About the Science

Disease models: COVID-19

Mortality data (cumulative) for the United States, acquired from Johns Hopkins, was used to fit the model. (See the model at the bottom of the page.) The model is valid for the original SARS-CoV-2 strain, but does not reflect the increased transmission possible with certain variants.

Published versions of the models or model modifiers can be found in the following papers:

Modifications to the model below were made to the following (in February, 2021):

COVID model flow diagram

This image was created to help visualize the progression of people between affected groups. Those who are susceptible can either end up being exposed or being vaccinated. A small portion of those vaccinated can lose immunity and end up in the exposed group. Exposed individuals can become symptomatic or asymptomatic. Asymptomatic people recover, while symptomatic people can end up recovered, hospitalized, or dead. Hospitalized people can either end up recovered or dead.

NOTE: Only susceptible individuals are eligible to be vaccinated in this model. Those who are exposed, infected, recovered, or dead cannot be vaccinated. This means that as more people are infected, less of the population can receive a vaccine.

Contact scaling with population size

A linear scaling system was used to create a multiplier that reflects an assumed city density. It takes the minimum city population (10,000) and the largest city population (25 million) and creates a linear multiplier between the two that goes from 0.85 to 1.75. This has the effect of reducing the spread of disease in smaller cities (below 1 on the scale) and increasing the spread of disease in larger cities (higher than 1 on the scale). This helps show the increased contact rates in dense metropolitan areas such as New York City or Shanghai. The contact scaling equation is as follows for a specific population A:

Using New York’s metro population of around 20 million people as an example, you get:

So the spread of COVID in New York among asymptomatic and symptomatic people is being multiplied by 1.57 from the base model to show the increase that comes with such a packed metropolitan area.

Parameter info

Vaccination rates

A wide range of daily vaccination rates are out there, depending on when and where you look. Though vaccination rates often start slowly after a vaccine is available, they often ramp up to higher rates as time goes on. As of the end of February, 2021, three months into vaccines being offered in the US, around 0.4 to 0.5% of the US population was being vaccinated every day. With the assumption that it will continue to ramp up a bit more, we selected 0.4% as an overall average for a realistic daily vaccination rate.

Natural herd immunity

When natural herd immunity is selected, we assume that no vaccinations are available; this is the most drastic and costly form of herd immunity, but represents the situation for many diseases before a vaccine can be developed and distributed.

Model equations and parameters

Here is the model we used and the related parameters:

COVID model equations
COVID model population descriptions
COVID model parameter descriptions
COVID model parameter values

Below are the formulas used to calculate the vaccinated R naught value (RcV, equation 0.1 below), the base R naught value (R0, equation 0.2 below), and the vaccinated herd immunity threshold (fv, 0.3 below).

COVID model vaccinated r naught equation
COVID model r naught equation
COVID model herd immunity equation

Learn the latest on the Ask A Biologist COVID-19 vaccines story page.