pandemic in 2020, many
studies have been published exploring different categories of models to assess
the status of the number of infections and/or to forecast its evolution in a
given population (Saldaña and Velasco-Hernández, 2021).

The availability of daily updated data sets on, e.g., the reported number of
infected cases and the reported number of deaths due to COVID-19 allowed data-
driven models to unveil several aspects of the pandemic. Nevertheless, two
different models designed to assess similar epidemiological variables may be
built based on distinct sets of assumptions. Such conceptual differences will
most certainly affect the interpretations and conclusions drawn from the
outcome of each model. That is particularly important to quantify the
underlying uncertainty associated with the forecast of each model (Hutson,
2020; Thompson et al., 2020; Gostic et al., 2020). The decision-making process
guided by information provided by any epidemiological model will always be
subject to uncertainty and risk. Hence, proper scrutiny is always necessary
before using outcomes from a given model, e.g., to support changes in NPIs or
to convey information to the public (Thompson et al., 2020).

The deterministic susceptible-infected-recovered (SIR) compartmental model
(Kermack and Mckendrick, 1927) and its variations were widely explored during
the COVID-19 pandemic. These models are based on a few simplifying assumptions
about the dynamics of the disease spreading in a population and are often more
user friendly to estimate, e.g., peaks of infections, the percentage of the
population that will get infected in scenarios with or without the adoption of
NPI measures (Morris et al., 2021). Unfortunately, many sources of uncertainty
affect the parameter estimation for those models, limiting their ability to
forecast the epidemic evolution. When more realistic assumptions are
introduced, the complexity of such models can quickly scale up, and extracting
reliable information from them also becomes cumbersome, mainly due to a large
number of parameters and the limited amount of available data. Uncertainty
related to the transition process between compartments can be managed, to some
extent, by stochastic epidemic models (Britton and Pardoux, 2019).

Rather than attempting to predict absolute figures in the number of infections
or deaths, SIR-type models and other epidemiological models can be used to
infer the relative outbreak potential of the infections over time by
estimating the so-called effective reproduction number $R(t)$, which provides
an important picture of the epidemic evolution over time. However, estimates
of $R(t)$ also need to be considered with care, especially because the real-
world data available for the estimation process may provide an incomplete
picture of the reality (Britton and Tomba, 2019; Gostic et al., 2020). Random
reporting delays, the backlog of reported infection data, a large number of
unreported asymptomatic infections, and a lack of an effective testing policy
are all examples of real-life issues that may influence the likelihood of the
data reflecting the underlying reality. Additionally, in each country, the
epidemic evolution is likely to be dependent on sub-national characteristics,
like demographics, and inter-regional migration dynamics.

Considering the many sources of randomness that may influence epidemiological
data, when using information from the surveillance of epidemiological
parameters to define NPIs, a proper assessment of uncertainty is necessary
(Morgan et al., 2021). In that sense, probabilistic Bayesian models, which
naturally are able to handle uncertainty, have an advantage. On the other
hand, NPI policies can be defined based on no specific epidemiological model,
but rather direct statistics obtained from the data on reported infections,
such as the evolution of its moving average (MA) over time. In fact, during
the COVID-19 pandemic, the weekly MA of reported infections was widely
reported by the media to picture the evolution of the pandemic.

In this paper, we present an analysis of the $R(t)$ estimates to characterize
the time evolution COVID-19 pandemic in Brazil. The investigation focuses on
evaluating the degree of similarity in the development of the pandemic over
time across the country by comparing national trends to the trends observed at
sub-national levels, such as geographic regions, and states. We highlight the
similarities and differences among estimates made with data collected at
different geographic levels: country, region, and state. The correlations
between national and sub-national $R(t)$ estimates are discussed, as well as
their implications when considering the design of NPI policies.

##  II Epidemic reproduction numbers

In this section, the fundamentals of epidemiological reproduction numbers are
briefly reviewed and a mathematical model for the effective reproduction
number $R(t)$ is discussed. Moreover, the Bayesian framework considered in
this work to estimate $R(t)$ from data on reported infections is also
presented.

###  II-A Basic reproduction number versus effective reproduction number

The basic reproduction number $R_{0}$ (R-naught) is arguably the most
important parameter to evaluate the potential for an infectious disease to
cause large epidemic outbreaks (Britton and Tomba, 2019). It is defined as the
average number of secondary infections an infected individual will generate in
a large population at the beginning of an infectious-disease outbreak, where
approximately all individuals are susceptible. Consequently, if $R_{0}<1$, the
disease will not have the potential to spread over a significant fraction of
the population, while if $R_{0}>1$ a large outbreak will almost certainly
occur. Therefore, for a given susceptible population, the $R_{0}$ value of an
infectious disease is a fingerprint of its epidemic potential. The assumption
of homogeneous mixing is usually applied to define $R_{0}$ in a few scenarios,
meaning that any infected individual in the population has the same
probability of transmitting the disease to any susceptible individual.

The importance of $R_{0}$ relates to the fact that it is a measure of the
potential for a disease to spread over a large susceptible population.
However, the assumptions used to derive $R_{0}$ may reflect well the
conditions at the beginning of an epidemic. Still, they tend to become less
realistic as the spreading of the disease evolves. Two main reasons
corroborate this fact. First, once an infectious-disease outbreak starts, non-
pharmaceutical or pharmaceutical measures can take place to control the
disease spreading. Consequently, the chances of a susceptible individual
becoming infected will change over time, e.g., depending on the severity of
the NPI measures. Second, even in the absence of NPI measures, complex
dynamics linked e.g. to the social structure in which the individuals are
inserted will influence the epidemic evolution. Therefore, knowing $R_{0}$ is
insufficient to guide the health authorities’ decisions in managing NPIs as
the epidemiological conditions change over time. Instead, the instantaneous
effective reproduction number $R(t)$ (Fraser, 2007; Cori et al., 2013; Flaxman
et al., 2020; Gostic et al., 2020) provides a better picture of an ongoing
epidemic.

The instantaneous effective reproduction number captures, to some extent, the
time-varying behavior of the disease spreading. Differently from $R_{0}$, the
definition of $R(t)$ does not assume that the entire population is
susceptible, i.e. it can be applied to any moment in time where an unknown
fraction of the population is can be infected. The value of $R(t)$ can be
interpreted as the average number of secondary infections that an infected
individual is expected to produce should the conditions at time $t$ remain
unchanged (Fraser, 2007). When $R(t)>1$, the number of newly reported
infections is likely to increase, whereas $R(t)<1$ suggests it is likely to
decrease. If $R(t)=1$, the number of new infections is likely to remain
stable. Moreover, the value of $R(t)$ is also related to how fast the number
of infections is increasing or decreasing. The main advantages of this
approach are the capability of $R(t)$ to track the time-varying epidemic state
and the fact that its estimation relies only on statistics from the number of
reported infections and a few specific characteristics of the disease (Cori et
al., 2013; Bettencourt and Ribeiro, 2008), such as the generation time
distribution or the serial interval (Britton and Pardoux, 2019). Hence, the
effective reproduction number provides useful information to assist decisions
by the health authorities during epidemic outbreaks.

Following (Fraser, 2007), $R(t)$ can be defined as

$R(t)=\frac{I(t)}{\int_{0}^{\infty}I(t-\tau)w(\tau)d\tau}$ (1)

where $I(t)$ is the number of infected individuals at time $t$ (calendar
time), and $w(\tau)$ is the generation time distribution, i.e., the
distribution of the new infections as a function of time of infection. A
discrete-time version of (1) is given by

$R\left(t_{i}\right)=\frac{I_{i}}{\sum_{j=1}^{n}w_{j}I_{i-j}}$ (2)

where $\left\\{w_{j}\right\\}_{j=1}^{n}$ is a discrete-time version of
$w(\tau)$. It is worth mentioning that the generation time distribution is
rarely known since the times of infection cannot be observed in practice.
Instead, the serial interval distribution, which corresponds to the
distribution of time intervals for symptoms onset between the infected and the
infectee is used as an approximation. It is important to emphasize that
uncertainties associated with the generation time distribution will affect the
uncertainty in estimating $R(t)$ over time. For instance, different variants
of SARS-CoV-2 may result in distinct generation time distributions and serial
intervals for COVID-19 (Hart et al., 2022; Pung et al., 2021). Nevertheless,
in the context of monitoring the spreading of a new disease, it is realistic
to assume that information about the generation time distribution of the
disease is limited. Here, as in (Flaxman et al., 2020), the generation time
distribution of COVID-19 is approximated by its serial interval distribution,
shown in Fig. 1.

Figure 1: Serial interval distribution/generation time distribution
$\left\\{w_{j}\right\\}_{j=1}^{n}$ assumed for COVID-19 (Flaxman et al.,
2020).

The definition in (2) provides a direct expression to estimate $R(t_{i})$ from
the data on reported infections $I_{i}$. The interpretation of $R(t_{i})$
becomes evident after simply rewriting (2) as

$I_{i}=R\left(t_{i}\right)\sum_{j=1}^{n}w_{j}I_{i-j}.$ (3)

Thus, from (3), the expected number of new infections $I_{i}$ at day $i$ is
given by the product of the effective reproduction number $R(t_{i})$ and the
total infectivity at day $i$. The total infectivity is the sum of the number
of new infections reported in the past days weighted by the generation time
distribution. The interval of past days considered in the sum depends on the
length $n$ of $\left\\{w_{j}\right\\}_{j=1}^{n}$. To simplify notation,
hereafter we refer to $\left\\{w_{j}\right\\}_{j=1}^{n}$ only as $\mathbf{w}$.
Moreover, we will use $I_{t_{1}:t_{2}}$ to denote
$I_{t_{1}},I_{t_{1}+1},\dots,I_{t_{2}}$, $t_{1}\leq\ t_{2}$.

In epidemiological modeling, reproduction numbers can also be directly
associated with expressions involving parameters of deterministic or
stochastic compartmental models, e.g., the infection and recovery rates used
in the SIR model (Britton and Pardoux, 2019). Thus, when fitting models with
epidemiological data, reproduction numbers can also be inferred from the
fitted parameters of each model. However, parameter estimation for such models
can be challenging, since it may require data that is not easily available,
other than just the number of reported infections (i.e., data on the number of
susceptible individuals, number of recovered individuals, etc.).

In general, the data collected during an epidemic outbreak is incomplete.
Particularly in the case of the COVID-19 pandemic, the number of daily
reported infections or deaths is commonly the only available information that
can be used to infer epidemiological parameters, such as reproduction numbers.
Other sources of data, such as the number of hospitalizations and the positive
test rates may also be available, depending on the country. During the
COVID-19 pandemic in Brazil, due to the lack of a standard policy for random
testing of a large and diverse population, as well as the absence of a unified
system to collect hospitalizations data all over the country, this information
was either unavailable or difficult to use (e.g. due to high variability).
Under such conditions, a model that relies only on daily reported infections
(2) becomes a useful tool to estimate the effective reproduction number.

###  II-B Estimating the effective reproduction number

In this paper, we consider the effective reproduction number estimator
proposed in (Cori et al., 2013), which requires only information about the
generation time distribution and the data on reported infections. This
estimator is derived under the following assumptions:

  1. 1.

The generation time distribution is independent of the calendar time.

  2. 2.

Transmission of the disease is described as a Poisson process, i.e., the
number of new infections at time $t_{i}$ is Poisson distributed with mean
$m=R(t_{i})\sum_{j=1}^{n}w_{j}I_{i-j}$. The likelihood of the number of new
infections $I_{i}$, given $R(t_{i})$ and the set of all previous observed
number of infections $I_{0:i-1}$, is written as

$P\left(I_{i}|I_{0:i-1},\mathbf{w},R(t_{i})\right)=\frac{\left(R(t_{i})\Lambda_{i}\right)^{I_{i}}e^{-R(t_{i})\Lambda_{i}}}{I_{i}!}$
(4)

where $\Lambda_{i}=\sum_{j=1}^{n}w_{j}I_{i-j}$.

  3. 3.

$R(t)$ is constant over time periods $[i-\tau+1;i]$ of length $\tau$, which is
indicated by the notation $R_{i,\tau}$. To simplify the inference process, a
prior distribution for $R_{i,\tau}$ is chosen such that it is a conjugated
prior for the Poisson distribution, i.e. a Gamma distribution.

Given the assumptions above, the likelihood of the incidence $I_{i-\tau+1:i}$
during this time period given $w$, $R_{i,\tau}$ and $I_{0:i-\tau}$ is

$P\left(I_{i-\tau+1:i}|I_{0:i-\tau},\mathbf{w},R_{i,\tau}\right)=\\!\\!\\!\\!\\!\\!\prod_{j=i-\tau+1}^{i}\\!\\!\\!\\!\\!\\!\frac{\left(R_{i,\tau}\Lambda_{j}\right)^{I_{j}}e^{-R_{i,\tau},\Lambda_{j}}}{I_{j}!}$
(5)

Adopting a Bayesian setting, by assuming a Gamma distributed prior for
$R_{i,\tau}$ with parameters $(a,b)$, i.e.,
$R_{i,\tau}\sim\Gamma\left(a,b\right)$, the posterior joint distribution of
$R_{i,\tau}$ is given in (6).

$\small
P\left(I_{i-\tau+1:i},R_{i,\tau}|I_{0:i-\tau},w\right)=P\left(I_{i-\tau+1:i}|I_{0:i-\tau},w,R_{i,\tau}\right)P\left(R_{i,\tau}\right)\normalsize$
(6)

where

$\small
P\left(R_{i,\tau}\right)=\frac{R_{i,\tau}^{a-1}e^{-R_{i,\tau}/b}}{\Gamma(a)b^{a}},\normalsize$
(7)

$P\left(I_{i-\tau+1:i}|I_{0:i-\tau},w,R_{i,\tau}\right)$ is given by (5) and
thus

$\small P\left(I_{i-\tau+1:i},R_{i,\tau}|I_{0:i-\tau},w\right)\propto
R_{i,\tau}^{a+\sum_{j=i-\tau+1}^{i}I_{j}-1}e^{-R_{i,\tau}\left(\sum_{j=i-\tau+1}^{i}\Lambda_{j}+\frac{1}{b}\right)}\prod_{j=i-\tau+1}^{i}\frac{\Lambda_{j}^{I_{j}}}{I_{j}!}.\normalsize$
(8)

Finally, it follows from (6) that the posterior distribution of $R_{i,\tau}$
is also a Gamma distribution, given by

$R_{i,\tau}\sim\Gamma\left(a+\sum_{j=i-\tau+1}^{i}I_{j},\frac{1}{\frac{1}{b}+\sum_{j=i-\tau+1}^{i}\Lambda_{j}}\right)$
(9)

The posterior mean $\mu$ and the posterior variance $\sigma^{2}$ of
$R_{i,\tau}$ can be calculated as

$\displaystyle\small\mu$
$\displaystyle=\frac{a+\sum_{j=i-\tau+1}^{i}I_{j}}{\frac{1}{b}+\sum_{j=i-\tau+1}^{i}\Lambda_{j}}$
(10) $\displaystyle\sigma^{2}$
$\displaystyle=\frac{a+\sum_{j=i-\tau+1}^{i}I_{j}}{\left[\frac{1}{b}+\sum_{j=i-\tau+1}^{i}\Lambda_{j}\right]^{2}}$
(11)

Therefore, given a time series of reported infections, the expressions in (10)
and (11) can then be readily used to obtain estimates for $R_{i,\tau}$. It
should be highlighted that, via numerical simulations, the authors in (Cori et
al., 2013) have verified that this estimator does not present significant
sensibility to underreporting of infection cases when the rate of
underreported cases is constant over time. This is an important feature to
consider when applying this estimator to COVID-19 data since there is a large
fraction of asymptomatic infections that do not get reported. Nevertheless, in
this work, there is no attempt to address the problem of underreporting rates
of COVID-19 data in Brazil, meaning that the reporting rates are assumed to
remain constant over time (Paixão et al., 2021). This assumption is also
corroborated by the fact that during the pandemic the was no broad testing
policy implemented by the Brazilian government aiming to evaluate e.g. the
number of asymptomatic infections.

##  III Assessing the reported data of the COVID-19 pandemic in Brazil

Monitored epidemiological parameters are often taken into account by the
health authorities when establishing NPI policies to control an epidemic. A
few of those parameters, such as $R(t)$, can be monitored periodically, e.g.
from the data on daily reported infections. Hence, it is important to
understand the caveats and pitfalls of using such indicators to implement
policy at the national and sub-national levels, particularly for large
countries such as Brazil. In this section, we focus on analyzing data-based
estimates of $R(t)$ in national and different sub-national levels of Brazil
and discuss its possible implications for the way NPI policies are
implemented. Three different scenarios that may be relevant for the
implementation of NPI policies are considered:

  1. 1.

Surveillance of epidemiological parameters based on nationwide data.

  2. 2.

Surveillance of epidemiological parameters based on statewide data.

  3. 3.

Surveillance of epidemiological parameters based on national geographic
regions.

###  III-A Data analysis

The data on reported COVID-19 infections is obtained from the public
repository (Cota, 2020), which aggregates the data reported by several state
health authorities in different geographic scales. For the analysis, we
consider the data on reported infections in the interval from 20 May 2020 to
20 May 2021. During this period, most of the Brazilian population was
unvaccinated and a number of NPI measures were adopted to contain the
transmission rate of the virus. Hence, data reported within this period should
be mostly independent of the vaccination process. In the ideal scenario, the
data on reported infections would not be affected by e.g. the underreporting
of cases, delays from infection to report, and the damming of reported data.
However, in reality, the available data is subject to many, if not all, of
those variables at once. The results presented in this paper were obtained by
analyzing the raw official datasets available with statistics from the
COVID-19 pandemic in Brazil. No particular pre-processing was applied to e.g.
fix or remove peaks of reported infections artificially generated from the
damming of reported data. Thus, emulating the conditions of how the
information would be available to the health authorities during the pandemic.
The estimator model assumes only a fixed generation time distribution for the
whole period analyzed. However, different virus variants may be linked to
different generation time distributions (Hart et al., 2022).

The $R(t)$ values are calculated using the estimator discussed in Section
II-B, assuming a weekly time window, i.e., $\tau~{}=~{}7$ days. The length of
the time window is chosen to average weekly seasonality effects in the
reporting. Wider windows (e.g. $\tau~{}=~{}15$ days) could be used for the
same purpose at the expense of increasing the lag in the estimation process.
In fact, it should be noted that, even though ideally $R(t)$ would track the
dynamics of the epidemic in real-time, random or unknown delays in between
infection and reporting will eventually introduce lags in the estimation
process, such that estimates on $R(t)$ in practice would reflect the state of
the epidemic some time in the past.

Whenever mentioned, the correlation coefficient $r_{xy}$ (Papoulis and Pillai,
2002) used to evaluate correlation between time series $x$ and $y$ is defined
as

$r_{xy}=\frac{\sum_{k=1}^{N}\left(x_{k}-\bar{x}\right)\left(y_{k}-\bar{y}\right)}{\sqrt{\sum_{k=1}^{N}\left(x_{k}-\bar{x}\right)^{2}}\sqrt{\sum_{k=1}^{N}\left(y_{k}-\bar{y}\right)^{2}}}$
(12)

where $N$ is sample size, $x_{k},y_{k}$ are the individual sample points
indexed with $k$, $\bar{x}=\frac{1}{N}\sum_{k=1}^{N}x_{k}$ is the sample mean
of $x$, and analogously for $\bar{y}$. The times series were named according
to the country name and standard Brazilian state abbreviation names, i.e.,
$x,y\in\left\\{\text{BR},\text{AC},\text{AL},\text{AP},\ldots,\text{SE},\text{SP},\text{TO}\right\\}$.
The correlation coefficient is a measure of linear dependence between two time
series.

In Fig. 2-6, the estimated mean of $R(t)$ (lines) and the $95\%$ credible
interval (shaded area) as a function of the calendar time is depicted for the
most populated state of each of the major geographic regions of Brazil.
Additionally, the estimated mean of $R(t)$ for the entire country is plotted
for comparison. Overall, the curves show similar trends, which indicates a
degree of similarity between the $R(t)$ at the state and the country level.
Nevertheless, in all five plots, significant discrepancies are observed in a
few time intervals. Such differences could be a result of pandemic evolution
scenarios varying over time from state to state and/or issues related to the
reporting data policy (delays, damming, etc.) performed by each state. Note
that, due to the large number of cases reported, the credible intervals become
significantly narrow around the mean, as shown in the insets in each figure.
The larger the population of the estate, the narrower the credible interval,
as expected from the estimation model. One should be careful when interpreting
the credible interval since it informs the range of uncertainty of the $R(t)$
estimates given the available data and the selected model. In other words, it
does not reflect the uncertainty associated with the model itself. In
particular, the model of (Cori et al., 2013) does not account for uncertainty
due to underreporting, reporting delay, and overdispersion in the case counts.

Figure 2: $R(t)$ evolution in the estate of Amazonas (AM). Lines indicate the
estimated mean of $R(t)$ and the shaded area delimits the $95\%$ credible
interval. Figure 3: $R(t)$ evolution in the estate of São Paulo (SP). Lines
indicate the estimated mean of $R(t)$ and the shaded area delimits the $95\%$
credible interval. Figure 4: $R(t)$ evolution in the estate of Rio Grande do
Sul (RS). Lines indicate the estimated mean of $R(t)$ and the shaded area
delimits the $95\%$ credible interval. Figure 5: $R(t)$ evolution in the
estate of Bahia (BA). Lines indicate the estimated mean of $R(t)$ and the
shaded area delimits the $95\%$ credible interval. Figure 6: $R(t)$ evolution
in the estate of Mato Grosso (MT). Lines indicate the estimated mean of $R(t)$
and the shaded area delimits the $95\%$ credible interval.

(a) Correlation coefficient between the moving average (7-day) of reported
COVID-19 infection cases for each state with respect to the time series
obtained by applying the same metric to the whole country.

(b) Correlation coefficient between the mean estimated $R(t)$ for each state
with respect to the time series obtained by applying the same metric to the
whole country.

Figure 7: Heat map of the correlation coefficients of state versus country.
Figure 8: Relative percentage of reported COVID-19 cases per state and
relative population per state compared the overall data for Brazil. Figure 9:
Per state comparison of the correlation coefficients MA and $R(t)$ of each
Brazilian state w.r.t the national trends. The error bars in the $R(t)$
correlation coefficients indicate the $95\%$ credible interval.

For comparison, the 7-day MA of reported cases is assessed. The MA time series
is obtained by summing the number of reported infections in the last 7 days
and dividing it by 7. In Fig. 7(a), a heat map is presented depicting the
correlation coefficient of the 7-day MA curve of reported infections of each
Brazilian state with the curve for the whole country. Overall, a relatively
high correlation is observed between the time series of the country and of the
states. For the interval analyzed, it is noted that the MA of reported
infections in Brazil is highly correlated with the moving average of the
states of São Paulo (SP) and Minas Gerais (MG). That can be linked to the fact
that those are the two most populated Brazilian states, accounting for roughly
32% of the Brazilian population. To better observe this trend, one can analyze
the bar plots in Fig. 8. The green bars represent the relative percent
contribution of each state to the overall number of infections reported in the
country. The orange bars show the percent fraction of the Brazilian population
living in each state, according to data obtained from (Inst. Bras. de Geo. e
Estatística - IBGE, 2022). Assuming that every state is uniformly affected by
the pandemic, it would be expected that the overall number of reported
infections per state to be roughly proportional to its population. The results
in Fig. 8 seem to support that hypothesis with a few exceptions. Notably, in
the state of RJ, 5% of the country reported infection cases were reported.
However, in the same period, RJ has reported close to 13% of the total number
of COVID-19 deaths in Brazil.

In Fig. 7 and in Fig. 9, a comparison of the correlation coefficients of the
MA and the estimated mean of $R(t)$ is presented. Credible intervals obtained
by sampling time series from the posterior distributions of $R(t)$ were
included in Fig. 9 to account for the uncertainty in the calculation of the
correlation coefficients between the times series of $R(t)$. Note that the
states with wider credible intervals are mostly states with small populations
(e.g. RR, AP, SE, RN), because those tend to be the ones with higher
uncertainty in the $R(t)$ estimation. The widest credible interval is observed
for CE, which is also the state with the lowest correlation with the national
trend in terms of $R(t)$. On the other hand, there are also a few small states
with high correlation and narrow credible intervals (e.g. DF, AL, AC, ES. RO).
Hence, for those states, the evolution of $R(t)$ presented a higher degree of
similarity with the overall national trend. In general, it would be always
expected some degree of bias of the time series of MA and $R(t)$ at the
national level towards the time series of the most populated regions. From
this perspective, correlations between the time series of the monitored
variables in the the least populated regions with the national trends are more
informative to evaluate whether national surveillance will provide an
effective description of the epidemic evolution in these areas.

For the time period analyzed, the correlation of the MA data from states with
the national data was clearly biased by the population of each state, whereas
the correlation in the estimated $R(t)$ time series was less affected. In
particular, for the states of MA, RR, AP, and AM, while $R(t)$ provides
reasonable correlation levels, practically no correlation was observed in the
moving average. Thus, when comparing national with state data, those results
suggest that the MA of reported cases and the $R(t)$ estimates may provide
different levels of agreement. Hence, such differences need to be considered
when defining triggers for NPI policies based on those epidemiological
surveillance indicators.

###  III-B Analysis per geographic region

Brazil is divided into five major geographic regions: North (N), Northeast
(NE), South (S), Southeast (SE), and Center West (CO). In Fig. 10 \- 14, the
estimated mean of $R(t)$ as a function of the calendar time is shown for each
major Brazilian geographic region and for the country as a whole. The results
show a similar degree of correlation as seen in Fig. 2-6. The S region is the
one with the highest divergence from the national curve. The similarities
among the estimated $R(t)$ curves indicate that the national trends find close
correspondence with regional trends.

Figure 10: $R(t)$ evolution in Southeast region (SE). Lines indicate the
estimated mean of $R(t)$ and the shaded area delimits the $95\%$ credible
interval. Figure 11: $R(t)$ evolution in Northeast region (NE). Lines indicate
the estimated mean of $R(t)$ and the shaded area delimits the $95\%$ credible
interval. Figure 12: $R(t)$ evolution in South region (S). Lines indicate the
estimated mean of $R(t)$ and the shaded area delimits the $95\%$ credible
interval. Figure 13: $R(t)$ evolution in Center West region (CO). Lines
indicate the estimated mean of $R(t)$ and the shaded area delimits the $95\%$
credible interval. Figure 14: $R(t)$ evolution in North region (N). Lines
indicate the estimated mean of $R(t)$ and the shaded area delimits the $95\%$
credible interval.

##  IV Conclusions

In this work, the evolution of the effective reproduction number as a function
of the calendar time was analyzed for the COVID-19 pandemic in Brazil.
Different scenarios with national and sub-national data on reported infections
were considered. For the time interval analyzed, higher correlation levels
between national and subnational data were observed for $R(t)$ estimates when
compared to the moving average of reported infections. Hence, these results
suggest that the $R(t)$ is a more consistent metric to monitor the rates of
disease transmission across distinct national levels than the moving average
of the number of reported infections. In terms of NPI policies, the results
indicate that epidemiological surveillance at the national level is likely to
be more effective for large population states, since they present high levels
of correlation with the national trends. In contrast, different levels of
correlation and credible intervals were found for states with smaller
populations, indicating that those are likely to benefit more by establishing
policies based on their local data. Finally, the use of more advanced
epidemiological models that include e.g. sub-national population dynamics
could be beneficial towards obtaining more robust estimators.

## Acknowledgment

The authors thank the Programa de Pós-Graduação em Engenharia Elétrica (PPGEE)
and Departamento de Engenharia Elétrica (DEE) both from the Universidade
Federal de Campina Grande (UFCG) for providing the research infrastructure
used in the course of this investigation. This study was financed in part by
the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil
(CAPES) - Finance Code 001.

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