Is it plausible that strict lockdowns made it more likely for the new variant of COVID to have emerged?

This is more plausible (second point below) and less plausible (third point below).

• Selective advantage The emergence of a new strain of a virus relates to the selective advantage (Gordo 2009). This selective advantage might be that the reproduction rate is relatively higher.

• Reproduction rate The selective advantage (in terms of relative growth rate) is based on the relative growth rate of the virus. Because the growth rate for viruses is not directly proportional to the reproduction rate (Wallinga 2009), this means that better selective advantage is not exactly the same as better reproduction rate (and that is just the selective advantage due to reproduction rate). Instead, it is more like related to the reproduction rate minus one. The relative (initial) growth rate of the mutated virus (made dimensionless by multiplying with generation time) gives the relative growth factor $$text{growthrate} times text{generation time} = overbrace{s}^{text{selective advantage}} = R_{mutant}- R_{other}$$ If you have some mutation that gives an improved effective reproduction factor $R_{mutant}/R_{other} = c$ then the difference $R_{mutant}- R_{other}$ will be larger for a larger $R_{other}$. This is not the case with a lockdown ($R$ will be typically lower) and in a situation with a specific ratio $R_{mutant}/R_{other} = c$ the relative growth of the mutant will be smaller if $R_{other}$ is lower

• What makes reproduction rate higher? There is a small catch with this second point (from which you could conclude that growth of new strains is higher without lockdowns). Another non-linearity is that the reproduction rate is not scaling linearly with the infectivity of the virus.

  For instance Ferrari et.al. (2006) model the probability of infection as an exponential function of the rate of transmission

  The susceptible nodes become infected in the next time-step with binomial probability 1−exp(−βI), where β is the rate of transmission across an edge, and I is the number of infected nodes to which the individual is connected.

  The infections are a function of time and proximity (and also the susceptibility and infectiousness of people), and this is not homogeneously distributed. This makes that a simple reduction of 'infectivity' by half is not at the same time also making the effective reproduction rate become halved.

  You could see the reproduction number being buffered in some sense. There is an excess/overflow of transmission in some parts of the network. When you are sick, then most likely your family members are gonna get sick as well. When the virus becomes half less transmissible, then it is not like only half your family members are gonna get sick. Reducing the transmissibility might have an effect for short-term contacts (which are important pathways as well), but it is of less influence for intensive contacts.

  This buffering makes that the transmissibility of the virus (the reproduction rate) is not so much controlled by the changes in the properties of the virus. The transmission is more controlled by the human traffic and contact network (and the fluctuations occurring within this network/traffic). If an average person sees, on average, only 3 people within the infectious period, then the basic reproduction number will be at most 3 and it can't suddenly make a big jump. You can not infect more than 3 people if you do not see more than 3 people, so the virus can become more transmissible but it won't influence the reproduction rate 'as much' (unless there is an entirely different mode of transport). (I am putting stress on 'as much' because there will be some influence, just like adding acid to a buffer will not have zero effect, but it just won't be the same)

  Any growth of small mutations in new strains is (initially) more likely to be occurring randomly. Viruses mutate a lot, and there will a lot of probability that there is a lucky sample (many others will be unlucky) that might accidentally reach some area with a fast growth rate and with little competition (isolation from competition is a good, but also likely, route for a non-competitive mutation to grow in number). This randomness becomes larger when the total numbers are smaller. If a new strain hits a (relatively) clean area then it will spread out without any/much competition and it can become big (especially if it is an area without much control) and spread over a large area unnoticed.

  This isolation might be more likely in a scenario with strict lockdowns, which makes the spread of the virus more chaotic (it's like rolling a dice a few times, you'll be more likely to get the same number with a high frequency in comparison to rolling the dice often; the lockdowns make the spread dependent on fewer dice rolls and give more chances to some mutation to slip through, reach some prosperous less lock downed area, and become more dominant)

So, in general there is no clear influence from a lockdown. In general the emergence of new variants relates to Fisher's fundamental theorem of natural selection

• "The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time."

The second point above (lower fitness/growth-rate results in lower natural selection) relates to the fact that it we reduce the fitness with a factor $c<1$ by a lockdown, than the variance of the fitness changes by a factor $c^2$. The change of the fitness will scale with $c^2$ and the relative change of the fitness will change with $c$.

The third point above relates to a breaking of the assumptions (there's no homogeneous spread of the virus). Breakdowns can create relatively separate/isolated regions and the spread may not be via a Malthusian growth model but instead more chaotic (with random branches like a thunder bolt). This makes it not unthinkable that a variant emerges quickly by a random effect and without much difference in selective advantage.

Some references

• Gordo, Isabel, et al. "Genetic diversity in the SIR model of pathogen evolution." PloS one 4.3 (2009): e4876.

• Wallinga, Jacco, and Marc Lipsitch. "How generation intervals shape the relationship between growth rates and reproductive numbers." Proceedings of the Royal Society B: Biological Sciences 274.1609 (2007): 599-604.

• Ferrari, Matthew J., et al. "Network frailty and the geometry of herd immunity." Proceedings of the Royal Society B: Biological Sciences 273.1602 (2006): 2743-2748.

Preview: The plausible scenario in which the lockdown could have helped a more contagious viral strain to spread is not the one that we have here.

Scenario: The more contagious viral strain has a selective advantage over the other strains. The speed with which such a strain pushes out the other strains out of the existence, i.e., the time that it needs to fix in the population, depends on the effective population size. In this respect, lockdown resulted in reducing the effective population size, which could have resulted in faster fixation of the more contagious strain. To understand this scenario it is important to keep in mind that the lockdown wasn't waterproof - the virus was still spreading, although at a lower pace.

Reality: We are not dealing with this situation, but rather with a viral strain that has emerged only recently, which is certainly far from having fixed in the population. One cannot even be sure that it has a selective advantage - it is more contagious, but may have other, yet unknown, shortcomings.