Didactic Redesign of Models in Epidemiology, With a Caveat on Probability and Latent Variables

Thomas Colignatus, Samuel van Houten Genootschap, The Netherlands

1. Introduction

The pandemic of SARS-CoV-2 (virus) and Covid-19 (disease) did not come about because of lack of knowledge in epidemiology but because of inadequate management of Public Health. Epidemiology exists for some 150 years, and what was once known by governments at the level of villages and nations now is rediscovered at the world level. Infectious diseases have always been a good motive for the rich to also protect the poor from becoming infectious. A world with increasing international contacts for persons and goods requires a fitting public health management system or we get the current chaos or worse.

The world is slowly adapting to the permanence of pandemics, not only for the current virus but also for the risk of new ones which the epidemiologists have been warning about for decades. Climate change and the growth of the world population – see Colignatus (2019) – increase the number of “hot spots” where such pathogens can adapt to the human host. Each new pathogen comes with the stages that we have seen for SARS-CoV-1&2, namely a period before it is discovered and its bio-medical properties analysed and a period during which it must be contained before a vaccine is developed. For SARS-CoV-2 the effort at a vaccine looks promising but the virus belongs to the common cold family for which immunity tends not to be retained, and perhaps the first vaccines will not do better. Evolutionary pressure will tend to make the virus less deadly but this would imply many victims before the deadly strains are culled.

The July issue of the Physics and Society Newsletter provides an excellent synopsis, and advised reading, of basic epidemiological modeling by Wiener (2020), while Goodman (2020) comments on physics and society. The APS (2020) has made a growing collection of articles available in the Physical Review journals. Mathematical biologist Cog (2020) reviews how physicists who want to look into epidemiology might help, and her warning is not to re-invent the wheel. I agree with her, except on didactics, that Heesterbeek et al. (2015) in Science of AAAS is advised reading. On the other hand, Richard Gill, the Leiden mathematical statistician with much experience in both the life sciences and physics, thinks that the traditional epidemiological modeling tradition is becoming too inflexible and that approaches as used in physics have more prospects, see his webpage Gill (2020).The appendix to this present article contains a caveat for physicists.

The purpose of this article is to review some aspects of the pandemic and focus on a redesign of the didactics of basic epidemiological modeling. Better didactics would help more people to understand these models. Understanding these models helps to see that the present problems of society and the pandemic do not reside in epidemiological modeling but rather in the governance of Public Health. My reference is to the situation in Holland and not the USA. The present article is a summary of Colignatus (2020c).

2. Choice of symbols and model format

As said, Wiener (2020) in the July edition of Physics and Society succinctly restates the S(E)IR(D) family of models, using the traditional symbols and equations as used in the epidemiological literature. Dynamic variables are the susceptibles S, infected or infectious I, exposed but not infectious E, recovered or removed R, and deceased D. Wiener (2020:3) also identifies a didactic problem: “By perhaps unfortunate convention R(t) is used to represent the dynamic variable for the removed fraction, and to represent a parameter.” Weisstein (2020a) at Mathworld states: "Note that the choice of the notation R₀ is a bit unfortunate, since it has nothing to do with R." A teacher would not want to accept this. The teaching material better be didactically sound. An argument might be that the models are going to be taught on a wider scale now, no longer within the small community of epidemiology where students can be trained to handle the illogic. Another problem is that R is “removed” in SIR but “recovered” in SIRD, since also D belongs to the “removed” in original SIR. In SIR I includes all infected and/or infectious units, but in SEIR it includes only part of the infected, because there are also the exposed E who are infected but not infectious. The tradition in epidemiology opts for using the same labels for new meanings. Instead, for didactics, when the model SIR is extended to SIRD or SEIR(D) then it is better to retain the meaning of what one has learnt for the original variables (including the dynamic properties), and introduce new variables for what is new.

Thus we better relabel the variables to SI(EY)A(CD). A search for possible names gave Acquitted as the best choice. The acquitted A = C + D are the cleared or deceased. This avoids the triple use of R for removed, recovered, and reproductive factor. The infected I = E + Y are the exposed and infectious. In SIA E = 0 so that I = Y.

In SIA(CD) we have A' = γ I, with γ the acquittal rate from infectiousness, with I' incidence, I prevalence and A cumulated prevalence. The format of ordinary differential equations (ODE) should not distract. The basic structure is given by the Euler-Lotka renewal equation. The deceased are a fraction of the acquitted, D = φ A, with φ the infection fatality factor (IFF). The latter is often called a “rate” but it is a factor. R0 is conventionally called the "basic reproduction number" but it is a factor too, since it is defined in relation to the number of the first infected units. The ODE format D' = μ I or D' = μ Y can be rejected since it turns the model into a course in differential equations, with the need to prove D = φ A which can already be stated from the start. The ODE format also causes distracting questions what μ might be and whether there is a difference between a lethal acquittal period and a clearing acquittal period, and how parameters values must be adapted when the acquittal rate γ changes.

We adopt the more common parameters β and γ with R₀ = β γ^H (rather than α and β by Wiener (2020), and with mathematical constant H = -1), and also simplify the equations so that the flow between the compartments is clearer from the start. Analysis will require substitutions but such can be done better when students grasp what is being substituted. The comparison in Table 1 considers SIA(CD) and not SEYA(CD).

Table 1. Tradition versus didactic redesign

Standard SIR(D)	Didactic redesign into SIA(CD)	Nr.
S’= – β S I	S’= – β S I	(1)
I’ = β S I – γ I	I’ = – S’ – A’ (N[0] = S + I + A)	(2)
R’ = γ I	A’ = γ I	(3)
(D’ = μ I and adapt (2))	(D = φ A, C = A – D)	(4)

The didactic redesign changes nothing on content. The redesign makes the model more accessible to a wider group of students and researchers from fields new to epidemiology. The redesign is not very relevant for students who can solve the ODE and handle the phase diagram as Wiener (2020) discusses, but education is not only for who need little of it. When experienced researchers from other fields embark upon epidemiology let me invite them to also express concern about the dark elements in the traditional S(E)IR(D) setup, and to encourage epidemiologists to consider the didactic redesign.

Colignatus (2020c) checks, perhaps over-thoroughly, that the SI(EY)A(CD) family of models indeed fits such unity in presentation, within a consistent programming environment and uniform presentation also for scenarios. For variable X we have Xq in numbers and Xp in proportion of the population at the start, N[0], a constant, as distinct from current population N = N[0] – D. Checks on examples from the literature show that results are reproduced so that nothing is changed on content indeed, and the examples highlight the use of the models.

3. Clarifying herd resistance

Another element for redesign of didactics concerns the notion of herd resistance. This can be explained best by reference to vaccination. The first infectious unit has an offspring of 1 * R₀ units, via direct infection of its contacts. When those contacts have been vaccinated with degree v in the range [0, 1], then this means that v R₀ would not be infected and (1 - v) R₀ would still be infected. The infection size will remain constant when (1 - v) R₀ = 1; it will grow then (1 - v) R₀ > 1; and it will reduce when (1 - v) R₀ < 1. Rewriting the latter gives a condition for the degree of vaccination to warrant that the infection will be reduced: v > 1 - 1 / R₀. Vaccinated units can be allocated to the compartment of the Acquitted too. The general condition Ap > 1 - 1 / R₀ can be called herd resistance.

The notion means that the herd survives and doesn't become extinct when that particular level is reached, even when there is no vaccination and when the infection is lethal for perhaps most units. The term “herd resistance” is accurate and much preferred above the inadequate term "herd immunity" that is common in the epidemiological literature. Experts know what they mean by the latter term, i.e. that only the herd survives, but policy makers and the general public associate "herd immunity" with protection for all units, which however is not what the notion means. Protection of all units can be a special outcome when vaccination is done in a well-mixed population before the onset of infection. But when vaccination is done during a raging epidemic then the compartment of the infectious I can already be large, with still many infectious units causing new infections, called “overshoot”, with the subsequent death toll. This source of confusion between experts and non-experts can be prevented by the use of the term "herd resistance".

In March 2020, RIVM (the Dutch CDC) mentioned R₀ = 2.5 and a "herd immunity" (i.e. herd resistance) of 60% as an aim for SARS-CoV-2. In SI(EY)A(CD) for a Dutch population of 17.4 million, a raging epidemic with R₀ = 2.5 with IFF = 1.5% can reach 60% with 156.000 deaths, and proceeds after 60% till the limit value of 89.3%, which would mean another 78,000 deceased, compared to 9,000 at the end of May. In Figure 1, taking R₀ = 2.5 on the horizontal axis, we can see the vertical value of 60% of so-called “herd immunity” and the limit value A[∞] ≈ 90% of a continuing epidemic. Thus RIVM applied a condition for a vaccination before onset to a situation with a raging epidemic. This was a serious break-down of communication between professional advisors and policy makers (and the general public). This is no small matter because RIVM got Dutch prime minister Mark Rutte (2020) to state the target in his address to the nation. Criticism soon caused a political redress from “aim” to “by-product” of national policy, apparently miraculously achieved without such death toll since the stated policy aim was to protect the public from the death by infection. The factual implication however is that Dutch academic epidemiology belongs to the top of the world but in this pandemic the official RIVM leadership on infectious diseases, in particular Jaap van Dissel and Aura Timen, has shown to be incompetent. This official leadership failed on more crucial aspects, starting with the slow response in January-March compared with the fast reaction in Japan, Germany and Greece. The RIVM leadership was not replaced and Holland also lost the Spring and Summer of 2020 for sound preparation for the Fall and coming Winter.

Figure 1. Limit outcome, herd resistance and so-called herd immunity

The issue can also be explained as follows. As said, Cog (2020) recommends Heesterbeek et al. (2015) in Science of AAAS, and I agree, except for the didactics. The latter authors use the effective reproduction factor Re = R₀ Sp, and then define: “herd immunity: state of the population where the fraction protected is just sufficient to prevent outbreaks (Re < 1).” This is a technical definition and authors are free to give such definitions. It may well happen that a domain of excellent research has some terminology that is less perfect but that insiders are familiar with. However, said definition cannot convince in logic and practice. The condition “prevent outbreaks” is not the same as Re < 1. The latter Re < 1 may occur also during an outbreak, namely when it starts waning. The condition “prevent outbreaks” doesn’t fit with an ongoing outbreak. The SARS-CoV-2 episode shows that epidemiologists have been using the condition Re < 1 during the outbreak, which is fine to indicate the waning of the epidemic, but they have also been using the term “herd immunity”, which is not fine since it suggests protection, which however only applies to the herd (that doesn’t become extinct), and which does not apply to a large number of units that will die from the infection during the “overshoot”.

The clear didactic solution is not to use the term “herd immunity” but use the term “waning epidemic” for Re < 1 and “herd resistance” for Ap > 1 - 1 / R₀. The extensive Colignatus (2020c) checks the various definitions in the epidemiological literature about “herd immunity” and conditions like an (asymptotic) steady state. Just stated clarity might seem remarkably simple compared to a huge epidemiological literature but the extensive consideration rejects this literature, as wanting in accuracy for communication with the non-initiated.

But even when a domain of research contains terminology that can be confusing for communication between official experts and policy makers, a Public Health condition remains that official experts should not be locked into their own terminology and try to teach the world to their own illogic, but they would need to remain open to the possible causes for misunderstanding and policy error, and be aware of the possibility of another 78,000 deaths and be clear about this instead of hide it.

4. Public health, life gain measures and loss of immunity

The objective of Public Health is to balance medical and economic issues, i.e. lives and livelihoods, see e.g. Acemoglu et al. (2020). Key criteria are the (quality adjusted) life-years gained (QALY gained) and the Incremental Cost-Effectiveness Ratio (ICER), defined as a gain divided by the cost for getting this gain. That the SI(EY)A(CD) family of models uses the death count and criterion of lives saved (lives extended) can be understood from their genesis in 1927, but modern Public Health requires developed life gain measures. In 2018 the (European) ECDC indeed calculated disability adjusted life-years (DALY) for infectious diseases, which can be seen as a useful first step. Teaching SI(EY)A(CD) models as they are indoctrinates (epidemiology) students into thinking in a mantra of lives saved only, which conflicts with Public Health application. It is better to extend these basic models with more life gain measures. At the minimum, given the aggregate D = φ A from the model and the assumption of homogeneity, we can use the average life expectancy (ALE ψ) at death, and find the total loss of life-years as ψ D, which then can be used for scenarios and the ICER. Mathematically it is only a minor additional parameter but it is somewhat remarkable that it isn’t much used in epidemiology.

The relevance is shown in the SARS-CoV-2 pandemic for which the infection fatality factor (IFF) differs so much per age group, with thus a different weighted average, especially when we include the lower life expectancy because of comorbidity. The Dutch RIVM model actually has different age groups and a matrix of their contacts, and they should be able to calculate life-years lost, but they don’t present these. The official argument (i.e. a comment by Van Dissel) is that lives saved and even more life-years saved are speculative, since it is not warranted how the epidemic would evolve without the interventions. But if the model records a death and the age (group) of death then age-specific life expectancy can be used, and the total loss would be crucial information for Public Health and its economics.

Rather than developing SI(EY)A(CD) models with age groups, Colignatus (2020c) opts for the use of both the population table and the life table. Any course in epidemiology or medical statistics has the life table as its core, with the notion of competing risks of death. Students using SI(EY)A(CD) models must be familiar with the life table too. Given the aggregate D = φ A from the model and the assumption of homogeneity, the fatalities can be allocated in proportion to the age groups using their specific factors, using population weights or life table weights. With these tables available within the modeling environment it becomes easier to run scenarios with assumptions on the age-specific IFFs too (rather than take ALE as a single fixed number for all scenarios or neglect it for that reason). Colignatus (2020c) has the IFF = 1.5% with infection prevalence in March 2020 as weights, with a higher prevalence in elderly home care. With population weights with uniform infection, the IFF drops to 1.1%.

An infection with the virus might not result in permanent immunity. It is a corona virus and such viruses are known for the common cold that has no immunity. Annual loss of immunity requires a life table re-computation of life expectancy with repeated application of the age group specific IFF. The 2018 all-cause death now becomes the 2020 other-cause death. The higher annual death toll is an acceleration of the mortality by comorbidity. It is statistically attributed to the virus rather than to comorbidity. This exercise shows that a 10% rise in annual cumulated prevalence (A) implies about a 0.5 year drop in life expectancy. The effect is relatively small since the fatalities are mostly in the higher age groups. If cumulated prevalence would rise to 60% as RIVM suggested, then Dutch life expectancy would reduce with 3 years, but the (by RIVM neglected) overshoot to 90% would reduce life expectancy by some 4.5 years in total. It seems a good suggestion to eradicate the virus.

Other life gain measures are fair innings, proportional shortfall and UnitSqrt. Colignatus (2003, 2020) is a discussion of life gain measures, with the suggestion that the UnitSqrt would be a compromise that likely many people would find interesting.

5. Wider context for democracy

We want to save lives and livelihoods but let us not forget fundamental insights about democracy and science & learning, see Colignatus (2020b). There is something fundamentally wrong in the relation between society in general and science & learning. For the democratic setup of each nation it is advisable to have both an Economic Supreme Court for the management of the state in particular and a National Assembly of Science and Learning for the flow of information also in this management. For the US electoral system there is the observation that a third of US voters have taxation without representation, see Colignatus (2020a) in the January edition of Physics and Society.

Apart from this general diagnosis, the SARS-CoV-2 pandemic comes with its own pecularities. We have seen national emergencies in which epidemiology with the SI(EY)A(CD) criterion of lives saved became dominant, including the objective to flatten the curve, prevent the break-down of the health care system, and remain within the capacity of Intensive Care Unit (ICU) beds: but all with neglect of life-years gained and the ICER. The economic cost has been huge while the life-years gained are relatively low, and we still don’t have proper statistics about the falling away of usual care that was postponed or rescinded because of priority for the lives saved related to the virus.

A rather reasonable point of view from epidemiology is that the lock-down prevented an explosive outbreak, and that the prevented death toll is huge, and by implication also the life-years saved. The ICER of the lock-down thus would compare favourably with conventional Public Health outcomes, see e.g. epidemiologists Bonten & Rosendaal (2020) (Dutch). The point however is that the lock-down in many countries was an emergency break, and its use was required because of the slow and inadequate response before. It is not quite an argument that a system works since the emergency break had to be applied. Germany, Greece and Japan had a decent system of early warning and source and contact tracing (SCT), see e.g. department director Saito (2020). Holland had years of budget cuts on SCT, with insufficient protests by the RIVM leadership, and when SCT was called upon in 2020 its managers stopped doing it when capacity was reached, in contradiction to their job description to do SCT; and only the dissenting provinces in the North decided to continue and if needed increase capacity and they indeed effected containment in their region.

The paper by epidemiologists Petersen et al. (2020) in The Lancet is highly problematic. For decades, these experts have been warning for a pandemic, they must have noticed that national warning and SCT systems were not in shape, but when SARS-CoV-2 arrived in November 2019 to January 2020 they were late on the uptake themselves, so that eventually the emergency brake had to be used in virtually all countries in the world. However, their article does not discuss that they themselves were late on the uptake, and that their scientific leadership on pandemics thus collapsed into the current chaos of this pandemic, with its death and economic misery. The authors are evasive about key criticism: (a) The authors recognise that SARS-CoV-2 of 2020 is “genetically closely related” to SARS-CoV of 2003. The naming is no coincidence. Also in characteristics we see the same virus. Yet the authors subsequently call it “new”, which switches to the language of identity, while the article concerns empirics and the closeness of charactistics. They suggest that SARS-CoV was eradicated in 2003 and “eliminated from the host [sic] reservoir”, while it clearly was not eradicated since it resurfaced in 2019-2020. The authors close their eyes for the return of the virus and they apparently have a vested interest in window-dressing their failure of not recognising it. The true story is that official epidemiology tended to neglect the virus after 2003. Perhaps the reason for this was that it did not reach the West in 2003. Remarkably, in November 2019, precisely when the virus returned to the attention of the world, Smith (2019) posed the very question whether it had been eradicated, and he comes out on the side that epidemiologists should stake out the claim that they had done so. Smith (2019) allows for caution: “The WHO’s consensus document on the epidemiology of SARS, published during the pandemic in 2003, stated “The eradication of SARS-CoV is unlikely if infection is zoonotic” (…), which was later found to be the case. Evolution has had time to generate variants in SARS CoV’s wildlife hosts [sic] during the past 13 years, so we may see a related virus emerge in the next decade or century. If we declare eradication, we run the risk of SARS someday re-emerging under similar conditions that gave rise to it originally, which has the potential to psychologically undercut any claim of eradication.” Now in 2020, the genome and bio-medical and epi characteristics show that the virus hasn’t been eradicated. (b) Petersen et al. (2020:table1) mention various characteristics but leave out the important ones of asymptomatic and/or aerosol transmission that cause a pattern of superspreading and clustering, which characteristics confirm that it is the same virus. Of these authors, it has been Koopmans who in Holland has been hindering advance during the Spring and Summer on reconstructing ventilation in schools, elderly care homes, trains and buses for the Fall and Winter. The argument by her team at Erasmus MC has been that aspects had not been proven empirically – concretely by the infection of a lab animal at least 1.5 meters at a distance, which experiment apparently wasn’t set up spontaneously – but Richard et al. (including Koopmans) (2020) misrepresent (which Nature’s peer review apparently allowed) the finding by Van Doremalen et al. (2020) who used a Goldberg drum and who conclude: “Our results indicate that aerosol and fomite transmission of SARS-CoV-2 is plausible, since the virus can remain viable and infectious in aerosols for hours and on surfaces up to days (depending on the inoculum shed).” See Colignatus (2020c) for references.

When the world of epidemiology is closely connected to a bureaucracy that will admit to no failure, then perhaps outside scientists might have the independence that is needed for criticism: but most would lack the competence in epidemiology. The ancient question remains: Quis custodiet ipsos custodes? The likely best answer by various philosophers of science has been: let democracy enable the forum of science & learning to perform, see Colignatus (2020b).

6. Conclusion

The SARS-CoV-2 pandemic highlights the need for good communication. We should avoid the re-invention of the wheel on epidemiology but we apparently can contribute to the redesign of the didactics of some basic epidemiological models. With sound command of modeling and empirical developments we can diagnose that the cause for the pandemic lies in governance. This puts the relation between democracy and science & learning into focus. The pandemic is only one of more issues that have caused the discussion in Colignatus (2020b) that advises the creation of a national assembly of science and learning.

7. Caveat

A caveat is:

(1) Professor of mathematical statistics Richard Gill has oft quoted Patrick Suppes (1963:334-335): “For those familiar with the applications of probability and mathematical statistics in mathematical psychology or mathematical economics, it is surprising indeed to read the treatments of probability even in the most respected texts of quantum mechanics. (...) What is surprising is that the level of treatment in both terms of mathematical clarity and mathematical depth is surprisingly low. Probability concepts have a strange and awkward appearance in quantum mechanics, as if they had been brought within the framework of the theory only as an afterthought and with apology for their inclusion.”

Physicists who look into epidemiological models with their underlying probability theory are advised to check modern probability theory too and check what Gill wrote on the issue.

PM. Gill argues that quantum events provide the empirical evidence that reality is probabilistic, while Colignatus (1981, 2007, 2011) argues that there are the three independent competing views of determinism, volition and randomness, and that there is no discerning experiment to decide what would be reality, since anything that happens can be described in any view.

(2) My own comment is that I am uncomfortable with the lack of logic in the texts intended for a general readership, like me, about Einstein’s relativity, see Colignatus (2005). There might be a historical explanation. In the 1800s, physics was much engaged with the notion of an aether, and researchers lost much time on this, till they decided that they would only work with observed variables for which measurement was defined. Apparently this katharsis came with a lot of emotions and a sense of loss, and the use of observed and measured variables has become a deeply engrained dogma in physics. If I understand it well, Einstein thus defined time as measured by clocks, and when those clocks moved at the speed of light then measurement became crooked, and Einstein solved this by suggesting that space itself curved. This uses the same word “space” for something new, which apparently is defined by the measurement outcomes. I have a hard time understanding what this means, for, when space is defined in Euclidean terms, then you cannot change that definition, and if you use another definition of space then I do not understand why you use the same word en what you are speaking about because my understanding of what space is has been given by Euclid. Drawings of masses that depress and make a curved plane are still pictures that use Euclidean space. It is not clear to me why you would use the same word for something else (though I have no problem with “n-dimensional Euclidean space” or that you might choose the surface of the Earth for your system of co-ordinates). For me, it makes more sense to hold that there can be measurement errors when the instrument of measurement has a Lorentz contraction. This suggests that stories for the general public might be written so that there still is Euclidean space but with measurement errors. I would like to read such a story that still fits Einstein’s formula’s and measurements. In economics and psychology it is common to have hidden or latent variables, i.e. variables that are not observed directly but that have relevance for a theoretical understanding. Also epidemiology has such variables. Thus, physicists looking into epidemiology preferably learn to deal with latent variables rather than start imposing the dogma of observed and measured variables and create new problems in communication.

8. References

Thomas Colignatus is the scientific name of Thomas Cool (cool at dataweb.nl), econometrician (Groningen 1982) and teacher of mathematics (Leiden 2008), Scheveningen, Holland, thomascool.eu. He worked at Erasmus Medical Center (Rotterdam 2002-2004) on (Markov) modeling for screening on the Human Papilloma Virus (HPV) concerning cervical cancer. He is secretary of the Samuel van Houten Genootschap, scientific bureau of the Social Liberal Forum, an initiative for a political party in Holland.

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