Fig 1.
Time series of Covid-19 in Brazil at the reported date on the national monitoring panel.
(a) new reported cases and (b) new reported deaths. Data from DATASUS [2].
Fig 2.
Stay-at-home index (ΔH) reported for the Brazilian population during the Covid-19 pandemic.
The dashed horizontal line highlights the baseline (ΔH = 0%). Data from Google Covid-19 Mobility Report [36].
Fig 3.
Time series of Covid-19 Intensive Care Unit (ICU) beds for adults in Brazil throughout the pandemic.
Data from DATASUS [39].
Fig 4.
Cumulative number of individuals vaccinated and fully vaccinated against Covid-19 in Brazil.
Data from Our World in Data [4].
Fig 5.
Temporal evolution of Covid-19 variants in Brazil.
Relative frequency (%) of different Covid-19 variants over time, based on data from the GISAID database [40].
Fig 6.
Temporal dynamics of Covid-19 in Brazil sourced from different datasets.
(a) New Covid-19 cases reported by onset symptoms date of Severe Acute Respiratory Syndrome (SARS) patients, sourced from DATASUS [43]. (b) New deaths of SARS patients reported by death date [43], new Covid-19 deaths reported in Mortality Information System by death date [3], and a 7-day rolling average of new Covid-19 deaths reported in Monitoring Panel by reporting date [2], all sourced from DATASUS.
Table 1.
Summary of Covid-19 data in Brazil (2020–2022).
Fig 7.
Effective reproduction number (Rt) for Covid-19 in Brazil.
The dashed horizontal line represents the reference value (Rt = 1) used to monitor epidemics. The Rt time series alternates between outbreak periods (solid line) and non-outbreak periods (dotted line). Nine outbreak periods, labeled from #0 to #8, were identified.
Fig 8.
Illustration of the Susceptible-Infected-Recovered-Dead-Susceptible (SIRDS) model.
Each compartment is denoted by a corresponding letter: S for Susceptible, I for Infected, R for Recovered, and D for Deceased. The model parameters include contact rate (β), recovery rate (γ), infection fatality probability (f), and immunity loss rate (ω).
Fig 9.
SIRDS model simulations across three years for different basic reproduction numbers (R0).
Each row corresponds to a specific R0 value. The charts on the left depict simulation outputs for the Susceptible (S), Infected (I), Recovered (R), and Deceased (D) compartments. On the right side, the charts display the observed effective reproduction number (Rt) over time, with a dashed horizontal line at the reference value (Rt = 1) used for epidemic monitoring.
Fig 10.
Comparative boxplots of effective reproduction number (Rt) similarity distributions in synthetic SIRDS outbreaks.
(a) Dynamic Time Warping (DTW) similarity for the first outbreak, contrasting synthetic samples with a change in basic reproduction number (R0) to their counterparts in synthetic samples without changing the R0. (b) Similarity between left and right sides for subsequent outbreaks in synthetic samples without changing the R0.
Fig 11.
Time series of Covid-19 Case Fatality Rate (CFR) in Brazil.
(a) General CFR calculated from cases reported in the Monitoring Panel [2] and deaths reported in the Mortality Information System [3]. (b) CFR calculated for Severe Acute Respiratory Syndrome (SARS) patients [43].
Table 2.
Model parameter bounds for optimization.
Fig 12.
Boxplots illustrating key metrics of the model optimization process for infection periods ranging from 8 to 20 days.
The boxplots depict: (a) the error in the objective function, (b) the number of iterations performed by the optimization algorithm, and (c) the number of objective function evaluations conducted by the optimization algorithm. In each boxplot, the lower and upper bounds represent the first and third quartiles, respectively. The horizontal line within the box indicates the median, while the whiskers extend to the minimum and maximum values within 1.5 times the interquartile range.
Fig 13.
Comprehensive analysis of simulation results for Covid-19 in Brazil.
(a) Model outcomes for an eight-day recovery period detailing the population compartments: Susceptible, Infected, Recovered, and Deceased. (b) Time series comparison between the effective reproduction number (Rt) estimated directly from reported Severe Acute Respiratory Syndrome (SARS) cases and Rt calculated by model simulations. (c) Time series comparison between new cases reported by health authorities and new infections in model simulations. (d) Time series comparison between new deaths reported by health authorities and new deaths in model simulations. (e) Time series comparison between cumulative cases reported by health authorities and cumulative infections in model simulations. (f) Time series comparison between cumulative deaths reported by health authorities and cumulative deaths in model simulations. Shaded regions depict the 95% Confidence Interval (CI).
Table 3.
Results for Covid-19 simulation with data from Brazil, Spain, United Kingdom, and United States.
Fig 14.
Time-varying model parameters fitted for Covid-19 in Brazil.
(a) Basic reproduction number (R0) varying with time (t). (b) Infection Fatality Rate (IFR) varying with t. (c) Days to loss of immunity (Ω) varying with t. Shaded regions depict the 95% Confidence Interval (CI).
Fig 15.
Comparison of cumulative Covid-19 infections simulated by our model, cumulative reported cases by health authorities, and serological prevalence during the early stages of the pandemic in various countries.
(a) Brazil: reported cases from DATASUS [2] and serological prevalence from Hallal et al. [10]. (b) Spain: reported cases from Our World in Data [4] and serological prevalence from Perez-Gómez et al. [11]. (c) United Kingdom: reported cases from Our World in Data [4] and serological prevalence from Public Health England [12–14]. (d) United States: reported cases from Our World in Data [4] and serological prevalence from Walker et al. [15] and Anand et al. [16]. Dashed lines are the simulated cumulative infections, and shaded regions depict the 95% Confidence Interval (CI).
Fig 16.
Forecasting Covid-19 dynamics in Brazil for different outbreaks.
Each row represents a distinct outbreak. Column (a) displays SIRDS simulation outputs for Susceptible (S), Infected (I), Recovered (R), and Deceased (D) compartments. Column (b) compares the effective reproduction number (Rt) from Section 3.2 with model simulations, including a dashed line at the reference value (Rt = 1). Column (c) contrasts the rate of new deaths per 100,000 inhabitants in a 7-day moving average from the Mortality Information System [3] with new deaths estimated by model simulations. Vertical dotted lines mark the 21st day inside the outbreak, the maximum fit date for forecasting the next 90 days. Shaded regions indicate the 95% Confidence Interval (CI).
Table 4.
Fit and prediction errors for the Covid-19 outbreaks in Brazil.
Table 5.
Percentual absolute error of Covid-19 deaths forecasting outbreaks in Brazil.
Fig 17.
Forecasting Covid-19 dynamics during the initial outbreak in Brazil.
Each line represents a day period within the first outbreak to mark the maximum adjustment date for forecasting the next 90 days, and the vertical dotted lines also highlight this date. Column (a) displays SIRDS simulation outputs for Susceptible (S), Infected (I), Recovered (R), and Deceased (D) compartments. Column (b) compares the effective reproduction number (Rt) from Section 3.2 with model simulations, including a dashed line at the reference value (Rt = 1). Column (c) contrasts the rate of new deaths per 100,000 inhabitants in a 7-day moving average from the Mortality Information System [3] with new deaths estimated by model simulations. Shaded regions indicate the 95% Confidence Interval (CI).
Fig 18.
Sensitivity analysis heatmap for perturbations of 1%, 10%, and 50% in optimized parameters with Covid-19 data in Brazil.
Each row corresponds to a specific parameter θk, where k denotes the parameter associated with a particular Covid-19 outbreak. When is mentioned, it represents an adjustment parameter for atypical outbreak k. The numerical values in each cell represent the elasticity measured for θ under a specific perturbation. The parameters include the initial quantity of infected population I(0), the breakpoint indicating the start of an outbreak (b), transition days between epidemic periods for fast transitions (τ), contact rate (β), infection fatality rate probability (f), and immunity loss rate (ω). Empty cells indicate simulations with errors due to invalid parameter values.