A recursive bifurcation model for early forecasting of COVID-19 virus spread in South Korea and Germany

Coronavirus disease (COVID-19) is a novel respiratory illness that originated in 2019 and spreads person-to-person. The first outbreak was reported in Wuhan, China, on December 31, 2019. COVID-19 belongs to the beta-coronavirus category, with a genome 96% similar to bat coronaviruses, though the exact zoonotic jump route remains unclear. The pandemic has had profound social and economic impacts, including high unemployment, halted economic activity, and international border closures, while prompting preventive measures like hand hygiene, masks, social distancing, and geographical closures. Despite temporary environmental benefits, lockdowns cause substantial societal costs, and the uncertain timing of the epidemic peak complicates planning. This motivates the need for accurate early forecasting models to inform timely decisions on interventions and to balance public health with economic considerations.

Prior studies on COVID-19 spread fall broadly into: (1) statistical analyses estimating key epidemic parameters such as basic reproduction number, doubling time, and serial interval; addressing untraced contacts, undetected international cases, and true infected counts; and using statistical reasoning and stochastic simulation; and (2) dynamic modeling, notably SEIR-based models to assess lockdowns, transmission, risk, quarantine effects, and time delays for incubation and recovery. The Richards growth model, an extension of the logistic model, has been used for single- and multi-phase outbreaks (e.g., SARS). The Richards model is defined by dP/dt = r P [1 - (P/K)^a] with a solution P(t) = K / [1 + e^{-r(t - t0)}]^a, reducing to the logistic model when a = 1. Recent work applied generalized logistic, Richards, and sub-epidemic models for short-term COVID-19 forecasts. However, accurate early forecasting from very early data remains elusive, which is crucial for decision-making. This paper proposes a new recursive bifurcation model and compares it with logistic and Richards models using data from South Korea and Germany.

The study models cumulative infected population P(t), emphasizing the log-transformed series to reveal bifurcation patterns observed in several countries (e.g., South Korea, Germany, U.S., France, Canada, Australia, Malaysia, Ecuador). The core is a recursive, cycle-wise Tanh-based relation for the (i+1)-th cycle: log(P_{i+1}) − log(P_{i−1} + 1) = (log(P_i + 1) − log(P_{i−1} + 1)) [1 + exp(−2π r_i (D − D_{i−1}))]^{-1}, where i indexes cycles, D is days since initiation, P_{i−1} is cumulative infected at the end of cycle i−1, r_i is the spread rate in cycle i, and D_{i−1} is the end day of cycle i−1. Adding 1 inside logs avoids singularity at P=0. For a single cycle (n=1), the equation reduces to a standard Tanh-form update. Estimating growth rate: The intrinsic spread rate r_t is estimated via linear least-squares using a linearized form: r_t (D_t − D_{t−1}) = −0.5 ln[ 2 / (1 + (log(P_{t+1}) − log(P_{t−1} + 1)) / (log(P_t + 1) − log(P_{t−1} + 1)) ) ]. Validation across cycles: Using r_1 from cycle 1, the model is validated in cycle 2 via a linear relation y = α z derived from the recursive equation; α ≈ 1 indicates consistency between cycles. Early forecasting algorithm (Table 1): - Step 1: Determine r_1 by least-squares fitting of the linearized equation. - Step 2: Recursively analyze cycles 2 to n−1. - Step 3: Set r_n = r_{n−1} for the current cycle n used for forecasting. - Step 4: Obtain an initial estimate β_n via linear least-squares from log(P + 1) − log(P_{n−1} + 1) = β_n [1 + e^{−2 θ_n (D_n − D_{n−1})}]^{−1}. - Step 5: Fit β_n, θ_n (slope control), and D_n using nonlinear Levenberg–Marquardt least squares with log(P + 1) = β_n [1 + (e^{−2 θ_n (D_n − D_{n−1})})^{−1}] + log(P_{n−1} + 1). - Step 6: Forecast future infected counts by propagating the fitted model. Inflection vs. cycle transition: The paper defines reference points relative to the inflection time t_i and observes that cycle transition time t_c often precedes t_i, enabling earlier forecasts than models relying on symmetry around the inflection point. Implementation details include identification of bifurcation via changes in tangential direction (automated detection discussed but not developed here) and the use of log-transformed cumulative counts for stability.

- South Korea cycle 1 spread rate: r_1 ≈ 0.106 (R^2 ≈ 0.9746). Predicted series closely matches true data in cycle 1. - Cross-cycle validation (South Korea cycle 2): α ≈ 0.968 (≈1), supporting reuse of r_1 in the next cycle. - Temporal relationship: For South Korea, the inflection point t_i occurred 12 days after the cycle transition t_c. For Germany, t_c occurred 38 days before t_i. This gap enables earlier forecasting using the bifurcation model. - Early forecast comparison at specified reference/forecast times: • South Korea (start Jan 20, 2020): T (cycle transition) = 28 days; t_i = 40 days; reference 0.9 t_i = 36 days; forecast 3.55 t_i = 142 days (Jun 12, 2020). True infected at 3.55 t_i: 12,051. - Simple logistic prediction: 3,560 (95% CI: 2,126–4,995); absolute relative error: 70.5%. - Richards prediction: 18,070 (95% CI: 115,497–151,697); absolute relative error: 49.9%. - Bifurcation prediction: 9,488 (95% CI: 4,468–20,144); absolute relative error: 21.3%. • Germany (start Jan 26, 2020): T = 30 days; t_i = 68 days; reference 0.8 t_i = 54 days; forecast 2.0 t_i = 138 days (Jun 12, 2020). True infected at 2.0 t_i: 187,226. - Simple logistic prediction: 109,400 (95% CI: 40,070–178,800); absolute relative error: 41.6%. - Richards prediction: 43,340 (95% CI: −68,990–155,700); absolute relative error: 76.8%. - Bifurcation prediction: 178,373 (95% CI: 63,316–502,508); absolute relative error: 4.7%. - Overall, the bifurcation model yields substantially lower early-forecast errors at reference points 0.8 t_i or 0.9 t_i compared with logistic and Richards models for South Korea and Germany. - Fitted parameter ranges for comparative models are provided, illustrating higher parameter uncertainty for Richards without good initial values, while the bifurcation model captures two-stage dynamics through cycle-wise fitting.

The study addresses the challenge of early forecasting by exploiting a bifurcation pattern in the log-cumulative infection trajectory, enabling parameter transfer (e.g., growth rate) from an earlier cycle to later cycles. Because the cycle transition precedes the inflection point, the recursive bifurcation model can make reliable forecasts earlier than methods that require data past the inflection point (e.g., symmetric sigmoidal fits like Richards). Empirical results on South Korea and Germany demonstrate that this approach reduces forecast error at early reference times compared with logistic and Richards models. The findings suggest that recognizing and modeling multi-stage (cycle-wise) epidemic growth improves early predictive performance, supporting more timely decision-making on interventions and resource allocation. The comparative analysis also highlights practical issues with the Richards model, including sensitivity to initial parameter guesses and parameter identifiability in early phases. The bifurcation model’s reliance on linear least squares for rate estimation and subsequent constrained nonlinear fitting offers robustness and efficiency for early-stage data.

The paper proposes a recursive bifurcation model for early forecasting of COVID-19 spread, combining cycle-wise linear estimation of growth rates with nonlinear least-squares fitting for current-cycle parameters. Applied to South Korea and Germany, the model outperforms logistic and Richards models for early forecasts (at 0.8 t_i and 0.9 t_i), leveraging the observed phenomenon that cycle transitions occur well before inflection points. This capability enhances early decision-making on public health interventions. Future research includes handling scenarios where the inflection point has not yet occurred (e.g., long, ongoing first waves), developing automated and robust bifurcation detection, and extending validation to additional countries and multi-wave settings.

- The approach relies on identifiable bifurcation (multi-cycle) patterns; when such structure is not apparent or the epidemic is in an extended pre-inflection phase (e.g., U.S. data as of Aug 3, 2020), early forecasting and subsequent validation are challenging. - Accuracy depends on data reliability and stabilization of reporting; parameter transfer between cycles assumes some consistency of intrinsic growth attributes. - Automated detection of cycle transitions is not developed here; identification was visual/heuristic. - Comparative models like Richards require good initial parameter estimates; while this is a limitation of the comparator, it also underscores sensitivity in early-phase nonlinear fitting, including for the proposed method’s nonlinear step.