[Paper Review] The D model for deaths by COVID-19

J. E. Amaro|arXiv (Cornell University)|Mar 30, 2020

COVID-19 epidemiological studies2 references4 citations

TL;DR

This paper proposes the D model, a simplified analytical framework derived from the SI epidemiological model, to describe and predict COVID-19 death trajectories. By assuming no recovery and modeling deaths as a delayed, scaled function of infections, the model fits data from China, Spain, and Italy with three adjustable parameters, enabling accurate short-term predictions of pandemic peak timing and plateau levels despite minimal assumptions.

ABSTRACT

We present a simple analytical model to describe the fast increase of deaths produced by the corona virus (COVID-19) infections. The 'D' (deaths) model comes from a simplified version of the SIR (susceptible-infected-recovered) model known as SI model. It assumes that there is no recovery. In that case the dynamical equations can be solved analytically and the result is extended to describe the D-function that depends on three parameters that we can fit to the data. Results for the data from Spain, Italy and China are presented. The model is validated by comparing with the data of deaths in China, which are well described. This allows to make predictions for the development of the disease in Spain and Italy.

Motivation & Objective

• To develop a fast, analytical model for predicting the trajectory of COVID-19 deaths without relying on complex numerical simulations.
• To validate a simplified model based on the SI framework—assuming no recovery—by fitting it to real-world death data from multiple countries.
• To provide a low-parameter, interpretable tool for early pandemic forecasting in countries with ongoing outbreaks.
• To assess the model's predictive power by comparing its fit to data from China, where the pandemic was already peaking, and applying it to Spain and Italy during active spread.

Proposed method

• The model derives from the SI equation, assuming no recovery (R=0), reducing the system to a single differential equation for infected individuals I(t).
• The solution to the SI equation is transformed into a logistic-type function for I(t), which is then used to model deaths D(t) as a delayed and scaled version of I(t).
• The D(t) function is expressed as D(t) = a * exp((t-t₀)/b) / (1 + c * exp((t-t₀)/b)), with three fit parameters: a (amplitude), b (time constant), and c (initial condition scaling).
• The parameters a, b, and c are fitted to actual death data from China, Spain, and Italy using nonlinear regression, without requiring knowledge of underlying biological parameters like transmission or recovery rates.
• The model assumes a fixed delay τ between infection and death and a constant mortality rate m, both absorbed into the fitted parameters.
• The model is validated on China’s data—where the curve had plateaued—before being used to predict the peak timing and final death count in Spain and Italy.

Experimental results

Research questions

• RQ1 Can a simplified analytical model based on the SI framework accurately describe the death trajectory of the COVID-19 pandemic in multiple countries?
• RQ2 How well does the D model, with only three parameters, fit real-world death data from China, Spain, and Italy?
• RQ3 Can the model reliably predict the peak of the pandemic and the final death count in countries still in the ascending phase of the outbreak?
• RQ4 What is the impact of assuming no recovery on the model’s predictive accuracy and applicability?

Key findings

• The D model fits the death data from China remarkably well, validating its use as a predictive tool for other countries.
• For Spain, the model predicted that the pandemic would reach its peak around day 21 of the outbreak, with the plateau expected by mid-April 2020.
• The model predicted that Italy would reach its pandemic peak in approximately 20 days from the start of data collection, similar to Spain.
• The model's three-parameter function captured the exponential rise and subsequent plateau in death curves across all three countries with high fidelity.
• The fit parameters a, b, and c were sufficient to describe the entire death curve without needing to estimate underlying biological parameters such as transmission rate or recovery time.
• The model demonstrated strong predictive capability for countries in the early phase of the pandemic, such as Spain and Italy, based on data from China alone.