Why the second wave of COVID-19?

 Here is a link to a post by Nic Lewis at judithcurry.com, "COVID-19: why did a second wave occur even in regions hit hard by the first wave?"

Some excerpts follow.

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Here are his key points.

The herd immunity threshold (HIT) depends positively on the basic reproduction number R0 and negatively on heterogeneity in susceptibility.

Since neither of the factors on which the HIT depends are fixed, the HIT is not fixed either.

R0 depends on biological, environmental and sociological factors; colder weather and the evolution of more transmissible strains likely both increase R0; more (less) cautious behaviour and social distancing / restrictions on mixing reduce (increase) R0.

Second waves were due primarily to changes in these factors increasing R0 and thus the HIT from below to above the existing level of population immunity.

Heterogeneity in susceptibility is partly biological, but social connectivity differences are key.

The effect of heterogeneity in susceptibility on the HIT can be represented by a single parameter λ.

λ will always exceed 1 (its level in a homogeneous population); pre-epidemic λ may be ~4. The higher λ is, the lower the HIT for any given R0.

The natural infection HIT is hence bound to be below the level of {1 – 1/R0} quoted by ‘experts’.

Government restrictions reduce λ as well as R0, so the HIT falls less than it would if λ were fixed.

The final size of an uncontrolled epidemic will substantially exceed the HIT, due to overshoot, so high reported seroprevalence levels can be consistent with a much lower HIT.

Here is his appendix, which attempts a layman's explanation.

The following discussion, which represents my semi-quantitative broad brush analysis of what has occurred, relates primarily to the progress of the epidemic in western Europe. However, it may also be somewhat applicable to the north east United States, where the epidemic took off only slightly later than in western Europe and where the seasonal variation in climate is also large.

In the initial stages of the first wave, which generally started in major cities, in early spring 2020, infections appear to have been doubling every three days or so prior to governments imposing restrictions or people becoming significantly more cautious. Depending on the assumed distribution of the generation interval (from one infection to those it directly leads to), that implies an R0 value of between 2 and 4.[14] I will assume a middle of the range R0 value of 3 for illustrative purposes. That would imply a HIT of 67% for a homogeneous population, reducing to 24% for a population with the highest degree of heterogeneity illustrated in Table 1, which might be expected to apply before people started behaving more cautiously and mixing less.

When people started mixing less, voluntarily or by government fiat, R0 would have reduced, but as discussed above λ will also have fallen. The combined effect of these changes can be visualised as moving diagonally upwards and leftwards in Table 1, from the green columns to the yellow columns and then to the salmon columns. The resulting reduction in the HIT would therefore be somewhat smaller than that implied by the reduction in R0 alone.

By late spring or early summer the first wave had largely faded, and it generally continued to decline after restrictions on mixing were at least partially relaxed. As summer progressed, people’s behaviour unsurprisingly returned closer to pre-epidemic norms. I will assume for illustrative purposes that the yellow columns (λ = 3) were representative of that period. Since by midsummer the epidemic appears to have been declining even where only a minor first wave had occurred, it seems that R0 must generally have declined to 1 or below, so that population immunity levels would everywhere have exceeded the HIT (which is only positive if R0 > 1).

As autumn arrived, infections and then serious illness started to rise again, although where testing was increasing the rise may have been exaggerated. It follows that R0 must have risen again, resulting in the HIT increasing to above the level of population immunity. An obvious explanation for the rise in R0 is seasonally reduced sun and cooler weather, with more contact occurring indoors, where almost all COVID-19 transmission appears to take place. A major increase in mixing among young people as school and, particularly, university terms started likely also boosted R0 and the level of infections in the autumn; young adults have generally had the highest incidence rates during the second wave.[15] In some places the rise in infections appears to have occurred slightly earlier, perhaps as a result of holidaymakers returning infected from areas where COVID-19 was more prevalent.[16]

Initially it seemed that some large cities where a significant proportion of the population had been infected in the first wave might be spared, but in most cases the increase in R0 evidently became sufficiently large to raise the HIT to above the level of population immunity. As a result of increasing infections, government-imposed restrictions were generally increased, which as well as reducing R0 will also have reduced the heterogeneity factor λ. This can be visualised as a move diagonally upwards from the yellow columns to the salmon columns. Those actions appear typically to have pushed Rt down to about 1, or slightly lower, which in the presence of a reasonable degree of existing population immunity implies an R0 level significantly above 1. With reduced heterogeneity, the existing level of population immunity causes a lesser reduction in Rt, relative to R0, but Rt will still be a smaller fraction of R0 than the proportion of the population that remains susceptible to infection.

In the UK, and possibly various other countries, a new lineage (B.1.1.7) of the SARS-CoV-2 virus has now emerged[17] and grown faster than existing ones, as discussed in a previous article[18]. Since writing that article, some further data has provided less indirect evidence that B.1.1.7 is 25–50% more infectious than pre-existing variants.[19] On the other hand, recent data from the regions where B.1.1.7 has become dominant suggests that it may now be growing no faster than other variants.[20] It has been suggested that the fast growth in the regions where B.1.1.7 now dominates may have been at least partly due to it spreading there in schools.[21] However, making for illustrative purposes the assumption that B.1.1.7 is actually 25–50% more infectious, R0 will have been increasing, perhaps typically reaching somewhere in the range1.5 to 2.0 once B.1.1.7 becomes the dominant variant, if R0 was previously in the 1.2 to 1.4 range.

Tougher restrictions that have been introduced in a number of countries in response to infection rates increasing, whether due to the spread of the B.1.1.7 lineage, to cold winter weather or to greater mixing, will have reduced population heterogeneity in social connectivity further. In these circumstances, is unclear whether existing levels of population immunity will suffice to prevent further growth of the B.1.1.7 lineage, or the rather similar one that has emerged in South African, even with severe restrictions being introduced. However, increased population immunity resulting from some combination of further spread of infections and vaccination programmes, the combination varying from one country and region to another, should bring COVID-19 epidemics under control within the next few months.