A modelling study on the vaccination against COVID-19 in the state of Rio de Janeiro, Brazil

9 The long-awaited roll-out of vaccination programmes against COVID-19 from across the globe has fuelled hope for a reduction in the incidence of cases and deaths, as well as the resumption of economic and social activities. Despite being the most effective measure to mitigate the pandemic, especially in regions where non-pharmaceutical interventions had been ineffective, many people suffered from the lack of efforts by government ofﬁcials to conduct vaccination. In Brazil, vaccination has always been cutting across party political and ideological lines, which have delayed the start of vaccination and brought the whole process into disrepute. Such disputes put the immunisation of the population in the background and create additional hurdles beyond the pandemic, mistrust and scepticism over vaccines. We conduct a mathematical modelling study to analyse the impacts of late vaccination and with slowly increasing coverage, as well as how harmful it would be if part of the population refused to get vaccinated or missed the second dose in the state of Rio de Janeiro, Brazil. The general framework we propose can be extended to analyse the epidemic situation in any region. Our results indicate that if the start of vaccination had been 30 days earlier, combined with efforts to drive vaccination rates up, about 18,000 deaths could have been averted. Furthermore, the slow pace of vaccination and the low demand for the second dose could cause a resurgence of cases as early as 2022.


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As of February 25, 2020, when the first case of infection with SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2) 12 was reported in Brazil 1 , the country has accumulated more than 18 million confirmed cases (https://ourworldindata. 13 org/covid-cases) and, on June 19, 2021, Brazil's death toll surpassed half a million. To date, upwards of ten per cent 14 of all cases in the world were identified in Brazil and, considering a seven-day rolling average, the country has had at least a 15 thousand deaths a day for more than 200 days since the onset of the epidemic. SARS-CoV-2 circulated undetected in Brazil 16 for more than a month 2 and, even after Brazil declared COVID-19 (coronavirus disease) a national public health emergency 17 on February 3, 2020 3 , the Brazilian government has managed the epidemic very loosely so far 4-6 , without a cooperative 18 effort and strategic planning to fight the pandemic. Brazil also faces many economic and socio-cultural challenges that affect 19 mitigation strategies, such as large disparity in the mortality rate in economically disadvantaged regions 7 , the uneven geographic 20 distribution of intensive care unit (ICU) beds 8,9 , and lack of investment and vulnerability of the health system 10 . Each federative 21 unit is self-governing for decisions regarding efforts to curb the spread of the disease 11 , which leads to inequalities, such as 22 unbalanced social distancing measures and lack of mass testing and viral spread tracking. 23 Despite the critical situation to contain the ensuing epidemic and the resurgence of cases (especially with the emergence of 24 new variants 12 ), Brazil had delays in starting the vaccination campaign, compared to other countries 13,14 , which took place 25 on January 17, 2021. Even with a slight increase in the pace of vaccination in recent weeks, vaccination efforts remain far 26 below what is required, with only 32.89% of the national population having received at least one dose by June 25, 2021. 27 In turn, the second dose began to be administered on February 5th, and since then only 11.91% of the population has been 28 immunised (https://ourworldindata.org/covid-vaccinations). To achieve full coverage of people aged 18 29 and over by the end of 2021, Brazil needs an average of 1.5 million doses of vaccine administered per day 15 . Currently, the 30 population benefits from vaccines from Pfizer-BioNTech, Oxford-AstraZeneca, Janssen, and Sinovac (the latter two approved 31 for emergency use up to the time of writing this paper).

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As it is a country with continental dimensions, the epidemiological situation in some states is particularly worrisome, due to especially regarding the federal government's rejection of vaccines from Pfizer in mid-2020 15, 21 , in addition to the rebuke 48 of the agreement signed with Sinovac 22 ; millions of people are also missing their second dose, either because of excessive 49 demand from the population concerning the national immunisation plan, where a number of doses may not have been reserved 50 for this purpose, or because of misinformation, assuming that just one dose provides the expected immunity 23, 24 ; temporary 51 interruptions of vaccination services, due to a lack of shots, logistical problems or absence of supplies (particularly active 52 pharmaceutical ingredient) [25][26][27] ; furthermore, there are on the one hand people who try to jump the queue to get vaccinated 53 early 28 , and on the other hand those who choose not to get vaccinated, seemingly motivated by political ideology 29 . 54 All these events potentially affect the Brazilian population, since they delay vaccination and bring into disrepute the actions 55 to help prevent the spread of COVID-19. Therefore, it is essential to investigate the likely consequences of such events and 56 circumstances regarding the burden of the epidemic. For this purpose, we conduct this study aiming at investigating the 57 following issues: 58 • What would be the influence of bringing forward or delaying the vaccination roll-out? 59 • How effective would a faster vaccination process be in mitigating the epidemic? 60 • How many deaths could have been averted if there had been more efforts to obtain and manage vaccines?

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• How harmful is the choice of part of the population for not getting vaccinated?

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• What is the effect of not taking the second dose of the vaccine on the population? 63 In this context, the objective of this work is to provide an analysis of scenarios related to the epidemic in Rio de Janeiro, in 64 order to answer the issues raised employing computational simulations whose results can be compared to the current situation 65 of the epidemic in the state.

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In this study, we consider that the target of individuals to be immunised in Rio de Janeiro is proportional to 80%, which also 68 corresponds to the number of inhabitants aged 18 years or over (https://www.ibge.gov.br/apps/populacao/ 69 projecao/index.html). Since the vaccination process is carried out with four vaccines, the vaccination rates associated 70 with each one of them is proportional to the number of doses granted to Rio de Janeiro by the Ministry of Health, as shown in 71 Fig. 1a. The simulations are conducted considering three scenarios related to the overall vaccination rate: the base scenario is 72 associated with the average vaccination rate at the time of writing this paper (considering the available data), that is, ν = 0.275% 73 of the population vaccinated per day. This corresponds to approximately 47,500 vaccinated individuals per day, which agrees 74 with the average of daily vaccinations. In two other hypothetical scenarios, we establish symmetric vaccination rates in relation 75 to the base scenario, with ν = 0.175% and ν = 0.375% of the population vaccinated per day. In this setting, approximately 76 30,200 and 64,800 individuals are vaccinated per day, on average, respectively. Figure 1b shows the frequencies of vaccination 77 rates taking into account both shots (single-dose vaccines count as second doses), given the cumulative number of individuals 78 vaccinated per day, which in turn is shown in Fig. 1c. For the base vaccination rate, the target vaccination coverage for the first 79 dose would be reached in approximately 290 days. This means that 80% of the population would have received at least the first 80 dose by November 2021, as supported by the prediction shown in Fig. 1c. As for the second dose, the prediction indicates that 81 the population would be immunised in the first months of 2022, respecting the interval between doses. For instance, adopting 82 ν = 0.275% means that nearly 0.136% of the population is vaccinated daily with the Oxford-AstraZeneca vaccine, on average.

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Additionally, the attributes related to each vaccine's efficacy and dosage (including the interval between doses) are listed in 84   Table 1.

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To perform the simulations, it is first necessary to infer the values of the free parameters of equation (3), whose model 86 outcomes best fit the regularised training data. Figure 1e shows the posterior distributions of parameters β 1 , . . . , β 4 , whose 87 statistics are detailed in Table 2. Of note, the posterior distributions can reasonably be expressed by normal distributions and 88 therefore the mean and MAP values (see Methods section) are quite similar. Figure 1d shows the behaviour of the function that 89 describes the transmission rate, given by equation (3), using the MAP values from Table 2, for the time period over which the   90 training data spam over. In the early stage of the outbreak, with more frequent contact between people and in the absence of 91 pharmaceutical interventions, the transmission rate was at a high level, gradually decreasing during the first wave of infections, 92 approximately until the end of July 2020. A further increase in the transmission rate led to the second wave, which remained at 93 a high level of transmission for months until its growth could be halted by the start of vaccination.

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Benefits and risks regarding the pace of vaccination The influence of the pace of vaccination on the mitigation of the 95 epidemic, in the matter of reducing the number of infected and dead individuals over time, is shown in Fig. 1f How the timing of vaccination roll-out affects disease mitigation Aiming to analyse how the epidemic would unfold in 108 Rio de Janeiro if vaccination had been rolled out at another time, we propose to consider hypothetical scenarios in which the 109 vaccination efforts get underway 10, 20 or 30 days before or after January 20, 2021. For each particular vaccination rate in this 110 analysis, we simulate the model for all combinations of proposed scenarios, whose outcomes are shown in Fig. 2a, concerning 111 the daily number of infected and dead individuals. In all results, left (←) and right (→) arrows denote anticipation or delay 112 in the start of vaccination, corresponding to the number of days that accompany the symbol, respectively. The grey shaded 113 area represents a six-month interval from the actual date the vaccination was started. For an arbitrary vaccination rate, visual 114 inspection of such results makes it clear that starting the vaccination campaign a few days earlier is beneficial both in terms of 115 "flattening the curves" and in terms of suppressing the epidemic. Take as an example the scenario in which ν = 0.275%. On  CI: 5,128), that is, a difference of more than 75% in relation to deaths 132 that would have been averted. 133 We also sought to directly relate the number of vaccinated and dead individuals, aiming to analyse the likely hardship 134 to the population when the start of vaccination is delayed, compared to the scenario in which vaccination had started earlier.

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Suppose vaccination had started on February 19, 2021, 30 days beyond the actual date, when 292 deaths (95% CI: 281-298) approximately 14% of the population. However, the worst-case scenario would bring out a far more ruthless possibility: even 140 with approximately 4,175,214 people immunised (95% CI: 4,174,177,014), deaths would peak in July 2021, reaching 141 703 deaths in a single day (95% CI: 607-789). This means that, despite having been vaccinated nearly 72% more people, 142 comparing both scenarios, a record-high daily death toll could have been reached, to a great extent driven by the late start 143 of vaccination. On the flip side, if there had been efforts to get vaccination started around December 21, 2020, even with 144 vaccination progressing at a slow pace (ν = 0.175%), the simulations in Fig. 2c show that deaths would peak at 182 (95% CI:

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Consider the transmissibility of SARS-CoV-2 in Rio de Janeiro in terms of the effective reproduction number, given by 147 equation (2). According to the simulations, in 2020 the effective reproduction number was only below the threshold R (t) = 1 148 between the end of June and September, as shown in Fig. 2d. In spite of that, in this period the lowest value reached was 149 R (t) = 0.977, at the end of July. Afterwards, the effective reproduction number was always above one, until the vaccination 150 started to take effect. We are interested in analysing the effective reproduction number, given the scenarios considered in 151 this work. So, at this point, assume that the vaccination had been brought forward by 30 days. On the same day as the concerns about the prevalence of the disease. Most public health experts admit that a herd immunity threshold is not attainable 163 (at least not in the foreseeable future) 30 , but immunising 50 to 90 per cent of the population could be enough to curb the 164 epidemic 31 . Aiming to analyse the adverse effects caused by people who are unwilling to be vaccinated, we simulate the model 165 considering only 50% vaccination coverage, to the detriment of the 80% coverage that we used to adopt. In this scenario, daily. In the same period, but assuming a slower vaccination rate, the daily number of confirmed cases would be 3,377 (95% CI: 171 2,793-3,942), when 34.51% of the population would be immunised. In addition, overall low vaccination coverage combined 172 with a lethargic immunisation program could raise the possibility of a resurgence of cases (and hence deaths) as early as 2022.

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After experiencing a reduction in the number of cases, to a large extent due to vaccination, in May 2022 there would be the 174 smallest number of infected individuals since the onset of the epidemic, 548 (95% CI: 320-826). However, in the following 175 months, the incidence of cases could increase again, reaching 867 new cases (95% CI: 415-1,512) per day by mid-August 2022.

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Such an epidemiological situation would resemble that which occurred in early May 2020, which could eventually be indicative 177 of a new outbreak.

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Thousands of people have also been missing their second dose of vaccine in Rio de Janeiro, further complicating a campaign 179 already marred by backwardness and supply shortages. To the best of our knowledge, there are still no studies that confirm the 180 overall efficacy of all vaccines used in Rio de Janeiro when only one shot is provided (except for the Janssen vaccine), although 181 some studies have already reported relevant results 32-36 . In the absence of such information, we assume two scenarios regarding 182 vaccine efficacies (see Table 1) when only the first shot is given, that is, efficacies are weakened proportionally to µ = 25% and 183 µ = 50%. Moreover, surveys show that around 14.5% of the Brazilian population somewhat disagree, strongly disagree or 184 remain neutral regarding vaccination 20 . Within this frame of reference, we also consider scenarios with low (α = 20%) and 185 moderate (α = 10%) demand for the second dose of vaccines (when applicable), as well as the best scenario in which α = 0%.

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Simulations for the number of dead individuals as of the actual day vaccination has started, combining factors associated with 187 parameters µ and α, are shown in Fig. 3d. Initially, assume that the first dose of vaccines would yield an efficacy proportional 188 to µ = 25% of the overall efficacy when both doses are given. In a scenario subject to slow vaccination, 50% of the eligible  197 More ambitious vaccination targets and avertable deaths All the aspects set out so far fall into the most dreadful scourge 198 of a pandemic, the ensuing deaths. Tallying deaths that could quite possibly have been averted, had there been more efforts for 199 an assertive vaccination campaign, means understanding the importance of having vaccines made available to the population as 200 early as possible. We attempt to infer deaths that could have been averted simply by having vaccination started days earlier or if 201 the daily rate of vaccination had been higher. Figure 3b shows the relationship between vaccinated individuals and cumulative 202 deaths over time. We simulate the model using the benchmark vaccination rate (ν = 0.275%) and compare the outcomes in the 203 context of a faster vaccination (ν = 0.375%), making allowance for different days for the start of vaccination from the day 204 it actually started. We recall that the right (→) and left (←) arrows correspond to the days of delay and anticipation of the 205 start of vaccination, respectively. In all simulations, the number of immunisations is the same, but the curves do not match, as 206 the pace and day of the start of vaccination change the day on which a certain vaccination coverage is reached. Simulations 207 show that presumably not-so-challenging measures, such as having anticipated the vaccination campaign roll-out by just 208 ten days, combined with an average vaccination rate approximately 36% faster, could have averted 7,300 deaths (95% CI: 209 6,683-7,775), whereas if the start of vaccination had also been delayed by 10 days, under the same circumstances, there could 210 have been 9,105 more deaths (95% CI: 8,268-9,803) in relation to the actual scenario; from a more optimistic, yet still realistic, 211 perspective on the vaccination roll-out, consider a 30-day advance on the date on which the campaign actually started. In this 212 framework, 18,041 deaths (95% CI: 16,659-19,110) could have been prevented, which represents 28.43% of the deaths (95% 213 CI: 27.46%-29.30%) that would have occurred since vaccination was started, assuming a vaccination rate equal to ν = 0.275%.

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When vaccination became available in Rio de Janeiro, 490,821 cases and 28,215 deaths had already been reported (on 215 that day, there were 4,015 new cases, with 189 deaths). Figure 3c shows that a hypothetical delay of 30 days in the start of 216 vaccination, when there were already 569,184 confirmed cases and 32,045 deaths (aware that these data already encompass compared to the worst-case scenario, assuming that ν = 0.275%. Such delays could also affect the prevalence of the disease, 223 causing the incidence of cases to be longer-lasting at high levels and, consequently, causing the peak of deaths to be shifted 224 forward. According to simulations, at the worst stage of the epidemic, there could be up to 394 deaths (95% CI: 368-413) a day 225 if there had been a 30-day delay in making vaccines available to the population, considering the benchmark vaccination rate, 226 shifting the peak two months ahead of what is expected without such a delay. Figure 3c supports the fact that delays in the start 227 of the vaccination campaign cause adverse effects that are more severe when the vaccination process is slower.  From the information in Fig. 1f, we see that bringing forward the vaccination in Rio de Janeiro could have anticipated 242 the epidemic peak, and also could have reduced the effective reproduction number to below the threshold R (t) = 1 on a date 243 before the actual scenario. On the other hand, had the start of vaccination been delayed, the adverse consequences could have 244 been disproportionately greater, as shown in Fig. 1f. The reason for this behaviour could be rationalised by an intrinsic feature 245 of the transmission, which in turn was captured by the simulations. One potential candidate is the mechanism of disease spread 246 following a power-law distribution 37, 38 , a behaviour that most epidemiology models can readily be modified to capture 39 .

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Although it is argued that power law is generally not appropriate for temporal spread 38 , some authors have employed this theory 248 to the spread of COVID-19 for both case and death data 40-43 , all relying on the evidence that disease containment measures 249 cause a sub-exponential increase of cases 44 . Another point to be brought out is that, although we have used fixed values for the 250 vaccine's efficacy, such values must be viewed with caution. Each efficacy is estimated considering different populations likely 251 to be subject to different prevalent variants and therefore any variability must be taken into account 45 . From a modelling point 252 of view, this could be addressed in future studies through population stratification.

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Spikes in the number of cases have also overwhelmed health services, largely due to a steep slowdown in vaccinations and 254 relaxed public health protocols 46 . Many municipalities in Rio de Janeiro are even worse off as they do not have ICU beds and 255 patients eventually need to be admitted to nearby hospitals. Even after the start of vaccination, for more than two months the 256 health care system continued to have an increase in the share of regular and ICU beds occupied, with peak occupancy in the first 257 week of April (see Extended Data Fig. 3), with 91.2% occupancy of ICUs and 80.9% of regular beds. These data are in line 258 with the simulations where the benchmark vaccination rate is adopted, which indicate the peak of daily infections on March 1, 259 2021 (see Fig. 1f). Shortage in vaccine supply also raises concern about the emergence of variants with potentially increased at workplaces, transit stations and shops in general, while moving to be more often in residential environments. However, 281 the population has always been resistant to most such measures, so much so that the number of people gathering indoors has 282 practically always been increasing since then. If we look at mobility with regard to the municipalities (see Extended Data 283 Fig. 4b), there seems to have been compliance with NPIs to some extent. At a time when there were 100,000 confirmed cases, 284 there were still far fewer visits to workplaces than at the beginning of the year, before the epidemic swept the country. More 285 recently, shortly after 1 million cases were confirmed (in mid-July 2021), vaccination has increasingly encouraged the return 286 to face-to-face work. Such factors combined with a slow-paced vaccination may mean that the resurgence of cases and the 287 emergence of new variants might not be so far on the horizon.

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The analysed scenarios reflect current knowledge about vaccination in Rio de Janeiro, from the perspective of available data.

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The persistence of such predictions depends to some extent on the confirmation of the hypotheses put forward. Particularly with 290 regard to vaccine hesitancy (whether for both doses or just the second), the inaction of certain people depends a lot on facts that 291 cannot be predicted. Despite this, social network posts can provide insight into attitudes and sentiments towards vaccination, Model description We extend the well-known SIR (susceptible-infected-removed) model 58 , aiming to incorporate the effects 303 of vaccination in the population. Initially, assume that β (t) is the transmission rate over time and γ is the removal rate. The 304 gain in the infective class (I) is at a rate proportional to the product of the contact rates and transmission probability between 305 infectives and susceptibles (S), that is, the rate of new incidences is given by β (t) S (t) I (t) /N, where N is the population size.

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In turn, the rate at which infected individuals move into the removed class (R) is given by γI (t). Of note, we also compute the 307 number of dead individuals, once infected, which are eventually moved into the dead class (D) at a rate of ρI (t), where ρ is the 308 death rate.

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Assume that both susceptible and infected individuals can be vaccinated (the latter are able to be vaccinated as they may be 310 asymptomatic). Considering that n vaccines can be administered in a population, individuals vaccinated with a given vaccine 311 i are moved into the corresponding vaccinated class (V i ) at a rate equal to ν i (S (t) + I (t)), where ν i is the vaccination rate 312 associated with vaccine i, for i = 1, . . . , n. Individuals remain in compartment V i for the period equivalent to the interval 313 between doses (when applicable), which is given by 1/τ i . After this period, vaccinated individuals are considered immune and 314 therefore moved into the removed class taking into account the efficacy of the corresponding vaccine, η i . If immunity is not 315 acquired with proper vaccination, vaccinated individuals may become susceptible again, whose class is fed back proportionally The model also covers two other aspects inherent to the vaccination process: first, part of the population eligible to be 318 vaccinated can choose not to take both doses of the vaccine (when applicable). In terms of vaccine efficacy, such individuals 319 have only partial protection, which we denote byη i , an impaired efficacy. In terms of the expected efficacy when both doses 320 are given,η i = µη i , where µ is the parameter that modulated the drop in efficacy; second, a number of eligible individuals 321 may decide not to get vaccinated. This portion of the population is denoted as α. Therefore, the rate of change of individuals 322 who take both doses of the vaccine (or the single-dose vaccine) is represented by the amount τ i η i (1 − α)V i (t), whereas for 323 those who take only the first dose (when two are foreseen), or choose not to get vaccinated, are expressed by τ iηi αV i (t).

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The susceptible class is also fed back proportionally to the value of α. The general description of the model is provided in 325 equation (1), the schematic representation is shown in Extended Data Fig. 1, and the summary of model parameters is given by 326 Extended Data Table 1.
Additionally, we employ the next-generation matrix method 59, 60 to derive the effective reproduction number expression, which 328 is given by For detailed derivation, refer to Supplementary Notes 1.

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Case incidence and vaccination data Daily data on confirmed cases and dead individuals due to COVID-19 in Rio 331 de Janeiro are divided into two subsets, from before and during vaccination. In the former, which we call the training 332 set, the data are in the range between March 10, 2020, the first day with at least five cases diagnosed, and January 19, The mean is often assumed to be zero (since the observed outputs can always be centred in order to have a zero mean).

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In a regression problem, the prior probability density of f (t) = ( f (t 1 ) , . . . , f (t p )) ⊤ has joint multivariate Gaussian 360 distribution f ∼ N (0, K (t, t, λ λ λ)), such that K (t, t, λ λ λ) is the covariance matrix (which is also noise-dependent), whose entries 361 are (K (t, t, λ λ λ)) i j = k (t i , t j , λ λ λ) +σ 2 δ i j , for i, j = 1, . . . , p, where λ λ λ is the set of kernel hyper-parameters and δ i j is the K (t, t, λ λ λ) +σ 2 I K (t, t * , λ λ λ) where I is the p × p identity matrix. Therefore, by deriving the conditional distribution 63 , the posterior predictive equation is 365 the multivariate Gaussian distribution As new pairs (t * , D * ) are incorporated into the regression problem, the mean set µ µ µ * is updated and adopted as the output of 369 the GPR model, whereas Σ Σ Σ * provides a measure of confidence in the estimate 66 . In this work, we adopt the RBF (radial basis Inference of model parameters Model outcomes are fitted to the training set using Bayesian inference. As the training data 375 is from the period prior to the start of vaccination, all model parameters that are associated with vaccination (α, ν i , τ i , η i , and 376η i , for i = 1, . . . , n) are set to zero at this point. In this setting, the model of equation (1) reduces to the SIR model (including 377 the dead class). As for the remaining parameters, we take the removal rate and the death rate as biological parameters, which 378 are equal to γ = 0.06 67 and ρ = 0.067 68 (a rate per day for both), respectively. In turn, regarding the transmission rate, we 379 adopt the functional form given by This specific choice is motivated by the fact that, in the particular time period over which the training data spam over, there 381 seems to be the incidence of two waves of infection. The contribution of the term associated with the negative exponential 382 would be able to represent the infection rate at an early stage when few individuals are immune, and the contact rate between 383 them leads to an increase in the incidence of cases until the peak of the first wave is reached. On the other hand, the contribution 384 of the term associated with the positive exponential would be related to a new increase in the infection rate after the event of the 385 first wave. Therefore, the parameters to be estimated are θ θ θ = (β 1 , β 2 , β 3 , β 4 ). p (θ θ θ) of the corresponding parameters (our prior belief about the distribution of θ θ θ), which we assume to be uniformly distributed, where 0 < ξ < 1 is a relative displacement. In this particular application, the prior distribution of β j is defined symmetrically 393 aroundβ j , for j ∈ {1, 2, 3, 4}, with ξ = 0.9. This strategy aims to bypass parameter identification problems 71 .

D(t)
as of Jan 20, 2021 ×10 2 μ = 50%, ν = 0.175% 1 2 / 0 2 / 2 1 2 4 / 0 3 / 2 1 0 3 / 0 5 / 2 1 1 2 / 0 6 / 2 1 2 2 / 0 7 / 2 1 3 1 / 0 8 / 2 1 1 0 / 1 0 / 2 1 1 9 / 1 1 / 2 1 2 9 / 1 2 / 2 1 0.0 0.6  Figure 3. Flawed vaccination policy and excess deaths. a, Model simulation where part of the population eligible to be vaccinated does not receive any dose. b, Ratio between the number of deaths given potential scenarios in which the start of vaccination is ahead of the actual date. Scenarios where vaccination would be implemented 10, 20, and 30 days before January 20, 2021 are considered, as well as two vaccination rates (ν = 0.275% and ν = 0.375%), and excess deaths are estimated. c, Variation in the cumulative number of deaths and the number of deaths at the peak of the epidemic curve (during vaccination) taking into account the start of vaccination on different days. The relative percentage amount of cumulative deaths is shown, as well as the month in which deaths would peak. d, Simulation considering that part of the population proportional to α does not take the second dose of the vaccine. Two scenarios are considered in which only the first dose of the vaccine has efficacy equivalent to µη, combined with two vaccination rates (ν = 0.175% and ν = 0.275%). Table 1. Characteristics of the vaccines used in the simulations, in terms of overall efficacy and interval between doses (when applicable). Of note, the recommended inter-dose interval for Pfizer-BioNTech vaccines is 21-28 days 75 . However, for countries that face a high incidence of COVID-19 cases and that have not yet achieved safe vaccination coverage rates, the World Health Organisation recommends that the interval between doses be extended to 12 weeks 76 , which has been adopted in all over Brazil.   Figure 1. Schematic representation of the model. The incidence of new cases is given by β (t) S (t) I (t) /N, where N is the population size. Both susceptible and infected individuals are vaccinated at a rate proportional to ν. Once infected, individuals are moved to the removed class at a rate proportional to γ, whereas the gain in the dead class is at a rate proportional to ρ. Vaccinated individuals can either be moved into the removed class or become susceptible again, respecting the interval between doses, 1/τ, when applicable. In the first case, immunised individuals are moved into the removed class proportionally to η, the overall vaccine efficacy. In the second case, the susceptible class is fed back proportionally to (1 − η). The portion of individuals who do not receive the second dose is equal to α. In this case, an overall impaired efficacy is assumed, given byη. In general, the model admits n classes of vaccinated, depending on the types of vaccines used.  The predicted average for each data set is adopted as training data in the Bayesian inference process. The shaded areas represent the 95% CIs of the GPR. We also plot the equivalent results for cumulative data in order to show the agreement between original and regularised data.