Mathematical modeling of COVID-19 in 14.8 million individuals in Bahia, Brazil

The study addresses the challenge of COVID-19’s rapid spread and its strain on healthcare systems, particularly in low- and middle-income countries such as Brazil. After the emergence of SARS-CoV-2 in late 2019 and rapid global dissemination, Brazil experienced widespread transmission from March 2020, with Bahia (population 14.8 million) as a representative state with marked health access inequalities. The research question focuses on quantifying how transmission dynamics—including the role of asymptomatic/non-detected infections—translate into hospitalization (clinical and ICU beds) and mortality demands, and how non-pharmaceutical interventions (NPIs) affect transmissibility and healthcare capacity. The purpose is to provide a locally informed modeling framework to project hospitalization needs, evaluate intervention impacts on transmission rates and effective reproduction numbers, and inform policy on timing, duration, and intensity of interventions needed to prevent health system collapse. The importance stems from guiding resource allocation and intervention strategies in a low-resource setting under evolving epidemic conditions.

Compartmental models (e.g., SIR/SEIR) have been widely used to study COVID-19 transmission, non-pharmaceutical interventions, asymptomatic transmission, social distancing, quarantine strategies, and post-epidemic scenarios. Prior works estimated high transmissibility (R0 ~2.2–2.5 in Wuhan) and highlighted the impact of NPIs on reducing spread. Studies have underscored the role of undocumented/asymptomatic transmission in facilitating rapid dissemination, examined hospitalization requirements, and assessed strategy impacts on epidemic trajectories and healthcare demand. The present work extends these approaches by incorporating asymptomatic/non-detected transmission, hospitalization flows (clinical and ICU), mortality, and time-varying transmission to capture policy and behavior changes, calibrated with local epidemiological and hospital data for Bahia.

Design: An 8-compartment SEIIHURD model with time-varying transmission was developed to capture COVID-19 dynamics including asymptomatic/non-detected infections and hospitalization flows in Bahia, Brazil. Data sources: State-level daily cumulative confirmed cases and deaths for Bahia, its capital Salvador, and the remaining 416 municipalities from the Secretary of Health of the State of Bahia (SESAB). State-level daily clinical and ICU bed occupancy totals (Bahia had 466 clinical and 422 ICU COVID-19 beds by May 2020). Administrative data from Instituto Couto Maia (ICM), Salvador, for 231 patients (Mar 23–Apr 16, 2020) to inform hospitalization-related parameters. Population estimates from IBGE. Initial analyses used data Mar 6–May 4, 2020; ex-post assessment extended to Sep 13, 2020. Model structure: Population is partitioned into S (susceptible), E (exposed), Ia (asymptomatic/non-detected infectious), Is (symptomatic infectious), H (hospitalized clinical beds), U (ICU), R (recovered), and D (deaths). Transmission from hospitalized compartments (H, U) was neglected due to containment. Patient flow allows transitions between H and U; ICU survivors transfer to H before recovery. Asymptomatic/non-detected infectivity is reduced by factor θ. Time-varying transmission: β(t) is a piecewise constant step function (Heaviside representation) changing at specified times to reflect NPIs and behavior changes. Pre-intervention β0 and subsequent βi values are estimated by fitting. Equations: Standard mass-action transmission with force of infection proportional to Is + θ Ia. Progression rates include κ (incubation), γa (asymptomatic duration), γs (symptomatic duration), γH (clinical stay), γU (ICU stay). Hospitalization proportion among symptomatics, ICU fraction, and death rates (μH, μU) govern H, U, D flows. Parameterization and estimation: Fixed parameter ranges informed by literature and ICM data (e.g., lengths of stay, death rates, transfer probabilities). Key parameters estimated via Particle Swarm Optimization (PSO) minimizing discrepancies across multiple series (daily cases and deaths for Bahia, Salvador, and remaining municipalities; clinical and ICU occupancy at state level). PSO settings: 300 particles, 1000 iterations, cognitive=0.1, social=0.3, inertia=0.9, 5-nearest neighbors (Euclidean). Confidence intervals via weighted non-parametric bootstrap (500 replicates). Reproduction numbers: R0 derived using the next-generation matrix for (E, Ia, Is), summing contributions of symptomatic and asymptomatic/non-detected transmission under β0. Rt estimated following Wallinga and Lipsitch using renewal equation, with both reported case series and model-simulated new infections to mitigate reporting fluctuations. Sensitivity analysis: Global variance-based Sobol method (SALib) with N=12,000 parameter combinations (total 120,000 simulations), uniform priors over plausible ranges. Total effect indices evaluated for Ia, Is, H, U, D over time to identify influential parameters and interactions. Scenarios: Counterfactual no-intervention; maintained NPIs (observed reductions); increased transmissibility among asymptomatic/non-detected (easing); immediate and critical interventions with varying reduction magnitudes (25%, 50%, 75%) and durations (7, 14, 30, 60, 90 days); periodic interventions (30 days on/30 days off). Ex-post validation: compare 30-day forecasts from model calibrated on May 4 to observed data; re-estimate β(t) through Sep 13, 2020 to capture further changes.

• Transmission reductions after NPIs:
  • Statewide Bahia: initial β0 = 1.28 (95% CI: 1.26–1.30); ~36% reduction to β1 = 0.92 (95% CI: 0.91–0.93) around April 2, 2020 (27 days after first confirmed case).
  • Salvador (capital): 54.7% reduction (around March 26, 2020).
  • Remaining 416 municipalities: ~40.6% reduction (around April 3, 2020).
  • Asymptomatic/non-detected infectivity factor (θ): Bahia 0.34 (95% CI: 0.33–0.35); Salvador 0.71 (95% CI: 0.69–0.72); remaining municipalities 0.62 (95% CI: 0.60–0.64).
  • R0 estimates: Bahia 2.25 (95% CI: 2.19–2.31); Salvador 3.56 (95% CI: 3.44–3.69); remaining municipalities 2.45 (95% CI: 2.36–2.55). Despite reductions, Rt remained mostly >1 during the period.
• Hospitalization demand and capacity:
  • Without interventions: clinical beds exhausted by April 24, 2020; ICU capacity exceeded by April 26, 2020.
  • With observed interventions: clinical bed exhaustion shifted to May 9, 2020; ICU capacity exceeded by May 13, 2020.
  • Observed (May 4): 240 clinical beds occupied (51.5% of 466); 176 ICU beds occupied (41.7% of 422).
  • Counterfactual no-NPI (May 4): clinical occupancy 6.5× higher (≈1,581 beds) and ICU 6.4× higher (≈1,131 beds).
  • NPIs reduced cases by ~7× and deaths by ~4× versus no-intervention by May 4.
• Role of non-detected infections:
  • Non-detected (asymptomatic/mild) cases contribute to a ~55.03% increase in R0.
  • Easing movement restrictions for asymptomatic/mild (modeled as +50% in their transmission) led in 20 days to: +50% cumulative cases, +37% deaths, +75% clinical bed demand, +87.5% ICU demand.
• Infection fatality ratio (IFR) and attack rate:
  • IFR estimated at 0.69% (95% CI: 0.67–0.71) for Bahia.
  • By May 4, 2020, estimated ~0.1% of Bahia’s population infected, consistent with seroprevalence findings.
• Intervention strategies:
  • Immediate 25% reduction in β for 30 days (from May 2) yields minimal delays (≈2 days for clinical collapse; ≈8 days for ICU collapse). A 50% reduction for 30 days or 75% for 14 days can delay bed exhaustion by ~40 days.
  • If interventions begin only at capacity threshold (May 9), milder reductions (25–50%) and/or short durations are insufficient for sustained relief; stronger, longer interventions needed for full recovery of hospital capacity (see Tables 1–2 in text).
  • Periodic interventions (30 days on/30 days off) with up to 50% reduction in β are insufficient to keep demand below existing capacity without concurrent capacity expansion.
• Ex-post validation and updated dynamics:
  • 30-day forecasts from the May 4-calibrated model captured reported cases, deaths, and ICU demand within 95% CIs. ICU capacity predicted to be reached May 13 vs actual May 24 (within CI). Clinical bed capacity predicted May 9 vs actual May 29.
  • Re-estimation through Sep 13, 2020 indicated an additional reduction in transmission around June 11, consistent with strengthened measures enacted in late May.

The findings show that while NPIs substantially reduced SARS-CoV-2 transmissibility in Bahia, the effective reproduction number remained above 1 during the initial phase, implying continued growth and persistent pressure on healthcare resources. Incorporating asymptomatic/non-detected transmission is crucial: these infections significantly elevate R0 and, if their mobility or contact rates increase, rapidly amplify cases, deaths, and bed demand. Model-based projections demonstrate that timing, intensity, and duration of interventions are critical for preventing the collapse of clinical and ICU capacities in a low-resource setting. Early, strong, and sufficiently prolonged reductions in transmission can meaningfully delay or prevent capacity breaches, whereas delayed or mild measures provide limited relief and may necessitate repeated interventions. Periodic intervention strategies alone are unlikely to maintain demand below fixed capacity, but they could be effective when combined with strategic capacity expansion. Ex-post comparisons validate the model’s short-term predictive utility for ICU demand under maintained NPIs, though longer-term clinical bed forecasts are more sensitive to unmodeled changes in hospitalization practices and parameters. Overall, the results guide policy on calibrated, data-informed NPIs and capacity planning to mitigate healthcare system overload.

This work presents a locally informed SEIIHURD model that integrates asymptomatic/non-detected transmission, hospitalization flows, mortality, and time-varying transmissibility to project COVID-19 dynamics and healthcare demands in Bahia, Brazil. The model quantifies NPI-driven reductions in transmission, shows the substantial impact of non-detected infections on R0 and hospital demand, and delineates intervention strategies (strength, timing, duration, periodicity) needed to avert system collapse. Short-term predictions proved useful for ICU planning, and re-estimation captured later declines in transmission. The framework is generalizable to other regions with re-calibrated parameters and local data. Future work should incorporate heterogeneous contact structures, granular spatial data, evolving clinical practices, and potential within-hospital transmission, and leverage broader testing/serosurveys for improved incidence estimation and robust medium- to long-term forecasts.

• Reliance on reported confirmed cases likely underestimates true incidence due to limited testing; although adequate for trend identification, it affects absolute projections.
• Homogeneous mixing and aggregation at the state level do not capture spatial/clustered transmission; heterogeneous models with finer granularity could improve accuracy.
• Hospital transmission (H, U) was not modeled; potential infections among healthcare workers and during aerosol-generating procedures are excluded.
• Hospitalization parameters may have evolved over time (e.g., admission criteria, length of stay), but updated data were not available to re-parameterize during the initial evaluation period.
• Long-term predictions are limited by changing human behavior and policy adjustments not fully encapsulated by stepwise β(t) changes; frequent re-calibration is necessary.