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In the CDC webpage about Covid-19's variants the next sentence appears:

This (the United Kingdom) variant spreads more easily and quickly than other variants.

In the ECDC paper about Covid-19's variants we can see that:

Several modeling studies corroborate the postulated increased transmissibility of VOC 202012/01 (the scientific name of the United Kingdom variant)

In a Nature article from last week written:

Several research groups suggested that B.1.1.7, which was first identified in the United Kingdom, was spreading more quickly than previous variants...

The probabilistic meaning of these sentences is not clear to me. Does this mean that the risk to be infected by the British variant is X times greater? E.g. if I was exposed to the British variant for 10 minutes, my chance of contracting is equal to the chance of contagion the original variant for 10 * X minutes?

Is there any nuance I miss?

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The studies referenced do not have any such specifics about transmission probability.

What they observe is that a variant is becoming more common. Modeling the spread gives a different R0 value for the variants vs the reference strains:

In line with previous modelling studies, the R0 for VOC-202012/01 was estimated as 75% more transmissible than other variants [35]. Davies et al. modelled data from three English regions to estimate that VOC 202012/01 is 56% more transmissible (95% credible interval (CrI): 50-74%) than previously identified SARS-CoV-2 variants [32]. Danish modelling studies performed in January 2021 estimated that VOC 202012/01 was 36% more transmissible than other variants [ECDC communication with Danish authorities].

(from your second reference)

They infer that the variant is more transmissible because the modeled number of people subsequently infected per infected person is greater. The reasons for this cannot be easily determined by the methodology they refer to, and there is no simple way to map this onto probability of infection given some exposure; they are just describing what happens in real-world conditions, on average.

What you can infer is that there are some types of contact that occur in the real world that wouldn't result in transmission with the old strain that do result in transmission of the variant. This could be because more infected people produce enough virus to spread to others, it could be because shorter contacts lead to infection, it could be because infected people have different behavior that leads to more contact with people to potentially get infected; you can't tell the difference from these data.

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