climateetc--2020-06-09--Dynamics of the Tropical Atmosphere and Oceans

Peter Webster ’ s magnum opus is now published : Dynamics of the Tropical Atmosphere and Oceans .
From the blurb on amazon.com : “ This book presents a unique and comprehensive view of the fundamental dynamical and thermodynamic principles underlying the large circulations of the coupled ocean-atmosphere system Dynamics of The Tropical Atmosphere and Oceans provides a detailed description of macroscale tropical circulation systems such as the monsoon , the Hadley and Walker Circulations , El Niño , and the tropical ocean warm pool .
These macroscale circulations interact with a myriad of higher frequency systems , ranging from @ @ @ @ @ @ @ that attend the low-frequency background flow .
A comprehensive overview of the dynamics and thermodynamics of large-scale tropical atmosphere and oceans is presented using both a “ reductionist ” and “ holistic ” perspectives of the coupled tropical system .
The reductionist perspective provides a detailed description of the individual elements of the ocean and atmospheric circulations .
The physical nature of each component of the tropical circulation such as the Hadley and Walker circulations , the monsoon , the incursion of extratropical phenomena into the tropics , precipitation distributions , equatorial waves @ @ @ @ @ @ @ holistic perspective provides a physical description of how the collection of the individual components produces the observed tropical weather and climate .
How the collective tropical processes determine the tropical circulation and their role in global weather and climate is provided in a series of overlapping theoretical and modelling constructs .
Following a detailed description of tropical phenomenology , the reader is introduced to dynamical and thermodynamical constraints that guide the planetary climate and establish a critical role for the tropics .
Equatorial wave theory is developed for simple and complex background flows @ @ @ @ @ @ @ moist processes .
The manner in which the tropics and the extratropics interact is then described , followed by a discussion of the physics behind the subtropical and near-equatorial precipitation including arid regions .
The El Niño phenomena and the monsoon circulations are discussed , including their covariance and predictability .
Finally , the changing structure of the tropics is discussed in terms of the extent of the tropical ocean warm pool and its relationship to the intensity of global convection and climate change.
” The complete table of contents are found here @ @ @ @ @ @ @ role in this was to help edit the final drafts of the chapters , so I have been through the entire book with a very detailed read .
Here is what stands out for me in the book .
First , the book is ‘ old school ’ in the sense of integrating observations and theory .
This approach is surprisingly rare these days in climate dynamics , with its heavy reliance on global climate model simulations .
The book has a very strong foundation in fluid dynamics and wave dynamics .
At @ @ @ @ @ @ @ are sufficiently clear to be followed by students , with additional details in the appendices .
Second , the book presents an underlying philosophy for approaching the understanding of tropical dynamics , integrating reductionist and holistic approaches .
Third , the book provides historical context for the development of our understanding .
Interesting historical snippets are provided , including biographical notes of key historical scientists .
Fourth , the above three elements integrate to provide insights into the process of the science of climate dynamics , not merely a recitation of our current @ @ @ @ @ @ @ diagrams/figures in the book , including many originally drawn schematics that are very effective at providing insights and supporting understanding .
An example is provided below : Figure 14.30 Schematic of the sequence of events in 1997–1998 .
( a ) The climatological alongshore winds off Sumatra ( E ) and the east African coast ( F ) .
The winds observed in the late summers and early autumns are denoted by G and H , respectively .
The right-hand panel shows the effect at the equator on the upper ocean induced by @ @ @ @ @ @ @ upwelling in the west .
Wind into and out of the plane of the page are denoted by the bull ’ s-eye and cross-hair symbols , respectively .
( b ) Distribution of the winds resulting from the anomalous SST gradient along the Equator and the changes in the SSH distribution .
( c ) Formation of the Ekman ridge in the central Indian Ocean and the forcing of westward-propagating downwelling equatorial Rossby waves to the west .
The right-hand panel shows the effect on the upper ocean near 5°S .
( d @ @ @ @ @ @ @ Ocean through enhanced mixing and coastal Ekman transports from stronger than average monsoon winds and through circulation changes associated with the weakening of the 1997–1998 El Niño .
The next section provides the complete text from the concluding chapter .
When I entered graduate school in the late 1960s , my advisor at the time had just completed a treatise entitled “ Some remaining problems in numerical weather prediction ” ( Charney 1967 ) .
Five problems were described and referred to as : • What is the relationship between the turbulent boundary @ @ @ @ @ @ @ How can steep gradients associated with fronts and topographic features be handled in models ?
• Do models correctly handle the cascade of energy between scales of motion ?
• How are convective processes and large-scale tropical circulations related ?
• What determines the structure , variability , and location of such preeminent tropical features as the ITCZ organized and maintained ?
Charney ’ s paper deeply depressed me !
Here I was , a brand new student in graduate school embarking on a career in tropical meteorology surrounded by bright and eager @ @ @ @ @ @ @ to know what they were doing , a faculty that was acknowledged as the world leaders in atmospheric science , and only a handful of questions remained .
Clearly all of these would be answered by the end of the semester .
This made me wonder whether I should have gone to medical school after all .
Now , over 45 years later , many new questions regarding the tropical system have arisen .
It is interesting , though , to determine what progress has been made in solving Charney ’ s list @ @ @ @ @ @ @ their solution .
Understanding complex natural phenomena has generally been undertaken by following a reductionist approach , whereby a phenomenon ’ s complex nature is reduced into individual components that are assumed to work together to produce observed structures .
Reductionism evolved from Rene Descartes ’ “ mechanical philosophy , ” whereby the universe is thought of as a complicated machine made up of identifiable components .
Essentially , reductionism aims to understand the nature of complex things by reducing them to the interactions of their parts , or to simpler or more @ @ @ @ @ @ @ complex system is nothing but the sum of its parts .
Chapters 6 and 7 adopted a reductionist approach in attempting to explain tropical variability in terms of fundamental linear equatorially trapped modes .
This modal structure formed the basic explanations of intraseasonal variability ( Chapter 15 ) and the coupled ocean–atmosphere ENSO and Indian Ocean Dipole ( Chapter 14 ) .
And , to some degree , the behavior of many of these large scale systems circulation systems may be identified as combinations of fundamental components of the ocean–atmosphere system .
However @ @ @ @ @ @ @ state of a complex system , we find that the reductionist approach does not help in the prediction of emergent ( or unforeseen ) phenomena .
If the climate system were purely a combination of linear modes , then prediction would be a much simpler endeavor , with the main challenge being to reduce the impact of inaccuracies in the initial fields that may introduce uncertainty into a prediction .
Yet extended weather and climate prediction has proven to be universally difficult .
For example , even though in Chapter 14 we pointed @ @ @ @ @ @ @ interaction are basically understood , each ENSO event ( the supposed sum of these parts or components ) is very different both in timing , duration , and amplitude .
Predictions of whether or not an El Niño or La Niña event will develop following the boreal spring show little skill .
Also , it is difficult to assess whether or not a La Niña will be followed by an El Niño or vice versa .
Once an ENSO event develops in the early boreal summer , it tends to follow its own @ @ @ @ @ @ @ somewhat different to other El Niño or La Niña events .
Similar differences between the characteristics of individual circulation events exist on subseasonal time scales as well .
As with ENSO , much of the structure of intraseasonal events features can be recognized in terms of fundamental equatorial modes whereby convection has been accounted for in some manner .
Each MJO has an individual character .
Even the canonical MJO , forming in the equatorial Indian Ocean , varies from case to case .
In Chapter 11 ( also Chapters 13 to 18 @ @ @ @ @ @ @ more holistic approach was necessary in order to understand the complex nature of the Earth system .
Holism claims that complex systems are inherently irreducible and are more than the sum of their parts , owing to chaos and nonlinearities .
Emergent behavior may arise from complex systems that can not be deduced from consideration of the components of the system alone .
Holism leads to “ systems thinking ” and possesses derivatives such as chaos and complexity .
This discussion grew from attempts to understand interactions between the extratropics and the tropics @ @ @ @ @ @ @ that extratropical waves had difficulty propagating through zonally symmetric easterlies toward the equator so that more complex explanations were necessary to explain extratropical–tropical interactions .
In Chapter 10 , it was found that a zonally symmetric Hadley Circulation could not explain fully the influence of the tropics on the extratropics or vice versa .
Yet progress was made toward understanding the interaction between the tropics and the extratropics by noting the interaction of two nonlinear systems , one being the divergent circulations transporting PV poleward and the other , the Rossby wave regime @ @ @ @ @ @ @ .
The predictability of complex systems can be described using concepts introduced by Hofstadter ( 1980 ) .
Simply stated , system predictive skill depends on its degree of complexity .
Three hierarchies of organization and disorganization are suggested : simple , complex , and tangled , where the simpler the complexity the greater the potential predictability .
By extension , the more complex the system , the less predictability the system possesses .
• A simple system possesses two components , A forcing B or two interacting bodies as in the classical @ @ @ @ @ @ @ interaction of a planet and its moon .
Variability of the predictive outcome arises only from the uncertainty of describing either A or B .
An example of a simple system is the lunar forcing of ocean tides and its high predictability .
• The introduction of a third component ( C ) produces a complex system and introduces uncertainty into how the three components ( A , B , and C ) interact .
First , initial conditions require a description of the system now extended to three components instead of two @ @ @ @ @ @ @ the system trajectory may be very different depending on the initial scale of each of the three components or its initial magnitude .
• The most complex system , tangled , may have multiple interacting components ( C , D , … , etc. )
.
The climate system itself is such an example , with interacting oceans , atmosphere , cryosphere , and land systems .
Specific circulation patterns in the climate system may be complex or tangled .
The existence of some predictability of an ENSO extrema , once it is @ @ @ @ @ @ @ complex and probably not tangled .
Similarly , the wider influence of ENSO is a complex system since some predictability is retained .
However , the lack of persistence or predictability across the boreal spring suggests that at longer time horizons , or at certain times of the year , the ENSO system would appear to be tangled and unpredictable .
Over the course of an annual cycle , though , the system moves from tangled to complex as it changes from frail to more robust , as discussed in Chapter 14 .
@ @ @ @ @ @ @ a number of factors : • The degree to which the initial conditions of the system are known .
• How well A and B are understood physically and represented by a model ( either empirical or numerical ) .
• How large and variable is component C ( or D and E … ) ?
Are they stochastic ?
Does one element or one process dominate over all others ?
For example , the solar system is a complex multi-body system but the Sun ’ s gravitation makes it ( almost ) @ @ @ @ @ @ @ % of the mass of the solar system and , thus , chaotic motion of the planets is rare .
Given these points , it is tempting to adopt a holistic approach to the prediction of tropical phenomena by resorting to complicated coupled ocean–atmosphere–land models .
However , given model formulation and initial data uncertainties , it is necessary to use a probabilistic approach in which the model is perturbed many times to produce an ensemble of forecasts .
It is also clear that a hierarchy of methods are needed to increase understanding @ @ @ @ @ @ @ and reductionist approaches .
If the components of a complex system can be identified and it can be determined that component C , for example , is more dominant at some stage of the prediction than another , one may be able to anticipate confidence in the results of the probabilistic forecast .
As discussed above , understanding of the interaction of the tropics and extratropics could only have been developed through a holistic or system approach .
The behavior of an individual component ( the collective divergent circulations or the recurring Rossby @ @ @ @ @ @ @ to a determination of the synergies between the tropics and extratropics .
Instead , we gained an understanding by applying the Haynes and McIntyre impermeability theorem that constrained the advection of a potential vorticity substance between the tropics and the extratropics or , specifically , across a latitude circle .
It could be that other difficult problems , such as why there is little difference in annual precipitation ( rate or volume ) of each hemisphere , will be understood through similar system constraints .
So , what can we say about the @ @ @ @ @ @ @ There has been substantial progress in the first two problems .
In 1967 , the grid point resolution of the earliest numerical weather models was hundreds of kilometers .
Now it is closer to 10 km and will possess greater resolutions and become cloud resolving in the near future .
The number of vertical levels has increased as well from only a few to over 50 in some operational models .
Topographic relief is incorporated directly through use of the sigma-coordinate system .
However , Charney ’ s third and fourth problems remain @ @ @ @ @ @ @ day .
Simply , we still are uncertain about how equatorial dynamics and convection interact and the degree of their mutual dependency .
With respect to the ITCZ , Section 13.1 offered six theories regarding the location of equatorial convection .
Although some are stronger than others , their number is an indication that closure on the issue has not yet been reached .
In addition , we have unearthed many new mysteries .
One is the discovery of enclaves of disturbances existing within tropics made up of families of convection ranging from @ @ @ @ @ @ @ .
In retrospect , Charney ’ s tropical problems were not solved by the end of the semester , nor by the end of the decade , and not even in the present time .
In fact , investigations of these problems have spawned many new exciting problems .
It seems that I was needlessly depressed in 1967 about the future opportunities in tropical meteorology .

by Judith Curry
Peter Webster’s magnum opus is now published: Dynamics of the Tropical Atmosphere and Oceans.
From the blurb on amazon.com:
“This book presents a unique and comprehensive view of the fundamental dynamical and thermodynamic principles underlying the large circulations of the coupled ocean-atmosphere system
Dynamics of The Tropical Atmosphere and Oceans provides a detailed description of macroscale tropical circulation systems such as the monsoon, the Hadley and Walker Circulations, El Niño, and the tropical ocean warm pool. These macroscale circulations interact with a myriad of higher frequency systems, ranging from convective cloud systems to migrating equatorial waves that attend the low-frequency background flow.
A comprehensive overview of the dynamics and thermodynamics of large-scale tropical atmosphere and oceans is presented using both a “reductionist” and “holistic” perspectives of the coupled tropical system. The reductionist perspective provides a detailed description of the individual elements of the ocean and atmospheric circulations. The physical nature of each component of the tropical circulation such as the Hadley and Walker circulations, the monsoon, the incursion of extratropical phenomena into the tropics, precipitation distributions, equatorial waves and disturbances described in detail. The holistic perspective provides a physical description of how the collection of the individual components produces the observed tropical weather and climate. How the collective tropical processes determine the tropical circulation and their role in global weather and climate is provided in a series of overlapping theoretical and modelling constructs.
Following a detailed description of tropical phenomenology, the reader is introduced to dynamical and thermodynamical constraints that guide the planetary climate and establish a critical role for the tropics. Equatorial wave theory is developed for simple and complex background flows, including the critical role played by moist processes. The manner in which the tropics and the extratropics interact is then described, followed by a discussion of the physics behind the subtropical and near-equatorial precipitation including arid regions. The El Niño phenomena and the monsoon circulations are discussed, including their covariance and predictability. Finally, the changing structure of the tropics is discussed in terms of the extent of the tropical ocean warm pool and its relationship to the intensity of global convection and climate change.”
The complete table of contents are found here [Table of Contents]
JC remarks
My minor role in this was to help edit the final drafts of the chapters, so I have been through the entire book with a very detailed read.
Here is what stands out for me in the book.
First, the book is ‘old school’ in the sense of integrating observations and theory. This approach is surprisingly rare these days in climate dynamics, with its heavy reliance on global climate model simulations. The book has a very strong foundation in fluid dynamics and wave dynamics. At the same time, the mathematical developments are sufficiently clear to be followed by students, with additional details in the appendices.
Second, the book presents an underlying philosophy for approaching the understanding of tropical dynamics, integrating reductionist and holistic approaches.
Third, the book provides historical context for the development of our understanding. Interesting historical snippets are provided, including biographical notes of key historical scientists.
Fourth, the above three elements integrate to provide insights into the process of the science of climate dynamics, not merely a recitation of our current understanding
Fifth, there are over 300 diagrams/figures in the book, including many originally drawn schematics that are very effective at providing insights and supporting understanding. An example is provided below:
Figure 14.30 Schematic of the sequence of events in 1997–1998. (a) The climatological alongshore winds off Sumatra (E) and the east African coast (F). The winds observed in the late summers and early autumns are denoted by G and H, respectively. The right-hand panel shows the effect at the equator on the upper ocean induced by increased upwelling in the east and decreased upwelling in the west. Wind into and out of the plane of the page are denoted by the bull’s-eye and cross-hair symbols, respectively. (b) Distribution of the winds resulting from the anomalous SST gradient along the Equator and the changes in the SSH distribution. (c) Formation of the Ekman ridge in the central Indian Ocean and the forcing of westward-propagating downwelling equatorial Rossby waves to the west. The right-hand panel shows the effect on the upper ocean near 5°S. (d) Subsequent cooling of the western Indian Ocean through enhanced mixing and coastal Ekman transports from stronger than average monsoon winds and through circulation changes associated with the weakening of the 1997–1998 El Niño.
The next section provides the complete text from the concluding chapter.
Chapter 19 Some Concluding Remarks
When I entered graduate school in the late 1960s, my advisor at the time had just completed a treatise entitled “Some remaining problems in numerical weather prediction” (Charney 1967). Five problems were described and referred to as:
…. certain islands of resistance which seem to hold out stubbornly in the face of all attacks.…
Broadly speaking these issues were:
What is the relationship between the turbulent boundary layer and synoptic scale variability?
How can steep gradients associated with fronts and topographic features be handled in models?
Do models correctly handle the cascade of energy between scales of motion?
How are convective processes and large-scale tropical circulations related?
What determines the structure, variability, and location of such preeminent tropical features as the ITCZ organized and maintained?
Charney’s paper deeply depressed me! Here I was, a brand new student in graduate school embarking on a career in tropical meteorology surrounded by bright and eager graduate students, all of whom seemed to know what they were doing, a faculty that was acknowledged as the world leaders in atmospheric science, and only a handful of questions remained. Clearly all of these would be answered by the end of the semester. This made me wonder whether I should have gone to medical school after all.
Now, over 45 years later, many new questions regarding the tropical system have arisen. It is interesting, though, to determine what progress has been made in solving Charney’s list of problems and how we have approached their solution.
Understanding complex natural phenomena has generally been undertaken by following a reductionist approach, whereby a phenomenon’s complex nature is reduced into individual components that are assumed to work together to produce observed structures. Reductionism evolved from Rene Descartes’ “mechanical philosophy,” whereby the universe is thought of as a complicated machine made up of identifiable components. Essentially, reductionism aims to understand the nature of complex things by reducing them to the interactions of their parts, or to simpler or more fundamental components. This implies that a complex system is nothing but the sum of its parts. Chapters 6 and 7 adopted a reductionist approach in attempting to explain tropical variability in terms of fundamental linear equatorially trapped modes. This modal structure formed the basic explanations of intraseasonal variability (Chapter 15) and the coupled ocean–atmosphere ENSO and Indian Ocean Dipole (Chapter 14). And, to some degree, the behavior of many of these large scale systems circulation systems may be identified as combinations of fundamental components of the ocean–atmosphere system.
However, when it comes to predicting the state of a complex system, we find that the reductionist approach does not help in the prediction of emergent (or unforeseen) phenomena. If the climate system were purely a combination of linear modes, then prediction would be a much simpler endeavor, with the main challenge being to reduce the impact of inaccuracies in the initial fields that may introduce uncertainty into a prediction. Yet extended weather and climate prediction has proven to be universally difficult. For example, even though in Chapter 14 we pointed out that the basic components of ocean–atmosphere interaction are basically understood, each ENSO event (the supposed sum of these parts or components) is very different both in timing, duration, and amplitude. Predictions of whether or not an El Niño or La Niña event will develop following the boreal spring show little skill. Also, it is difficult to assess whether or not a La Niña will be followed by an El Niño or vice versa. Once an ENSO event develops in the early boreal summer, it tends to follow its own trajectory, which may be similar or somewhat different to other El Niño or La Niña events.
Similar differences between the characteristics of individual circulation events exist on subseasonal time scales as well. As with ENSO, much of the structure of intraseasonal events features can be recognized in terms of fundamental equatorial modes whereby convection has been accounted for in some manner. Each MJO has an individual character. Even the canonical MJO, forming in the equatorial Indian Ocean, varies from case to case.
In Chapter 11 (also Chapters 13 to 18), it was argued that a more holistic approach was necessary in order to understand the complex nature of the Earth system. Holism claims that complex systems are inherently irreducible and are more than the sum of their parts, owing to chaos and nonlinearities. Emergent behavior may arise from complex systems that cannot be deduced from consideration of the components of the system alone. Holism leads to “systems thinking” and possesses derivatives such as chaos and complexity. This discussion grew from attempts to understand interactions between the extratropics and the tropics. In Chapter 9 it was argued that extratropical waves had difficulty propagating through zonally symmetric easterlies toward the equator so that more complex explanations were necessary to explain extratropical–tropical interactions. In Chapter 10, it was found that a zonally symmetric Hadley Circulation could not explain fully the influence of the tropics on the extratropics or vice versa. Yet progress was made toward understanding the interaction between the tropics and the extratropics by noting the interaction of two nonlinear systems, one being the divergent circulations transporting PV poleward and the other, the Rossby wave regime, returning the PV to the tropics.
The predictability of complex systems can be described using concepts introduced by Hofstadter (1980). Simply stated, system predictive skill depends on its degree of complexity. Three hierarchies of organization and disorganization are suggested: simple, complex, and tangled, where the simpler the complexity the greater the potential predictability. By extension, the more complex the system, the less predictability the system possesses.
A simple system possesses two components, A forcing B or two interacting bodies as in the classical “two-body” problem such as the interaction of a planet and its moon. Variability of the predictive outcome arises only from the uncertainty of describing either A or B. An example of a simple system is the lunar forcing of ocean tides and its high predictability.
The introduction of a third component (C) produces a complex system and introduces uncertainty into how the three components (A, B, and C) interact. First, initial conditions require a description of the system now extended to three components instead of two thus adding further uncertainty. Further, the system trajectory may be very different depending on the initial scale of each of the three components or its initial magnitude.
The most complex system, tangled, may have multiple interacting components (C, D, …, etc.). The climate system itself is such an example, with interacting oceans, atmosphere, cryosphere, and land systems.
Specific circulation patterns in the climate system may be complex or tangled. The existence of some predictability of an ENSO extrema, once it is initiated, suggests that the system is complex and probably not tangled. Similarly, the wider influence of ENSO is a complex system since some predictability is retained. However, the lack of persistence or predictability across the boreal spring suggests that at longer time horizons, or at certain times of the year, the ENSO system would appear to be tangled and unpredictable. Over the course of an annual cycle, though, the system moves from tangled to complex as it changes from frail to more robust, as discussed in Chapter 14.
Hence predictability of a system depends on a number of factors:
The degree to which the initial conditions of the system are known.
How well A and B are understood physically and represented by a model (either empirical or numerical).
How large and variable is component C (or D and E …)? Are they stochastic? Does one element or one process dominate over all others? For example, the solar system is a complex multi-body system but the Sun’s gravitation makes it (almost) a stable system as it represents 98% of the mass of the solar system and, thus, chaotic motion of the planets is rare.
Given these points, it is tempting to adopt a holistic approach to the prediction of tropical phenomena by resorting to complicated coupled ocean–atmosphere–land models. However, given model formulation and initial data uncertainties, it is necessary to use a probabilistic approach in which the model is perturbed many times to produce an ensemble of forecasts. It is also clear that a hierarchy of methods are needed to increase understanding and predictive capability, including both holistic and reductionist approaches. If the components of a complex system can be identified and it can be determined that component C, for example, is more dominant at some stage of the prediction than another, one may be able to anticipate confidence in the results of the probabilistic forecast.
As discussed above, understanding of the interaction of the tropics and extratropics could only have been developed through a holistic or system approach. The behavior of an individual component (the collective divergent circulations or the recurring Rossby waves families) could not have led to a determination of the synergies between the tropics and extratropics. Instead, we gained an understanding by applying the Haynes and McIntyre impermeability theorem that constrained the advection of a potential vorticity substance between the tropics and the extratropics or, specifically, across a latitude circle. It could be that other difficult problems, such as why there is little difference in annual precipitation (rate or volume) of each hemisphere, will be understood through similar system constraints.
So, what can we say about the problems Charney laid out in 1967? There has been substantial progress in the first two problems. In 1967, the grid point resolution of the earliest numerical weather models was hundreds of kilometers. Now it is closer to 10 km and will possess greater resolutions and become cloud resolving in the near future. The number of vertical levels has increased as well from only a few to over 50 in some operational models. Topographic relief is incorporated directly through use of the sigma-coordinate system. However, Charney’s third and fourth problems remain “islands of resistance” to this day. Simply, we still are uncertain about how equatorial dynamics and convection interact and the degree of their mutual dependency. With respect to the ITCZ, Section 13.1 offered six theories regarding the location of equatorial convection. Although some are stronger than others, their number is an indication that closure on the issue has not yet been reached. In addition, we have unearthed many new mysteries. One is the discovery of enclaves of disturbances existing within tropics made up of families of convection ranging from diurnal through synoptic and biweekly to intraseasonal.
In retrospect, Charney’s tropical problems were not solved by the end of the semester, nor by the end of the decade, and not even in the present time. In fact, investigations of these problems have spawned many new exciting problems. It seems that I was needlessly depressed in 1967 about the future opportunities in tropical meteorology.
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climateetc--2020-05-29--When does government intervention make sense for COVID-19?

I showed in my last article that inhomogeneity within a population in the susceptibility and infectivity of individuals would reduce the herd immunity threshold , in my view probably very substantially , and that evidence from Stockholm County appeared to support that view .
In this article I will first provide other evidence pointing to such population inhomogeneity being very considerable .
I will then go on to consider how the overshoot of infections beyond the herd immunity threshold could be reduced .
I ’ ll start with a recap .
The basic reproduction ratio of an epidemic , R @ @ @ @ @ @ @ average , each infected individual infects at the start of the epidemic .
If R exceeds one , the epidemic will grow , exponentially at first .
But , assuming recovered individuals become immune , the pool of susceptible individuals shrinks over time and the current reproduction ratio falls .
The proportion of the population that have been infected at the point where the current reproduction ratio falls to one is the ‘ herd immunity threshold ’ ( HIT ) .
Beyond that point the epidemic is under control , and shrinks .
@ @ @ @ @ @ @ the HIT will be .
I used an R value of 2.4 , the baseline value used in the influential Imperial College model ( Ferguson20 [ 1 ] ) .
Standard simple compartmental models of epidemic growth , which assume a homogeneous population , imply that the HIT equals { 1 – 1/R } .
For R =2.4 , they imply the HIT is 58 % .
For R value of 3 , which is towards the upper end of most estimates , the HIT is 67 % .
These naïve , unrealistic @ @ @ @ @ @ @ of 60–70 % for COVID-19 often cited by epidemiologists quoted in the mainstream media .
There is no doubt that inhomogeneity within a population in the susceptibility and infectivity of individuals will reduce the HIT .
I cited the Gomes et al .
[ 2 ] paper as showing this and I adopted , with some modifications , its susceptible – exposed – infectious – recovered ( SEIR ) compartmental model ( Figure 1 ) .
I also adopted its gamma probability distribution for population inhomogeneity that arose from varying social connectivity – @ @ @ @ @ @ @ in contact with ) other people , which affects both susceptibility and infectivity .
The gamma distribution can represent the existence of a small number of highly connected “ superspreaders ” with a very high susceptibility and infectivity , together with a far larger number of people who have a much lower connectivity .
I used illustrative coefficients of variation ( CV ) – a measure of the extent of inhomogeneity – of 1 and 2 in my article for inhomogeneity related to social connectivity .
Those levels are consistent with the evidence @ @ @ @ @ @ @ SEIR 4-compartment epidemiological model diagram .
Initially all individuals are susceptible .
A tiny number are seeded with infection at the start of the epidemic .
Exposed individuals are susceptibles who have been infected , but who remain uninfectious until a probabilistic latent period has expired .
Once they become infectious they remain so for a probabilistic infectious period and then become ‘ recovered ’ – which includes some who are still ill and may die , and some who have died while infectious .
In the standard model version , the rate @ @ @ @ @ @ @ product of the numbers of infectious and susceptible individuals .
In the modified model , these numbers are weighted by respectively the infectivity and susceptibility of each of the individuals involved , both of which vary between individuals according to their social connectivity .
Individual infectivity and susceptibility also vary separately , with factors specific to each .
Other evidence regarding the effects of population inhomogeneity Another recent paper , Britton et al .
[ 4 ] , also shows that varying social connectivity will lower the HIT for COVID-19 .
They use @ @ @ @ @ @ @ simpler probability distribution , with the population divided into only three segments , with arbitrarily chosen social mixing levels , giving rise to a smaller CV of 0.56 , and assume R =2.5 The result is a reduction of the HIT from 60 % to 46 % .
They point out that it is only the disease-induced HIT that is reduced ; the HIT for vaccination is unaffected by population inhomogeneity .
It is becoming evident that , in addition to individuals ’ general resistance to infection varying , around half the population @ @ @ @ @ @ @ COVID-19 due to previous encounters with other coronaviruses .
[ 5 ] [ 6 ] [ 7 ] Variation in susceptibility related to resistance to COVID-19 infection is therefore an important factor .
A 20th May preprint paper , McGeoch and McGeoch , [ 8 ] which divides the population into only two parts , considers variability in susceptibility that is related only to resistance to infection , and not to social connectivity .
In my model , such variability was included in the probabilistic factor that reflected non-social connectivity related variability in @ @ @ @ @ @ @ CV values of 0.42 and 0.95 [ 9 ] to represent that factor , while the CV of their susceptibility distribution is 0.6 .
[ 10 ] They find a significant reduction in the projected HIT and final infected proportion , but a smaller one than in my model .
That is to be expected , because their model omits the social connectivity factor , which affects both the susceptibility and infectivity of each individual .
A recent working paper [ 11 ] from the US National Bureau of Economic Research reviews models @ @ @ @ @ @ @ simple and complex , and their policy implications .
It has a whole section on heterogeneities that are not included in standard simple compartmental models , limiting their realism .
However , it is not very complimentary about more complex models .
Regarding the highly influential , complex Ferguson20 model , it says regarding how it treats the effects of policy intervention : “ The changes in contact rates assumed in this model are never justified and , in fact , appear to be entirely arbitrary and in some cases clearly inaccurate ” @ @ @ @ @ @ @ simplistic and very limited treatment of uncertainty in Ferguson20 .
As I stated in my original article , the Ferguson20 model appears to account for inhomogeneity in susceptibility arising only from a very limited set of factors , with only a modest resulting impact on the growth of the epidemic .
Although their model does account for substantial inhomogeneity in infectivity , using the same gamma distribution as I did , in their case inhomogeneity in infectivity appears to be uncorrelated with inhomogeneity in susceptibility , and thus has a negligible effect on @ @ @ @ @ @ @ the overshoot beyond the herd immunity threshold Although inhomogeneity can greatly lower the herd immunity threshold , the ultimate proportion of the population that becomes infected will exceed the HIT , since further infections occur after the HIT is reached .
Although such infections are continuously diminishing , if the epidemic is unimpeded they have a major impact on its ultimate size .
In the examples I gave , I used a R value of 2.4 .
On that basis , I showed that the final infected proportion is about 1.5 times the @ @ @ @ @ @ @ and about twice the ( far lower ) HIT if the population is inhomogeneous in the way that I modelled .
Figure 2 shows the moderate inhomogeneity case that I illustrated , for which the HIT is 24 % ( against 58 % for a homogeneous population ) but the final infected proportion is 43 % ( down from 88 % ) , a lesser reduction .
The reason for the large overshoot of the HIT is that there are still many infectious individuals at the time the HIT is reached .
Figure @ @ @ @ @ @ @ model with R =2.4 and a population of 1 million with CV=1 common factor inhomogeneity in susceptibility and infectivity and also unrelated multiplicative inhomogeneity in susceptibility with a CV of 0.42 .
The latent and infectious periods are 3 and 4 days respectively .
Government intervention at an early stage appears to have been designed mainly to avoid health systems being overwhelmed , but the subsequent paths of the epidemics show that in most cases it was unnecessarily strong for that purpose .
Moreover , as Figure 3 shows in the homogeneous population @ @ @ @ @ @ @ the epidemic , with the effect of reducing R from 2.4 to 0.8 , and maintaining it for six months , merely delays the progress of the epidemic , with the final infected proportion barely reducing , from 88 % to 86 % .
Figure 3 Epidemic progression in an SEIR model with a homogeneous population , where R =2.4 until a lockdown is imposed ( dotted red line ) at day 30 after which R =0.8 until lockdown is ended 180 days later ( dotted green line ) .
The latent and @ @ @ @ @ @ @ .
The effect of an early imposed , long lockdown is also minor in the heterogeneous population case ( Figure 4 ) .
The ultimate proportion infected falls by slightly under 5 % , from 43 % to 41 % – still far above the HIT level .
Figure 4 Epidemic progression in an SEIR model with an inhomogeneous population , where R =2.4 until a lockdown is imposed ( dotted red line ) at day 30 after which R =0.8 until lockdown is ended 180 days later ( dotted green line ) @ @ @ @ @ @ @ inhomogeneity are as in Figure 2 .
Moreover , intervention can have dangerous longer term effects in relation to infections .
[ 13 ] Absent vaccination becoming available and providing long-lasting immunity , the virus is likely to resurge in the future if herd immunity is not reached in the original epidemic , and vulnerable people may repeatedly be at risk if not totally isolated .
However , government intervention at a later stage , as the HIT is approached , could enable the overshoot to be greatly reduced .
Suppose the intervention @ @ @ @ @ @ @ 0.8 , is instead delayed until the HIT is being approached .
As Figure 5 shows , applying a short lockdown ( 30 days ) later , hugely enhances the reduction in eventual total infections , compared with an early intervention lasting six times as long .
The final infected proportion falls from 43 % to 27 % , rather than only to 41 % .
The reason is that dramatically slowing the infection as the HIT is approached greatly reduces the number of active infections as the HIT is crossed , and @ @ @ @ @ @ @ at which infections decline thereafter .
Figure 5 Epidemic progression in an SEIR model with an inhomogeneous population , where R =2.4 until a lockdown is imposed ( dotted red line ) at day 53 , after which R =0.8 until lockdown is ended 30 days later ( dotted green line ) .
The latent and infectious periods and inhomogeneity are as in Figure 2 .
If the population were prepared to obey a lockdown for 60 days at that stage , and its timing were perfect , it would potentially be feasible @ @ @ @ @ @ @ HIT .
Figure 6 shows this case .
To three significant figures , the final infected proportion equals the HIT .
Figure 6 Epidemic progression in an SEIR model with an inhomogeneous population , where R =2.4 until a lockdown is imposed ( dotted red line ) at day 52 , after which R =0.8 until lockdown is ended 60 days later ( dotted green line ) .
The latent and infectious periods and inhomogeneity are as in Figure 2 .
The take home lessons are , first , that imposing stricter restrictions @ @ @ @ @ @ @ to prevent a health system being overwhelmed is likely to have little impact on the proportion of the population that is eventually infected , in the absence of a vaccine becoming available before restrictions are relaxed .
And secondly , that a well-timed imposition of strict restrictions for a fairly short period as the herd immunity threshold is approached can hugely reduce the overshoot of the eventually infected proportion above the HIT .
States that imposed strict restrictions early on and then relaxed them may find their populations unwilling to see such measures @ @ @ @ @ @ @ states that introduced milder restrictions and are in reality pursuing a herd immunity strategy may find the imposition of strict restrictions for a short period bracketing the crossing of the HIT to be an attractive option .
In either case , the serious illness and fatalities associated reaching the eventual level of infections can be very greatly reduced if elderly and vulnerable people are shielded from infection , as discussed in an earlier article .
[ 14 ] [ 1 ] Neil M Ferguson et al.
, Impact of non-pharmaceutical interventions ( NPIs @ @ @ @ @ @ @ demand .
Imperial College COVID-19 Response Team Report 9 , 16 March 2020 , https : //spiral.imperial.ac.uk:8443/handle/10044/1/77482 [ 2 ] Gomes , M. G. M. , et al .
Individual variation in susceptibility or exposure to SARS-CoV-2 lowers the herd immunity threshold .
medRxiv 2 May 2020. https : //www.medrxiv.org/content/10.1101/2020.04.27.20081893v1 [ 3 ] My lower CV=1 level was also the lower level illustrated by Gomes et al.
, but I used a more conservative upper level , of CV=2 , than their choice of CV=3 .
That was partly because , unlike Gomes @ @ @ @ @ @ @ inhomogeneity of susceptibility factor that was unrelated to social connectivity .
However , I cited two COVID-19 studies [ 3 ] [ 3 ] that gave best estimates for the social connectivity CV of marginally above 3 .
Gomes et al .
also provided some arguments in favour of connectivity-related CV being as high as 3 , and argued that such CV probably exceeded 1 .
They showed that for other infectious diseases CV varies between 1.8 and 3.3 , and they said , referring to a SARS-CoV study : “ the way @ @ @ @ @ @ @ infectiousness stems from higher connectivity with other individuals , who may be susceptible .
This would support the scenarios displayed in Figure 2 , with CV = 3 for exposure to infection.
” [ 4 ] Britton , T. , Ball , F. and Trapman , P. , 2020 .
The disease-induced herd immunity level for Covid-19 is substantially lower than the classical herd immunity level .
arXiv preprint arXiv:2005.03085 .
[ 5 ] Grifoni , A.et al Targets of T cell responses to SARS-CoV-2 coronavirus in humans with COVID-19 disease and unexposed @ @ @ @ @ @ @ Braun , J. , Loyal , L. , Frentsch , M. , Wendisch , D. , Georg , P. , Kurth , F. , Hippenstiel , S. , Dingeldey , M. , Kruse , B. , Fauchere , F. and Baysal , E. , 2020 .
Presence of SARS-CoV-2 reactive T cells in COVID-19 patients and healthy donors .
medRxiv preprint https : //www.medrxiv.org/content/10.1101/2020.04.17.20061440v1 .
[ 7 ] Le Bert , N. et al.
, 2020 .
Different pattern of pre-existing SARS-COV-2 specific T cell immunity in SARS-recovered and uninfected individuals .
bioRxiv @ @ @ @ @ @ @ Malcolm W McGeoch and Julie McGeoch , 2020 .
COVID-19 Propagation and Mortality in a Two-Part Population .
medRxiv preprint doi:10.1101/2020.05.17.20104356 [ 9 ] These CVs resulted from using zero ( log ) mean lognormal distributions with log standard deviations of 0.4 and 0.8 .
[ 10 ] However , with their model having only two population segments with different susceptibility rather than a continuous distribution the CV may not be directly comparable with that in my model .
[ 11 ] Avery , C. et al.
, 2020 : Policy Implications of @ @ @ @ @ @ @ Perspectives and Opportunities for Economists .
NBER Working Paper No .
27007 [ 12 ] A highly susceptible but averagely infectious person is more likely to be removed from the susceptible pool early in an epidemic , reducing the average susceptibility of the pool .
However , no such selective removal occurs for a highly infectious person of averagely susceptibility .
Therefore , as Gomes et al .
point out , variability in susceptibility lowers the HIT , but variability in infectivity does not do so except to the extent that it is @ @ @ @ @ @ @ 13 ] See this interview with Professor Senetra Gupta of Oxford University : https : //unherd.com/2020/05/oxford-doubles-down-sunetra-gupta-interview/ [ 14 ] A sensible COVID-19 exit strategy for the UK , Nicholas Lewis , 26 April 2020 Originally posted here , where a pdf copy is also available

By Nic Lewis
Introduction
I showed in my last article that inhomogeneity within a population in the susceptibility and infectivity of individuals would reduce the herd immunity threshold, in my view probably very substantially, and that evidence from Stockholm County appeared to support that view. In this article I will first provide other evidence pointing to such population inhomogeneity being very considerable. I will then go on to consider how the overshoot of infections beyond the herd immunity threshold could be reduced.
I’ll start with a recap. The basic reproduction ratio of an epidemic, R 0 , measures how many people, on average, each infected individual infects at the start of the epidemic. If R 0 exceeds one, the epidemic will grow, exponentially at first. But, assuming recovered individuals become immune, the pool of susceptible individuals shrinks over time and the current reproduction ratio falls. The proportion of the population that have been infected at the point where the current reproduction ratio falls to one is the ‘herd immunity threshold’ (HIT). Beyond that point the epidemic is under control, and shrinks.
The higher R 0 is, the greater the HIT will be. I used an R 0 value of 2.4, the baseline value used in the influential Imperial College model (Ferguson20[1]). Standard simple compartmental models of epidemic growth, which assume a homogeneous population, imply that the HIT equals {1 – 1/R 0 }. For R 0 =2.4, they imply the HIT is 58%. For R 0 value of 3, which is towards the upper end of most estimates, the HIT is 67%. These naïve, unrealistic values probably account for the HIT range of 60–70% for COVID-19 often cited by epidemiologists quoted in the mainstream media.
There is no doubt that inhomogeneity within a population in the susceptibility and infectivity of individuals will reduce the HIT. I cited the Gomes et al.[2] paper as showing this and I adopted, with some modifications, its susceptible – exposed – infectious – recovered (SEIR) compartmental model (Figure 1). I also adopted its gamma probability distribution for population inhomogeneity that arose from varying social connectivity – different rates of mixing with (being in contact with) other people, which affects both susceptibility and infectivity. The gamma distribution can represent the existence of a small number of highly connected “superspreaders” with a very high susceptibility and infectivity, together with a far larger number of people who have a much lower connectivity. I used illustrative coefficients of variation (CV) – a measure of the extent of inhomogeneity – of 1 and 2 in my article for inhomogeneity related to social connectivity. Those levels are consistent with the evidence.[3]
Figure 1. SEIR 4-compartment epidemiological model diagram. Initially all individuals are susceptible. A tiny number are seeded with infection at the start of the epidemic. Exposed individuals are susceptibles who have been infected, but who remain uninfectious until a probabilistic latent period has expired. Once they become infectious they remain so for a probabilistic infectious period and then become ‘recovered’ – which includes some who are still ill and may die, and some who have died while infectious. In the standard model version, the rate of new infections is proportional to the product of the numbers of infectious and susceptible individuals. In the modified model, these numbers are weighted by respectively the infectivity and susceptibility of each of the individuals involved, both of which vary between individuals according to their social connectivity. Individual infectivity and susceptibility also vary separately, with factors specific to each.
Other evidence regarding the effects of population inhomogeneity
Another recent paper, Britton et al.[4], also shows that varying social connectivity will lower the HIT for COVID-19. They use, for illustrative purposes, a much simpler probability distribution, with the population divided into only three segments, with arbitrarily chosen social mixing levels, giving rise to a smaller CV of 0.56, and assume R 0 =2.5 The result is a reduction of the HIT from 60% to 46%. They point out that it is only the disease-induced HIT that is reduced; the HIT for vaccination is unaffected by population inhomogeneity.
It is becoming evident that, in addition to individuals’ general resistance to infection varying, around half the population may well have pre-existing partial immunity to COVID-19 due to previous encounters with other coronaviruses.[5] [6] [7] Variation in susceptibility related to resistance to COVID-19 infection is therefore an important factor.
A 20th May preprint paper, McGeoch and McGeoch,[8] which divides the population into only two parts, considers variability in susceptibility that is related only to resistance to infection, and not to social connectivity. In my model, such variability was included in the probabilistic factor that reflected non-social connectivity related variability in susceptibility. I used probability distributions with CV values of 0.42 and 0.95[9] to represent that factor, while the CV of their susceptibility distribution is 0.6.[10] They find a significant reduction in the projected HIT and final infected proportion, but a smaller one than in my model. That is to be expected, because their model omits the social connectivity factor, which affects both the susceptibility and infectivity of each individual.
A recent working paper[11] from the US National Bureau of Economic Research reviews models for the spread of COVID-19, both simple and complex, and their policy implications. It has a whole section on heterogeneities that are not included in standard simple compartmental models, limiting their realism. However, it is not very complimentary about more complex models. Regarding the highly influential, complex Ferguson20 model, it says regarding how it treats the effects of policy intervention:
“The changes in contact rates assumed in this model are never justified and, in fact, appear to be entirely arbitrary and in some cases clearly inaccurate”
They are also strongly critical of the simplistic and very limited treatment of uncertainty in Ferguson20.
As I stated in my original article, the Ferguson20 model appears to account for inhomogeneity in susceptibility arising only from a very limited set of factors, with only a modest resulting impact on the growth of the epidemic. Although their model does account for substantial inhomogeneity in infectivity, using the same gamma distribution as I did, in their case inhomogeneity in infectivity appears to be uncorrelated with inhomogeneity in susceptibility, and thus has a negligible effect on the HIT.[12]
Reducing the overshoot beyond the herd immunity threshold
Although inhomogeneity can greatly lower the herd immunity threshold, the ultimate proportion of the population that becomes infected will exceed the HIT, since further infections occur after the HIT is reached. Although such infections are continuously diminishing, if the epidemic is unimpeded they have a major impact on its ultimate size. In the examples I gave, I used a R 0 value of 2.4. On that basis, I showed that the final infected proportion is about 1.5 times the HIT if the population is homogeneous, and about twice the (far lower) HIT if the population is inhomogeneous in the way that I modelled. Figure 2 shows the moderate inhomogeneity case that I illustrated, for which the HIT is 24% (against 58% for a homogeneous population) but the final infected proportion is 43% (down from 88%), a lesser reduction. The reason for the large overshoot of the HIT is that there are still many infectious individuals at the time the HIT is reached.
Figure 2. Epidemic progression in an SEIR model with R 0 =2.4 and a population of 1 million with CV=1 common factor inhomogeneity in susceptibility and infectivity and also unrelated multiplicative inhomogeneity in susceptibility with a CV of 0.42. The latent and infectious periods are 3 and 4 days respectively.
Intervention early on
Government intervention at an early stage appears to have been designed mainly to avoid health systems being overwhelmed, but the subsequent paths of the epidemics show that in most cases it was unnecessarily strong for that purpose. Moreover, as Figure 3 shows in the homogeneous population case, imposing a lockdown early in the epidemic, with the effect of reducing R 0 from 2.4 to 0.8, and maintaining it for six months, merely delays the progress of the epidemic, with the final infected proportion barely reducing, from 88% to 86%.
Figure 3 Epidemic progression in an SEIR model with a homogeneous population, where R 0 =2.4 until a lockdown is imposed (dotted red line) at day 30 after which R 0 =0.8 until lockdown is ended 180 days later (dotted green line). The latent and infectious periods are as in Figure 2.
The effect of an early imposed, long lockdown is also minor in the heterogeneous population case (Figure 4). The ultimate proportion infected falls by slightly under 5%, from 43% to 41% – still far above the HIT level.
Figure 4 Epidemic progression in an SEIR model with an inhomogeneous population, where R 0 =2.4 until a lockdown is imposed (dotted red line) at day 30 after which R 0 =0.8 until lockdown is ended 180 days later (dotted green line). The latent and infectious periods and inhomogeneity are as in Figure 2.
Moreover, intervention can have dangerous longer term effects in relation to infections.[13] Absent vaccination becoming available and providing long-lasting immunity, the virus is likely to resurge in the future if herd immunity is not reached in the original epidemic, and vulnerable people may repeatedly be at risk if not totally isolated.
Intervention at a later stage
However, government intervention at a later stage, as the HIT is approached, could enable the overshoot to be greatly reduced. Suppose the intervention, again reducing R 0 from 2.4 to 0.8, is instead delayed until the HIT is being approached.
As Figure 5 shows, applying a short lockdown (30 days) later, hugely enhances the reduction in eventual total infections, compared with an early intervention lasting six times as long. The final infected proportion falls from 43% to 27%, rather than only to 41%. The reason is that dramatically slowing the infection as the HIT is approached greatly reduces the number of active infections as the HIT is crossed, and the lockdown also greatly increases the rate at which infections decline thereafter.
Figure 5 Epidemic progression in an SEIR model with an inhomogeneous population, where R 0 =2.4 until a lockdown is imposed (dotted red line) at day 53, after which R 0 =0.8 until lockdown is ended 30 days later (dotted green line). The latent and infectious periods and inhomogeneity are as in Figure 2.
If the population were prepared to obey a lockdown for 60 days at that stage, and its timing were perfect, it would potentially be feasible virtually to eliminate the overshoot of the HIT. Figure 6 shows this case. To three significant figures, the final infected proportion equals the HIT.
Figure 6 Epidemic progression in an SEIR model with an inhomogeneous population, where R 0 =2.4 until a lockdown is imposed (dotted red line) at day 52, after which R 0 =0.8 until lockdown is ended 60 days later (dotted green line). The latent and infectious periods and inhomogeneity are as in Figure 2.
Conclusions
The take home lessons are, first, that imposing stricter restrictions early in an epidemic than are necessary to prevent a health system being overwhelmed is likely to have little impact on the proportion of the population that is eventually infected, in the absence of a vaccine becoming available before restrictions are relaxed. And secondly, that a well-timed imposition of strict restrictions for a fairly short period as the herd immunity threshold is approached can hugely reduce the overshoot of the eventually infected proportion above the HIT. States that imposed strict restrictions early on and then relaxed them may find their populations unwilling to see such measures reintroduced. However, the populations of states that introduced milder restrictions and are in reality pursuing a herd immunity strategy may find the imposition of strict restrictions for a short period bracketing the crossing of the HIT to be an attractive option. In either case, the serious illness and fatalities associated reaching the eventual level of infections can be very greatly reduced if elderly and vulnerable people are shielded from infection, as discussed in an earlier article.[14]
Nicholas Lewis 29 May 2020
[1] Neil M Ferguson et al., Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand. Imperial College COVID-19 Response Team Report 9, 16 March 2020, https://spiral.imperial.ac.uk:8443/handle/10044/1/77482
[2] Gomes, M. G. M., et al. Individual variation in susceptibility or exposure to SARS-CoV-2 lowers the herd immunity threshold. medRxiv 2 May 2020. https://www.medrxiv.org/content/10.1101/2020.04.27.20081893v1
[3] My lower CV=1 level was also the lower level illustrated by Gomes et al., but I used a more conservative upper level, of CV=2, than their choice of CV=3. That was partly because, unlike Gomes et al., I also had a inhomogeneity of susceptibility factor that was unrelated to social connectivity. However, I cited two COVID-19 studies [3] [3] that gave best estimates for the social connectivity CV of marginally above 3. Gomes et al. also provided some arguments in favour of connectivity-related CV being as high as 3, and argued that such CV probably exceeded 1. They showed that for other infectious diseases CV varies between 1.8 and 3.3, and they said, referring to a SARS-CoV study:
“the way authors describe superspreaders is suggestive that higher infectiousness stems from higher connectivity with other individuals, who may be susceptible. This would support the scenarios displayed in Figure 2, with CV = 3 for exposure to infection.”
[4] Britton, T., Ball, F. and Trapman, P., 2020. The disease-induced herd immunity level for Covid-19 is substantially lower than the classical herd immunity level. arXiv preprint arXiv:2005.03085.
[5] Grifoni, A.et al Targets of T cell responses to SARS-CoV-2 coronavirus in humans with COVID-19 disease and unexposed individuals. Cell doi:10.1016/j.cell.2020.05.015
[6] Braun, J., Loyal, L., Frentsch, M., Wendisch, D., Georg, P., Kurth, F., Hippenstiel, S., Dingeldey, M., Kruse, B., Fauchere, F. and Baysal, E., 2020. Presence of SARS-CoV-2 reactive T cells in COVID-19 patients and healthy donors. medRxiv preprint https://www.medrxiv.org/content/10.1101/2020.04.17.20061440v1.
[7] Le Bert, N. et al., 2020. Different pattern of pre-existing SARS-COV-2 specific T cell immunity in SARS-recovered and uninfected individuals. bioRxiv preprint https://doi.org/10.1101/2020.05.26.115832
[8] Malcolm W McGeoch and Julie McGeoch, 2020. COVID-19 Propagation and Mortality in a Two-Part Population. medRxiv preprint doi:10.1101/2020.05.17.20104356
[9] These CVs resulted from using zero (log)mean lognormal distributions with log standard deviations of 0.4 and 0.8.
[10] However, with their model having only two population segments with different susceptibility rather than a continuous distribution the CV may not be directly comparable with that in my model.
[11] Avery, C. et al., 2020: Policy Implications of Models of the Spread of Coronavirus: Perspectives and Opportunities for Economists. NBER Working Paper No. 27007
[12] A highly susceptible but averagely infectious person is more likely to be removed from the susceptible pool early in an epidemic, reducing the average susceptibility of the pool. However, no such selective removal occurs for a highly infectious person of averagely susceptibility. Therefore, as Gomes et al. point out, variability in susceptibility lowers the HIT, but variability in infectivity does not do so except to the extent that it is correlated with variability in susceptibility.
[13] See this interview with Professor Senetra Gupta of Oxford University: https://unherd.com/2020/05/oxford-doubles-down-sunetra-gupta-interview/
[14] A sensible COVID-19 exit strategy for the UK, Nicholas Lewis, 26 April 2020
Originally posted here, where a pdf copy is also available
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