The COVID-19 (SARS-CoV-2) Uncertainty Tripod in Brazil: Assessments on model-based predictions with large under-reporting

The COVID-19 pandemic is definitely the global crisis of our time. A Chinese scientist first identified the SARS-CoV-2 virus in humans in Wuhan, in the province of Hubei, China by December 2019. This virus causes severe acute respiratory syndromes which can become potentially fatal. By the end of June, the World Healthy Organization (WHO) estimated that the number confirmed cases was already reaching the order $10,000,000$, with over $490,000$ confirmed deaths.

Much more than presenting drastic effects on health systems, social and economical backlashes are already felt by many countries; this is especially evident in countries with larger social inequalities, such as Brazil. The effects of the virus on populations with poorer access to health systems and sanitation facilities are strikingly harder (Kabir et al., 2020; San Lau et al., 2020; Khachfe et al., 2020). The city of São Paulo shows a very illustrative example of these differences: the city hall released a technical note by the end of April¹¹1Secretaria Municipal da Saúde de São Paulo. Boletim Quinzenal Covid-19. April 30, 2020. Available at: https://www.prefeitura.sp.gov.br/cidade/secretarias/upload/saude/PMSP_SMS_COVID19_Boletim%20Quinzenal_20200430.pdf. stating that the observed mortality rate is $10$ times larger in neighborhoods of the city with worse social conditions and precarious housing.

In this paper, we consider the Brazilian COVID-19 pandemic context, as detailed by Werneck & Carvalho (2020). Brazil is currently facing many issues due to the SARS-CoV-2 contagion, such as the advance of the virus to farthest western cities, away from urban areas, where medical care is somehow less present. The country has $26$ federated states, which have been choosing different social distancing measures since mid-March²²2Throughout this paper, the Year/Month/Day notation is used.. Even though a strong public health system is available in Brazil, many states have been exhibiting near-collapsing conditions since May, with over $95\,\%$ of Intense Care Unit (ICU) hospital beds occupied with COVID-19 patients (Brazilian Federal Medicine Council, 2020). Furthermore, we note that the SARS-CoV-2 is currently posing great threat to indigenous communities, such as the Yanomami and Ye’kwana ethnicities³³3The Brazilian Socioenvironmental Institute (ISA, Instituto Socioambiental, see https://www.socioambiental.org/en) has released a technical note Ferrante & Fearnside (2020) which warns for the contagion of COVID-19 of up to $40\,\%$ of Yanomami Indigenous Lands, amid the states of Amazonas and Roraima and a long the border between Brazil and Venezuela, due to the presence of approximately 20000 illegal mining prospectors. Datasets regarding the COVID-19 spread amid indigenous communities are available in https://covid19.socioambiental.org.. Clearly, the situation is border-lining.

The first official death due to the SARS-CoV-2 virus in Brazil was registered in March $17$, while the first case was officially notified in February $26$. Through inferential statistics, Delatorre et al. (2020) acknowledge the fact that community transmission has been ongoing in the state of São Paulo since the beginning of February (over one month before the first official reports). This points to empirical evidence that the true amount of infected individuals, and possibly registered deaths, are actually very under-reported.

Due to absence of mass testing, Brazil is currently only accounting for COVID-19 patients with moderate to severe symptoms⁴⁴4As well as those deceased due to the SARS-CoV-2 virus.. People with mild or no symptoms are not being accounted for, since, in the beginning of the pandemic, the Ministry of Health oriented to stay home and not to seek medical help if one only displayed mild symptoms. Added to this fact, the SARS-CoV-2 virus shows itself as an asymptomatic contagion for a large number of individuals. For these reasons, the scientific community has been warning for a possibly huge margins of underestimated cases in Brazil (Silva et al., 2020; Rocha Filho et al., 2020; Delatorre et al., 2020). Some studies, such as EPICOVID19 (2020a), point out to the presence of over $700\,\%$ of under-reported cases. Furthermore, the daily reports (“datasets”) disclosed by the Brazilian Ministry of Health⁵⁵5These datasets comprise the number of infected and deceased patients at the given day., only give an impression of the virus contagion in the past, since, in average, a person exhibits acute symptoms only $20$ days of the moment of infection. Through statistical procedures, Paixão et al. (2020) have recently confirmed the empirical evidences that the margin of under-reported cases is quite large in the majority of Brazilian states. The state of São Paulo seems to be the one with less uncertainty regarding the number of deaths, because the data also partially incorporates those deceased due to severe acute respiratory syndromes even without COVID-19 testing.

Therefore, it seems evident that to account for uncertainty is essential to formulate adequate public health policies to address the SARS-CoV-2 virus spread. With respect to this context, this paper intends to model the role of uncertainty in this pandemic in such way that decision makers are able to plan more coherent, and adherent to reality, policies. It is worth mentioning that propositions to address the pandemic through recurrent social isolation periods has been recently assessed through optimal control in (Morato et al., 2020).

Table 1 summarizes the estimates available in the literature regarding the under-reporting levels and true mortality rates for Brazil. In this Table, we evidence the percentage increase of under-reports w.r.t. to reported deaths and cases, as evaluated by prior studies. Most of these works show that, in average, the number of infected individuals could be $3$ to $14$ times higher. Some studies from other countries point out that this number could reach up to $30$⁶⁶6Naomi Martin. Mass. official coronavirus count is 218, but experts say true number could be as high as 6,500. The Boston Globe. March 17, 2020. Available at: https://www.bostonglobe.com/2020/03/17/metro/mass-official-coronavirus-count-is-197-experts-say-true-number-could-now-be-high-6000/.. According to technical news-pieces disclosed by UOL (2020) and Globo (2020), the amount of under-reports in terms of deaths due to the SARS-CoV-2 virus ranges from 17% to 122%. Alves et al. (2020) and Gaete (2020) estimate the real mortality rate estimated for Brazil to be approximately 1.08% to 1.11%, but estimates from the research group EPICOVID19 (2020c, b) from random samples of the Brazilian population point out that this mortality could be ranging from 0.42% to 0.97%. Studies from other countries, such as Khan (2020) and Baud et al. (2020) show that this true mortality rate varies roughly from $1$ to $5\%$, but (Mallapaty, 2020) states that recent research are converging to the estimation of the true mortality rate to be between 0.5 to 1%. (Perez-Saez et al., 2020) find the true mortality rate to be approximately 0.64% for Geneva, Switzerland.

Bearing in mind the previous discussion, the main motivation of this paper is to present an adapted Susceptible-Infected-Recovered (SIR) model which inherently takes into account these uncertainty levels, considering the Brazilian COVID-19 context. Furthermore, the motivation is also to assess the role of uncertainty in the predictions cast with such models. We denote uncertainty as the amount of sub-notification with respect to infected and deceased individuals. Our approach comprises the following ingredients:

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  Firstly, we propose a new modelling scheme that incorporates a dynamic decaying parameter for the viral transmission rate. The dynamic decaying parameter for the transmission rate, adapted from Morato et al. (2020), considers that the government applies contagion mitigation measures (such as social isolation and incentives to use masks, which we refer as “pandemic policies” henceforth), which decreases the contagion spread dynamics.

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  Secondly, we develop an uncertainty measure with respect to infected and deceased individuals. The uncertainty is embedded to the optimization procedures used to determine the epidemiological parameters, in order to correct underestimates of infected and deceased individuals.

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  Thirdly, we use this adapted SIR models to make predictions for the Brazilian scenario regarding several different uncertainty sets, with short and long term forecast spans. By this, we are able to illustrate the role of under-reporting in the model response curves. Specifically, we study its effect upon peak of infections (in terms of amplitude and time shift), upon the total number of deaths, observed mortality rate, and model epidemiological parameters.

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  Finally, considering the uncertainty tripod, i.e. the strong link between under-reporting of infected and deaths and the true mortality rate, we extrapolate and suggest an alignment of the observed and the true mortality rate to infer on the level of uncertainty present on the measurements.

We note that this paper relates to a previous paper by the Authors (Bastos & Cajueiro, 2020), wherein SIR-like models present short and long term outlooks for Brazil. The previous work covered only the data from an early stage of the contagion, until March 30, 2020. Furthermore, the Authors used parametric variations of the parameters to introduce the uncertainty in the identified model, differing from the approach proposed herein.

Figure 1 shows the SIR model forecasts back in March 17, with respect to real data, considering variations for the transmission parameter $\beta$. We note that parametrically changing the transmission parameter is not enough to make the model fit to the real data. Therefore, a time-varying parameter is added to model the effect of pandemic policies, which meddle with the viral transmission rates. Such adapted model is able to account for the population response to governmental enacted policies. In practice, the pandemic policy can be understood as a feedback of endogenous variables, also depending non-observed time-varying factors (people can simply decide to relax quarantine measures, even if the a social isolation policy is still enacted). Through our simulations, we are able to replicate the real data with a fair amount of similarity, as shown in the sequel.

We organize this paper as follows. Section 2 presents the proposed SIR-adapted epidemiological model, which incorporates a new dynamic variable that describes the response of the population to enacted pandemic policies. In this section, we also discuss uncertainty modelling, in terms of the available datasets. Section 3 details the parameter estimation procedure, which inherently includes uncertainty. Section 4 shows the results in terms of parameter estimation and forecasts for the COVID-19 pandemic in Brazil. We present a through discussion on the achieved results and concluding remarks in Section 5.

Recent literature (Peng et al., 2020; Kucharski et al., 2020b) has demonstrated how the the infection rate and evolution dynamics of the SARS-CoV-2 virus can be adequately described by Susceptible-Infected-Recovered kind models. In this Section, we present the classical SIR model due to Kermack & McKendrick (1927), the new dynamic variable which models the population’s response to isolation policies (enacted by local governments) and discuss some remarks on data uncertainty.

The Brazilian Ministry of Health provides (daily) two useful data time-series which are used to evaluate (i.e. identify) the SIR model parameters: i) the total number of infected individuals, denoted

and ii) the total number of deaths, which is $D(t)$. In this paper, we use data from February 25, 2020, to May 31, 2020.

Parameters $\beta$, $\gamma$ and $\rho$ are computed according to the procedure adapted from Bastos & Cajueiro (2020), considering that the period for which pandemic policies had not yet been formally implemented (this period comprises the weeks from February 25, 2020 until March 22, 2020). We find that the recovery rate $\gamma$ is found to be roughly constant at $0.150876$, which is consistent with other findings in the literature (Read et al., 2020; Nishiura et al., 2020). We maintain this value for $\gamma$ throughout the following identification steps. Note that this parameters is inherent to the disease and time-invariant.

In order to estimate the parameters of the SIRD+$\psi$ and SIRASD+$\psi$ models, considering the remainder of the available data (from March 22, 202 onward), we proceed by performing a Least-Squares identification procedure, with respect to different levels of uncertainty ($\tilde{q}_{D}$ and $\tilde{q}_{I}$). These uncertainty levels are directly embedded to the datasets prior to the actual identification. Minimizing the square-error between the integrated variables and their real values is in accordance with regular identification methodologies (Bard, 1974; Brauer et al., 2019). Then, we proceed by following an hierarchical procedure, as done in Bastos & Cajueiro (2020): firstly, we estimate parameters of the SIRD model, then, assuming that the infected individuals in the SIRD model are the symptomatic individuals in the SIRASD model, we estimate the remaining parameters of the SIRASD model.

Our identification procedure also includes constraints ¹⁰¹⁰10If more restrictive bounds are used, we mention them explicitly. to the possible parameter values: $\beta,\beta_{S},\beta_{A}\in[1/20,2]$, $\gamma,\gamma_{S},\gamma_{A}\in[1/14,1/2]$, $\rho,\rho_{S}\in[0.001,0.2]$, $\alpha\in[0,1]$ and $\psi_{\infty}\in[0.0,1.0]$. These limits are in accordance with those presented by Werneck & Carvalho (2020) and Bastos & Cajueiro (2020).

Globally, the identification procedure for the SIRD+$\psi$ model resides in solving the following minimization problem:

We stress, once again, that a large number of different SIRD+$\psi$ and SIRASD+$\psi$ models are obtained through the identification procedures detailed in the prequel. Each one of these models is identified for different level of uncertainty ($q_{D}$ and $q_{I}$).

In this Section, we present the main results of our paper, which comprise the role that sub-notification uncertainty plays in the model-based predictions of the COVID-19 contagion, harshly affecting the outlooks for its evolution spread in Brazil. To have the available data with such large amounts of uncertainty is a very troublesome issue, since public health policies are currently derived through the available datasets, meaning that these policies may be equivocate, such as reducing social isolation policies before adequate time, and thus lead to possibly unwanted phenomena.

The following results comprise predicted forecasts with the SIRD+$\psi$ and SIRASD+$\psi$ models; they have been obtained using Python software.

In this paper, we discussed the Brazilian COVID-19 pandemic scenario, the effects of under-reporting regarding the number of infections and deaths due to the SARS-CoV-2 virus in the country. Through two SIR-like adapted models, which include the population’s response to measures of control of the pandemic (such as social isolation, the use of masks, etc.) and deaths due to the disease. Furthermore, we make a set of possible forecasts and evaluate how uncertainty meddles with the COVID-19 pandemic prediction curves. Specifically, we analysed how uncertainty affects the infection peak displacement in time and its amplitude, epidemiological parameters, the observed and real mortality rate evolution, and how infection and deaths uncertainty and the true mortality rate could be related.

We used two SIR-like models, SIRD$+\psi$ and SIRASD$+\psi$, which differ in two aspects. First, the SIRD$+\psi$ model considers that asymptomatic and symptomatic parameters are equal, that is, $\beta_{S}=\beta_{A}=\beta$ and $\gamma_{S}=\gamma_{A}=\gamma$ and that the mortality rate is applied in the whole infected population, while the SIRASD$+\psi$ model distinguishes symptomatic and asymptomatic parameters and applies the mortality rate to the symptomatic only. Second, the initial condition of the SIRD$+\psi$ model suggests that the infected class of individuals is actually split as $100q_{I}\%$ symptomatic and $100(1-q_{I})\%$ asymptomatic, while the initial condition of the SIRASD$+\psi$ model considers that the proportion of one symptomatic per $\displaystyle\frac{(1-q_{I})}{q_{I}}$ asymptomatic individual (as gives Eq. (9)).

It is a fact (and heavily discussed by recent literature) that the available datasets disclosed by the Brazilian Ministry of Health have large sub-notification margins. Brazil currently conducts roughly $4000$ tests per million inhabitants, which is one of the lowest rates in the world (Volpatto et al., 2020). Likewise, the number of deaths may be underestimated for the same reason. According to (COVID-19 Brasil Research Team, ), only $8\,\%$ of infections are reported; the real number of infected individuals are possibly up to $30$ times bigger than what is being disclosed by the authorities (Scabini et al., 2020; Volpatto et al., 2020).

Herein, we have tried to expose some essential insights regarding sub-notification and how to incorporate them to pandemic models; below, we summarize the main findings of this paper, enlightening the key points:

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  Since the spread of the SARS-CoV-2 virus is inherently complex and varies according to multiple factors (some which are possibly unmodelled and external), exact forecasts of the pandemic dynamics are not viable. Therefore, the correct procedure should be based on a recurrent model tuning (via identification), always taking into account the uncertainty margins. This measure would allow to present more coherent forecasts as time goes, since the uncertainty margins tend to decrease as the pandemic ceases (and as more testing is done). We remark that, in this paper, the uncertainty margins are assumed constant along the forecast horizons in order to tune/estimate the model parameters. In future works, the Authors plan in exploring the possible differences obtained when implying a dynamic behaviour for the uncertainty margins (asymptotic and decreasing).

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  The simulation forecasts indicate that the amount of uncertainty influences directly on the date of the infection peak, on the number of deaths and on the mortality rate of the decease.

  Higher levels of infection uncertainty anticipates the infected peak in time. Considering no death uncertainty, the symptomatic peak at July 31, 2020 for $\tilde{q}_{I}=7$ is anticipated to July 3, 2020 for $\tilde{q}_{I}=30$ (see Table 3), almost a month difference. This is followed by the corresponding decrease (increase) in the peak amplitude of symptomatic (asymptomatic) individuals.

  Higher levels of death uncertainty cause increases in the observed and true mortality rates. Considering a fixed amount of infected uncertainty ($q_{I}$ or $\tilde{q}_{I}$ is constant), the number of deaths and the mortality rate are higher if the death uncertainty is higher (see Figures 8 and 11). Also, considering a fixed amount of death uncertainty ($q_{D}$ or $\tilde{q}_{D}$ is constant), the number of deaths and the mortality rate are lower if the infected uncertainty is higher.

  Furthermore, there is a direct relationship between the uncertainty level and the observed mortality rates, which are in fact time-varying (Figure 11). The instant mortality rates are calculated using the number of deaths and the cumulative number of infected, while the true mortality rate is its asymptotic value. So depending in which stage of the epidemic we are at, estimations using population samples could vary, not only by the method itself, but also due to the natural evolution of the proportions of infected and deaths.

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  Higher uncertainty levels of asymptomatic individuals causes a decrease in the epidemiological parameters $\beta_{A}$ and $\gamma_{A}$ (Table 7). There are two possible interpretations for the transmission parameter: (i) according to (Kemper, 1978), asymptomatic individuals presumably have more contacts, since they do not have any symptoms and, therefore, do not do a self-induced quarantine; and (ii) according to (Robinson & Stilianakis, 2013), the transmission of the infection is more readily on symptomatic individuals due to physical signs of illness (coughing, sneezing, etc.), which outweighs this first factor. Since $\beta_{A}>\beta_{S}$, our results support the first hypothesis.

  That said, the transmission parameter $\beta_{A}$ is approximately 0.47 for $\tilde{q}_{I}=5$ and 0.45 for $\tilde{q}_{I}=30$, corresponding to 2.3% to 6.8% higher than the symptomatic transmission parameter¹²¹²12Considering $\beta_{S}=0.44$, then $\beta_{A}=0.47$ corresponds to $\displaystyle 0.03/0.44\approx 6.8\%$ and $\beta_{A}=0.45$ corresponds to $\displaystyle 0.01/0.44\approx 2.3\%$., meaning that asymptomatic individuals are more likely to transmit the disease since the greater number of contacts over-weights the probability of transmitting the disease due to physical signs of illness.

  The duration of the infection $1/\gamma_{A}$ is approximately $1/0.21\approx 4.8$ days for $\tilde{q}_{I}=5$ and $1/0.17\approx 5.9$ days for $\tilde{q}_{I}=30$, corresponding to 12% to 28.4% less than the average time to recovery of symptomatic individuals ($1/\gamma_{S}=1/0.15\approx 6.7$ days), meaning that the infectious period of the asymptomatic is smaller than the symptomatic one. This is consistent with other pandemic, such as the H1N1 pandemic (see Table 1 from Robinson & Stilianakis (2013) for $\gamma_{A}$ and $\gamma_{S}$, and corresponding references).

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  We find that the effect of under-reporting of the number of infected and deaths is related to the true mortality rate, which we call “COVID-19 under-reporting tripod”. Considering that there are three variables with uncertainty (the under-reporting of infected individuals, under-reporting of deaths, and the true mortality rate), if two of them are known (or measured empirically), the other can be inferred assuming a constant amount of uncertainty. Alternatively, if one is known, it is possible to infer a path for the other two. This is shown in Figure 10.

  This approach allows to align the observed to the true mortality rate in order to find the population uncertainty. (Mallapaty, 2020) mentions that many studies are estimating the true mortality rate to be in the range of 0.5-1%. Assuming that this is the true mortality rate, the number of times more infected than reported is about: 8-16 considering the deaths reported by the Ministry of Health; 10-20.5 considering 25% more deaths than reported; 12-24.5 considering 50% more deaths than reported; and 15.5-30 considering 100% more deaths than reported. Alternatively, considering 9 times more infected than reported, the true mortality rate varies from 0.87% to 1%, resulting in 0 to 13% more deaths than reported; if we consider 12 times more deaths than reported, the true mortality rate range is 0.67%-1%, and the death uncertainty range is 0-50%. Other analysis can be inferred directly from Figure 10. These results are consistent with estimated and empirical findings presented in Table 1.

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  We note that the level of uncertainty plays a significant role in shifting the Susceptible-Infected transmission curves of the contagion. Uncertainty not only influences the effective transmission rate per infectious contact, but also the recovery and mortality rates associated to the virus. It is possible to make a parallel between this paper and the recent study by (Hébert-Dufresne et al., 2020), which investigates how macroscopic patterns and social reinforcement of interacting contagions through SIR-like models; it is shown how social reinforcement can directly meddle with the contagion transmission rate along the spread regime. The results point to very similar behaviours of the SIR curves as to the influence of the uncertainty levels. We note, nonetheless, that uncertainty also shifts and increases the peak of these curves, which does not happen in the demonstrations presented in (Hébert-Dufresne et al., 2020).

We must stress, once again, that the used models have a series of limitations (such as unmodelled phenomena and disregarded transient behaviours), the available datasets are very imprecise, and also that the forecasting / model-based prediction problem has a lot of associated sensibility: it is composed of several coupled nonlinear differential equations, which heavily rely on initial conditions (and also contour factors due to the time-series behaviour implied via $\psi(t)$). Furthermore, the pandemic dynamics may vary abruptly if more intense health policies are adopted (or dropped) on the future. Therefore, we must recall that the reader that the results presented in this paper are qualitative. Our intentions in showing long-term predictions is not to provide perfect accurateness regarding the number of infection and deaths, but to show relevant phenomena regarding the levels of uncertainty. If more testing is performed, for example, the uncertainty levels tend to decrease and, thus, the forecast should also change. Also, if the elderly people were to be isolated from the virus in a more effective way, we would expect to see a decrease in death rate, even if the number of infected is increasing, since the mortality rate for this group of people is considerably higher than for people younger than 65 years, as shown by (Perez-Saez et al., 2020).

The Authors truly hope that the proposition herein formalised can serve to help determining adequate public health policies for Brazil.

D. O. C. and J. E. N. acknowledge the financial support of CNPQ under respective grants $302629/2019-0$ and $304032/2019-0$.