What the usual COVID-19 models leave out

Models of COVID-19 spread reflect a division of the population into groups such as: susceptible, exposed, infected, contagious, recovered, and deceased. An example are the various SEIR models, where S, E, I, and R stand for Susceptible, Exposed, Infected, and Recovered, respectively. These SEIR models and similar ones are widely used and quoted by the media, politicians, and representatives of institutions such as the CDC and WHO.

SEIR models typically incorporate relationships such as the following.

1. The rate of increase in Infected persons is proportional to the number of Exposed persons, less the decline in the number of Infected persons, due to recovery or death.

2. The rate of increase in Exposed persons is proportional to the product of the number of Susceptible persons and the number of Infected persons divided by the current population size, less the decline in Exposed persons who become Infected.

3. The rate of decrease in Susceptible persons is the negative of the rate of increase of Exposed persons.

4. The rate of increase in Recovered persons is proportional to the number of Infected persons.

These kinds of relationships imply that the rate at which a disease spreads can be reduced by:

1. Reducing the likelihood that a Susceptible person meets an Infected person.

2. Reducing the likelihood that transmission occurs if a Susceptible person meets an Infected person.

3. Reducing the number of Susceptible people.

4. Reducing the number of Infected people.

For example, lockdowns and social distancing work by reducing the likelihood that a Susceptible person will meet an Infected person, and wearing masks works by reducing the likelihood that transmission occurs if a Susceptible person meets an Infected person.

In these SEIR models, the number of Susceptible people is reduced to the extent that they become exposed and either infected or prove to be immune.

What is left out of all these models is allowance for the fact that Recovered persons can become Susceptible persons if their immunity lapses. If this happens before the disease runs its course, the rate of increase in Infected persons is increased and the pandemic worsens compared to if there is long-term immunity.

Here are two extreme examples that illustrate the impact of immunity.

Case 1: Immunity lasts forever.

Since a Recovered person cannot become a Susceptible person, the number of Susceptible persons is lower than it might otherwise be. Other things equal, this reduces the rate of increase in Exposed persons from what it would otherwise be, which reduces the rate of increase in Infected persons from what it would otherwise be. The disease runs its course with minimum loss of life and dies out over a reasonable time.

Case 2: Immunity lasts one day.

Since a Recovered person becomes a Susceptible person almost immediately, the number of Susceptible people never declines much (unless everyone gets the disease at once and recovers at once). Other things equal, this increases the rate of increase of Exposed persons from what it might otherwise be, which raises the rate of increase of Infected persons from what it would otherwise be. The disease may never die out and the loss of life is maximized.

Cases 1 and 2 suggest that policies that do not reflect the nature of immunity are unlikely to be efficient. COVID-19 models that fail to address immunity should not be trusted to provide good guidance to decision makers who set policy – but that is exactly what is happening.

For the techies, an example of an SEIR model is: