[Paper Review] Estimation of population infected by Covid-19 using regression Generalized logistics and optimization heuristics

Mario Villalobos-Arias|arXiv (Cornell University)|Apr 2, 2020

COVID-19 epidemiological studies3 references4 citations

TL;DR

This paper proposes using generalized logistic (GL) and Gompertz growth models to estimate and predict cumulative COVID-19 infections using early epidemic data. It applies nonlinear optimization heuristics to fit these flexible curves to case data from China, South Korea, Costa Rica, Italy, and Spain, achieving R² values above 0.998, enabling reliable prediction of epidemic peak and final case count from as few as 20–35 days of data.

ABSTRACT

In this work, a proposal for the estimation of the populations using logistic curve fitting is presented. This type of curve is used to study population growth, in this case population of people infected with the Covid-19 virus; and it can also be used to approximate the survival curve used in actuarial and similar studies in Spanish: En este trabajos se presenta una propuesta para la estimación de la poblaciones usando ajuste de curvas del tipo logística. Este tipo de curvas se utilizan para el estudio de crecimiento de poblaciones, en este casos población de personas infectadas por el virus Covid-19; y también se puede utilizar para aproximar la curva de supervivencia que se utiliza en estudios actuariales y otras similares

Motivation & Objective

• To improve upon standard logistic regression for modeling cumulative COVID-19 case counts by increasing curve flexibility.
• To address the limitations of the SIR model, which requires precise parameter estimation from limited early data.
• To develop a method that predicts epidemic peak and final case count using only early epidemic data (20–35 days).
• To compare the performance of generalized logistic and Gompertz models in fitting real-world case data from multiple countries.
• To validate the predictive power of these models using data from China, South Korea, Costa Rica, Italy, and Spain.

Proposed method

• Uses the generalized logistic (GL) function: P(t) = M / (1 + e^(-at + b))^α, which introduces flexibility via an additional shape parameter α.
• Applies the Gompertz function: G(t) = a * exp(-b * exp(-ct)), a three-parameter model with inherent flexibility for epidemic curve fitting.
• Employs nonlinear optimization heuristics (e.g., simulated annealing) to estimate model parameters due to the nonlinearity of the GL and Gompertz functions.
• Normalizes case data by dividing by the maximum observed count to improve numerical stability during fitting.
• Validates model fit using R² and visual comparison between observed data and predicted curves.
• Incorporates prediction intervals with a 7% error margin based on data uncertainty.

Experimental results

Research questions

• RQ1 Can generalized logistic and Gompertz models accurately fit early cumulative case data from multiple countries?
• RQ2 To what extent can these models predict the epidemic peak and final case count using only 20–35 days of data?
• RQ3 How do the predictive performances of GL and Gompertz models compare with standard logistic regression in terms of R² and curve fidelity?
• RQ4 Does the inclusion of an additional shape parameter (α) in the GL model significantly improve fit compared to standard logistic regression?
• RQ5 Can the models reliably estimate the inflection point (onset of decline in daily cases) from early data?

Key findings

• The generalized logistic model achieved an R² of 0.99858 for Costa Rica using 35 days of data, with a predicted peak of 886 total cases.
• The Gompertz model for Costa Rica yielded an R² of 0.99854 and predicted a final case count of 887, closely aligning with the GL model.
• For Italy, the Gompertz model achieved an R² of 0.99968, predicting a final case count of approximately 261,052.
• Spain’s Gompertz fit achieved R² = 0.99941, predicting a final case count of 468,495 with data up to March 31, 2020.
• The models converged toward similar peak predictions over time, with Gompertz and GL predictions differing by less than 10 cases by April 1, 2020.
• The models successfully captured the inflection point in daily cases, indicating the onset of decline, even with limited data.