![Distribution of generation times. Our data was described by the Weibull distribution with mean = 0.89, median = 3.1 days, and peak value 2.3 days. Dates of symptom onset with intervals of exposure for both source and recipient (when available) were collected in [13] in order to select the best distribution.](https://www.researchgate.net/publication/341635043/figure/fig1/AS:895258778927104@1590457734280/Distribution-of-generation-times-Our-data-was-described-by-the-Weibull-distribution-with_Q320.jpg)
Available via license: CC BY-NC-ND 4.0
Content may be subject to copyright.
1
Quantifying Effects, Forecasting Releases, and Herd Immunity
of the Covid-19 Epidemic in S. Paulo – Brazil
S. Celaschi
CTI Renato Archer
www.cti.gov.br
May, 2020
Important Note: This preliminary report has not been certified by peer review. It should not be relied
on to guide health-related behavior, and should not be reported in news media as established
information.
Abstract: A simple and well known epidemiological deterministic model was selected
to estimate the main results for the basic dynamics of the Covid-19 epidemic breakout in the city
of São Paulo – Brazil. The methodology employed the SEIR Model to characterize the
epidemics outbreak and future outcomes. A time-dependent incidence weight on the SEIR
reproductive basic number accounts for local Mitigation Policies (MP). The insights gained
from analysis of these successful interventions were used to quantify shifts and reductions on
active cases, casualties, and estimatives on required medical facilities (ITU). This knowledge
can be applied to other Brazilian areas. The analysis was applied to forecast the consequences
of releasing the MP over specific periods of time. Herd Immunity (HI) analysis allowed
estimating how far we are from reaching the HI threshold value, and the price to be paid.
Introduction
This work aims to shed some extra light in understanding the dynamics of the COVID-
19 epidemics in Brazil, particularly in the city area of São Paulo with its 12.2 million
inhabitants. The first patient in Brazil was tested positive in São Paulo who had returned from
Italy. Since then, were officially confirmed 233.142 cases, and 15,644 deaths in Brazil (May, 16
2020), and 61,183 cases, and 4,688 deaths in S. Paulo. The public response to the pandemic has
been the introduction of mitigation policies to ensure quarantine social distancing, such as
closing schools, restricting commerce, and home office. No country knows the true number of
people infected with COVID-19. All is known is the infection status of those who have been
tested. The total number of people that have tested positive – the number of confirmed cases – is
not the total number of people who have been infected. The true number of people infected with
COVID-19 is much higher. The number of those who have been tested positive (by May 1st) in
Brazil, 0.46/thousand, is very low compared to other countries [1] (Fig. 1).
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
2
Figure 1. Data (relative to May, 1st) collected from www.ourworldindata.org [1].
The response to a new pandemic, such as Covid-19, can be based on four major actions:
1) surveillance and detection; 2) clinical management of cases; 3) prevention of the spread in
the community; and 4) maintaining essential services. Actions across the four pillars
complement and support one another. In principle if the virus is left to infect people without any
containment measure, the population may acquire immunity in one semester or less. However,
hospital intensive care services would lack the capacity to deal with the sudden, large inflow of
severely ill people resulting in a very large number of deaths. Mitigation strategies aim to slow
the disease, and to reduce the peak in health care demand. This includes policy actions such as
social distancing, lock-down determination, and improved personal and environmental hygiene.
Studies as this presented here, consistently conclude that packages of containment and
mitigation measures are now days an effective approach to reduce the impact of the Covid-19
epidemic. Epidemiological models are commonly stochastic, diffusive-spatial, network based,
with heterogeneous sub-populations (meta-population approaches) [2, 3, 4]. However, the
parameters of dynamical and deterministic models, such as SIR and SEIR, are more directly
related to and interpretable as physical processes [5, 6]. On the other hand, deterministic models
impose restrictive analysis, once the dynamics of the host population and the virus are not
deterministic. The population has free will, and the virus undergoes “random” mutations.
The intent of this work was to build a simple epidemiological tool to estimate the main
results for the basic dynamics of the Covid-19 epidemic breakout. The methodology employed
is the application of the deterministic and discrete SEIR Model to characterize the Covid-19
0,1
1
10
100
Test per 1,000 people
Total Covid-19 Tests per 1,000 people
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
3
outbreak in São Paulo – Brazil. The model accounts for the following 5 groups: Susceptible,
Exposed, Symptomatic infected, Asymptomatic infected, and Removed or recovered. A time-
dependent incidence weight on the SEIR basic reproductive number R(t) was used to account
for dynamical transmission behavior to model and quantify local Mitigation Policies (MP). The
insights gained from analysis of these successful interventions can be used to predict results for
the MP of other Brazilian regions.
Recent published data [7, 8] from January, 1st, 2020 to May 8, 2020 was used to adjust
all the model parameters aiming to forecast the evolution of the COVID-19 pandemic in the city
of Sao Paulo - Brazil. The model provides predictions of the time series of infected individuals
and fatalities in the area studied. Simulations of midi-term scenarios of the epidemic outbreak
were done dependent on the level of confinement policies. Forecasts show how confinement
policy alters the pattern of contamination, and suggest the existence of post epidemic periods.
Application of mathematical models to disease surveillance data can be used to address
both scientific hypotheses and disease-control policy questions [9]. Such models have been used
to estimate the demand for hospital beds, ICU days, number of critical equipments, and the need
and required extent of governmental intervention. A model is only as good as the assumptions
put into it. Clearly, there are phenomena of the COVID-19 epidemic that are not yet
understood. Models are constantly being updated and improved.
The SARS-CoV-2 is a membrane protected, single-stranded RNA virus. It is commonly
referred to by the name of the disease it causes, which is COVID-19. The incubation period is
defined as the time between infection and onset of symptoms. It is estimated as the time
between exposure and report of noticeable symptoms. Currently, the incubation period for
COVID-19 is somewhere between 2 to 5 days after exposure [13]. More than 97 percent of
people who contract SARS-CoV-2 show symptoms within 12 days of exposure. For many
people, COVID-19 symptoms start as mild symptoms and gradually get worse over a few days.
Other large and unknown fraction of exposed people is asymptomatic. The story of the COVID-
19 outbreak is ongoing. Our knowledge of this novel virus is in a state of flux. Every week
seems to bring additional important medical and epidemiological information.
COVID-19 has a latent or incubation period, during which the individual is said to be
infected but not infectious. Members of this population in this latent stage are labeled as
Exposed (but not infectious) here on. The model with the Susceptible, Exposed, Symptomatic,
Asymptomatic, and Removed groups is our SEIR Model. During the initial 20 days exponential
phase of growth, the SP data shows that in 2.3 ± 0.1 days the number of symptomatic infected
people doubles. During this initial exponential period, the model confirms the predicted values
for the incubation, and immunization periods of time.
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
4
The SIR and our SEIRD Models
The SIR model [5] is one of the simplest compartmental models, and many other
models are created from this basic formulation. The model consists of three compartments: S for
the number of susceptible, I for the number of infectious or active cases, and R for the number
of recovered, deceased (or immune) individuals. This model is reasonably predictive for
infectious diseases that are transmitted from human to human. In epidemics or pandemic
outbreaks, the numbers of susceptible, infected and recovered individuals varies with time (even
if the total population size remains constant). For a specific disease in a specific population,
these functions may be worked out in order to predict possible outbreaks and bring them under
control. For many important viral infections, there is a significant incubation period during
which individuals have been infected but are not yet infectious themselves. During this period
the individual is in a new compartment E (for exposed) as in the SEIR model.
SIR Model. Arrows label the transition rates between compartments.
Our S-E-Ia-Is-R version of the SEIR model [10] describes the spread of a disease in a
population split into five nonintersecting groups:
(S) Susceptible: The population that can be exposed to the disease;
(E) Exposed: Group in the latent or incubation stage exposed to virus but not infectious;
(Is) Symptomatic Infected: Group of individuals who are infected, and account to the
official number reported cases;
(Ia) Asymptomatic Infected: Group of individuals who are infected, and do not account
to the official number of cases reported;
(R) Removed: Group of individuals who are recovered from the disease.
Due to the evolution of the disease, the size of each of these groups change over time
and the total population size N is the sum of these groups
N = S(t) + E(t) + Ia(t) + Is(t) + R(t) = Constant
At the initial exponential outbreak, let βo be the average number of contacts (per unit
time) multiplied by de probability of transmission from an infected person. Let β(t) = βa(t) =
βs(t) be the infection rate that models temporal Mitigation Policies (MP). Let
(t) quantify the
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
5
MP in terms of social distancing plus the use of personal protective equipment (PPE). Let
o be
the rate that exposed individuals get infected. Let
a and
s be the removed rates, which are the
rates that infected individuals (symptomatic and asymptomatic) recover or die (only
symptomatic), leaving the infected groups, at constant per capita probability per unit of time.
Let
a being the fraction of asymptomatic individuals. Let N be the susceptible population
considered in the study. Based on these definitions, we can write the model as:
(4)
(5)
constrained to the following initial conditions: S(0) = N, E(0) > 0, Ia(0) = 0 , Is(0) = 0, R(0) = 0.
By the time this study was conducted, information on the number of asymptomatic
infected individuals was unknown. So, some parameters could not be determined in order to
fully apply our S-E-Ia-Is-R version of the SEIR model. Later on, this restriction may be removed
once the required information on Covid-19 asymptomatic hosts will be reported by testing an
expressive fraction of the population. The following assumptions are established in order to
continue the analysis: β(t) = βa(t) = βs(t),
=
a =
s, and
a = ½, and total number of infected
people will be I(t) = (1-
a) Is(t) +
aIa(t). The set of ordinary differential equations (Eqn. 1 to 5)
is reduced to the standard SEIRD model,
(6)
(7)
(8)
(9)
t
Ist( )
d
d
st( )Ist( )1a
o
E t( )
t
R t( )
d
d
at( ) Iat( ) st( ) Ist( )
(1)
(2)
(3)
t
S t( )
d
d
at( )Iat( )S t( )
N
st( ) Ist( )S t( )
N
t
E t( )
d
d
at( ) Iat( ) st( ) Ist( )
S t( )
N
oE t( )
t
Iat( )
d
d
at( )Iat( ) ao
E t( )
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
6
(10)
An extra equation (Eqn.10) has been added to account for fatalities to complete the SEIRD
Model. As before, S(t), E(t), I(t), R(t), and F(t) are respectively daily numbers of susceptible,
exposed, infected or active cases, removed or recovered individuals, and fatalities (deaths). S(t)
+ E(t) + I(t) + R(t)+ F(t) = N = Constant. The constant N assumption is very restrictive, and
limits model’s coverage. Releasing this assumption goes beyond the scope of this study. RN(t) =
β(t)/
o . RN(t=0) = Ro, is defined as the Basic Reproductive Number. This number quantifies the
expected number of new infections (these new infections are sometimes called secondary
infections) that arise from a typical primary case in a completely susceptible population N,
where all individuals are susceptible.
Methodology
Official data, from March 1st to May 15, 2020, provided by the Ministry of Health of
Brazil [7], and Sao Paulo government [8] was considered to estimate part of the epidemiological
parameters that govern the dynamics established by Eqs. (6), (7), (8), and (9). By the lack of
information about the asymptomatic individuals, the mortality rate in the model is evaluated
over the symptomatic ones. All model parameters were estimated by minimizing the mean
squared quadratic errors.
A key parameter in deterministic transmission models is the reproductive number Ro,
which is quantified by both, the pathogen and the particular population in which it circulates.
Thus, a single pathogen, like the SARS-CoV-2, will have different Ro values depending on the
characteristics and transmission dynamics of the population experiencing the outbreak. When
infection is spreading through a population that may be partially immune, it has been suggested
to use an effective reproductive number R, defined as the number of secondary infections from a
typical primary case. Accurate estimation of the R value is crucial to plan and control an
infection [11].
The methodology to estimate R follows: The exponential growth rate of the epidemic, r
was obtained from the early stages of the epidemic in Sao Paulo, such that the effect of control
measures discussed later will be relative to post stages of this outbreak. This assumption is
implicit in many estimative of R. The growth rate r = 0.31 ± 0.02 of infected people was
estimated applying the Levenberg-Marquardt method [12], to data of symptomatic infected
people (Fig. 2), during the first 15 days of exponential growth according to the expression I(t) =
Io.exp(-r.t).
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
7
Figure 2. Exponential fitting of the initial growth of symptomatic infect people. The growth rate r
= 0.31 ± 0.02 was estimated applying the Levenberg-Marquardt method.
The time period required to double the number of symptomatic cases is straightforward given
by ln(2)/r = 2.3 ± 0.1 days. The basic Reproductive Number Ro = 2.53 ± 0.09 was estimated
according to; “In an epidemic, driven by human-to-human transmission, whereas growing
exponentially, in a deterministic manner, the incidence I(t) can be described by the Renewal
Equation”, or the Lotka–Euler equation [13, 14]:
(11)
Where (τ) is the mean rate at which an individual infects others a time after being infected
itself. Substituting into Equation (11) an exponentially growing incidence, I(t) = Io.exp(r.t),
Io=1, gives the condition,
(12)
where
(13)
ɷ(τ) is the generation time distribution, i.e. the probability density function for the time
between an individual becoming infected and their subsequent onward transmission events. Ro
is the basic reproduction number. If the exponential growth rate r and the generation time
distribution ɷ(τ) have been estimated, Ro is readily determined from Eqn. (12), as
(14)
S(t) = 1.73xe0,31.t
R² = 0,974
0
50
100
150
200
250
0 5 10 15
Symptomatic Infected S(t)
Days after March 1st
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
8
The term that appears in the right-hand side of this equation is the Laplace transform of the
integrand function. More specifically, it is known as the Momentum Generating Function of this
distribution. As in [13], a normalized Wiebull generation distribution is adopted (Eqns. 15, 16
and 17), with mean = 0.89, median = 3.1, and peak value at 2.3 days (Fig. 3).
(15)
(16)
(17)
Figure 3. Distribution of generation times. Our data was described by the Weibull distribution
with mean = 0.89, median = 3.1 days, and peak value 2.3 days. Dates of symptom onset with intervals of
exposure for both source and recipient (when available) were collected in [13] in order to select the best
distribution.
In short, the values of the parameters governing SEIRD model are:
(t),
,
o,
, and N.
The SEIRD dynamics is constrained to the following initial conditions S(0) = N, E(0) > 0 , I(0)
= 0, R(0) = 0. March 1st 2020 was considered the first day (day zero) to model the epidemic
outbreak at the city of Sao Paulo. As proposed by Bastos et al. [10], the temporal impact of the
confinement policy was considered weighting the initial transmission factor
o by ψ(t) fitted to
data, and adjustable to allow releases of the MP (Fig. 4a) [7,8]. This leads to factor as a
temporal function
(ψo, to, t1, t2,
, t), where ψo, to, t1, t2, and
are values set by the MPs
considered in this report.
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
9
(a) (b)
Figure 4. Quantification of the MP application on the city of S. Paulo by Social Distancing (SD)
measures (a), Reproductive Basic Number R(t) modeling the Social Distancing effective change after
March 22nd (b). Blue point and dashed line represent the progressive return to Ro after day 100.
Accordingly, R(t)=ψ(t).Ro becomes dependent on the confinement policy (Fig. 4b). The
effectiveness of this policy may be quantified by a social distancing factor defined as SD = 1 –
ψ(t). Furthermore, by lack of information about asymptomatic hosts, a value of 50% of the
exposed hosts was assumed.
Discussion and Forecasts
The fittings of data on symptomatic infected individuals, and fatalities to the SEIRD
model are show in Fig. 5. The fitting values are: Ro = 2.53,
o = 0.9,
o = 0.339, ψo = 0.525,
= 0.017,
o = 0.5, and t1 = 22. The fitting to the infected individuals (logarithmic scale)
presents standard deviation SD = 0.05 and root mean square RMS = 0.07. The exponential rate
of incubation 1/
o was assumed here as 2 days. Important to mention that the model was also
applied assuming 1/
o= 4 days [13], and fixed values for Ro = 2.53, ψo = 0.525,
= 0.017,
o =
0.5, and t1 = 22. The new fittings to data, preserving Ro (
o = 2.445, and
= 0.923), led
essentially to the same results.
Figure 5 shows reductions in exponential growths of infected and death rates after the
third week. This reduction clarifies the effectiveness of the confinement policy, when social
distancing took place. In fact, after day 22, the data shows reduction in the tax of transmission.
So, was defined as the initial date to estimate the parameter ψo, keeping all model parameters
as previously estimated [10]. The transmission rate β(t) is reduced to approximately 52% of its
original value βo, leading to a MP of 48% in accordance to the data shown in Figure 4. Final
values for asymptomatic infected individuals and fatalities, at day 190, are respectively 110,000,
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
10
and 10,500. Taking into consideration all the hosts, the average rate of lethality is 4.5 ± 0.5 %,
in agreement to recent published data [14a].
(a)
(b)
Figure 5. Official data on symptomatic infected individuals (a), and fatalities (b) are shown by
black circles, and diamonds respectively. Error bars account for the RMS values on the SEIRD
parameters. The fitting values to SEIRD model shown by red dashed line and error bars are: Ro = 2.53
± 0.09,
o = 0.913 ± 0.018,
= 0.316 ± 0.01,
= 0.017, ψo = 0.525 ± 0.03,
o = 0.50 ± 0.02, and tSD = 22
days. The mean fitting to the infected individuals (logarithmic scale) presents standard deviation SD =
0.05 and root mean square RMS = 0.07. Data points for the last 10 days (inserts) were added after the
model was complete, granting confidence on outcomes.
The major results obtained for the epidemic of COVID-19 in the city of S. Paulo,
maintaining the MP for the entire period, are: At least 4% of susceptible persons; Ro = 2.5, the
average number of symptomatic people infected by a sick person; Projected lethality rate ~ 5 %;
Projected symptomatic infected hosts 110,000: Projected fatalities 9,800, keeping 48% SD
constant.
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
11
Figure 6. Data on the daily number of new symptomatic infected compared to predictions of the SEIRD
model (a). The large data scattering on day to day-plus differences comes from the available reported
cases. The number of Intensive Care Units (ICU) was estimated to be proportional to the weekly number
of symptomatic infected (b). Permanence of SD after day 22nd was assumed. Error bars account for the
RMS values on the SEIRD parameters.
Figure 6 presents data on the daily number of new symptomatic infected, compared to
as predicted from modeling. The large data scattering on day to day-plus differences comes
from the available reported cases. The number of Intensive Care Units (ICU) was estimated to
be proportional to the weekly number of symptomatic infected. The proportionality constant in
Eqn. (18) was determined as the ratio of occupied ICU beds (SARG - COVID-19) to the
number of new infected during the 17th week, where SARG = 8469 [7].
(18)
Prediction on the average weekly number of intensive care units (ICU) is also included
in Fig. 6b by open blue circles. Error bars account for the RMS values on the SEIRD
parameters. The maximum weekly number of ICU beds for COVID-19 is predicted to be
approximately 24,000 by the first week of June, 2020.
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
12
Regarding fatalities, average data on daily new casualties is compare to model
prediction on Fig.7a. Do to large data scattering; a 5 days weight moving average was applied to
data on Fig. 7a. Keeping the MP, as modeled after March 22nd, the maximum number of new
daily fatalities is predicted to happen by the last week of May 2020. To compare this outbreak
regarding other countries, we present in Fig 7b the number of fatalities per thousand inhabitants
recently published [16].
(a)
(b)
Figure 7. (a) Data on daily new fatalities is compare to model prediction. Keeping the MP as
modeled after March 22nd. The RMS values to SEIRD model are shown by black error bars The maximum
number of new daily fatalities is predicted to happen by the last week of May 2020. (b) Number of
fatalities per thousand inhabitants reported from other countries. The fatality raise in the city of S. Paulo
is shown for comparison.
The number of susceptible individuals N = 500,000, which represents 4.1% the
population of Sao Paulo city, was considered to be the minimum number to fit the available data
on new symptomatic hosts. As early mentioned, the constant N assumption restricts our
analysis, and forecasts. Since the number of sub notifications is a reality well accepted, we
present in Table 1 suggested results folding N by a factor of three.
1,E-05
1,E-04
1,E-03
1,E-02
1,E-01
1,E+00
1,E+01
0 15 30 45 60 75
Fatalities per Thousand
Inhabitants
Days after 10 Deaths
SP-Brazil
Brazil
USA
Italy
UK
Doubles 1 day
Doubles 2 days
Doubles 3 days
doubles 4 days
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
13
Table 1. Folding N by a factor of 3, peaks the active cases2.5 weeks later, and raises the number
of immune hosts, ITU units, and fatalities by the same factor.
In order to quantify the MP imposed at the city of Sao Paulo, a sequence of 3 plots
presented by Fig. 8 demonstrate the effects of SD progressive releases imposed by state
regulations. The sequence shows the daily numbers of additional symptomatic infected, and
deaths. Social distancing of 48% was set as a constant from day - 22nd on. Selected values for
beginning a linear progressive SD release (t2), ending it (tend) to a final value of 24% are shown
in Figs. 8a, 8b, and 8c. The sequence forecasts additional daily numbers of symptomatic
infected (red line), and additional deaths (black line). The prospected accumulated additional
fatalities for t2 = {90, 100, 110}, and tend = t2 + 30 days are respectively 48, 36, and 32%.
(a)
(b)
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
14
(c)
Figure 8. The sequence of figures demonstrates effects of releasing the 48% social distancing set
constant from day - 22nd. Selected values: t2 - beginning progressive SD release; tend - end SD release,
final SD = 24%. The sequence forecasts additional daily numbers of symptomatic infected (red line), and
additional deaths (black line).
The analysis can also be applied to forecast the consequences of releasing the MP over a
longer period of time. Figure 9 illustrates this simulation by a linear and progressive MP release
starting by the end of the third month, and ending ten months later, when the reproductive
number returns to its initial value Ro. Surprisingly or not, the model suggests an “endemic”
outbreak of Covid-19 as shown in Fig. 9 by the presence of the second peak ~11 months after
the first one. The number of infected individuals is estimated to be 55% lower compared to the
first outbreak. This is a result of a partial reduction of 58% on the initial number of susceptible
individuals. As recently published by US National Library of Medicine, National Institutes of
Health, agencies worldwide prepare for the seemingly inevitability regarding the COVID-19, to
become endemic [15].
Figure 9. Result of the MP release over a longer period of time. Simulation by a progressive SD
release, starting by the end of the third month, and ending ten months later, as shown by R(t).
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
15
Herd immunity analysis
Acquired immunity is conquered at the level of the individual, either through natural
infection with a pathogen or through immunization with a vaccine. Herd immunity stems from
the effects on individual immunity scaled to the level of the whole population. It is referred to
the indirect protection from infection conferred to susceptible individuals when a sufficiently
large proportion of immune individuals exist in a population. Depending on the prevalence of
existing immunity to a pathogen in a population, an infected individual propagates the disease
through susceptible hosts, following effective exposure to infected individuals as described by
the SEIR model. However, if a percentage of the population has acquired some immunity level,
the likelihood of an effective contact between infected and susceptible hosts is reduced, and the
infection will not transmitted by this path. The threshold proportion of susceptible persons
required for transmission is known as the, or critical proportion Pc [15].
A relevant measure to evaluate the social cost of achieving global SARS-CoV2 herd
immunity is the use of the Causality Rate (CR), defined as the proportion of deaths caused by a
certain disease among all infected individuals. Now days in Brazil many Covid-19 cases are not
reported, especially among asymptomatic hosts or individuals with mild symptoms, the CR will
inherently be lower due to sub-notifications. It is important to remember that was established
50% asymptomatic hosts to the SEIR Model in this study. Massive serological testing will be
required to better determine how many individuals have been infected, how many are immune,
and how far we are from reaching the herd immunity threshold.
Within all those limitations, we can estimate a value for the herd immunity threshold Pc.
Under the deterministic SEIR model, Pc = 1 – 1/Reff, i.e., herd immunity threshold depends on a
single parameter, the effective basic reproduction number Reff [14]. Where Reff = (1 + r/
o).(1 +
r/
o), r is the rate of the initial exponential growth, o the exposed rate and o the rate to be
removed from the symptomatic infected group [14]. Since the onset of SARS-CoV-2 spread,
studies have estimated the value of Reff in the range of 1.1 < Re ff < 6.6 [16]. From the previous
values determined for r, o, and o, we obtained Reff = 3.0 ± 0.3, and Pc = 0.67 ± 0.03. Again, in
this study, Reff is restricted to symptomatic hosts only, i.e., 50% of the exposed ones [17]. As a
result, the herd immunity threshold will be a.Pc = 0.34 ± 0.03, and at least 35% of the
population considered here remains to be immunized. As commented before the number of
COVID-19 notifications do not include the asymptomatic hosts or individuals with mild
symptoms. As a result the number N of susceptible was fold by a factor of three, representing
12.3% of the Sao Paulo population. An extra fraction of 35% or, in the total, over 2 Million of
exposed individuals to SARS-CoV-2 are required to cross herd immunity threshold at the city of
S. Paulo.
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
16
Finally, given that the CR of COVID-19 estimated here is 0.23%, and preserving a
factor of three fold in the number N of susceptible hosts, 44,000 is estimated as the number of
people who could potentially die from COVID-19, whilst the population naturally reaches herd
immunity. This number is difficult to be accepted, so, before a new vaccine becomes available
reinforcements of mitigation policy become imperative.
References
[1] https://ourworldindata.org/covid-testing. Page visited on May 1st 2020.
[2] Zlojutro, A., Rey, D. & Gardner, L., “A decision-support framework to optimize border
control for global outbreak mitigation”, Nature Sci Rep 9, 2216 (2019).
[3] Wang, L., Wu, J.T., “Characterizing the dynamics underlying global spread of epidemics”.
Nature Commun 9, 218 (2018).
[4] Ruiyun Li, Sen Pei et. Al., “Substantial undocumented infection facilitates the rapid
dissemination of novel coronavirus (SARS-CoV-2)”, Science 01: Vol. 368, Issue 6490, pp. 489-
493, May 2020
[5] Roosa, K. and Chowell, G., “Assessing parameter identifiability in compartmental dynamic
models using a computational approach: application to infectious disease transmission
models”, Theoretical Biology and Medical Modelling (2019) 16:1
[6] Kissler, S. M., Tedijanto, C., Goldstein, E., ET. Al., “Projecting the transmission dynamics
of SARS-CoV-2 through the postpandemic period”, Science 14 Apr 2020.
[7] “Painel de casos de doença pelo coronavírus 2019 (COVID-19) no Brasil pelo Ministério
da Saúde”, https://covid.saude.gov.br/. Page visited on May, 3 2020.
[8] Coronavirus casos em SP https://www.seade.gov.br/coronavirus/. Visited on May, 18.
[9] Grassly, N., Fraser, C. “Mathematical models of infectious disease transmission”. Nature
Rev Microbiol 6, 477–487 (2008).
[10] Bastos S. B., Cajueiro, D. O., “Modeling and forecasting the Covid-19 pandemic in
Brazil” Research Gate, March 2020.
[11] Anderson R.M., May R.M. Oxford Science Publications; Oxford, UK: 1992. Infectious
Diseases of Humans: Dynamics and Control; p. 768. ISBN-10: 019854040X. ISBN-13: 978-
0198540403.
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
17
[12] Gill, P. R.; Murray, W.; and Wright, M. H. "The Levenberg-Marquardt Method" §4.7.3 in
Practical Optimization. London: Academic Press, pp. 136-137, 1981.
[13] Ferretti, L., Wyman, C., “Quantifying SARS-CoV-2 transmission suggests epidemic control
with digital contact tracing”, Science 31 Mar 2020.
[14] Wallinga, J., Lipsitch, M., “How generation intervals shape the relationship between
growth rates and reproductive numbers”, Proc Biol Sci. 2007 Feb 22; 274(1609): 599–604.
[14a] Li., L., et. al., “COVID
19 patients' clinical characteristics, discharge rate, and fatality
rate of meta
analysis”, J. Med. Virol., 12 March 2020. https://doi.org/10.1002/jmv.25757.
[15] Hunter, P., "The spread of the COVID-19 coronavirus: Health agencies worldwide prepare
for the seemingly inevitability of the COVID-19 coronavirus becoming endemic”, EMBO Rep.
2020 Apr 3;21(4): e50334.
[16] Hasell, J., et. al., “Research and data to make progress against the world’s largest
problems” https://ourworldindata.org/. Visited on May, 1st.
[17] Kwok, K., Lai, F., “Herd immunity – estimating the level required to halt the COVID-19
epidemics in affected countries”, J. Infect. 2020 Mar 21, 2020, preprint.
. CC-BY-NC-ND 4.0 International licenseIt is made available under a
is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 25, 2020. .https://doi.org/10.1101/2020.05.20.20107912doi: medRxiv preprint
