Growth in Variant of Concern - Revision history

Alex: /* Estimating transmission advantage using variant counts only */

2026-10-21T22:50:57Z

Estimating transmission advantage using variant counts only

← Older revision		Revision as of 16:50, 21 October 2026
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	A simpler version of this is to combine the estimates weighted by their inverse variance, and this tends to give more-or-less similar answers to the above likelihood method (effectively it's a special case of the likelihood method where we've replaced likelihoods by normal approximations). Under usual Poisson assumptions, the variance of $$\log(AD/(BC))$$ is approximately $$A^{-1}+B^{-1}+C^{-1}+D^{-1}$$, which leads to this estimate for the overall log cross ratio:		A simpler version of this is to combine the estimates weighted by their inverse variance, and this tends to give more-or-less similar answers to the above likelihood method (effectively it's a special case of the likelihood method where we've replaced likelihoods by normal approximations). Under usual Poisson assumptions, the variance of $$\log(AD/(BC))$$ is approximately $$A^{-1}+B^{-1}+C^{-1}+D^{-1}$$, which leads to this estimate for the overall log cross ratio:
	\[		\[\frac{\sum(A_i^{-1}+B_i^{-1}+C_i^{-1}+D_i^{-1})^{-1}\log((A_iD_i)/(B_iC_i))}{\sum(A_i^{-1}+B_i^{-1}+C_i^{-1}+D_i^{-1})^{-1}},\]
	\frac{\sum(A_i^{-1}+B_i^{-1}+C_i^{-1}+D_i^{-1})^{-1}\log((A_iD_i)/(B_iC_i))}{\sum(A_i^{-1}+B_i^{-1}+C_i^{-1}+D_i^{-1})^{-1}},
	\]
	where $$i$$ ranges over different subgroups (typically geographic). Note that such a decomposition would fix the problem with the example above: if $$(A_1,B_1,C_1,D_1)=(100,1,100,2)$$ and $$(A_2,B_2,C_2,D_2)=(1,100,5,1000)$$ then the overall estimate for the cross ratio will be exactly $$2$$ simply because each component is equal to $$2$$.		where $$i$$ ranges over different subgroups (typically geographic). Note that such a decomposition would fix the problem with the example above: if $$(A_1,B_1,C_1,D_1)=(100,1,100,2)$$ and $$(A_2,B_2,C_2,D_2)=(1,100,5,1000)$$ then the overall estimate for the cross ratio will be exactly $$2$$ simply because each component is equal to $$2$$.

Alex at 23:06, 13 August 2021

2021-08-13T23:06:23Z

← Older revision		Revision as of 17:06, 13 August 2021
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	# if the amount of "suppression" (in the sense of the number $$s$$ above) is correlated with the proportion of one variant over the other.		# if the amount of "suppression" (in the sense of the number $$s$$ above) is correlated with the proportion of one variant over the other.

	<nowiki>[This section was mostly written earlier in May 2021 which explains why it is slightly defensive in nature arguing for a high transmission advantage.]</nowiki>		<nowiki>[This section was mostly written earlier in May 2021 which explains why it is slightly defensive in nature arguing for a high growth/transmission advantage, since this was a point of contention with other scientists at the time downplaying the likelihood that B.1.617.2 was going to be a major problem.]</nowiki>

	The first of these objections isn't too hard to work around provided the bias is variant-independent. Non-independence of that type is akin to multiplying up each of $$A$$, $$B$$, $$C$$, $$D$$ by the same constant, and can be dealt with just by widening the confidence interval accordingly. Note also that there is a limit to how big such a multiplier could be because we are currently (date of writing: 27 May 2021) sequencing the majority of viable PCR-confirmed cases in the UK, and it is thought we are finding about 40% of all cases (we will return to this point).		The first of these objections isn't too hard to work around provided the bias is variant-independent. Non-independence of that type is akin to multiplying up each of $$A$$, $$B$$, $$C$$, $$D$$ by the same constant, and can be dealt with just by widening the confidence interval accordingly. Note also that there is a limit to how big such a multiplier could be because we are currently (date of writing: 27 May 2021) sequencing the majority of viable PCR-confirmed cases in the UK, and it is thought we are finding about 40% of all cases (we will return to this point).

Alex at 03:02, 4 August 2021

2021-08-04T03:02:08Z

← Older revision		Revision as of 21:02, 3 August 2021
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	Written in May 2021		Written in May 2021

			''Added note in July 2021'': the point of the second section is to try to get a modelling advantage, in terms of better accuracy and less lag, by combining two different things, the variant counts and the total case counts. The assumption is that the variant counts are only a sample, a subset, of the total case counts, and they are delayed by around two weeks, so "total case counts" is bringing something extra to the party. The method's advantage can only last as long as there are a significant number of each variant type, but by July almost all cases in the UK were Delta so this method becomes similar to standard binomial (or, in general, multinomial) fitting of the variant types.

	The objective here is to try to understand to what extent the variant of concern B.1.617.2 (first found in India, now known as Delta) is growing in the UK, both in its own terms and in comparison to B.1.1.7 (first found in Kent, now known as Alpha). Different methods seemed to be giving different answers and this is an attempt to reconcile them.		The objective here is to try to understand to what extent the variant of concern B.1.617.2 (first found in India, now known as Delta) is growing in the UK, both in its own terms and in comparison to B.1.1.7 (first found in Kent, now known as Alpha). Different methods seemed to be giving different answers and this is an attempt to reconcile them.

Alex at 11:18, 11 June 2021

2021-06-11T11:18:06Z

← Older revision		Revision as of 05:18, 11 June 2021
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	Written in May 2021		Written in May 2021

	The objective here is to try to understand to what extent the variant of concern B.1.617.2 (first found in India) is growing in the UK, both in its own terms and in comparison to B.1.1.7 (first found in Kent). Different methods seemed to be giving different answers and this is an attempt to reconcile them.		The objective here is to try to understand to what extent the variant of concern B.1.617.2 (first found in India, now known as Delta) is growing in the UK, both in its own terms and in comparison to B.1.1.7 (first found in Kent, now known as Alpha). Different methods seemed to be giving different answers and this is an attempt to reconcile them.

	Most indicators considered here suggest a substantial transmission advantage of B.1.617.2 over B.1.1.7, estimated as about 60-70% country-wide, or an extra growth rate of about 10-11% per day, though possibly lower in certain places like London. Here "transmission advantage" is taken to mean the extra biological transmissibility of B.1.617.2 together with any immune-escape advantages it may have at the current level of vaccinations and past infections. The model used here finds typical R numbers of 0.9 for B.1.1.7 and 1.5 for B.1.617.2, though they vary around the country and are notably lower in some areas. In particular, Bolton seems to have crushed the outbreak, presumably by means of intensive "surge" tactics, and Wales, without any particular country-wide surge tactics, has a considerably lower virus growth rate than the rest of the UK, maybe in part due to the significantly higher vaccination levels in that nation.		Most indicators considered here suggest a substantial transmission advantage of B.1.617.2 over B.1.1.7, estimated as about 60-70% country-wide, or an extra growth rate of about 10-11% per day, though possibly lower in certain places like London. Here "transmission advantage" is here taken to mean the extra biological transmissibility of B.1.617.2 together with any immune-escape advantages it may have at the current level of vaccinations and past infections. In other words, "transmission advantage" is taken to mean $$R_t(\text{B.1.617.2})/R_t(\text{B.1.1.7})$$, and not, for example, $$R_0(\text{B.1.617.2})/R_0(\text{B.1.1.7})$$. The model used here finds typical R numbers of 0.9 for B.1.1.7 and 1.5 for B.1.617.2, though they vary around the country and are notably lower in some areas. In particular, Bolton seems to have crushed the outbreak, presumably by means of intensive "surge" tactics, and Wales, without any particular country-wide surge tactics, has a considerably lower virus growth rate than the rest of the UK, maybe in part due to the significantly higher vaccination levels in that nation.

	This analysis is predicated on the idea that there is a single multiplicative constant, the ratio of the (time-varying) R-numbers for B.1.617.2 and B.1.1.7, that is independent of suppression measures or immunity. This seems plausible as an approximation, though it is not biologically guaranteed.		This analysis is predicated on the idea that there is a single multiplicative constant, the ratio of the (time-varying) R-numbers for B.1.617.2 and B.1.1.7, that is independent of suppression measures or immunity. This seems plausible as an approximation, though it is not biologically guaranteed.

Alex: /* Model details */

2021-06-11T11:14:12Z

Model details

← Older revision		Revision as of 05:14, 11 June 2021
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	\]		\]

	Likelihood/distributions (and definitions of intermediate terms), where $$i=0,\ldots,N-1$$ or $$i=0,\ldots,N$$, and $$N=$$(number of days):		Likelihood/distributions (and definitions of intermediate terms), where $$i=0,\ldots,D-1$$ or $$i=0,\ldots,D$$, and $$D=$$(number of days), $$N=D+2g_\text{timescale}$$:
	\[		\[
	\begin{equation*}		\begin{equation*}
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	\]		\]

	A bit about the motivation for these choices: growth rates (or reproduction numbers) for each variant can change over time, so it is desirable to allow for some time-variation in these. Keeping the growth rate of the two variants separated by a universal constant ($$h$$) seems like a plausible assumption which allows us to make decent predictions. The formula for $$g_i$$ is a kind of low-pass filtered Brownian motion (the variance of the Fourier coefficients being in a certain proportion). This parametrizes a nice space of smooth functions that may possibly work better than, for example, splines because it is low dimensional and hopefully characterises the kind of "wiggles" that are likely to occur. The parameter space being of relatively low dimension makes it fast and robust to analyse. The beta-binomial is used as an "overdispersed" binomial distribution. We don't try to model $$r_j+s_j$$, only $$r_j, s_j$$ conditioned on $$r_j+s_j$$. The reason for this is that we don't want/need to get into predicting what proportion of cases are sequenced, because it's volatile and not relevant to inferring the true proportion of variants, assuming there is no bias relating overall number of variant counts to the proportion of the new variant. Doing it this way has the benefit that we can use recent, incomplete, variant counts alongside older, complete ones.		A bit about the motivation for these choices: growth rates (or reproduction numbers) for each variant can change over time, so it is desirable to allow for some time-variation in these. Keeping the growth rate of the two variants separated by a universal constant ($$h$$) seems like a plausible assumption which allows us to make decent predictions. The formula for $$g_i$$ is a kind of low-pass filtered Brownian motion (the variance of the Fourier coefficients being in a certain proportion). This parametrizes a nice space of smooth functions that may possibly work better than, for example, splines because it is low dimensional and hopefully characterises the kind of "wiggles" that are likely to occur. The parameter space being of relatively low dimension makes it fast and robust to analyse. We choose $$N>D$$ to remove boundary/periodicity effects since the space of parametrized functions is periodic in $$N$$ plus a drift term. The beta-binomial is used as an "overdispersed" binomial distribution. We don't try to model $$r_j+s_j$$, only $$r_j, s_j$$ conditioned on $$r_j+s_j$$. The reason for this is that we don't want/need to get into predicting what proportion of cases are sequenced, because it's volatile and not relevant to inferring the true proportion of variants, assuming there is no bias relating overall number of variant counts to the proportion of the new variant. Doing it this way has the benefit that we can use recent, incomplete, variant counts alongside older, complete ones.
	<nowiki>[In fact we use the negative binomial and beta-binomial for training, but variance-scaled-Poisson and variance-scaled-Binomial with corresponding parameters for runs. Writeup to be completed later.]</nowiki>		<nowiki>[In fact we use the negative binomial and beta-binomial for training, but variance-scaled-Poisson and variance-scaled-Binomial with corresponding parameters for runs. Writeup to be completed later.]</nowiki>

Alex: /* Model details */

2021-06-04T13:49:37Z

Model details

← Older revision		Revision as of 07:49, 4 June 2021
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	A bit about the motivation for these choices: growth rates (or reproduction numbers) for each variant can change over time, so it is desirable to allow for some time-variation in these. Keeping the growth rate of the two variants separated by a universal constant ($$h$$) seems like a plausible assumption which allows us to make decent predictions. The formula for $$g_i$$ is a kind of low-pass filtered Brownian motion (the variance of the Fourier coefficients being in a certain proportion). This parametrizes a nice space of smooth functions that may possibly work better than, for example, splines because it is low dimensional and hopefully characterises the kind of "wiggles" that are likely to occur. The parameter space being of relatively low dimension makes it fast and robust to analyse. The beta-binomial is used as an "overdispersed" binomial distribution. We don't try to model $$r_j+s_j$$, only $$r_j, s_j$$ conditioned on $$r_j+s_j$$. The reason for this is that we don't want/need to get into predicting what proportion of cases are sequenced, because it's volatile and not relevant to inferring the true proportion of variants, assuming there is no bias relating overall number of variant counts to the proportion of the new variant. Doing it this way has the benefit that we can use recent, incomplete, variant counts alongside older, complete ones.		A bit about the motivation for these choices: growth rates (or reproduction numbers) for each variant can change over time, so it is desirable to allow for some time-variation in these. Keeping the growth rate of the two variants separated by a universal constant ($$h$$) seems like a plausible assumption which allows us to make decent predictions. The formula for $$g_i$$ is a kind of low-pass filtered Brownian motion (the variance of the Fourier coefficients being in a certain proportion). This parametrizes a nice space of smooth functions that may possibly work better than, for example, splines because it is low dimensional and hopefully characterises the kind of "wiggles" that are likely to occur. The parameter space being of relatively low dimension makes it fast and robust to analyse. The beta-binomial is used as an "overdispersed" binomial distribution. We don't try to model $$r_j+s_j$$, only $$r_j, s_j$$ conditioned on $$r_j+s_j$$. The reason for this is that we don't want/need to get into predicting what proportion of cases are sequenced, because it's volatile and not relevant to inferring the true proportion of variants, assuming there is no bias relating overall number of variant counts to the proportion of the new variant. Doing it this way has the benefit that we can use recent, incomplete, variant counts alongside older, complete ones.
	<nowiki>[In fact we use the negative binomial and beta-binomial for training, but ~~quasi~~-Poisson and ~~quasi~~-Binomial with corresponding parameters for runs. Writeup to be completed later.]</nowiki>		<nowiki>[In fact we use the negative binomial and beta-binomial for training, but variance-scaled-Poisson and variance-scaled-Binomial with corresponding parameters for runs. Writeup to be completed later.]</nowiki>

	The log likelihood, including the prior for the parameters (and omitting some constant terms) is then		The log likelihood, including the prior for the parameters (and omitting some constant terms) is then

Alex: /* Combining variant counts with case counts */

2021-06-04T13:48:48Z

Combining variant counts with case counts

@@ Line 179: / Line 179: @@
 NB: the confidence intervals in the above table underestimate the uncertainty, because the model relies on assumptions which may not be completely correct.
-NB2: The above estimates are not independent of each other because, in this mode, the model is using information from around the country to inform the more local estimates.
+NB2: The above estimates are not independent of each other because, in this mode, the model is using information from around the country to inform the more local estimates. We can illustrate this by the following two different runs that estimate parameters for the regions of England.
 = Model details =

Alex: /* Combining variant counts with case counts */

2021-06-04T13:29:02Z

Combining variant counts with case counts

← Older revision		Revision as of 07:29, 4 June 2021
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	\|England \|\| UK \|\| COG-UK \|\| 74% (70% - 79%) \|\| 0.88 (0.78 - 0.98) \|\| 1.53 (1.36 - 1.71)		\|England \|\| UK \|\| COG-UK \|\| 74% (70% - 79%) \|\| 0.88 (0.78 - 0.98) \|\| 1.53 (1.36 - 1.71)
	\|-		\|-
	\|England \|\| LTLAs \|\| Sanger \|\| 72% (62% - 83%) \|\| 0.95 ~~(1.00 - 1.20)~~ \|\| 1.49 (1.27 - 1.73)		\|England \|\| LTLAs \|\| Sanger \|\| 72% (62% - 83%) \|\| 0.95 \|\| 1.49 (1.27 - 1.73)
	\|-		\|-
	\|Scotland \|\| UK \|\| COG-UK \|\| 74% (70% - 79%) \|\| 0.96 (0.81 - 1.14) \|\| 1.68 (1.39 - 1.99)		\|Scotland \|\| UK \|\| COG-UK \|\| 74% (70% - 79%) \|\| 0.96 (0.81 - 1.14) \|\| 1.68 (1.39 - 1.99)
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	\|Wales \|\| UK \|\| COG-UK \|\| 74% (70% - 79%) \|\| 0.81 (0.62 - 1.07) \|\| 1.42 (1.08 - 1.87)		\|Wales \|\| UK \|\| COG-UK \|\| 74% (70% - 79%) \|\| 0.81 (0.62 - 1.07) \|\| 1.42 (1.08 - 1.87)
	\|-		\|-
	\|London \|\| London LTLAs \|\| Sanger \|\| 44% (24% - 68%) \|\| 0.92 ~~(0.92 - 1.20)~~ \|\| 1.29 (1.12 - 1.57)		\|London \|\| London LTLAs \|\| Sanger \|\| 44% (24% - 68%) \|\| 0.92 \|\| 1.29 (1.12 - 1.57)
	\|-		\|-
	\|London \|\| England \|\| Sanger \|\| 71% (64% - 78%) \|\| 0.83 (0.69 - 1.00) \|\| 1.41 (1.17 - 1.71)		\|London \|\| England \|\| Sanger \|\| 71% (64% - 78%) \|\| 0.83 (0.69 - 1.00) \|\| 1.41 (1.17 - 1.71)
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	NB: the confidence intervals in the above table underestimate the uncertainty, because the model relies on assumptions which may not be completely correct.		NB: the confidence intervals in the above table underestimate the uncertainty, because the model relies on assumptions which may not be completely correct.

			NB2: The above estimates are not independent of each other because, in this mode, the model is using information from around the country to inform the more local estimates.

	= Model details =		= Model details =

Alex: /* Combining variant counts with case counts */

2021-06-02T10:17:22Z

Combining variant counts with case counts

← Older revision		Revision as of 04:17, 2 June 2021
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	\|Wales \|\| UK \|\| COG-UK \|\| 74% (70% - 79%) \|\| 0.81 (0.62 - 1.07) \|\| 1.42 (1.08 - 1.87)		\|Wales \|\| UK \|\| COG-UK \|\| 74% (70% - 79%) \|\| 0.81 (0.62 - 1.07) \|\| 1.42 (1.08 - 1.87)
	\|-		\|-
	\|London \|\| London LTLAs \|\| Sanger \|\| 41% (31% - 52%) \|\| 0.93 (0.92 - 1.12) \|\| 1.29 (1.20 - 1.55)		\|London \|\| London LTLAs \|\| Sanger \|\| 44% (24% - 68%) \|\| 0.92 (0.92 - 1.20) \|\| 1.29 (1.12 - 1.57)
	\|-		\|-
	\|London \|\| England \|\| Sanger \|\| 71% (64% - 78%) \|\| 0.83 (0.69 - 1.00) \|\| 1.41 (1.17 - 1.71)		\|London \|\| England \|\| Sanger \|\| 71% (64% - 78%) \|\| 0.83 (0.69 - 1.00) \|\| 1.41 (1.17 - 1.71)

Alex: /* Combining variant counts with case counts */

2021-06-02T10:15:56Z

Combining variant counts with case counts

← Older revision		Revision as of 04:15, 2 June 2021
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	\|London \|\| England \|\| Sanger \|\| 71% (64% - 78%) \|\| 0.83 (0.69 - 1.00) \|\| 1.41 (1.17 - 1.71)		\|London \|\| England \|\| Sanger \|\| 71% (64% - 78%) \|\| 0.83 (0.69 - 1.00) \|\| 1.41 (1.17 - 1.71)
	\|-		\|-
	\|Bolton \|\| England \|\| Sanger \|\| 64% (59% - 69%) \|\| 0.62 (0.50 - 0.78) \|\| 1.02 (0.82 - 1.28)		\|Bolton \|\| England \|\| Sanger \|\| 72% (62% - 83%) \|\| 0.59 (0.47 - 0.75) \|\| 1.02 (0.81 - 1.30)
	\|}		\|}