Mathematical model of COVID-19 intervention scenarios for São Paulo—Brazil

The World Health Organization declared COVID-19 a pandemic in March 2020. By November 2020, over 52 million cases and 1.2 million deaths had occurred worldwide. Brazil, and particularly the state of São Paulo, experienced high incidence and mortality. Despite social distancing (SD) measures, estimates in May 2020 placed São Paulo’s effective reproduction number Rt at approximately 1.46, indicating uncontrolled transmission. Demographic factors (e.g., a substantial older population), environmental considerations, and socio-economic challenges may alter intervention effectiveness in Brazil relative to other regions. The purpose of this study is to model different COVID-19 SD intervention strategies for São Paulo to determine optimal approaches—considering strategy type, SD magnitude and timing, and personal protection levels—to control the current peak and avoid a second wave.

The paper references global evidence that widespread SD and personal protective measures (PPM) can mitigate transmission, reports from Imperial College London on Brazil’s Rt, and modeling work proposing exit strategies combining SD with testing and contact tracing. Prior studies have emphasized mask efficacy (estimated 50–90% depending on type and fit) and the importance of hygiene and distancing. Related modeling for the US suggested stepping-down strategies could be effective and reduce overall SD time. The study builds on SEIR-type models adapted for COVID-19.

The authors developed an eight-compartment SUEIHCDR model (Susceptible, Unsusceptible, Exposed, Infected, Hospitalized, Critical, Dead, Recovered), extending SEIR with COVID-19-specific factors. A time-varying protection rate α moves individuals from susceptible to unsusceptible, capturing personal protective measures (PPM) such as masking and hand hygiene. Transmission is modulated by social distancing (SD), inferred from mobility data (Apple Maps and Google Community Mobility). Equations define flows between compartments, including incubation (1/γ), infectious period (1/δ), hospitalization (1/ε), ICU stay (1/ζ), and outcomes (recovery or death). The effective reproduction number Rt is derived as a function of SD, α, and infectiousness parameters. The state of São Paulo is modeled as a single, homogeneously mixing system, with sensitivity analyses addressing this assumption. Data and preprocessing: Daily cases and deaths (7-day running averages) for Brazil and São Paulo were used. To account for under-reporting, corrections were applied: cases multiplied by 25.9 and deaths by 1.9 (Brazil) and 1.7 (São Paulo). Mobility time series were low-pass filtered (Butterworth 4th order, 0.09 Hz) to estimate SD from combined Apple Driving and Google category indices. Parameter fitting: A custom MATLAB global optimization (Monte Carlo and multiple local minima searches) fitted model coefficients within literature-informed bounds (WHO and other publications). Initial conditions (E0, I0, H0, etc.) were optimized within ranges relative to inferred values. Confidence intervals (95%) were obtained from 300 runs with 1% parameter perturbations; results averaged over 5000 runs. Scenario design and optimization: From June 1, 2020 forward, future SD and protection were manipulated under three strategies: (1) Stepping-down (initial SD, then halved each of the next three windows, returning to initial value on the fourth, repeated), (2) Intermittent (alternating SD with no SD), and (3) Constant SD. Window sizes of 40, 60, and 80 days were tested. A DOE (SOBOL and Latin Hypercube; ~2000 individuals) explored SD maxima (0–75%) and protection (20–95%), with subsequent constraints to realistic ranges (SD 15–40%, protection 50–70%). A multi-objective genetic algorithm (MOGA) minimized: total critical cases over ICU threshold (ICU_E) over the analysis horizon, ICU_E in the first peak (ICU_E1), ICU_E in the second peak (ICU_E2), and SD burden. Sensitivity analyses examined (a) the SD–protection tradeoff and (b) the impact of localized pockets of transmission on outcomes and strategy rankings.

• Model fit: The model accurately fit corrected 7-day running average daily cases and deaths for Brazil and São Paulo. On May 11, 2020, estimated SD was 41% (Brazil) and 52% (São Paulo), protection 59–64% (Brazil) and 60–65% (São Paulo). Estimated Rt: 1.33 (Brazil) and 1.31 (São Paulo). IFR: 0.46% (Brazil) and 0.53% (São Paulo). Attack rate: 2.2% (Brazil) and 3.5% (São Paulo). Periods (Brazil/SP): latent 1.1/0.5 days, infectious 14.4/9.2 days, hospitalization 4/4.1 days, ICU 10/9.2 days.
• Optimal strategy and timing: Across constrained SD (30–70%) and protection (50–70%), the stepping-down SD strategy was optimal, with an 80-day window best controlling both first and second peaks.
• ICU threshold impacts: With current SD and protection maintained, stepping-down with an 80-day window contains the first peak but a second peak is likely if SD subsequently drops. Scenarios where SD was dropped or oscillated below current levels yielded ~20,000 critical cases over ICU threshold. An 80-day stepping-down strategy reduced critical cases over ICU threshold by ~90,000 versus 80-day intermittent and constant strategies; by ~70,000 versus a 40-day window; and by ~30,000 versus a 60-day window.
• Lockdown scenario: Maintaining protection and enforcing a 60-day lockdown (SD 75%) followed by a 40-day stepping-down strategy could contain the first peak and substantially reduce total cases through October 2020; however, if protection declines post-lockdown, a large second peak may occur.
• Protection increases enable lower SD: Maintaining current SD well below lockdown levels, increasing the proportion strictly adhering to protective guidelines by ~5–10% (to 65–70% protection) enables containment of first and second peaks when combined with a 60–80-day stepping strategy. After 60 days, a 60–80-day stepping strategy yields ~30,000 fewer critical cases over ICU threshold than an 80-day intermittent strategy and ~100,000 fewer than constant SD; compared to a 40-day window, 60–80-day windows reduce critical cases by ~40,000.
• Scenario with lower SD and higher protection: If protection increases by 10% and SD drops from 52% to 40%, the optimal 80-day stepping-down strategy yields ~30,000 critical cases over ICU threshold, ~11,000 fewer than intermittent and ~100,000 fewer than constant SD; compared to 40-day and 60-day windows, 80-day reduces by ~50,000 and ~15,000, respectively.
• Protection vs SD on second peak: Protection levels exert greater influence on the second peak than SD. If protection falls to 50% of its current value while SD remains at 52%, the pandemic is not contained.
• Additional findings: The stepping-down approach can reduce total SD time by ~6.5% over two years while maintaining control, potentially mitigating economic and social costs.

The study shows that, as of mid-May 2020, São Paulo’s Rt remained above 1, implying the first peak had not been contained under then-current SD and protection levels. The model indicates that increasing adherence to protective behaviors (facemasks, hand hygiene, distancing, avoiding crowds) to 65–70% can control the first peak even without additional ICU capacity or treatment improvements. Alternatively, brief stringent SD (e.g., 75% lockdown) could suppress the first peak but risks a large second peak if lifted too early or if protection declines. To avoid a second wave, the analysis identifies a stepping-down SD strategy with an 80-day window as optimal, outperforming constant and intermittent strategies by reducing critical cases over ICU thresholds and overall SD burden. This approach allows limited transmission that builds immunity without overwhelming healthcare resources and reduces the total time under SD. Elevated protection levels can substitute for higher SD magnitudes, offering potential economic and psychological benefits. Conversely, reductions in protection can precipitate a second peak that surpasses ICU capacity, increasing mortality risk due to strained healthcare systems. The findings underscore the central role of personal protection in pandemic control, especially where testing and contact tracing capacity are constrained. While focused on São Paulo, the framework is applicable to other regions.

A multi-objective optimization of an extended SEIR (SUEIHCDR) model indicates that a stepping-down social distancing strategy with 60–80-day windows, particularly 80 days, is optimal for São Paulo to control the first peak and prevent a second peak. Achieving and maintaining high protection adherence (approximately 65–70%)—an increase of 5–10% over observed levels—enables reducing SD over time while keeping critical cases within ICU capacity. Compared with intermittent and constant SD, stepping-down significantly decreases ICU exceedance and overall SD time. The approach is generalizable to other locales, and the results highlight that strengthening protective behaviors may be the most effective and sustainable means to manage COVID-19 transmission. Future work should test this framework in other settings and integrate evolving treatment, vaccination, and resource capacity data.

Key limitations include: (1) Using mobility-derived SD as a proxy for contact rates; while correlated, the relationship is complex and influenced by changing protective behaviors (α). Noise in mobility data may affect estimates. Sensitivity analysis showed that decreases in SD magnitude correlated with compensatory increases in protection levels. (2) Assuming São Paulo as a homogeneous mixing system; localized transmission pockets may alter dynamics. Sensitivity analyses with varying isolated population fractions and reduced SD impact produced similar qualitative conclusions; projections may overestimate if substantial disease-free regions exist or underestimate if pockets do not alter SD effects. (3) Assuming a cross-country constant age-adjusted IFR (with Iceland-based case under-reporting correction); differing comorbidity profiles could affect IFR, though this has limited impact on ICU occupancy outcomes central to the study. (4) Constraints in Brazil’s testing and tracing capacity and data quality may introduce uncertainties. Despite these, the main conclusions about optimal strategies and the primacy of protection were robust in sensitivity analyses.