key: cord-0861761-wh2hphln
authors: Celaschi, S.
title: Quantifying Effects, Forecasting Releases, and Herd Immunity of the Covid-19 Epidemic in S. Paulo, Brazil
date: 2020-05-25
journal: nan
DOI: 10.1101/2020.05.20.20107912
sha: b78b4f010169814c97f4a02ef5b1a893a04b27e9
doc_id: 861761
cord_uid: wh2hphln

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 S. 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.

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 license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20107912 doi: medRxiv preprint

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.

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;

(I s ) Symptomatic Infected: Group of individuals who are infected, and account to the official number reported cases;

(I a ) 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) + I a (t) + I s (t) + R(t) = Constant At the initial exponential outbreak, let β o be the average number of contacts (per unit time) multiplied by de probability of transmission from an infected person. Let β(t) = β a (t) = β s (t) be the infection rate that models temporal Mitigation Policies (MP). Let (t) quantify the . CC-BY-NC-ND 4.0 International license It 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/10.1101/2020.05.20.20107912 doi: medRxiv preprint 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:

constrained to the following initial conditions: S(0) = N, E(0) > 0, I a (0) = 0 , I s (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-I a-I s-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) = β a (t) = β s (t),  =  a =  s , and  a = ½, and total number of infected people will be I(t) = (1- a ) I s (t) +  a I a (t). The set of ordinary differential equations (Eqn. 1 to 5) is reduced to the standard SEIRD model,

CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20107912 doi: medRxiv preprint (10) An extra equation (Eqn.10) has been added to account for fatalities to complete the SEIRD

are respectively daily numbers of susceptible, exposed, infected or active cases, removed or recovered individuals, and fatalities (deaths). S(t) 

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 R o , 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 R o 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 license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. 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 R o = 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) 

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. R o is the basic reproduction number. If the exponential growth rate r and the generation time distribution ɷ(τ) have been estimated, R o is readily determined from Eqn. (12) , as . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20107912 doi: medRxiv preprint

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) . [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 1 st 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 (ψ o , t o , t 1 , t 2 , , t), where ψ o , t o , t 1 , t 2 , and  are values set by the MPs considered in this report.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Accordingly, R(t)=ψ(t).R o becomes dependent on the confinement policy (Fig. 4b) .

The effectiveness of this policy may be quantified by a social distancing factor defined as SD = 1ψ(t). Furthermore, by lack of information about asymptomatic hosts, a value of 50% of the exposed hosts was assumed.

The So, was defined as the initial date to estimate the parameter ψ o , keeping all model parameters as previously estimated [10] . The transmission rate β(t) is reduced to approximately 52% of its original value β o , 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 license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20107912 doi: medRxiv preprint 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] . . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. 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. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. In order to quantify the MP imposed at the city of Sao Paulo, a sequence of 3 plots . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. 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 R o . 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] . . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20107912 doi: medRxiv preprint

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 P c [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 P c .

Under the deterministic SEIR model, P c = 1 -1/R eff , i.e., herd immunity threshold depends on a single parameter, the effective basic reproduction number R eff [14] . Where R eff = (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 R eff in the range of 1.1 < R eff < 6.6 [16] . From the previous values determined for r,  o , and  o, we obtained R eff = 3.0 ± 0.3, and P c = 0.67 ± 0.03. Again, in this study, R eff is restricted to symptomatic hosts only, i.e., 50% of the exposed ones [17] . As a result, the herd immunity threshold will be  a .P c = 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 license It 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/10.1101/2020.05.20.20107912 doi: medRxiv preprint 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.

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