# How to estimate the change of rate of COVID-19 infections when accounting for vaccine effectiveness and increased transmissibility of mutated strains?

The world has reasonably accurate numbers of the transmissibility of the strains of COVID-19 that have ravaged humankind.

Now, two changes are afoot:

1. Some wealthy regions have access to limited quantities of vaccines (although most poor regions continue to be without any access to vaccines). Each vaccine has its own scientific estimation of efficacy.

2. New mutated strains of COVID-19 have been discovered with significantly higher levels of transmissibility.

What is the correct way to take into account those two changes when projecting case rates?

For example, if you have a vaccine that is 80% efficacious against a new strain, but that new strain is 80% more transmissible than the original strain (and that new strain becomes dominant, which is scientifically most likely), what is the estimated net outcome in case rates (removing other variables)? Is it zero? To keep things simple, let us assume that the new strain will account for all new cases.

As another example, with the same conditions and removal of variables as above, if the vaccine is 65% efficacious against the new strain, but that new strain is 80% more transmissible than the original strain, what is the estimated net outcome in case rates?

• Are you really interested in just case transmission rates? What if someone gets the virus, but has no symptoms and incurs no costs as a result? Have you defined "case" more carefully? – StatsStudent Feb 2 at 3:29
• @StatsStudent Yes, I am really interested in transmission rates. If someone gets the virus, but has no symptoms or financial costs, that is still an infection case. – RockPaperLz- Mask it or Casket Feb 2 at 3:31
• Purely statistical methods are a (spectacularly) poor choice for addressing such questions: you need an epidemiological model of transmission, infection, vaccination efficacy, and case identification. – whuber Feb 2 at 10:21
• @whuber I partially agree, except that I'm not interested in publishing results. I'm only interested in this estimation for personal understanding. – RockPaperLz- Mask it or Casket Feb 2 at 13:24
• I am very open to those possibilities -- but I also am aware of so many of those "shades of gray," and have so much experience with this, that I'm trying to help you out by saving you some wasted effort (or worse, developing and communicating misleading results). But I appreciate that you don't value that and therefore won't continue my efforts to help you. I do, however, want to make sure other readers of this thread are not similarly misled. – whuber Feb 2 at 13:45

### You are not gonna get an answer

Exactly predicting the influence of effects such as mutated strains is extremely difficult. Even governments with lot's of mathematicians and big computers working for them are not well capable of making good predictions (or even bad predictions).

Predicting the course of Covid is like predicting the weather more than 14 days ahead of time but with very inaccurate models and inaccurate measurements.

• Models that are inaccurate about details. Like

• How do people move around, where do contacts take place, what is the influence of news and politics, etc.
• In which ways does the virus transmit? What are effects of measures on this (masks, distancing, ventilation, hand-washing, vitamin-D and exercise, etc.)
• Measurements of the current weather (state of the virus) that are inaccurate

• What are the current epidemiological parameters? How do they vary as function of weather (effects of temperature, humidity, wind), age of the people, health of the people, immunity due to vaccines, immunity to having had the disease (and how long this lasts), location (variations in population density, human behavior, state of health, age, partial herd immunity).

• What is different for the new strain? People believe that several strains have a selective advantage, but this is mostly based on simple measurements about the total cases (and theory about the changes in the proteins).

How is the strain better or more fit? Does it spread easier or faster? Is less viral load needed or does it create a high viral load? Does it survive better in some environments? Is it having some (partial) resistance to immunity in some people (how much and in which people?)? Is its incubation time shorter or is it's infectious phase longer? Etc.

### Intuition of the mechanics by means of a game

If your goal with this question is just to gain some insight into the underlying theoretical considerations then you might get some improvements in your intuition.

In this part of the question, I am gonna give some intuition by replacing viruses with dice. I am actually currently making a board game that includes the aspect of virus mutations, and now even my 5-year-old son "understands" the aspects of the mechanics and aspects of probability for an epidemic.

The game mimics transmission by rolling several dice. When you roll a six then an infection occurs. When you roll 2/3 or more sixes then the newly infected person gets a headache as well (2 when it's an adult, 3 for kids).

Let's see what the probability is for rolling at least 1 six.

Note: It is not linear! Reducing transmissibility rate by $$x \%$$ (changing the number of dice, the viral load) may not be the same reduction in transmissibility probability per contact.

In this game, the virus variant will increase the number of dice used in the infection by 'two dice' (in a real-world the variant might also increase the rate of infection, the viral load, in some way).

Already for this very basic game, it is very difficult to compute the effect of the variant (the two extra dice). There are several ways in which the effect of higher infectivity will have different effects that vary due to inhomogeneities.

• When you change the number of dice from 10 to 12 then the effect (probability increases a bit from 84% to 89%) is a lot different from when you change the number of dice of 1 to 3 (Increase from 17% to 42%, more than doubled).

There are several ways in which increased infectivity will have inhomogeneous effects. The effect of increased infectivity is for this game the largest when the infectivity is low (I believe the same will be true for the real world). When the probability of infection is high, for instance inside families, then people are gonna get sick anyway (at least with high probability).

(The same is true for the interventions/measures like mask-wearing that are in the opposite direction as the virus variant; For these as well the effects will be variable and depend on the situation. There is another question here where this came across as well.)

• The types of spread and the population are inhomogeneous. Some areas are more crowded than others, have more complex networks with bigger knots, etc. Because of these, you may have different effects of the increased transmissibility of the variant (the increase of dices).

It is wrong to think of some virus mutation to increase the R(t) value (the reproduction number) or the transmissibility rate or transmissibility probability.

Those figures eventually are just based on an estimated/observed growth rate that is rescaled to express the reproduction scale.

The growth rate might is not the same as transmissibility, and transmissibility is not the same as what changed in the virus mutant. The underlying mechanisms are complex, inhomogeneous, and change from place to place. The growth rate, or other related statistics, is just what is observed at the surface, but it is something inside the machinery that has changed.

### In short

The effect of a virus mutant is not easily determined.

Let alone that the effect can be predicted/extrapolated. But what people (politicians and media) do is just extrapolate the observed effects of the mutant to other populations (without considering the mechanics and the idea that the effects may be different for different situations).

For example, in South East England scientists observe some increase of a variant. Subsequently, it became the dominant strain in London.

Then people start to use the same figures (like R value increases such and such) and plug it into their own simple model as if this effect on the surface (a particular increase of the R value) is gonna be exactly the same.

I believe that this is not the right way to handle the lack of information. The correct way is to gather more/richer/better data (and that is what people did afterward, and they found in many other places that the variant is not as strongly fast-spreading, or sometimes they found no effect at all.).

a vaccine that is 80% efficacious against a new strain, but that new strain is 80% more transmissible

...

what is the estimated net outcome in case rates

... Is it zero?

Typically the vaccine/immunity is gonna have more effect.

In the basic SIR model, the effective reproduction number is often compared with the basic reproduction number along with the fraction of people that are not immune (non-susceptible) (the ratio of the total susceptible people $$S$$ over the total number of people $$N$$)

$$R(t) = R_0 \cdot S/N$$

So vaccination is gonna reduce that number $$S$$ and mutations are gonna increase $$R_0$$. However, the number $$R_0$$ does not increase linearly with an increase of transmissibility, it will be less.