[Paper Review] A simple mathematical model for the evolution of the corona virus

Stefan Tappe|arXiv (Cornell University)|Mar 20, 2020

COVID-19 epidemiological studies6 references4 citations

TL;DR

This paper proposes a simple two-parameter mathematical model for forecasting COVID-19 deaths using logarithmic transformation and piecewise-defined functions, with convex growth before governmental interventions and concave stabilization afterward. The model accurately fits Chinese data and predicts Italy’s death trajectory, showing that earlier interventions could halve total deaths, making it accessible for students and public health planners alike.

ABSTRACT

The goal of this note is to present a simple mathematical model with two parameters for the number of deaths due to the corona (COVID-19) virus. The model only requires basic knowledge in differential calculus, and can also be understood by pupils attending secondary school. The model can easily be implemented on a computer, and we will illustrate it on the basis of case studies for different countries.

Motivation & Objective

• To develop a mathematically simple yet effective model for predicting the evolution of COVID-19 deaths using only basic calculus.
• To enable practitioners and students with minimal mathematical training to understand and apply the model.
• To estimate the impact of governmental interventions by modeling the transition from exponential to saturated growth in death counts.
• To extend the model to estimate infected populations and infection probabilities using known mortality and reporting rates.
• To validate the model on real-world data from China and predict outcomes for countries like Italy, Germany, and the USA.

Proposed method

• Model the number of deaths D(t) as an exponential of a logarithmic function L(t), i.e., D(t) = exp(L(t)), to linearize growth dynamics.
• Define L(t) piecewise: on [t₁, T₁], use a power function with concavity parameter β ∈ (0,1] to capture accelerating growth.
• On (T₁, ∞), model L(t) using a logistic-type function L(t) = L(T₁) + λ(1 − exp(−ν(t − T₁))) to represent decelerating growth and stabilization.
• Estimate parameters β and λ using early death data and assume λ is shared across countries with similar intervention rigor, e.g., λ = ln(3.25) for China and Italy.
• Use the time T₁ as 17 days after governmental measures, based on the average incubation-to-death period.
• Derive estimates for infected individuals I(t) using I(t) = D(t+17)/(μ·κ), with μ = 0.01 and κ = 5 as standard values.

Experimental results

Research questions

• RQ1 How can a simple mathematical model with only two parameters capture the evolution of COVID-19 deaths across different countries?
• RQ2 What is the impact of governmental intervention timing on the final number of deaths, as predicted by the model?
• RQ3 Can the model accurately fit historical death data from China and reliably predict future trends in countries like Italy?
• RQ4 How does the concavity parameter β relate to the growth pattern of deaths before and after interventions?
• RQ5 To what extent can the model estimate the true number of infected individuals and infection probabilities using only death data?

Key findings

• The model fits Chinese death data well with β = 0.6 and λ = ln(3.25), showing a clear transition from convex to concave growth around February 10, 2020.
• For Italy, the model estimates β = 0.65 and uses the same λ = ln(3.25), predicting a peak death count of approximately 13,000 by late April 2020.
• A scenario analysis shows that advancing governmental measures in Italy by three days (to March 25 instead of March 28) would reduce total deaths by roughly half.
• The model predicts that countries with β > 0.6, such as Germany and the UK, may experience less concave growth, suggesting more sustained death increases.
• The model estimates that the true number of infected individuals in Italy could be up to 50 times higher than reported cases, based on μ = 0.01 and κ = 5.