Accurate early forecasting of COVID-19 spread is critical for informed decision-making regarding lockdowns and other public health interventions. The unpredictable nature of the virus's peak makes early forecasting crucial. This paper addresses this need by developing a novel recursive bifurcation model. The model analyzes the cumulative number of infected individuals, a key metric for assessing the extent of the pandemic's impact. While many countries exhibited a logistic or sigmoid pattern in their infection curves, the logarithm of infected populations often reveals more informative patterns, such as bifurcations. These bifurcations represent shifts or changes in the rate of infection, potentially reflecting the impact of interventions or changes in transmission dynamics. By leveraging this bifurcation, the model aims to improve the accuracy of early forecasts compared to existing models such as the logistic growth model and the Richards model. The study focuses on South Korea and Germany as case studies, countries that exhibited clear bifurcations in their infection data. The results are compared against these existing models to demonstrate the proposed model's improved forecasting capabilities in the early stages of an outbreak. The limitations of the approach and suggestions for future research are also discussed.

Existing studies on COVID-19 spread have largely focused on statistical analyses (estimating parameters like reproduction number and doubling time) and dynamic modeling (primarily using the SEIR model to assess interventions and transmission dynamics). The Richards model, an extension of the logistic growth model, has also been employed for fitting epidemic curves. However, a highly accurate early forecasting model, crucial for timely interventions, remained elusive. The Richards model, while flexible, often struggles with accurate prediction before the inflection point of the infection curve. This paper seeks to address this gap by proposing a novel recursive bifurcation model.

The core of the proposed model is a recursive Tanh function applied to the logarithm of the cumulative infected population. This function describes the infection trajectory within each cycle of the overall spread process. The model recursively solves the equation, starting from the first cycle and progressing to subsequent cycles. The virus spread rate (r) is a key parameter within each cycle and is estimated using linear least-squares fitting. For each cycle, a linear equation is formulated, relating the spread rate to the change in the logarithm of infected population over time. The equation is then solved using least-squares fitting to determine the spread rate (r). Once the spread rate in cycle 1 (r1) is established, it is used to analyze subsequent cycles. The validity of using r1 for subsequent cycles is tested by comparing the model's prediction to the actual data using Equation 6a and 6b. A key aspect is the identification of the bifurcations or cycle transition points in the logarithm of the infected population. This is done through visual identification in the provided figures, though the authors note an algorithm could be developed to automate this process. For forecasting, an algorithm is developed to first determine the spread rate in cycle 1 (r1), then recursively analyze subsequent cycles. In the final cycle, the spread rate is assumed to be equal to the previous cycle's spread rate (rn = rn-1). An initial value for the logarithm of the infected population in the final cycle is estimated via linear least-squares fitting. Finally, a nonlinear Levenberg-Marquart least-squares fitting procedure is employed to determine the parameters (βn, θn, and Dn) in the forecasting equation (Equation 8). This equation then allows for the prediction of future infected population values. The methodology is applied to South Korea and Germany's data, and the results are compared to those obtained using the simple logistic growth model and the Richards model.

The study found that the bifurcation points in the logarithm of infected population data often precede the inflection points of the standard infection curves. This temporal difference offers a significant advantage to the proposed recursive bifurcation model for early forecasting. The model's performance was evaluated by comparing its predictions to those of the logistic growth model and the Richards model at different time points relative to the inflection point. The model was evaluated at 0.8Ti and 0.9Ti, where Ti represents the inflection point of the cumulative infected population curve. The predictions were then compared to the actual data at various future time points. Results from South Korea and Germany showed that the recursive bifurcation model provided significantly more accurate early forecasts (at 0.8Ti and 0.9Ti) compared to the logistic growth model and the Richards model, as indicated by the narrower 95% confidence intervals of prediction error. Specifically, in the South Korean data analysis, the 95% confidence intervals for the bifurcation model at the reference point of 0.9Ti were significantly narrower compared to the other models. At 3.55Ti (a later time point) the relative error was substantially lower for the bifurcation model (21.3%) compared to the logistic growth model (70.5%) and the Richards model (49.9%). Similarly, the German data showed the superiority of the bifurcation model, with a relative error of 4.7% at 2.0Ti, against 41.6% for the logistic growth model and 76.8% for the Richards model. The parameter estimations for each model (logistic growth, Richards, and the bifurcation model) are detailed in Table 4, along with the 95% confidence bounds. The bifurcations, visually identifiable in the infection curves, appear to capture key transition points in the spread dynamics, allowing for better prediction, particularly in the early stages of the outbreak. The inflection point (ti) and the cycle transition point (tc) were observed to be significantly different (ti appeared later than tc in the case of South Korea and much later in Germany), highlighting the advantage of this model for early forecasting.

The findings demonstrate the superior performance of the recursive bifurcation model for early forecasting of COVID-19 spread, especially when compared to existing models such as the simple logistic growth model and the Richards model. The ability to identify and leverage the bifurcation points, which often precede the inflection points, is a key strength. This characteristic makes the model particularly useful for making timely decisions based on early data. The recursive nature of the model allows for iterative improvement of forecasts as more data becomes available. The temporal difference between the bifurcation point and the inflection point suggests that the model is sensitive to changes in transmission dynamics that are not always immediately reflected in the overall growth curve. This added sensitivity contributes to improved accuracy. The success in South Korea and Germany supports the model's applicability in various contexts, provided the presence of identifiable bifurcation points. This is a significant finding with clear implications for public health policy and resource allocation.

This paper successfully developed a recursive bifurcation model for early forecasting of COVID-19 spread. The model demonstrates improved accuracy compared to existing models, especially in the crucial early stages of an outbreak. The identification and utilization of bifurcation points in the data proved to be key to this improved accuracy. While the model performed well in South Korea and Germany, the case of the United States presents a challenge, highlighting the need for further research into scenarios where inflection points are not yet apparent. Future work should focus on developing robust algorithms for automatic bifurcation detection and extending the model to handle situations where clear bifurcations may be absent or difficult to identify, thus improving its generalizability. The model's potential for informing timely public health interventions makes it a valuable tool for managing future outbreaks.

The model's accuracy relies on the presence of clearly defined bifurcation points in the data. The absence of these points, as seen in the early United States data, limits the applicability of the model. Additionally, the model's performance might be affected by data quality and the reliability of infection case counts. The automatic identification of bifurcation points, currently done visually, needs further algorithmic development. Lastly, the model assumes a certain degree of homogeneity in transmission dynamics within each cycle, which may not always hold true in real-world scenarios.