CITE THIS NOTEBOOK: Redesign of didactics of S(E)IR(D) -> SI(EY)A(CD) models of epidemics by Thomas Colignatus. Wolfram Community JUN 15 2020.
ORIGINAL ARTICLE: Thomas Colignatus, Samuel van Houten Genootschap, Didactic redesign of models in epidemiology, with a caveat on probability and latent variables, Forum on Physics and Society, Volume 49, Number 4, (2020). WordPress article.
Reference to exit strategies for the economic crisis. A nasty virus, asking for eradication. Could the virus be used as its own vaccine for some? How immune will one be? Didactics on modeling of the SARS-CoV-2 (covid-19) pandemic of 2020, with the example case of Holland. Prevent or prepare for the next pandemic, perhaps already this Fall or Winter.
This notebook also available on https://zenodo.org/communities/re-engineering-math-ed/about/

Abstract

This notebook and package redesign the didactics of the classic SIR, SIRD and SEIRD models. A first step is to relabel to SI(EY)A(CD). The acquitted A = C + D are the cleared or deceased. This avoids the triple use of R for removed, recovered and reproductive number. The infected I = E + Y are the exposed and infectious. The basic structure is given by the Euler-Lotka renewal equation. The format of ordinary differential equations (ODE) should not distract. The deceased are a fraction of the acquitted, D = φ A, with φ the infection fatality factor (IFF). 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 what 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. The didactic redesign is discussed with the example of the SARS-CoV-2 pandemic. The common formula on herd immunity 1 - 1 /
R
0
assumes a steady state, but unless infections are zero then they actually proceed in a steady state, and thus not with the promised protection. SI(EY)A(CD) has only an asymptotic steady state so that this formula does not even apply. For SI(EY)A(CD) a notion of “near herd immunity” might be 95% of the limit values. For SARS-CoV-2, RIVM (the Dutch CDC) has mentioned
R
0
= 2.5 and herd immunity of 60%, presumably using another type of model. In SI(EY)A(CD) an infection with
R
0
= 2.5 proceeds after 60% till the limit value of 89.3%, which, with IFF = 1.5%, would mean another 78,000 deceased in Holland, compared to 9,000 at the end of May. For Public Health, it is important to balance medical and economic issues. A better understanding of the SI(EY)A(CD) family of models helps to gauge exit strategies for the pandemic and its economic crisis. A possible strategy is to eradicate the virus: with test, test, and test it would be possible to put positively tested persons in quarantine till they have cleared. Another possible strategy is that the vulnerable (elderly and comorbid younger) are put into quarantine while the less vulnerable are infected (in cohorts dictated by ICU capacity), effectively using the virus as its own vaccine, for a period of 12-16 months until there is a proper vaccine for the vulnerable compartment of society. It is remarkable that these scenario’s are so little discussed in policy making circles, where there seems to be a preference for a lock-on-off approach, that is risky and prolongs the economic crisis. Epidemiology exists for longer than a century and there have been many warnings about the risk of pandemics. Lessons learnt at the level of cities and nations are now learnt at world level. 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 and a National Assembly of Science and Learning. We want to save lives and livelihoods but let us not forget fundamental insights about democracy and science & learning.

Keywords and codes

Journal of Economic Literature (JEL) codes: P16 (political economy), E66 (macro-economics, general outlook), I18 (public health), Q56 (sustainability), D60 (welfare economics), A12 (general economics and teaching, relation to other disciplines)
Medical Subject Headings (MeSH) (IDs): D011634 (Public health), C000656484 (COVID-19), D004813 (Epidemiology), D000066949 (Biobehavioral sciences), D015233 (Models, Statistical), D000075082 (Proof of Concept Didactics), D017145 (Models, Educational), D004778 (Environment and public health), D000072440 (Public health systems research), D040381 (Education, public health professional), D006285 (Health planning), D004989 (ethics)
MSC2010: 92D30 (Epidemiology), 92C60 (Medical epidemiology), 91B64 (Macroeconomic theory), 91B76 (Environmental economics), 00-01 (Introductory exposition)

Disclaimer and declaration of interest

Thomas Colignatus is the scientific name of Thomas Cool, econometrician (Groningen 1982) and teacher of mathematics (Leiden 2008), in Scheveningen, The Netherlands. He worked at Erasmus Medical Center (Rotterdam 2002-2004) on (Markov) modeling for screening on the Human Papilloma Virus (HPV) concerning cervical cancer. The article has used software of The Economics Pack. Applications of Mathematica, which has been developed by the author since 1993, which is proprietary software, available at a price, and not falling under the CC license for the PDF. The targeted publication venue is not a journal but a book publisher.
(c) Thomas Colignatus 2020
CC BY-NC-ND

Table of contents


1. Introduction

1.1. What this notebook and package do

1.1.1. Didactic redesign

This notebook and package redesign didactics for the S(E)IR(D)  SI(EY)A(CD) epidemic models.
- There is no new finding on content. There is no new analysis. There are no new parameter estimates.
- There is only an effort at better didactics of what already is known in the field.
- Readers of this notebook may be researchers from other fields than epidemiology or they may be first-year students from a general background. Readers will have at least a first-year student understanding of mathematics and they will be open to the use of Mathematica. Readers will require a discussion that uses mathematics.
A readership with such competence should have little problem with the standard didactics of the S(E)IR(D) models. Thus, an effort at improving these didactics can only have a very marginal effect. However, advancement comes in little steps. The pandemic will cause an interest by researchers from other areas and new students. They can be served by clearer exposition, with the benefit of computer algebra and a computable environment. One hope is that this notebook also reaches conventional teachers: they may not adopt all suggestions but can find some alerts.
PM. A new element may be the distinction in quarantine classes with a colouring scheme in Chapter 4 that could contribute to better communication.

1.1.2. There already is an abundance

In addition:
- There is an abundance of SI(EY)A(CD) modeling. Important is the Berger, Herkenhoff and Mongey (2020) (March 29) paper, that adapts the SEIRD  SEYCD model to test-dependent states of quarantine. Their code in Python is available. These authors are not epidemiologists but they present their paper as a proof of concept for the exit scenario of test, test, and test. See Tabarrok (2020) for an independent appreciation that this exit scenario is cost-effective. Acemoglu et al. (2020) (May) employ a SIRD  SICD model with at least three age compartments, and plot GDP & Death policy frontiers. See another application in Python at Idmod.org, called “Emod”.
- Such abundance also exists for Mathematica, see the list of references. For SARS-CoV-2 (Covid-19), the Kaurov (2020) collection of computational publications must be mentioned. Appendix D shows that one does not need this present notebook and package of mine. Who knows the S(E)IR(D) models and Mathematica will find it simple and straightforward to create such model in Mathematica and run a WhenEvent policy intervention. This present notebook and package only provide structure in use, though at the cost of a learning curve. We only assume now that (such) experienced readers are willing to look at the issue from the angle of didactics.
- There are other approaches to epidemic modeling than the SI(EY)A(CD) models. A fine discussion has been given by 3Blue1Brown (2020). See the interview by Spinney (2020) with Karl Friston on models derived from physics. This kind of modeling is supported by Gill (2020).

1.1.3. Overview of this Introduction

This Introduction proceeds with (i) the overview of didactic redesign, (ii) the context of the SARS-CoV-2 pandemic and exit strategies, (iii) the overview of the notebook itself.
PM. Overall, this notebook has a lot of repetition. Not only the models are much alike, but also the analytical properties and the explanations of the routines, and the discussion of the aspects of the SARS-CoV-2 pandemic (e.g. in Holland and elsewhere). The assumption has been that readers might skip sections (e.g. jump to their model of interest), so that such repetition is beneficial.

1.2. The renaming S(E)IR(D)  SI(EY)A(CD)

The S(E)IR(D) abbreviation stands for the family of epidemic models that have these compartments or phases (switched on or off):
Susceptible - Exposed - Infectious - Cleared or Deceased
The conventional S(E)IR(D) setup uses the label “R” in three different meanings: Removed in the SIR model, Recovered in SIRD and SEIRD, and overall for the (basic) reproductive factor
R
0
. Also, all models use the “I” for the infectious compartment, but some also for “Infected”. SEIR(D) includes the exposed, as the infected but not infectious yet. Since all models claim to add up to the total population, and the exposed did not occur in SIR or SIRD, this implies that SEIR(D) would describe a different population.
This family of models has been presented in this manner for half a century. Within the computer algebra environment of Mathematica it appeared that there is a clear advantage of a better nomenclature. Unique names help the identification of variables, their parameters and the application of associated routines. When we discuss the same population then our analysis should not be cluttered with issues on the administration of the compartments. While humans are flexible, computers are a bit less so, and when humans work with computers then humans become rather inflexible too.
This notebook and package use:
- The Acquitted A instead of the Removed.
​- The Cleared C instead of the Recovered.
​- So that the A compartment can be split in A = C + D or the Cleared or Deceased.​
​- The Infected I can be infectious or exposed but not infectious yet.
- The Infectious Y.
​- So that the I compartment can be split in I = E + Y or the Exposed and Infectious.​
​- So that the SIA and SICD models have E = 0 and I = Y, or that their infected I are also the infectious Y.
Thus we get the abbreviation SI(EY)A(CD). For all variables V we distinguish Vq for the levels and Vp for the proportions w.r.t. the population N[0]. For plotting we can also plot the current population N = N[0] - D.
The decision to choose these names in this manner was taken with quite some hesitation and consideration but would seem to be optimal. A practical argument for this compartmentalisation and nomenclature is that the limit properties of SI(EY)A(CD) are determined by I (i.e. the sum I = E + Y) just like in the SIA model, without the need for relabeling in SEYA(CD). The introduction of the exposed compartment has only effect for the short run behaviour. For the (crucial) limit property there really is a compartment I = E + Y, apart from the temporary distinction in the phases of E and Y. The package allows the user to adopt the conventional nomenclature, including the possibility of using Y = E + I instead. The underlying programming structure however is as above, and there is no reason to hide it and not take advantage of it.
With a standard SEYCD model, an unhindered pandemic in Holland would have given roughly the following path, i.e. neglecting a change in parameters due to the intervention in March. These figures have only a didactic meaning, see below for a comparison to RIVM figures (the Dutch CDC). The first death was reported on March 6 but according to the model there were actually already 6 that day. The eventual total death toll could have been around 250,000, with 95% reached by May 22, and the remaining 5% spreading over the remainder of the year. Only 2% of the population would not have been infected at one stage.
Table legend: S = Susceptibles (uninfected), I = Infected (exposed or infectious), D = Deceased, C = Cleared, N = N[0] - D remaining population, A = C + D (the acquitted). Percentages (p) are w.r.t. the onset, i.e. N[0]. (Table taken from below.)
Out[]//TableForm=
pDate
Day
Sp
Ip
Dp
Cp
Np
Ap
Onset
2020-02-23
0
100.0
0.0
0.0
0.0
100
0.0
1st Death
2020-03-06
12
100.0
0.0
0.0
0.0
100.0
0.0
1st Intervention
2020-03-18
24
99.9
0.1
0.0
0.0
100.0
0.0
Top if free
2020-04-21
58
25.0
40.3
0.5
34.1
99.5
34.7
Ap = 50%
2020-04-25
62
13.5
36.5
0.7
49.2
99.2
50.0
95% Ap[∞]
2020-05-22
89
2.4
4.5
1.4
91.7
98.6
93.1
Limit if free
∞
∞
2.0
0.0
1.5
96.5
98.5
98.0
The levels (q) of the relevant compartments are as follows. (Table taken from below.)
Out[]//TableForm=
pDate
Day
Sq
Iq
Cq
Dq
Onset
2020-02-23
0
17.4×
6
10
100
0
0
1st Death
2020-03-06
12
17.4×
6
10
1290
391
6
1st Intervention
2020-03-18
24
17.4×
6
10
19438
6356
97
Top if free
2020-04-21
58
4.4×
6
10
7.×
6
10
5.9×
6
10
90455
Ap = 50%
2020-04-25
62
2.4×
6
10
6.3×
6
10
8.6×
6
10
130500
95% Ap[∞]
2020-05-22
89
419710
778037
16.×
6
10
243034
Limit if free
∞
∞
344995
0
16.8×
6
10
255825
ReadMeSIA["Symbol"]
Out[]=
N[0] = S + E + Y + C + D, with I = E + Y, is the startup population of units.N = N[0] - D is the current population, mainly used in plotting.S are the Susceptible.E are the Exposed but not infectious yet. (They might become diseased.)Y are the Infectious. The (currently) Infected are I = E + Y, while C + D are have-beens.A = C + D are the Acquitted, or the Aftermath. (A avoids confusion with R0. A is conventionally denoted as R that may cause that confusion.)C are the Cleared. (Avoid the confusing use of R as "Recovered".)D are the Deceased.I = E + Y are the Infected. In SIA E = 0 and I = Y.O = N[0] - S - I are "the other", so that these are all SIO models.For variable V: Vq is the level and Vp = Vq / N[0] (and explicitly not divided by N).
In[]:=
?ReadMeSIA
Out[]=
Symbol
ReadMeSIA[] gives an overall explanation of the SIA` package. See there for: ReadMeSIA["Terminology"]ReadMeSIA["Symbol"]ReadMeSIA["Period"]ReadMeSIA[model] for models SIA, SICD and SEYCDReadMeSIA[SEYCD, Simplify]
In[]:=
ReadMeSIA["Terminology"]
Out[]=
This terminology applies to the SICD model:SICD improves on SIA by distinguishing A = C + D, for D deceased and C cleared. (-Sp') is the *incidence* of infections at moment t. Ip is the *prevalence* of infections at moment t. 1 - Sp = I + C + D is the *cumulated prevalence* at moment t (i.e. the haves and hads). 1 - Sp[∞] = Ap[∞] in both SIA and SICD is the *limit prevalence*, since Ip[∞] = 0. Cp is the cumulated survival factor at moment t (but the Sp also count as living). Dp is the *cumulated mortality factor* at moment t (mortality ~ population). Cp[∞] is the limit cumulated survival, and part of the burden of disease (infection). Dp[∞] is the *limit (specific) Infection Mortality Factor* (IMF) (prevalence of death). Dp[∞] / Ap[∞] = Dp[∞] / (Cp[∞] + Dp[∞]) is the *limit Infection Fatality Factor* (IFF). MuSICD[] is the *(instantaneous) Infection Fatality Rate* (IFR) (fatality ~ infection). MuSICD[] = GammaSIA[] * PhiSICD[] in the model, with PhiSICD[] the IFF. In addition for SEYCD: Ip = Ep + Yp subdivides into Exposed (but not infectious) and Infectious.

1.3. Infection Fatality Factor (IFF) and symptomatic Case Fatality Factor (sCFF)

The conventional discussion of the SI(EY)A(CD) model family is oriented at the ordinary differential equations (ODE). In particular, the development of the deceased compartment is modeled as:
D’ = μ I in SICD
​D’ = μ Y in SEYCD
This causes the question what μ stands for. What happens if the parameter γ for the infectious period 1/γ changes ? Would there be a distinction between a “lethal infectious period” and a “clearing infectious period” ? Maugeri et al. (2020) have the better format D’ = γ sCFF, for the symptomatic case fatality factor (sCFF). A change in γ will not affect the sCFF and the limit property of the model. The Maugeri et al. (2020) formulation still relies upon the ODE format.
It appears that the C and D compartments are mere proportions of the acquitted A department. The formulation as an ODE is only mathematical convenience of modelers who have adopted this format but it is not the essential property of the model as such. A discussion of these epidemic models should not be confused with a course on differential equations. Assuming that didactics is targeted at the relevant property, we get this structure, with φ the infection fatality factor (IFF):
A’ = γ I in SIA(CD)
​A’ = γ Y in SEYA(CD)
​D = φ A
C = (1 - φ) A = A - D
This immediately clarifies that there is no distinction between a lethal infectious period and a clearing infectious period: both compartments have the common 1/γ infectious period. (Some readers might want to check Appendix A.)
The package allows the conventional format too, and such properties can then derived, even though it is also mathematically more elegant to directly specify what the model does.
This notebook and package use an Infection Fatality Factor (IFF) of 1.5%, based upon reweighing the Verity et al. (2020) and Ferguson et al. (2020) estimates, applying the Dutch age composition and inserting a special assumption on the age 70-79 group, see Colignatus (2020cd), part of which is reproduced below in section 2.7. In testimony for Dutch Parliament, RIVM (2020e) mentions 9,000 deaths, and, using tests on antibodies that cause an expected value of the prevalence of infected (current and past) of 650,000 (3.74%), arrives at a crude IFF ≈ 1.38%, not far of the 1.5% used in this notebook (that is directed at didactics).
PM. Not irrelevant: these models for infections are also used for symptomatic phases, which is also the case for the Goh (2020) “epidemic calculator”. Thus below we also discuss the distinction between the infection generation interval and the symptomatic serial interval. The sCFF has a higher value than the infection fatality factor (IFF) since sCFF excludes the infections without symptoms, which is important in the SARS-CoV-2 case.
PM. For antibody testing, see Slot et al. (2020), who also mention a 9.5% prevalence in the hardest hit areas, and “that the protective effect of SARS-CoV-2-specific antibodies is not yet known”.

1.4. The meaning of parameters
R
0
, α, β and γ

1.4.1. Euler 1767

Leonhard Euler in 1767 already formulated a “renewal equation” for population dynamics, and the SI(EY)A(CD) model appears to be a reformulation in terms of (ordinary) differential equations (ODE). Relevant papers are Fine (2003), Svensson (2005, 2007), Breda et al. (2012) and Champredon, Dushoff & Earn (2018). The latter paper is quite recommendable, though they regrettably do not refer to Svensson while a comparison of results would have been informative. Delamater et al. (2019) clarify that George MacDonald who introduced the term of “reproductive number” (factor) in the epidemiological literature in the 1950s used
Z
0
. See Heesterbeek (2002) for an overall brief history.
The structure that seems best for didactics can be found in the Wallinga & Lipsitch (2006) paper. For understanding of the SI(EY)A(CD) model family we better now specify the latter main structure here at the outset. Let us mention two models that do not apply - the fixed period and the normal distribution - and one model that does apply: the exponential distribution (to be distinguished from exponential growth).

1.4.2. Constant generation interval

1.4.3. Normal distribution

1.4.4. The SI(EY)A(CD) family

Epidemiology uses these notions from demographics for the SI(EY)A(CD) family of models. See the formulas and the plot of the growth rates in Section 5.7 on the SIA model. The demographic model assumes exponential growth of the population. This will only work in the first and final phases of an epidemic. For a long period, the development is distinctly not exponential. However, the first and last phases can still be used for estimating the relevant parameters.
The relevant distinction within the SI(EY)A(CD) family is:
Rewriting from periods to the parameters for the rates:
Each case has its relationship between the reproductive number and the growth rate r.
(ad 1) β / γ = 1 + r / γ or r = β - γ
(ad 2) β / γ = (1 + r / α)(1 + r / γ)
For the RIVM figures, assuming the SEYCD model:
PM 4. It is useful here to refer to the same result as in Chapter 10. The PDF of the random variable that is the sum of two exponentially distributed variables is (and observe the symmetry):
When we substitute this pdf in the Moment Generating Function as indicated by Wallinga & Lipsitch (2006), then (while remembering that this only holds for part of the process in SEYA(CD), namely the exponential phases):

1.4.5. Empirical distributions

Wallinga & Lipsitch (2006) mention the important case when the R[t] can be taken from the moment generating function of an empirically observed distribution of interarrival times. Wallinga has been affiliated with the RIVM, the Dutch counterpart of the US CDC, and has a special professorship in Leiden. He is also co-editor of the “Handbook of infectious disease data analysis”. Our chapter 2 below will copy the RIVM graph of such R[t]. See section 5.2 for terminology, with β[t] / γ[t] for the “dynamic reproduction factor”, while the “effective reproduction factor” R[t] also corrects for the level of susceptibles Sp[t].
A comment from theory is that it would remain important to distinguish different types of infectious paths. (i) For a longer horizon, there are births and background deaths. (ii) When there is symptomatic disease, there may be asymptomatic transmission, see Chapter 10, and this may have complications for estimation and source and contact tracing. (iii) When one path systematically has short intervals and another path has systematically longer intervals, then the joint distribution is not stable, and one might better distinguish two compartments. A distinction may be that the elderly have faster symptoms and clear later. (iv) The RIVM R[t] graph is estimated on hospital admissions but over time in Holland the empirical value of this series has not been constant. There was a learning-effect about the virus, with a shift away from hospitalisation, with deaths no longer tested and not officially recorded as “SARS-CoV-2 deaths” but showing up as “surplus deaths” in the monthly statistics.

1.4.6. Interpretation of γ, also for estimation

The interpretations of “infectious period” and rate of acquittal are adequate for an application for e.g. such natural infectious period (a “real infectious period”). The issue may become more complicated when we change our assumptions and the model might not quite apply like we might suppose.
The intervention is dramatically effective. The death toll quickly tops at its limiting value of some 7000 persons (including those of before the intervention).
(The following graphs are copied from section 16.13.)
However, let us focus on the lower ranges. To the left of the intervention, the plotted lines overlap. At the intervention and some few days after it, we see a surge of clearings and deaths. With γ changing from 1/10 to 2, the model presumes that there is an acceleration in acquittals indeed.
Now that the graphs have been shown it will be easier to discuss the issue. The point is that the SI(EY)A(CD) model concerns compartments and not necessarily events. It is true that the intervention with the quarantine reduced the “infectious period” from 10 days to 1/2 day. And it is true that the Infection Fatality Factor (IFF) proportion of those acquitted belongs to the compartment of the (soon to be) deceased. However, the intervention did not speed up the process of dying. Given the new “infectious period” from the intervention, the infected are properly allocated to their compartment but we should not assume that this “infectious period” still would be a “natural or real infectious period” (ending e.g. in death). ( A consideration might be to use D’ = μ I and not change μ, so that the change in the “infectious period” is borne by C’ = λ I. However, this implies changing the IFF. The intervention cannot change this. A given proportion of the infected belongs to the compartment of the deceased.) Thus, entry into the compartment of the deceased does not yet indicate the day of actual death. The estimation on the observed death toll requires that we take into account that the model speeds up the allocation to this compartment, so that we need a longer period from entry into the compartment to actual death.
The conclusion is unavoidable that the SI(EY)A(CD) model has a parameter γ that must be treated with some care. The reading of 1/γ as the “infectious period” would be unproblematic for the natural process. When this interpretation is changed, in the context of an intervention, then 1/γ is rather regarded as a normalising constant and the interpretation as “infectious period” should not be taken too literally with connotations from the other context.

1.5. Benefit of better didactics on SI(EY)A(CD) epidemic models. Understanding herd immunity

1.5.1. General understanding and communication

The 2020 pandemic (again) proves the relevance of the SI(EY)A(CD) models for the common understanding of infectious epidemics and their impact on the economy. The Goh (2020) “epidemic calculator” allowed perhaps millions of people a better understanding, and clarified for them the notion of “flattening the curve” to remain within the capacity of the health system. Pueyo (2020) provided clarity for many too. There is common use of the model in the literature (see the references). The model appears to be of key importance for a world at mercy of the pandemic, not only for education, but also for communication between research communities and for the media and general public. It stands to reason, but is not guaranteed, that better didactics would be beneficial for such understanding and communication.
The SI(EY)A(CD) model remains too simple as a foundation for practical policy making (when we have better models, see e.g. Diekmann et al. (1990) and Grant (2020)). However, the model remains relevant for general understanding, and such general understanding very much concerns issues of strategy too.

1.5.2. Interpreting official reports

RIVM - the Dutch equivalent of the US CDC - provides the national reservoir of experience in infectious diseases and must have employed more complex models than SI(EY)A(CD) for its substantiation to the Dutch government that there was a deadly pandemic. The international reports, the WHO PHEIC of January 30, and the growing alarm in Dutch hospitals themselves were factors too of course. The point remains that, against this backdrop, the SI(EY)A(CD) family of models can still be regarded as a bedrock for our understanding of epidemics.
The RIVM policy of not mentioning an implied death toll seems rather a consequence of the attitude of avoiding speculation and “sticking to the facts”. However, it is actually better to mention these implications, because death toll estimates make for a more transparant discussion about the “value of a statistical life”. There may be a difference between the world of Public Health (including economics) and the world of Medicine (without economics), with RIVM too much influenced by the latter.

1.5.3. Remarkable statistical fits

1.5.4. Limit values

1.5.5. Understanding herd immunity

RIVM informed the Dutch Prime Minister about the notion of “herd immunity” which information the PM used in his television speech of March 16. Rutte (2020) did not mention a specific number but RIVM circulated that herd immunity would be around 60% of the population, see the interview by journalist Tweebeeke and RIVM department head Van Dissel (2020). There are at least four possibilities how to arrive at such notion with such value, see Chapter 6.
(b)Potentially, 40% of society (7 million) will remain in permanent quarantine, so that 60% (10.4 million) would be the less vulnerables. This would work, see below, but it is remarkable that RIVM does not discuss the issue in these terms.
(abcd) In sum, the possibility cannot be excluded that Dutch RIVM was itself the cause for confusion here, and that it was reticent in discussing the implied death counts for such policy scenario’s. For journalists, it might be that it requires knowledge of mathematical modeling to know when those models are not needed when IFF values can applied directly.
PM. The three (inadequate) scenario’s as proposed by RIVM: https://www.rivm.nl/en/novel-coronavirus-covid-19/dutch-response-to-coronavirus or https://web.archive.org/web/20200527143138/https://www.rivm.nl/en/novel-coronavirus-covid-19/dutch-response-to-coronavirus.

1.5.6. Relevance of better didactics

A presumption of better didactics is that it would be beneficial for the general discussion, at least within the wider community of researchers and policy makers, when such notions, as just discussed, would be well-understood.
PM. In a paper that got through peer-review, Ioannidis (2020) (my italics): “Focusing on protecting susceptible individuals may be preferable to maintaining countrywide lockdowns longterm.” Likely he means “protecting vulnerable individuals”. With the terminology of epidemiological models, protection of all susceptibles requires a national lockdown.

1.6. A nasty virus, asking for eradication. A challenge anyway

1.6.1. A world at risk but also badly managed

John Snow (1813-1858) is considered one of the founders of modern epidemiology, when he deliberately searched for and identified a Soho water pump in the London cholera epidemic of 1854. We are more than 150 years later. It is not for nought that the world has set up the WHO and created the notion of an official warning of a public health emergency of international concern (PHEIC). Recent more general warnings were e.g. from HIV / AIDS 1981-present, SARS 2003, MERS 2012-present, the Mexican “swine flu” 2009, Ebola 2013-2016 and 2018-2020 and still risky, and Zika 2015-2016. Microsoft founder Bill Gates is credited for a 2015 TED talk warning for “the next outbreak”. Piot (2018) is more recent and essentially more qualified. The UK government provided for a study how to respond to pandemics, which study was duly used by Singapore for the current SARS-CoV-2 outbreak. However, it was not used by the UK itself. The USA response team was dismantled from the US White House a few years ago. Something has gone badly wrong in the last decade. The world in the last few years shows a huge pandemic mismanagement, likely caused by fundamental misunderstandings, while the issue was already clearly spelled out in the literature decades ago.
Observations are: (1) The health community has a hierarchical setup and mindset while the economic community is more liberally minded. When top health officials do not strongly criticise policy makers - because they have to work with them on a daily basis - then the health community apparently also subdues criticism. But economics is not perfect either. (2) What are severely lacking in the democratic setup of each nation are both an Economic Supreme Court and a National Assembly of Science and Learning, see Colignatus (2020a).

1.6.2. Prevent future pandemics

Future pandemics are better prevented. The BBC Horizon (2020) broadcast can be much recommended for its overview of the issue and essential information, including the discussion of the WHO 2018 warning on pandemics and the identification of risk "hotspots". At roots, the lessons already learnt on epidemics ages ago are rediscovered for the world-scale without world-governance, and with scientists already warning for decades. Mankind is advised to make some fundamental changes, towards much lower world population and environmentally sustainability, including less contact with wildlife and their pathogens, but also in the information and decision making structure in society, see Colignatus (2020ab).
There is a perspective that SARS-CoV-2 might only mean a reduction of life-expectancy of 1%, see Richardson & Spiegelhalter (2020). For Dutch readers there is Jaspers (2020). From this perspective, there is only a limited effect, and there might be no need for more involved changes. However, the other view is that we got only lucky now. And perhaps we are not so lucky actually, since the virus appears to be a nasty one.

1.6.3. A nasty virus, asking for eradication

SARS-CoV-2 is a nasty virus that attacks many parts of the body, see the column by Collins (2020), the director of the US National Institutes of Health (NIH), and the interview with Longley by BBC Horizon (2020) (May 26). For Dutch readers there is De Visser (2020). A possibility is also that the body is cleared of infection but that the immune system is out of control and starts to attack the own body.
Draulans & Piot (2020), is an interview with Piot, who suffered the virus, and who says:
"Many people think COVID-19 kills 1% of patients, and the rest get away with some flu-like symptoms. But the story gets more complicated. Many people will be left with chronic kidney and heart problems. Even their neural system is disrupted. There will be hundreds of thousands of people worldwide, possibly more, who will need treatments such as renal dialysis for the rest of their lives. The more we learn about the coronavirus, the more questions arise. We are learning while we are sailing. That's why I get so annoyed by the many commentators on the sidelines who, without much insight, criticize the scientists and policymakers trying hard to get the epidemic under control. That's very unfair."
Viruses mutate fast, and are apt in exchanging parts of their RNA, and we would not want that this "multi-system attack property" is spread amongst viruses. Perhaps it is still feasible to choose for eradication. The example of smallpox, eradicated, and polio, though still with problems, is encouraging, see WHO GPEI & Heymann (2020). This notebook will not arrive at a choice, and only mention the options: either eradication or find a way to live with the virus till there is a vaccine. There is an overlap in measures: a discussion about “herd immunity” might also fit a scenario of eradication.
Piot also states: “Let’s be clear: Without a coronavirus vaccine, we will never be able to live normally again. The only real exit strategy from this crisis is a vaccine that can be rolled out worldwide. That means producing billions of doses of it, which, in itself, is a huge challenge in terms of manufacturing logistics. And despite the efforts, it is still not even certain that developing a COVID-19 vaccine is possible.”

1.6.4. Under normal conditions the virus can no longer be contained

If we want to eradicate SARS-CoV-2 then special conditions must apply. Under normal conditions it can no longer be contained in the world. The reasons for the latter are standard from an introductory course in infectious disease - see e.g. BCM (2020).
- getting a vaccine takes time (perhaps 1.5 years) … and then give it to 7.5 billion people on the planet
- health care is under strain, will perform less well, thus there will be breaches and ever newer infections
- the virus already shows mutations and is likely to continue to do so: every new mutation would require a swift reaction for new containment – but the health care system already is under strain
- special for SARS-CoV-2: the pandemic cannot be suppressed because of the asymptomatic infectious.
Kissler et al. (2020) at Harvard mention already two strains and suggest that intermediate periods of national lockdowns might be needed (“lock-on-off”). The logic is impeccable, and worthy of more than a century of research in epidemiology. The logic however leads to the absurd conclusion that we must destroy our economy in order to save lives, while the destruction of the economy also costs lives - not included in their study. Epidemiologists have lived too long in a world that accepted models without consequences for the economy.
Less developed nations lack the medical systems of the developed world, will be open to the virus, and will form breeding grounds for new waves and mutations. Developed nations might think that they might insulate themselves by containment and suppression, but it is dubious that they can insulate themselves forever. We can expect waves of new infections indeed, like with the flu, but then 10 times more infectious / deadlier than the flu, with the risk of shorter intervals because of faster mutations. The Northern hemisphere now benefits from the Summer, but in Autumn the reduced health because of the common cold and flu will combine with SARS-CoV-2, causing increased joint mortality. The Summer of 2020 should rather not be wasted.
There are more than 7.5 billion people on the planet. Not all will comply with quarantine or get the eventual vaccine in time. Viruses mutate. This virus derives from the common cold strain and people may lose immunity after a period. The economic price of national lockdowns is high and it seems that the virus is “not terrible or deadly enough” for people to bear that price, causing much indecision and delay perhaps till it is too late again.

1.6.5. Return to some normalcy, or consider what urgently requires improvement

With the pandemic still continuing, the medical world faces the challenge to restore care to the level of before the pandemic. Gupta (2020) reports that some 40% of normal care has fallen away because of the focus on SARS-CoV-2 and the fear amongst patients of getting infected in the hospital. De Rek & Goudsmit (2020) is an interview with virologist Jaap Goudsmit (involved in finding HIV) who comments that the emergency brake of the national lockdown contained the problem but that normal care ought to be restored, and: (i) there is a lack of data, (ii) reliance on models means also a reliance upon assumptions, (iii) there is too little attention for scenario’s (the three scenario’s presented by RIVM are not enough). See https://jaapgoudsmit-over-corona.nl/
The economy faces the similar challenge, see Baldwin (2020ab), Baldwin & Weder di Mauro (2020), Berger et al. (2020) and Acemoglu et al. (2020).
A core question is: if some return to normalcy would be possible today, in May through September 2020, then why wasn’t it considered in January ? Was it really necessary for Western Europe to first have this wave and national lockdowns from February to May, before its CDCs, governments and citizens would accept the idea that “normalcy” would require some adaptation ? If CDCs have only an advisory role, what happened at the government Departments of Health ? Indeed, it is reported that Taiwan, Hong Kong, Singapore and South Korea learned their lessons by the SARS-CoV-1 epidemic in 2003. Would this mean that countries can only learn from disasters happening on their own territory ? And if this would be the sad truth indeed, then what have countries learned from the present episode ? (It is said, however, that Singapore used the pandemic scenario developed in London, that London itself did not use. Thus Singapore can learn, and what is so special about Singapore ?)
RIVM, the Dutch counterpart of the US CDC, issued a health warning in January. RIVM (2020) advised the Ministry to give alert status A to SARS-CoV-2, which e.g. allows mandatory quarantine. This status was assigned. (https://www.rivm.nl/meldingsplicht-infectieziekten/welke-infectieziekten-zijn-meldingsplichtig) However, when a general practitioner doctor (GP) reports a case, the case is only a “suspected case”, and there is no legal obligation for microbiological testing, and somehow this then isn’t done. Thus, Dutch statistics, with the official death toll of 5748 at the end of May 2020, are based upon hospital admissions. RIVM’s modelers estimated the dynamic reproductive number R[t] as getting below 1 by March 16, but based upon such hospital admissions where such testing is done. At the end of May, CBS Statistics Netherlands reported a total of 9000 “surplus death”, i.e. above the normal level expected from earlier years. Thus some 3000 cases (likely mostly reported) were left untested. Why doesn’t RIVM make sure that all suspected cases are tested and included in the official statistics ? If making sure that statistics are collected is not in their job description, then in whose job description is it ?
Originally there was source and contact tracing by GGD but this was later repealed when the number of cases rose, instead of hiring more people to do the job. In May, the GGD director stated that they would be capable to take up source and contact tracing again. The media and eventually Parliament found it problematic that he mentioned some 2-3 hours per case, and that he intended to use of a letter of warning. Germany uses some 16 hours per case. A letter of warning might be used for a sexually transmitted infection while SARS-CoV-2 requires direct telephone calls.
If Holland had clammed down the infections from January like Taiwan did, and e.g. forbidden Carnival, and quarantined the provinces of Brabant and Limburg where they had the Carnival festivities anyway, then the economy would not have needed to lock down, and 9000 deaths by the end of May could have been prevented. Comparison with Malaysia Airlines MH17 with 298 deaths in 2014 might be seen as somewhat dramatic, but we are speaking about thirty of such disasters, caused by home-grown failure of the protective system.
So, if RIVM is only advisory, what prevented the Dutch Ministry of Health from seeing the notion of a pandemic in proper perspective, even while there had been ample warnings about pandemics before, and while they are the professionals on public health ? If we can understand what is blocking insights at government Departments of Health the world over, then perhaps we can find out what is blocking the insight on preventing pandemics in the first place. Yes, indeed “the world” is a big vague place outside of your own territory, but are you sure that you can really continue to neglect it ?

1.6.6. While there is no vaccine: options for an exit strategy

In all cases, there remains the challenge for the upcoming period till there would be a vaccine.
As an economist, I find it remarkable that the Dutch Prime Minister Mark Rutte (2020) (March 16) spoke about "herd immunity", first as one of the aims of the Dutch partial lockdown but later corrected to only a consequence and by-product of the real policy aim of flattening the curve. A crucial question is: if the government allows the virus to spread in a more natural way amongst the less-vulnerables, as a by-product, then why not use the virus as its own vaccine ? Apparently there is only the problem of responsibility - so to speak "the (legal) blame" - of who causes an infection, especially when it goes wrong: either some "natural cause", or the person self or a vaccinator. This issue of responsibility can better be discussed openly with an evaluation of the options, risks and costs involved. Also with an eye on future pandemics (that we rather ought to prevent though). Obviously, herd immunity is an objective in any vaccination programme, and vaccines differ from the dangerous pathogens: so the properly phrased question is whether this virus for this target subgroup of the less-vulnerables is like such vaccine and thus not the dangerous pathogen as it is for the vulnerables. One cannot avoid the impression that the Dutch government thought so, as it allowed the virus to spread (though asking for self-quarantine).
Another element is: a vaccine test on humans also requires deliberate infection. McKie (2020) reports in May about the Van Bunnik et al. (2020) paper: “UK scientists want to infect volunteers with Covid-19 in race to find vaccine. Trials could be speeded up by using risky strategy of deliberately introducing the virus”. Critical researcher Riley at Edinburgh University: “Challenge studies are done for many diseases but only when strict criteria are followed. Firstly, the virus should be really well studied and its clinical behaviour understood in detail. It should also be incapable of causing severe illness in healthy individuals, or there should be a highly effective drug to clear the infection. None of these criteria are met for Covid-19, and I would be very concerned to hear challenge studies were being planned.” The latter reasoning is within the well-defined world of medical testing and its protocols. It neglects the phenomenon that the lifting of lockdown will cause the virus to spread again anyhow, and that the economy collapses if we do not lift the lockdown.
There are four main scenario’s on the table. All scenario’s rely upon some notion of infection status. Chapter 4 distinguishes actually 11 compartments with own risk profiles, and the compartments are indicated by a colouring scheme. An inspiration was my suggestion of having a “passport” for the status on sexually transmitted infections, see Colignatus (2004). At issue is not whether such compartments might exist, since any scenario uses their existence. At issue is whether it is feasible or even desirable to give an official role for such compartments.
(1) The current policy by the Dutch government: muddle through, wait and see, prepare ICU capacity for the Autumn, apparently especially for the vulnerable compartment that has only a 50% of survival at the ICU (see below “ad 1”).
(2) Lock-on-off. The “Harvard study”, or Kissler et al. (2020). RIVM commented on it favourably, but also called it risky, and it need not be the policy of the Dutch government. Kissler et al. (2020) point to the need to have (recurrent) lock-on-offs to “flatten the curve” and remain within the capacity of the health system. They suggest that over two years some “herd immunity” can arise as a by-product as well. Thus, they allow infections, but shift the blame to “natural causes”. They might also think that it is impossible to shield the vulnerables by quarantine so that there indeed would be a high demand for the ICU. Remarkably, RIVM (2020ef) in May still discuss only the Harvard study but not the two following options that should already have been obvious in February too.
(3) Eradication. There is the test, test, and test approach, and then quarantine anybody who tests positive. Berger, Herkenhoff and Mongey (2020) provide a model-based proof of concept. Tabarrok (2020) supports the cost-effectiveness. Suppression fits the observation by Draulans and Piot (2020) that an infection may cause more damage than thought. It may be that the use of a SI(EY)A(CD) model is required for the proof of concept, but one might also argue that the method would be logical by itself too. The death toll could likely be calculated directly without a model run. In Holland in March, RIVM stated that there was a lack of tests. However, Alex Friedrich, head of microbiology and infection diseases of the University of Groningen medical center (UMCG), states that RIVM made a deliberate decision, and he arranged a testing strategy for Groningen, see 1V EenVandaag (2020) (April 17). RIVM misinformed the public about the options. At the end of May, the government and RIVM are making tests available for the whole country. Tweebeeke & Sietsma (2020) (June 10) report that elderly home care managers are distressed and furious that the much higher available capacity for testing has not been used at a much earlier moment.
(4) Segment and shield. A policy is to distinguish a compartment of vulnerables - say 60+ and those with comorbidity - and a compartment of less vulnerables - the younger others though perhaps still with some hidden risk. Van Bunnik et al. (2020) (May 5) call this “segmenting and shielding”, see also Sample & Mason (2020). The issue is discussed by Colignatus (2020bcd) (March 31) and with a full model by epidemiologists De Vlas & Coffeng (2020) (April 12) at Erasmus MC, while Frijters (2020) (March 21) has much by implication. A non-quantitative discussion is by Eichenberger et al. (2020) (March 25), who also point to the rising frustration by the immune compartment when they would be subjected to a string of lock-on-offs. Criticism is: (a) Sridhar & Rafiei (2020) argue that it is practically impossible to shield the vulnerables. Government policies like in the UK and Holland already promised such shielding but actually only achieved this by the national lockdown. Adamik et al. (2020) alert us that families magnify the number of contacts. The proper question is: Have we done enough to achieve the required shielding ? (b) There is the protest that deliberate infection runs against the rules for vaccination. This is better seen in the light of the Draulans and Piot (2020) observation that it is a nasty virus that can have chronic effects. If there were no nasty effects then the principle on vaccination would actually be adhered to (which is also why the idea originated).
We arrive at the latter scenario without the SI(EY)A(CD) models. Thus, our praise for SI(EY)A(CD) is guarded. We could have arrived at such compartmentalisation also in February, right after the WHO declaration of the PHEIC. A modeler’s question obviously is whether these two sections in the population can be put into the model, and how would they interact ? To some extent it is remarkable and even amazing that RIVM in its reports to Dutch parliament hasn’t much reported on these two structural compartments, and still presents only a single R[t], as if persons from the two compartments could freely mingle, and as if it should not be obvious that there is quite a distinction between the elderly and comorbids who would always have to be quarantined and the others for who contact reduction is less relevant.
Consider the following figures about the Dutch population, latest age-group data of 2019, using the level of 17.4 million of 2020. For SARS-CoV-2 there are 9000 “surplus deaths” including 5748 official deaths, of which 177 below the age of 60, of which none in the ages 9-14. Let us allocate the 3252 = 9000 - 5748 unofficial deaths in proportion. The distinction of the chronically ill and multi-morbid are by Van Oostrom et al. (2011). Multi-morbidity is defined as having at least 2 chronic afflictions. Their age-group specific prevalences are applied to the age-groups of 2019. It is remarkable that the younger vulnerables have such a low SARS-CoV-2 death score, but they do not tend to live in care homes.
Let us identify the vulnerables as the 60+ and the younger chronically ill (which includes the multi-morbid), thus 4.5 + 3.4 ≈ 7.9 million (45%). Then the less-vulnerables are 9.5 million (55%).
Let us assume that the vulnerables are put into quarantine but that 1% of quarantines will be breached, with the group-specific IFF of 5.1% (observed for the 60+), meaning 7.9 million x 1% x 5.1% ≈ 4029 deaths. Potentially, all these deaths have tried the ICU with a 50% chance of survival (but still this net outcome). Regrettably there is no good indication for the IFF of the younger vulnerables.
At issue is not the choice between lockdown versus “let the pandemic run its course”. At issue is a sound (and not only preached) quarantine of the vulnerable and possibly deliberate (self-) infection of the less-vulnerable with sufficient medical backup if something goes wrong (because it remains risky to predict who would be less-vulnerable). This does not yet include the observation by Peter Piot on chronic effects of infection.
The scheme is not without risks, for example when people cross the quarantine barriers. This idea thus requires close discussion. See Van Bunnik et al. (2020) for their version of “segment and shield” and self-infection, see also McKie (2020). The idea to deliberately speed up the path to the limit values provides a way to cope with the economic recession, which otherwise might develop into worse than the Depression of 1930-1941. The world has some luck that the SARS-CoV-2 pandemic hits an identifiable compartment in society. In another situation, when potential victims cannot be identified so easily, it would be much harder to think of a solution approach (other than test, test, and test). The sobering idea is the observation by Peter Piot that we actually may be in such a much harder situation, with a burden of disease that still must be diagnosed.
Above compartmentalisation could have been done in February with the information available at that time. Why were these policy options not formulated in this manner ? The potential answer is: RIVM might be rather a “Bureau of Infection Statistics” and not an “Infection Planning Bureau”. Though there are “the three scenario’s”, also on the RIVM website, it still seems as if they do not have a tradition of making such scenario’s, and if they do not wish to speculate about the number of deaths as if such discussion would make it seem as if they are responsible for those deaths. RIVM advises, they report and trace infections and advise on vaccinations rather than “plan” infections in the sense of advising on organising who gets what. Vaccinations are designed to protect the vaccinator from accusations of wrong-doing. By consequence, at the end of May, Holland has 9000 deaths, but precisely in the category of the vulnerables, the very group that RIVM advised to protect. Risk aversion behaviour caused the risk to actually happen (to others than RIVM itself). And while Holland in May somewhat recovers from the first wave, above scenario with another 9336 deaths is still in the waiting, at least if we want “segment and shield” as an exit strategy for the economic crisis.
We will return to the issue in Chapter 2. In all cases, a discussion of these aspects requires an underlying understanding of epidemics. This translates as a basic understanding of models of epidemics.

1.6.7. Information that is missing now about such scenarios

Information that is missing now, and that we would need for informed decision making, is the following.
Common effect measures are lives extended (“lives saved”) and (quality adjusted) life-years gained. Colignatus (2003) (2020b) proposes the Unit-Sqrt compromise: though mortality and qaly’s might no longer be needed, it still would be useful to calculate them to clarify the compromise. The issue is complicated by the Piot (2020) implication of chronic effects (an argument for eradication and an argument against deliberate infection: but there is uncertainty about the effect size), and we may include the increased awareness of the Post Intensive Care Syndrome (PICS). Such effects do not affect the death count but reduce the qaly’s and increase economic costs. PM. Such effect measures require the extension of the modeling with the hospital sector, not included in the present SI(EY)A(CD) model family. Models with such enlargement are Goh (2020) and Berger et al. (2020) and Acemoglu et al. (2020). Relevant is Bodenstein, Corsetti, Guerrieri (2020), for the distinctions in economic sectors, and that rational policies, including lockdowns, are preferable above chaos. For the present modeling, I give priority on didactics above including such enlargements.
While the alternative scenarios of eradication (assuming that asymptomatic infection allows this) versus “segment and shield” (with quarantine for the vulnerables and cohort-wise deliberate infection for the less vulnerables) seem rather well defined, it is still unclear what the current government policy is (except that it starts to look like the Harvard lock-on-off), and it needs to be defined if we want to be able to compare outcomes.
For the base scenario, the first half of the year 2020 forms a sunk cost. We would be interested in an evaluation but there is no use in crying over spilled milk (except for restructuring the social decision making process). We better start from the present state of the economy and health system, and work from there. Since government policy can be expected to be erratic, we need a stable base scenario. The best selection is a path that avoids global warming and reduces the risks of future pandemics. It would be rather irrational to assume a path towards global warming with new pandemics, and then try to micro-manage the case of 2020 on this path. Indeed, it makes sense to assume a structural change, and then evaluate policies about SARS-CoV-2 in this context. See Colignatus (2020fg) for the Tinbergen & Hueting approach in the economics of National Accounts and ecological survival.
The required information is in the following table. We would gauge the costs not only in sizeable percentages of national income and national debt, but also in terms of unemployment, private debt and bankruptcy, and measures of inequality. We see a strange combination of both epidemiological modeling that reckons in days and macro-economic modeling that concerns the medium term (5 years) and the long term (the year 2100). A new topic is how the standard economic models would need to be adapted to the new circumstances - while it seems that we did not yet finish the discussion on including the financial crisis. PM. Mortality rates are w.r.t. the population (as opposed to the IFF that is w.r.t. the infected). PM. S&S = “segment and shield”.

1.7. Summary of a complex objective

In summary, this notebook has a complex objective:
(a) Redesign of didactics about SI(EY)A(CD) in order to reduce confusion.
(b) Document the latter within the computer algebra environment of Mathematica.
​(c) Middle of the road: we may assume novice readers but do not aspire at an introductory course for novices, and allow for refreshment for modelers.​
​(d) Make this modeling and documentation very accessible for others, also for variants.
(e) Use SARS-CoV-2 parameter estimates and scenarios as examples for such SI(EY)A(CD).
(f) Place the discussion within the literature, with real data and commentary about the quality of the data.
(g) Allow for some creativity along the way, like the use of colour-coding for (quarantine) compartments.
(h) Give the context of above exit strategies of eradication or using the virus as its own vaccine.
(i) Identify what information is missing, which we need when we want to make informed decisions.
(j) Regard the pandemic as only an example of a much more fundamental issue, see Colignatus (2020a).
SI(EY)A(CD) is only a very basic model. It is adequate only for outlining the exit strategies. It allows us to show that the current (partial) lockdown is wise as an emergency brake - for Holland - given the otherwise deadly outbreak, with a rough indication of how many deaths would have occurred otherwise. It allows us to observe that said exit strategies were already feasible in February, so that the lockdown in March was an emergency brake indeed. It also allows us (in a later notebook) to evaluate the (quality adjusted) life-years gained, since we should not focus on the mortality count only.

1.8. Structure of this notebook

The SIA, SICD and SEYCD models are discussed sequentially in Chapters 5-9. Each opens with a first practical chapter on model structure and how to run the software and proceeds with a second chapter on analytical properties (supported by other routines). There is no analytical chapter on SICD since the model merely splits A = C + D.
The SI(EY)A(CD) model family is formulated for infections but is also applied for symptomatic disease. Chapter 10 thus discusses the distinction between the infection generation interval and the symptomatic serial interval. There is no suggestion of new didactics on this, and the discussion here is only provided for completeness, taking advantage of the best presentations that I have seen elsewhere.
Chapter 11 compares with four empirical studies, reproducing their model within the present environment. Those studies tend to use symptomatic disease, for the obvious reasons that patients turn up when they have symptoms and that (lab) testing is not easily arranged. Chapter 11 does actual model runs, while some such comparisons are already mentioned in Chapter 10.
This present notebook is not a developed “course”. There are no student syllabus and teacher’s manual. This notebook serves both such purposes now. The assumption is that new students look at all material and that experienced readers (teachers) might take bigger steps. In some sense, the Introduction chapter is a teacher’s manual. Novice readers will understand little about this Introduction but can return to it after reading the subsequent chapters. PM. This notebook and package target didactics, and do not target education. Still, while writing, I have been increasingly assuming that this notebook might also be used by first year students under guidance of their teachers. Supplementary introductory comments for this have been put in Appendices A, B and C. Appendix A is advisable and B & C may be useful.
Appendix A discusses the basic Success versus Failure model, with exponential decay of Success, and with Failure collecting the fall-out. The model is basic for understanding the relationship between factor and rate, the relation between rate and period = 1/rate, the notion of a half-life, and for the comparison of dynamic events and the exponential probability distribution. Appendix B refers to my background on queueing theory and related packages in The Economics Pack. Epidemiological models with interarrival periods have a conceptual overlap with some queueing theory. The distinction between exposed and infectious phases concerns sequential processing but the distinction between cleared and deceased is not parallel processing but merely proportional allocation. Appendix C refers to other survival analysis packages in The Economics Pack. Appendix D shows that one does not need this notebook and package. For who knows the S(E)IR(D) models and Mathematica, it is simple and straightforward to create such model in Mathematica and run a WhenEvent policy intervention with it. The present notebook and package only provide for some structure, and intended didactics. The present notebook and package also confront the user with a learning curve which might slow down needlessly. Mathematica is a wonderful computer algebra environment but has a learning curve of its own. There is always the issue that when some routine doesn’t (quite) do what one wishes, or doesn’t work for other applications than shown in the example, then it might be easier to adapt or trace errors when you have written the routine yourself. For example, the SI(EY)A(CD) model family has extensions with Birth and background death, that allow for annual dynamics in infection processes (and a real steady state), and likely more relevant notions of herd immunity: this is not implemented here. A user faces the question whether to try to adapt what has been presented here or find alternatives.
These Chapters on SI(EY)A(CD) are preceded by the following three chapters. Chapter 3 is on the Survival`SIA` package, its limitations, and repeats the aims in didactics and application, but now directed at the level of programming. Chapter 4 discusses compartments and quarantine colour coding. There are more compartments in the real world than used in SI(EY)A(CD). For a policy discussion on SARS-CoV-2 it is relevant to distinguish between the more vulnerables and the less vulnerables, but these compartments are not in the models that we look at. Thinking about quarantine caused me to design a colour scheme anyway. This scheme was at least useful for the current plotting of SI(EY)A(CD) graphs.
Finally there is Chapter 2, following next, about the Dutch timeline and some issues of strategy and policy making. The Dutch performance is actually not so well. While the WHO issued its PHEIC on January 30, i.e. announced that there was a pandemic, the month of February was basically not used in Holland. The period of the (partial) lockdown since March 18 has passed much in waiting and without developing clarity about an exit strategy. Is the world going to try to exterminate SARS-CoV-2 or are we going to accept and perhaps even target some (potentially confused notion of) herd immunity or .... ? Such issues of strategy however are not in the SI(EY)A(CD) models themselves. This notebook and package were written partly to clarify that some points ought to be clear at the foundational level. The focus was on didactics and there has been no effort at economic modeling.

1.9. About the author

1.9.1. Caveat

The caveat is that I am an econometrician (Groningen 1982) and teacher of mathematics (Leiden 2008) and no medical doctor or epidemiologist. If SARS-CoV-2 is a virus that humanity cannot live with - like with smallpox that was eradicated in 1980, see WHO GPEI & Heymann (2020) - for example because the virus has risky properties also for future mutations, then the current low prevalence might be a good start for eradication via surveillance and quarantine for the infected. This might come at the economic price of national lockdown and the burden of disease because of the deterioration of common health care. It might be unwise to target for herd immunity, even though such might be seen as part of eradication. It depends upon the properties whether stable immunity can be created indeed. Corona viruses of the “common cold” tend to be forgotten by the immune system. At first I did not think that containment / suppression / eradication was really a feasible option - Taiwan would remain an island - but the Piot (2020) warning and the Berger et al. (2020) paper and Tabarrok (2020) commentary caused me to change views. For now, it is best to present and discuss scenario’s, and subsequently enquire for medical views.

1.9.2. Work related to medicine

In 2002-2004 I collaborated at Erasmus Medical Center on the modeling of the Human Papilloma Virus (HPV) as the cause of cervical cancer. I gave a presentation on MISCAN modeling at a gynaecology conference in Paris and had a poster at the 4th IHEA world congress at San Francisco on public health economics. My background in modeling and also logistics was relevant because diseases may look like a Markov logistics process with stages and transition probabilities. There can be the same issues of test reliability, criteria of lives-saved or life-years-gained, and cost-effectiveness of screening and treatment. I also followed the discussion about the SARS epidemic of 2003. My period at Erasmus MC was too short to allow for publishing peer reviewed papers but let me mention two working papers of that period, and actually another one of 2019, all with software that eventually was included in The Economics Pack. Applications of Mathematica, see Colignatus (1995, 2020e). Currently I am in a phase towards retirement, and consider the purpose of the Pack as shifting towards didactics and education.
(1) Colignatus (2003): On the value of life. This compares the lives-saved and life-years-gained measures, and develops a compromise: a “unit-square-root” measure, that regards each life as 100% and takes the square root of the relative gain. It compares somewhat with the “fair innings” notion but I rather tend to regard it as the “fair outings”. The relation to SARS-CoV-2 is discussed by Colignatus (2020b). Discussion of the value of life meets with the problem of the “cricket and the ant”. If there would be an ideal scoring measure for ants, then what to do with a cricket that has been living irresponsibly and in distress wants to be treated like an ant ? The “unit-square-root” method would apply for a general class of deserving persons, consisting of generally responsible behaviour, a grace period up to 25 years where youngsters might need to learn from accidents, and the occasional insurable glitches for ages 25+. I have no developed ideas for chronic irresponsible behaviour of the 25+ ages who burden the public health system. However, the latter discussion should not be confused with the earlier topic of the valuing normal decent lives (e.g. the lives of doctors).
(2) Colignatus (2004): Modifying behaviour with a passport for sexually transmitted infections (STI). At that time there was no HPV-vaccine yet. An option was to manage human behaviour. The status of infection can be recorded in the medical dossier: free (green) or carrier (red). While children can gets warts, an assumption might be that children start out uninfected by the harmful HPV variants (status green). When adults meet and want to get into a serious relationship – in the sense of sharing their germs – then they can show each other their status of infection in their medical dossier and discuss the implications. From this working paper, we may take the idea of recording the status of infection, and using colour coding for clear communication. For SARS-CoV-2, it is better to use “red” (alarm, or hungry in Chinese restaurants) for the barrier between zones and groups. This is discussed by Colignatus (2020c). PM. Due to the national lockdowns, we may also see a reduction in STIs, and one may wonder whether behaviour oriented policies should not have been more prominent in the past (catching HIV before it hibernates).
(3) Colignatus (2019): Population size and climate change. Encourage women to delay having their first child till they are age 30+, in order to provide a crucial contribution in fighting climate change, especially in the crucial decade 2020-2030. Colignatus (2020a) discusses the gap between demographers and researchers on climate change.

2. Timeline and assumptions of the Dutch case

2.1. Introduction

This Chapter assumes that the reader has knowledge about the SI(EY)A(CD) models. Readers new to the subject are advised to reserve this chapter for later. This especially holds if you already had problems with the Introduction.

2.2. Properties of SARS-CoV-2 and some support for the Dutch policy statement of 2020-03-16

SARS-CoV-2 can be deadly for (i) younger persons with comorbidity and (ii) the elderly (60+) (often with comorbidity). This group can be called “the vulnerable”.
For younger and healthier persons, SARS-CoV-2 may compare to the common cold - though this issue is not clear-cut, see section 1.6.3 that refers to Draulans & Piot (2020) with the warning about chronic effects. With respect to short term fatality, this group can still be called “the less vulnerable”. There is always a risk that such person is misclassified and actually appears to be vulnerable.
In an address to the nation of March 16, Dutch prime-minister Mark Rutte (2020) formulated the policy:
(1) Protect the vulnerable
(2) Remain within the capacity of the health system, and in particular the intensive care units (ICU)
(3) Allow the less vulnerable to build up herd immunity.
At first it was presented as if herd immunity was a policy objective. In discussion (also) with Dutch parliament Rutte clarified that the 3rd aspect was only a consequence of the first two aspects.
Ever more measures were imposed, short of a total lockdown of Holland that would require permits for leaving one’s home. Eventually, reports were focusing on the hope that the basic reproduction factor was getting below 1. However, in sum, containment / suppression hasn’t been an longer term policy objective (yet).
In the press conference on April 21 2020, the policy was (re-) formulated:
(4) Wait and see, collect data and policy options, because we are still much in the dark
(5) Develop instruments for better surveillance, like an “app”, for a potential long term policy objective of containment and suppression
(6) Restart the health system on normal care, which had deteriorated by the focus on the pandemic
(7) Release a few restrictions, like allowing young children to go to school and participate in sports.
Official Dutch policy is much guided by the lack of good data (see below), but it also seems that the policy options are not fully developed yet. The ship sails between the cliffs of a deadly outbreak and economic collapse (with additional collapse of the health system). This situation is not different at the beginning of June.
The following discussion basically supports points (1) - (3) above, and allows for the notion of deliberately using the virus as its own vaccine, namely for the less vulnerable group, while still remaining within the confines of the health system (ICU) and protecting the vulnerable. The issue is wide open however, given the nasty properties of the virus that are asking for eradication.

2.3. Relating to the RIVM data and parameters (1)

Footnote: This first date seems a result of later back-tracing. It was not widely published at the time (did not stop Carnival).
RIVM presents the following graphs as input and result from their model on R[t]. Their approach differs from our present simple models. The inference that R[t] dropped below 1 on March 16 can only hold for the official death counts; additional CBS data show a spike of some 3000 additional “surplus deaths” in April. Many patients, supported by their families and GPs, decided to no longer report to the hospital, since they were afraid of the prospect of dying alone and unconsciously in an ICU bed. It is fully unclear what this means for the dynamics of the process and the estimate. PM. See section 5.2 on the terminology of the effective R[t].
PM. Documentation of the RIVM modeling: https://www.rivm.nl/en/novel-coronavirus-covid-19/calculations/calculation-models-are-public-and-accessible

2.4.1. Choice of latency α

Anderson et al. (2020), Supplement 3, suggest a value of 1 day for the latency of the Exposed. There are not really competing alternatives. Thus α = 1 / (1 Day) = 1 too.

2.4.3. Choice of γ

For this notebook we want to have a value of the infectious period 1/γ such that the natural process of clearing has been observed without confounding and intervention, so that we can subsequently insert the assumption that this clearing period would also be the (natural) infectious period.
Kretzschmar et al. (2020) (May 15) take an infectious period of 1/γ = 10 days, though with variable infectiousness, which doesn’t quite fit the assumptions of a constant rate in SI(EY)A(CD). But a constant rate can be seen as a special case of a variable rate.
For SIA, we have the following equations for the growth rate r = β - γ:
For SEYCD we have the following equations for the growth rate, for us with α = 1:
For SIA:
For SEYCD with α = 1:
Rocklöv et al. (2020) also take an incubation period as the parameter for their “exposed” category, suggesting that their “SEIR” model does not concern infections but symptomatic disease. Their α = 1 / (5 days) = 0.2. For infections, we took α = 1.
We can agree that 1/γ = 10 is a conservative assumption w.r.t. the (longer SIA) estimates by Li et al. (2020) and Joseph Wu et al. (2020), in the sense of a faster indication of the risk. This gives us the γ that we are looking for.
PM 1. The distinction between infection and symptom will be discussed in Chapter 10. Let us first develop the models with a focus on infection only, so that we are in a better position to understand the complication of symptoms (especially for parameter estimation).
PM 2. SIA, SICD and SI(EY)CD assume that the mean clearing period is also the mean infectious period. Therefor the naming and value-assignment of 1 / γ causes some difficulty. A clearing period is biologically given and cannot be easily changed. It is a property of the person rather than the process of infection, that can involve (self-) quarantine. A choice of a mean clearing period of 10 days seems acceptable for a younger population, if the virus would only circulate there. The period would be longer for elderly persons who do not cure as fast, but they would get sick and soon be less in contact with others. Overall, the actual clearing is not relevant for us, since we do not look (yet) at the further process towards (self-) isolation or hospitals. For the present modeling exercise, it suffices that the period of infectiousness has stopped after the mean value of 10 days.
PM 3. On May 15 2020, the European CDC reports for the general public: “The infectious period may begin one to two days before symptoms appear, but people are likely most infectious during the symptomatic period, even if symptoms are mild and very non-specific. The infectious period is now estimated to last for 7-12 days in moderate cases and up to two weeks on average in severe cases.” https://www.ecdc.europa.eu/en/covid-19/questions-answers. However, they do not provide a reference. It may well be that their “infectious period” doesn’t quite apply to the kind of modeling that we intend. Potentially they refer to the 99% confidence value required to keep people in quarantine, while the SI(EY)A(CD) model requires a mean value.

2.4.4. Consequence for β

NB 1. There is a difference between the internationally applied “full lockdown” (people may no longer leave their home, except for one person who gets the necessary supplies, and the personal distance is 2 meters) and the Dutch “partial lockdown” (in which people might on occasion go for a stroll but keep a distance of 1.5 meters).
NB 2. The Adamik et al. (2020) study of the MOCOS groups is important. They warn that the number of contacts should rather be based upon households, since household members infect each other. They do a micro-simulation rather than this SI(EY)A(CD) model. PM. They also nicely phrase the dichotomy between “infect nobody” and “infect towards herd immunity”: “Mitigation of a novel infectious disease with the aim to reach herd immunity is a classical textbook concept in epidemiology and has been successfully applied in the past, foremost in the case of novel influenza strains (...). The idea is simple: in the absence of a vaccination for a novel infectious disease one tries to flatten the incidence curve to such an extent that the daily number of cases that require medical assistance is kept below the capacity of the health care system. The long term goals are to obtain a sufficiently large fraction of the population that has become infected and to reach herd immunity which would lead to a less severe or even subcritical second outbreak wave. On the other hand, an extinction strategy would aim at introducing sufficient contact reductions to keep the epidemic subcritical and not lifting these restrictions until the disease becomes extinct.”
The first plot below gives the contour lines of the acquitted A as a function of the basic reproduction factor. See a discussion in Section 6.10 below. Observe that the 50% level (62 days) does not have to be in the legend, and that there cannot be a number of days for the limit value because this would be infinite. (Potentially, though, we might say that the infection is exterminated when the value of the infected units drops below 1 or 0.5 (rounding error), but this does not work well for our interpolations.)
(PM. The plot uses a the current variables. In this case, first a SI(EY)CD model with a latency of 1 day was run, which means a delay of about 14 days w.r.t. the SIA(CD) without latency. The model has I = E + Y = Exposed + Infectious, so that the variable I of infected can directly be compared to the I in SIA(CD) (with E = 0).)

2.5. Choice of Infection Fatality Factor (φ = IFF)

Verity et al. (2020) find an Infection Fatality Factor (IFF) for China of 0.66%. Ferguson et al. (2020) adapt to the UK population profile and find 0.9%. Applying their age-group specific fatality factors (that they call a “ratio”, their Table 1) to the Dutch age composition causes a Dutch infection fatality factor of 1.27%. Holland has on average an older society.
However, we will look at this in section 2.7 in more detail. A summary is: The Dutch data are quite unreliable, with underreporting, lack of testing, testing focused in hospital workers, changes in practices (whom to send to a hospital), and after March 15-18 the national intervention. However, the age-group of 70-79 seems most reliable, since they tend to still participate in society but do not work at hospitals. For this age-group, Ferguson et al. have an infection fatality factor of 5.1% (that they call a “ratio”). When we apply this to the Dutch deaths in this age-group (certified for SARS-CoV-2 with the current reporting practices) then the prevalence of infections for them (on some n days before April 2) was 0.535%. When we assume this to be the national infection prevalence, then we get an age profile of infections and deaths that is not unreasonable, see Colignatus (2020cd), with also adoption of the Ferguson IFF for the 80+. In that case, the Dutch IFF is 1.45% instead of 1.27%. This is considerably higher than the UK 0.9%, but Holland is apparently not only an older but also a more densely populated country after all. This translates into a SI(EY)CD model input parameter, rounded to φ = 0.015 = PhiSICD[] = IFF, or the Infection Fatality Rate IFR = 0.015 * γ = 0.0015.

2.6. A timeline with scores from a SEYCD model without intervention

With these parameters for the standard SEYCD model, an unhindered pandemic in Holland would have given the following path, i.e. neglecting a change in parameters due to the intervention on March 12-15 2020, with an assumed delay of 3 days, thus March 18.
The death toll could have been at 250,000, with 95% reached by May 22, and the remaining 5% spreading over the remainder of the year. Only 2% of the population would not have been infected at one stage.
Table legend: S = Susceptibles (uninfected), I = Infected (exposed or infectious), D = Deceased, C = Cleared, N = N[0] - D remaining population, A = C + D (the acquitted). Percentages (p) are w.r.t. the population at the time of onset, i.e. N[0].
The levels (q) of the relevant compartments are as follows.

2.7. Dutch data are unreliable, but the 70-79 age group might be stable

While epidemiologists and these modelers emphasize the uncertainty in the data, they are also the group whom we expect to provide us with estimates anyhow, whatever those uncertainties. Thus, paradoxically, we may be so focused on finally receiving some reliable information that we might perhaps not listen carefully enough to them about their warnings about these uncertainties. Some statisticians, who are (now) at some distance of involved modeling, can be more candid, see Richardson & Spiegelhalter (2020) (Royal Statistical Society) and Gill (2020) (emeritus Leiden) for example. It remains useful to identify some uncertainties.
The Dutch data are unreliable because:
(1) Tests were hardly available, and symptomatic cases were not trusted because of a partly concurrent flu episode. There was some surveillance in the beginning, but this was stopped because of the lack of tests, and the authorities decided to “monitor hospital admissions and the death counts”, assuming the connection between the infections, symptoms and deaths, but relying on people reporting to hospitals and apparently underestimating those who did not report to hospitals (the “surplus deaths”).
(2) Available tests were restricted to health system workers and hospital patients. The ratio "hospitalised / (confirmed infections)" thus contains confirmed infections amongst (younger) health workers.
(3) As indicated, overall deaths, also from non-hospitalised cases, also including normal deaths, were not tested. Later there appeared to exist sizeable "surplus deaths". The mixture of data components changed, as normal hospital services stagnated, and the national lockdown changed the economic process.
(4) In the beginning, GP doctors were unfamiliar with the disease, and were more inclined to send patients to the hospital, when they considered that the symptoms were more severe than a flu. Over time, when it became clear that elderly patients would suffer longer periods of isolation at the ICU, and with a lower survival probability, patients themselves, their families and GPs became more reluctant on hospitalisation. Holland has a somewhat stronger notion on the quality of life and the acceptance of death when life reaches its end. However, the change in attitude within this episode still makes for mixed data. When hospitals and ICUs reported lower arrivals, then this does not mean by itself that the infections were waning. The infected vulnerables have become more reluctant.
(5) There have been reports about outbreaks in home-care for the elderly. Personal protection equipment (PPE) and tests for care workers there were hardly available. Those patients might no longer be hospitalised because of (4).
(6) New tests are arriving and thus more infections will be reported merely because of this availability. There is also a planned uptake on surveillance now that this wave seems to be ending (May 20).
(7) Obviously, there is the distinction between pre-lockdown, the lockdown phase, and the now partial release. Modeling these phases becomes an even more complex issue.
My impression is that the data on the 70-79 age group are the most reliable. They still are relatively healthy and partake in society, they do not work in the health care system, and they are still in the phase that hospitalisation is regarded as a better option than accepting the end. Using their data as the “canary in the mine” indeed gives a profile also for the other age groups that looks reasonable and comparable with international data. See Colignatus (2020cd) for the argument and calculation.
Let us reproduce some aspects. Colignatus (2020c) states about Ferguson et al. (2020) (their Table 1) and these Dutch data of April 2:
"The Imperial College estimates give problems for the Dutch data. With 121 deaths in the Dutch 60-69 age group, the London age-specific IFF gives 5500 infected in the population while their "symptomatic cases per hospitalised" gives 7663 symptomatic cases in the population, which is too much since we are assuming that the flu season is over. Holland has 29 hospitalised children of age 0-9, and the London symptoms / hospital ratio for this group gives 29000 symptomatic children in the Dutch population, which would create panic if true. Looking the issue over, I cannot find a match. It must be remarked that the Dutch "reported number of cases" is rather useless, because of the lack of tests, and their preferred application to medical personel rather than patients. Also the death count is understated since non-hospitalised deaths are not tested. See Table 1 below."
“However, the Dutch 70-79 age group may be used as canary in the mine. The number 2951 of “reported cases” will be accurate for this group, since they do not belong to medical personel. These patients will have some symptoms (like “feeling really sick”) and not be tested for nought. The reported number of 2951 means only 0.19% of the whole age group. The Imperial College IFF for this group gives an estimate that 8137 would be infected, or a share of 0.005346 or 0.5%. We arrive at the problem that we are not in the steady state. Either these elderly “infected but non-patients-yet” have a stronger immune system or they are due to arrive at the hospital at a later moment. With lack of other information, we can still presume that this is the overall prevalence of infection (haves and have-beens) in Holland. When we apply this prevalence to the whole population, then we get age-group specific ratios of hospitalisation and IFF that show the same pattern as in China and the London research group. Especially relevant is the “hospitalised per infected ratio” (H/I). See Table 2 below. NB. This uses IFR and CFR, namely as “rates” while it actually are factors IFF and sCFF (symptomatic case fatality factor).”
Colignatus (2020d) performs the same analysis, now with data of April 19. The table 2 at that moment is as follows. Given the IFF of 0.051 for the age-group of 70-79, the implied infection prevalence of the 70-79 group is 20824 / 1522110 = 0.0136807, and this is applied to the groups younger than 70 years. The pattern of the age group specific IFFs is acceptable. The overall IFF of 0.0145 has been rounded for this notebook and package as 0.015.

2.8. Timeline of the Dutch case, to determine onset and when β changed

Reference to the time line on wikipedia (a portal and no source) may be made.
The WHO confirmed on January 12 2020 that there was a novel virus in Wuhan. The European Centre for Disease Prevention and Control (ECDC) judged the risk of introduction and spread in Europe as low. They might have been little aware that Northern Italy has workshops with cheap Chinese immigrant workers who produce luxury “Made in Italy” products (while Southern Italy has unemployment). It later appeared that Lombardy had had privatisations in the health system that caused a slow uptake of alarm.
Dutch RIVM (2020) discussed the situation on January 27, and formulated a strict "case definition": (i) symptoms, and either from an infected area or having been in contact with a certified infected person, (ii) neglect others. Cases falling under this definition however had to reported with priority A.
On January 30, WHO Director-General, Tedros Adhanom, declared the PHEIC or pandemic. Holland was rather lacklustre in responding. While January was lost to the whole world, February was also lost to Holland, see Hendrickx & Modderkolk (2020). China collected medical supplies from all over the world, and the Dutch emptied their stocks and sent an airplane load to China on February 10, which effectively meant that those medical supplies and personal protection equipments (PPE) were in short supply when Holland needed them later. The February 21 conference at the Dutch royal academy of sciences KNAW (2020) had presentations that were correct on scientific content - they indicated the general lack of knowledge about the virus - but they also showed the failure by the RIVM agency for public health: namely, that while Aura Timen (RIVM) mentions the PHEIC, RIVM apparently neglects the cause for alarm given by the Asian data. Later the Leiden expert on Korea, professor Remco Breuker (2020), suggests that RIVM might be an expert on epidemiology but no expert on Asia. Another problem of course lies in communication. If RIVM had advised a lockdown on February 1, the Dutch population would not have understood. It is advisable to first have a national training on the situation, and broadcast reports about the situation in other countries, to generate public understanding for the WHO declaration of the pandemic. Within 10 days (before the plane leaves).
The Northern part of the country benefitted from an earlier Winter holiday and people returning from ski holidays earlier. On February 23-25 there was carnival in Southern Holland, with people participating who had returned from ski holidays and Northern Italy. For Holland, the first infection was confirmed for February 27, for a man having been in Italy. (RIVM states February 10 as the date of first recorded infection, but this seems a result of later back-tracing and was not widely published at the time.) The first official death was reported on March 6. According to the model run shown above, that day actually had six deaths in total: according to the model five were not officially recorded as SARS-CoV-2 cases. The policy of surveillance of individual cases soon broke down, since the available tests were urgently required within the health care system itself, to protect health care workers.
The government then opted for herd immunity, which was supported by RIVM (Jaap van Dissel), see Tweebeeke & Van Dissel (2020). Prime Minister Mark Rutte warned about the economy, and that parents would not be able to go to work when their children could not go to school. Keulemans (2020) is an interview with Jaap van Dissel (RIVM) published on March 12. Van Dissel indicates that RIVM has only limited responsibility, namely advising about measures and not having responsibility in managing them, and that RIVM did not protest against limited measures for the South only - apparently allowing for a spread to other parts of the country. A quote: “If you would lock down the country then the economy stops.” This neglects partial solutions, ranging from quarantining Southern Holland to the whole range mentioned by RIVM at the end of May. Van Dissel showed little concern about the death counts in other nations and likely implied by RIVM models, and he has tended to in general to reject such modeling projections. Later that day, March 12, the government however extended the measures for the entire country. It was only on Sunday March 15 that cafes and restaurants and schools were closed, after strong advice by the Dutch federation of medical specialists (FMS), but not so advised by RIVM.
Mark Rutte (2020) is the prime-minister's speech of March 16. He indicates three options: (1) free reign and overburdening the health care system, with many deaths as a result, (2) total lockdown, with a permit system for leaving one’s home, for the coming two years, effectively closing down the economy, (3) his choice: (a) to protect the vulnerable, (b) to remain within the capacity of the health care system, (c) to allow the less vulnerable to build up herd immunity. The third option (3abc) became government policy. It is exactly the policy suggested in the original version of this present notebook, but this notebook is much stricter on quarantine barriers and deliberate use of the virus as its own vaccine. Since March, it has appeared that the virus is nastier than I thought, see Collins (2020) and De Visser (2020) and Draulans & Piot (2020). Herd immunity may be the wrong tactic, and eradication a better one.
Tweebeeke & Van Dissel (2020) is the interview that discusses the prime-minister's speech. It shows RIVM's support for the government policy towards herd immunity. Remarkably, Van Dissel suggests that children would have no role in spreading viruses, and that this would neither show from the Asian data. His statement is in contradiction with views by epidemiologists if not everyday experience that children / schools have a key role in spreading viruses. He however states that the role of children in spreading SARS-CoV-2 is being investigated now. Later in May, there is the report by German researchers, Jones et al. (2020) (including Christian Drosten).
Over the course of April it appeared that the protection of the vulnerable wasn’t well maintained in the home-care for the elderly. Personal protection equipment (PPE) supplies were directed towards hospitals and ICUs. They were not supplied to home-care institutes and workers. This is curious, since it is the system of care that sends patients to the ICU. There are reports about a distinction between “medicine” and “care”, and a pecking order in social status, also in attention by policy makers.
RIVM stated that there was a lack of tests. However, Alex Friedrich, head of microbiology and infection diseases of the University of Groningen medical center (UMCG), states that RIVM made a deliberate decision, and he arranged a testing strategy for Groningen, see 1V EenVandaag (2020) (April 17). At the end of May, RIVM is making tests available for the whole country. RIVM misinformed the public about the options for testing.
The number of SARS-CoV-2 confirmed deaths on April 2 was 1339 and on April 18 was 3600, and this rise predominantly was in the 60+ age group, apparently not-well-protected, even though it was the government-stated policy target. These are the official numbers only.
Also, there is a rise in general mortality, higher than what would be "normal" in this period like in the past, the “surplus deaths”. This rise in mortality includes persons who have not been tested on the virus, because they were not hospitalised. RIVM has estimated that this rise is twice as high as the officially recorded number of SARS-CoV-2 deaths. Thus the April 2-18 period may have seen a rise of some 4000 deaths, predominantly in the 60+ age with comorbidity group.
PM. With a low prevalence, and regional heterogeneity, testing the general population is costly. However, testing the deaths (the regular average of 12,000 per month) comes with a higher prevalence, and thus is relatively cheap, and can be targeted on indications by GPs.
On April 21, the prime-minister announced that the government will kick the can further down the road, with another reprise at the end of May. However, children can go to elementary school again at the end of current holidays, on May 11. RIVM suggests that they tested that children were infected via adults and not via other children. (The finding is against common sense and other epidemiological reports. One hopes that RIVM did not test only during this quarantine phase.) A personal report is that traffic intensity rose significantly in the days after April 21, which suggests that many people regard the April 21 press statement as an indication of a reduced need for quarantine. However, one might assume that people may be aware of their personal risk, and behave accordingly, though with greater likelihood with neglect of the risk of others.
Blok & Stronks (2020) (May 6) of an action group for containment, protest against the Dutch policy of mitigation, that allows gradual spreading of the virus, and the path towards release of the lockdown, with the prospect of lock-on-off cycles.
On June 3, the bloodbank Sanguin reports, based upon blood samples amongst blood donors (age 18-75), that 5-6% of the population has antigens. This outcome must be biased, and RIVM itself combines such finding with its own Pienter project.
RIVM views and the latter research are now contested by Willem Engel (https://viruswaanzin.nl), but in dubious fashion, see Van Erp (2020). Why doesn’t Engel publish a paper on the issue ?
PM. Above timeline gives events that are specific for the development of the infection itself, dating from the presumed onset at the carnival of February 23, and assuming unmitigated spreading. There are some other points in time that are relevant to mention as well. In the “didactic estimation” below we will only take account of the intervention on day 24.
(a) Delay in response, from earliest indication to PHEIC to the closing of Dutch schools: this period remains uninformative since we do not have other data about the role of schools.
(b) Assume 100 infections for carnival February 23 and unmitigated spreading. The first official death was recorded on March 6. There is a break in the β[t] by the closing of schools on March 15 with an overall delay of say 3 days for societal implementation.
(c) The press statement of April 21 that some people interprete as a loosening of quarantine. Elementary schools start up again from May 11, though modestly, like with half-classes per day.

2.9. Relating to the RIVM data and parameters (2)

2.9.1. Introduction

RIVM calculation models are public and accessible, see: https://www.rivm.nl/en/novel-coronavirus-covid-19/calculations/calculation-models-are-public-and-accessible
The above discussed the situation in March 2020, when we selected the parameters for this didactic version of the SI(EY)A(CD) model. Subsequently, new data have come available, and RIVM, the Dutch CDC, provided evidence to Dutch Parliament also on new modeling. Above, we already compared these findings with the parameter values in this notebook and package. In this section, we summarily discuss some reports.
There are a RIVM report from January 27 and subsequently slides from March to May with presentations at Dutch Parliament by RIVM head of the department for control of infectious diseases, professor Jaap van Dissel. We will not discuss all of these. These reports have different objectives than merely reporting on modeling but we can trace some aspects on this. The slides must be read in the context of listening to the spoken presentation. I still find it remarkable that the slides do not provide clear definitions: they could be included for reference without the need to read them aloud. Below I try to determine what some figures or graphs may mean, and mainly arrive at questions. I wonder whether the Members of Parliament would have surplus information.
RIVM & the Outbreak Management Team (OMT) (2020) (May 6 online) gives the Dutch view on asymptomatic transmission. Our own Chapter 10 below provides two common definitions. (i) The “serial interval” for symptomatic disease. (ii) The “generation interval” for infections. This was also discussed in the Introduction, subsection 1.4.

2.9.2. RIVM & the Outbreak Management Team (OMT) (2020) on source and contact tracing

In conventional language, “contact” means any type of contact. For epidemic “source and contact tracing” for a particular subject (“index”), the term “contact” is redefined, and there is the terminological difference between “source” (subject and unknown infector) and “contact” (subject infecting sinks). (Dutch weblink: https://lci.rivm.nl/COVID-19-bco.) There is also a difference between asymptomatic transmission (the infector is asymptomatic) and the incubation period (the infectee is asymptomatic).
(1) Incubation is generally taken as 6 days. In order to trace the infector of the subject, i.e. “source tracing”, contacts (in the colloquial sense though) in the six days before the onset of symptoms must be traced - or actually till the source has been found.
(2) RIVM & OMT (2020) discuss asymptomatic transmission and the consequences for “contact tracing” (i.e. that the subject can infect others). They arrive at the same conclusion of the European CDC (ECDC). The asymptomatic period can be 1-3 days. For surveillance it would suffice to back-trace contacts (recipients) to 2 days before onset of symptoms with the subject.
("Dit betekent dat voor effectieve bron- en contactopsporing (met of zonder app) contacten tot 1-3 dagen voor de eerste ziektedag moeten worden gerapporteerd en opgespoord. Het ECDC neemt hiervoor in haar rapport over contactonderzoek twee dagen voor de eerste ziektedag (ECDC 2020). Voor de Nederlandse praktijk sluiten wij aan bij het advies van de ECDC. ") https://lci.rivm.nl/covid-19/bijlage/onderbouwing-a-pre-vroegsymptomatische-transmissie
The reasoning is unclear.
(2a) If it is acknowledged that asymptomatic transmission can occur 1-3 days earlier than why stop at 2 ? RIVM & OMT only refer to the ECDC guideline and apparently do not question it ? (If it is a matter of costs, specify those: what is the price of eradication ?)
(2b) RIVM refers to model studies, cluster studies, and cross-sectional studies. Only the first two could in theory support a 1-3 period of asymptomatic infectiousness on content. The cross-sectional studies indicate asymptomatic infectedness but this differs from infectiousness. RIVM & OMT (2020) notably refer to Kimball et al. (2020). Their key finding is: “Among 23 (30%) residents with positive test results, 10 (43%) had symptoms on the date of testing, and 13 (57%) were asymptomatic. Seven days after testing, 10 of these 13 previously asymptomatic residents had developed symptoms and were recategorized as presymptomatic at the time of testing.” A key conclusion is that 3 / 23 = 13% remains asymptomatic. Supposedly the infection clears after a while. Nevertheless, the relevant issue is whether they were also infectious. This, however, was not looked at in this study. Perhaps the viral load might say something about this, but this is unclear to me. However, for the issue of contact tracing, the safest assumption is that these 13% asymptomatic infected persons would also be infectious. Thus, if contact tracing hits upon an asymptomatic infected person, then contacts over the full infectious period of this person must be traced, in our assumptions 10 days (though perhaps the growth and waning of the viral load might give an indication how long this person has been infectious). This notion is relevant precisely because we have seen that new outbreaks can occur via asymptomatic carriers. It is unclear why RIVM has no remark on this.

2.9.3. RIVM January 27

At the aggregate level, the generation interval is also the transmission period, which, in the assumptions of this notebook, consists of an exposed period of 1 day and an infectious period of 10 days, thus a total of 11 days. The value stated by RIVM also fits ranges mentioned in the literature but does not fit our choice of parameters. There is no reason to assume that the Dutch in February and the beginning of March were inclined to self-quarantine, as much as the Chinese people were at the time of some of their reports. Also, later an incubation of 6 days is mentioned: RIVM apparently allows that the incubation period is longer than the generation interval. Perhaps Chapter 10 should drop the assumption that for the aggregate the transmission interval = generation interval = incubation + infectious period = serial interval ? It may just be that we are using a SI(EY)A(CD) model while RIVM is not. RIVM later reports a "generation time" of 3-5 days, and their website later in the year refers to the Ganyani et al. (2020) estimate in which researchers at RIVM partook: this study is discussed in our section 10.6. A key comment by Ganyani et al. (2020) is: "This means that our estimates do not necessarily reflect the natural epidemiology of COVID-19, but instead reflect what is observed in the presence of these intervention measures." Instead, for the present notebook, we want to have the "natural epidemiology" first, before we consider an intervention. Ganyani et al. (2020) mention that the interventions cause a larger number of observations of negative serial intervals for individual cases, i.e. that the infectee develops symptoms before the infector does. This is okay for individual cases. However, Ganyani et al. (2020) (table 1 and 2) also have the aggregate effect that the generation interval and the serial interval have the same duration. This does not take away the large difference between 5 (RIVM) and 11 (us), and is suggestive of a distinction between “first infection” (starting a generation) versus “generation mean”.

2.9.4. RIVM March 25

We assume the exposed period as 1 / α = 1 day. Thus, the RIVM “generation time” with value 4-5 days differs even more from the “generation interval” in our assumptions.
With an incubation of 6 days, we have 5 = 6 - 1 days of asymptomatic infectiousness. RIVM would have a symptomatic period of 12.1 - 6 = 6.1 days. Since RIVM did not provide definitions or discussion, it is unclear where they stand.
(2) RIVM concentrates on “flattening the curve”, to remain within the capacity of the ICU, even to the effect that they do not provide any indication of the potential number of deaths. Basically, RIVM avoids any discussion about the effect measures of “lives saved” and (quality adjusted) “life years gained”. Policy makers, the “Outbreak Management Team”, the medical world and RIVM have converged on the idea that medical conditions of patients imply treatment at an ICU so that this capacity must be respected. The government and RIVM had presented the “three scenario’s” but in qualitative format, and there is no effort at developing more options, in also quantitative manner.
RIVM reproduces the Chinese CDC summary (Wu & McGoogan (2020) in JAMA), keeps the numbers and changes the diagnoses: 81% “mild” becomes “little affected or without symptoms”, 14% “severe” becomes “seriously sick”, and 5% “critical” becomes “fatal”. While the Wu & McGoogan (2020) write “critical” and also explicitly state a case fatality of 2.3%, the RIVM interpretation is bizarre. Potentially, though, RIVM finds the CDC estimate unreliable, with its selection of mostly symptomatic patients reporting at hospitals. It is better to mention this.
RIVM oppose this Chinese "pyramid" with a finding at hospitals in Breda and Tilburg of 97% little affected and 3% seriously sick, and apparently no deaths. We can wonder why this is so relevant because in the next slide RIVM mentions 276 deceased.

2.9.5. RIVM April 8

(2) Remarkably, the first slide still has the Breda and Tilburg message that there would be no deaths, and a few slides later that there are 2101 deceased, and again a few slides later there is the report of the “surplus deaths”. The first slide has become somewhat like a “logo”.
(3) Again RIVM concentrates on “flattening the curve”, to remain within the capacity of the ICU, but they do not provide any indication of the potential number of deaths.
(4) Slide 17 shows the effective R, in which the first day of symptomatic disease is traced back from the Osiris database, apparently February 27, which allows us to infer that the GP-data are not in this database. The effective R got lower than 1 on March 23. Slide 18 shows the measures that were gradually taken in the period of February 27 to March 16.
The number of implied deceased, using IFF = 1.5%:

2.9.6. RIVM April 22

(1) RIVM & Van Dissel (2020c), April 22, still have the first “logo” slide, and correct with a later slide with 3916 deaths. Osiris mentions 34 thousand patients who tested positive on SARS-CoV-2, of which 33% are care workers, given the lack of tests and the policy that only patients and care workers are tested.
Keep in mind that Infection = Exposed + Asymptomatic + Symptomatic, but that we might have to make a distinction between “merely symptomatic” and “disease that might cause hospitalisation”.
(3) Slide 16 has the claim that the effective reproduction factor came below 1 on March 16, i.e. the first Monday when schools were closed. Apparently RIVM adapted the estimate. The estimate refers to Wallinga & Lipsitch (2006).
Contact tracing apparently allows a chart of “what age group infects what age group” (slide 20).
RIVM reports about a study done on infectious spread under and by children, but this study was done in April after schools had closed, and RIVM accurately reports that there is “selection bias”. We can be very skeptical about this study. I am much amazed that RIVM presents this biased finding as “conclusions”.
(4) Tests on blood donors: On average 3.6% of the population has antigens. This is 1% for persons below 20 years and 4.2% for those older. We may presume that there has been correction for sampling selection; it leads too far now to look into the referred study (F van der Klis et al, IIV-CIb/RIVM). The issue returns on May 20, and then allows an estimate of the Infection Fatality Factor (IFF).

2.9.7. RIVM May 7

RIVM & Van Dissel (2020d), May 7, still have the first “logo” slide, corrected later with 5204 deaths.
The levels at the ICU have dropped so much that RIVM concentrates on giving scientific blessing to the government plan of phasing out lockdown measures such that a second wave ought to be avoided.

2.9.8. RIVM May 20

(1) RIVM & Van Dissel (2020e), May 20, no longer have the first “logo” slide. The death toll is 5715, not counting the “surplus deaths”. Osiris counted 44249 patients, and thus a Hospital Fatality Factor (HFF) of 12.9%. In the chart on slide 9, RIVM puts Holland below instead of above Spain with 12%. Below, for lack of other data, we will take the sCFF as this HFF.
(2) Slide 9 gives the “surplus deaths” as 9000, which means some 3300 cases more than the official number 5715. There is no information yet about the composition of these surplus deaths. There may also be flu victims, deaths from reduced normal care, and there will be fewer traffic accidents (normally 65 per month).
Research on blood samples, now a combination of blood donors and a RIVM panel called “Pienter”, shows that 3-4% of Dutch inhabitants have antigens. RIVM mentions 650,000 that “have been” infected, which means that they took 3.735%.
RIVM arrives at an Infection Fatality Factor of 1.3%, and it is somewhat remarkable that they do not round to 1.4%.
(3) RIVM actually provides this (incomplete) table that is focused on hospitals.
We have: IFF = D / I = D / H * H / I, but there are different death counts, and the H / I ratio must be corrected for this.
We can combine the Osiris data and the infections from the blood testing.
We would like to have the symptomatic Case Fatality Factor (sCFF) too. We have IFF = D / I = D / sC * sC / I = sCFF * sCPI with the symptomatic Case Per Infection (sCPI). Above we had an indication that it might be 0.49 but this was dubious.
(4) However, let us suppose that the distribution of infections copies the distribution of deaths, or that we can use the IFF to backcast the infections. Let us suppose that the symptomatic death rate as observed for hospitals (HFF) roughly applies to the total of 9000 deaths, with sCFF = HFF, so that we can backcast the symptomatic development. Then we can fill in two blanks.
And we can now compare with the Chinese CDC “pyramid”, see Wu & McGoogan (2020). Though they also have confirmed cases, their selection has been biased by mostly symptomatic patients, though confirmed by microbiological testing.
Thus we now have the decomposition IFF = sCFF * sCPI with sCFF = 12.9% (HFF = 9000 / 69684) and sCPI = 69684 / 650000 = 10.7%.
(5) The claim on the number of ICU admissions that has been prevented has now risen to 44000. RIVM does not specify what percentage this is, and thus we search for a statistic on the current cumulative number of ICU admissions. For May 20 this is 2847, see https://www.stichting-nice.nl/. Thus we can do the same algorithm as above. It is actually here on May 20, slide 16, where it is stated that the ICU form 0.35% of all diseased per day. RIVM states that 16 /100,000 new diseases per day mean 40 hospital admissions and 10 ICU admissions per day, in the latter ratio of 1/4. Less rounded numbers are 154 / 100,000 disease new cases, 384 hospital and 96 ICU admissions, still with the latter 1/4. With a population of 17.4 million, the 0.00154 figure translates into 26796 cases per day, of which then 384 = 1.4% hospital and 96 = 0.358 ICU admissions. It still are rounded numbers. The implied figure of 1.4% “hospital case per disease” differs much from what we found above, as 9.34% “severe” or even the 6.8% “hospital cases per infection”. Something happened to the definition of “disease” and it is unfortunate that the slides do not give a definition. (It is even unclear whether the ICU belong to the hospitals or are a separate category.) However, we can take 1/4 as the ratio and presume that 9.34 / 5 of the critical might be at ICU of which more than 50% actually dies.
(8) Slide 13 has this curious diagram of the periods between the stages. Incubation has been reduced from 6 to 5 days, and instead of “symptoms” there are “complaints” (“klachten”) (which I would regard as a major distinction). In the recording, Van Dissel states that the seriously ill have their own shorter path from complaints (“klachten”) to ICU (9.2. days). The total period from infection to hospital admission is 5 + 7.7 + 7 = 19.7 days, suggesting a one-decimal accuracy. However, when RIVM estimates the effective reproductive factor R[t], then they can back-trace hospital admissions to infections 14 days earlier (as apparently recorded in the Osiris database). I would say that 14 differs from 19.7. It may well be that various visits to the GP (“huisartsbezoek”) do not result into a hospital admission, so that the averages in this scheme are much influenced by either the healthier who eventually clear by themselves or those who prefer to die at home rather than unconscious in an ICU bed. This scheme is more confusing than informative, though the period for back-tracing likely will be accurate (as accurate as the database entry system).
(9) The pyramid in above diagram refers to the original Chinese CDC summary that 81% mild, 14% severe, and 5% critical (that RIVM calls “fatal”). RIVM has drawn a dashed line to indicate that there are many asymptomatic infectious, i.e. “below the radar of symptoms”. In the video presentation, Van Dissel very shortly remarks that most experts think that the pandemic cannot be suppressed because of these asymptomatic infectious. Singapore and South Korea thought that they had the infection under control but were confronted by recurrences caused by such carriers. Never-ending surveillance and contact tracing are required.
(10) RIVM then discusses Kissler et al. (2020), the Harvard study, referring to it as “Marc Lipsitch et al, Science, 2020”, and subscribe to the findings. A method of lock-on-and-off could respect the capacity of the health care system, and would gradually result into a build-up of herd immunity of some 55% over two years. The fact that antigens have been found in Dutch blood samples is encouraging for the assumption of immunity, but it is not clear how long this will last. (Sanguin stated independently that antibodies need not say much about immunity. Corona viruses cause the common cold and the immunity system tends to forget them.) The method requires close monitoring of developments (with said delay of 14 days between onset and hospital admission), and comes with a risk of a “Fall peak” (this Fall and Winter) and overlap with the flu. Apparently “herd immunity” still is an appreciated by-product (see our deconstruction in section 1.5.5 and Chapter 6).
(11) RIVM suggests a gradual release of lockdown. There are distinctions between individual / group, inside / outside, local / regional. There are systems for “early warning” and “surveillance”.

2.9.9. RIVM May 25

2.10. A partial result of this exercise

2.10.1. A result for us

This exercise is partly useful to determine what if we hadn’t had a lockdown. Thus, using β = 4 and γ = 1/10, what would the death toll have looked at the end of the year ? Many people would agree that a temporary lockdown in March 2020 has been sensible, if only to consider the options. Dutch society saved some time to consider the options. But did we really consider those options, and, what are the options actually ? As said, this discussion cannot be settled with a didactic exercise, but one can discuss what information would have to be improved in order to make such discussion fruitful.

2.10.2. A list of questions for others

The government and RIVM allowed the virus to spread, first formulated as an objective but quickly rephrased as a by-product of the policy of flattening the curve. Allowing a virus to spread might be based upon medical criteria and the properties of the virus, like the common cold, but it has a distinct legal flavour of avoiding responsibility for the spread. If the virus is innocent then why not use it as its own vaccine ? The situation is a conundrum in terms of public health. Such questions should have been addressed with the WHO PHEIC, but are still waiting for an answer. The health system allowed events to evolve such that the emergency brake of a lockdown was required, so that such questions were sidetracked, but such questions still need attention.
It is sobering that the Dutch lockdown in the middle of March was an emergency brake. Given the statement of RIVM (2020) of January 27, then, when the WHO declared the PHEIC on January 30, the proper course would have been to clam down on the infection, as in Taiwan, Hong Kong, Singapore and South Korea. This would have required intensive education of the Dutch population, within 10 days, but the scenes from Wuhan and Italy would have helped. The plane that left Holland on February 10 with medical supplies and personal protection equipments collected for China, would have stayed here.
The above criticises RIVM statements about herd immunity and the role of children. There was the collapse of conventional hospital care. It seems that RIVM was focused on the end-of-pipe ICU beds, with prominent doctors, and with neglect of up-stream homes of the elderly, with less status, even while RIVM stated that the policy was targeted at protecting the vulnerables. RIVM only advises, but still.
Sridhar and Rafiei (2020) indicate that it is almost impossible to shield the vulnerables, except by a national lockdown. I have my doubts whether we actually tried enough. However, RIVM in the early phase called for such protection as if it were possible without a lockdown. In the later phase RIVM provided various measures for social distancing, but rather late and even now, with RIVM supporting the release of lockdown measures, it seems that they trust that the vulnerables can be shielded. We can only assume that this is evidence based, so, what is this evidence ?
There are questions about source and contact tracing. Critical questions arose in the media while one would assume that RIVM should have spotted the problem itself earlier (even when it is GGD that has been assigned that task).
RIVM allowed that deaths were only counted for hospitals, and that suspected cases at home were not tested, even though RIVM knows that this distorts the information. We discussed the estimate of the dynamic R[t] on hospital data only, with unreliable uncertainty because of the changing attitude w.r.t. hospitals and the peak in “surplus deaths” in April.
There are questions about the balance of medical needs and economic needs. The provinces of Brabant and Limburg were not put into quarantine with the argument that a lockdown would also lock down the economy, but the lock down of two provinces does not lock down the entire nation. When RIVM had manoeuvred the process into the funnel of a national lockdown, then it seemed as if discussion with economists was superfluous, but such discussion would have been needed at all times, even in January, even now. Though it is useful to remark that this notebook looked at the didactics of SI(EY)A(CD) modeling with the example context of SARS-CoV-2 for Holland, and this notebook does not discuss the didactics of economic models for this context.
PM. The above basically concerned the RIVM department of infectious diseases. RIVM does more than this, and RIVM also does many more things not. In 2004, minister of Economic Affairs L.J. Brinkhorst and public health and environment secretary P. van Geel wrote to Dutch Parliament that RIVM had been given funds to calculate each five years the figure of environmentally Sustainable National Income (eSNI) according to the definition by Hueting, see Brinkhorst & Van Geel (2004). This was done till 2005. Thereafter the calculations stopped, and it is not clear why. See Colignatus (2020f).

3. The package

3.1. Caveat: This is didactics. The package only hints at the real world

The above already indicated:
(a) We want to focus on SARS-CoV-2 but we will not get far. SIA, SICD and SI(EY)CD have few parameters, and we should not expect that these few parameters are sufficient to describe the world. It is educational that such a simple model already conveys major conceptual aspects about a pandemic, but an increased confidence in one’s understanding of pandemics should not cause overconfidence.
For example, these basic models have only 1 class of susceptibles. In our own discussion there are rather two classes (SI(EY)CD2): vulnerables and less-vulnerables. Modeling with more stages and more age-groups likely requires SI(EY)CD20.
In other words: If and when elements from this notebook and package are used for real world applications then those must be documented for themselves.
(b) The notebook and package present both the conventional S(E)IR(D) setup and a didactic redesign to a new format SI(EY)A(CD). It you want to, you can use this package in the traditional S(E)IR(D) format. In all cases, there is no change in conceptual content or material findings. There is only the suggestion that the (underlying) notions are expressed more clearly and more accessible, extended now with user-friendly programming for an environment of computable writing. The principle is: Do not program to others what you do not want to be programmed to you.
There are obvious drawbacks to this effort at restating what is already known. An experienced reader might wonder why this redesign is needed: there is a learning curve for this notebook and package, with nothing gained on new understanding. A novice reader might benefit from a better understanding of models and programming as provided here, but then still faces the literature with its conventions. This notebook and package thus are a rather dubious endeavour. The proof is in the eating of the pudding. My expectation is that many readers will find that the reading has been worthwhile. (For teachers: When one doesn't adopt the redesign itself then this discussion at least highlights issues in didactics.)

3.2. Design / Redesign of didactics

The following repeats the overview of the choices from didactics, as stated in the Introduction, but relates those choices now to some programming particulars, like the choice of options and the use of variable names.

3.2.1 Avoid using R for compartments, and instead use A and C

PM. The symbol R is also avoided in the “traditional versions” that the package recognises. However, Rq, Rp, and Rg are available, and users may set Rq = Aq, Rp = Ap and Rg = Ag. (This would have to be done after each clearing of the variables, and one might write a small routine that both clears and redefines.)

3.2.2. Have D = φ A as an explicit proportion

Section 7.3. The conventional format of the SICD (SIRD) model has D’ = μ I. This causes the question what μ means, how to calculate its value, and what happens if the overall γ would change. The best didactic format is to give the actual model D = φ A. The deceased are a mere proportion of the acquitted, with φ = Infection Fatality Factor (IFF). The conventional format is oriented to Ordinary Differential Equations (ODEs) and it is an exercise in mathematics to show that this translates in the proportions. It is clearest however to state what the model entails, and regard the education about ODE as another issue.
PM. The “traditional version” in the package has D’ = μ I.

3.2.3. One single conceptual model SIA with two kinds of splits: A = C + D and I = E + Y

SIA is the conceptual model that has two kinds of nuance splits, whence the full name is SI(EY)A(CD):
(a) Section 7.3. The acquitted A can be split into the Cleared and Deceased, thus A = C + D.
(b) Section 8.3. The infected = infectious I can be split into the Exposed and Infectious, either I = E + Y (didactically clearest) or traditionally Y = E + I (in which I puts emphasis on being infectious).
Within the new didactic format, the two splits are such, that the properties of the S, I, and A compartments remain the same (taking I = E + Y). This warrants the use of a single set of Options[SIA] that control the functioning of all model formats. The routines for SIA, like finding the top of I = E + Y and the limit values, apply to all variants, since the same assumptions apply.
PM. The “traditional version” in the package has Y = E + I.
PM. For simplicity, no background births and deaths have been included, though they obviously effect conclusions about waves of infections and a steady state.

3.2.4. Separate models but joint options

Good programming requires that a new model comes with its own set of options: but this applies only when we really have different models. The SIA, SICD and SI(EY)CD models are so alike that it would be unwise to have different sets of options. Having different sets of options would require checks on e.g. the same size of the population etcetera, and this better be avoided.
It would have been possible to construct SICD by first run SIA and then split A into C and D, but we follow the convention (historical development) to present SICD as a separate model indeed. This has the advantage that all variables are available in both input and output. We include A = C + D in the model, for this purpose. For SEYCD we also include I = E + Y.
(It may be remarked that the design of this package started out with separate options and the more traditional format “SEICD”. This appeared to be too complex, for the reasons given. Thus the setup was simplified, and using the routines showed that this was beneficial. An example is the run on the intervention, in which the same set of Options[SIA] can be adapted to the new regime.)
PM. Still, applications of the options can differ. The user can call the routine SIA[Options, Check] for some diagnostics on the settings. For example, the parameters GammaSICD[] and MuSICD[] are used in the traditional setup of the models, and it may be useful to check that they add up to GammaSIA[] for the SIA.

3.2.5. Joint set of parameters and variables

The models not only have a joint set of parameters but also a joint set of variables Sq[t], Iq[t], Aq[t], ... for the levels and Sp[t], Ip[t], Ap[t], ... for the proportions. The routines defined for the variables of SIA can be used by SICD and SEYCD. The routines defined for C and D for the SICD model can also be used by SEYCD.
While variables Sq[t], Iq[t], Aq[t], ... have specifications that are also agreeable for computational writing, the package also provides for translation to shorter symbols , ℐ, , ... for the human eye.

3.2.6. Distinction between infections and symptomatic disease

The distinction between infection and symptomatic disease is fundamental in this discussion. Chapter 10 gives the conventional distinction between the generation interval and the serial interval. This notebook and package have no contribution to this discussion. There is only the advantage that the discussion has been rephrased within the confines of above redesign. A basic finding is that many reports in the literature apply the S(E)IR(D)  SI(EY)A(CD) model to symptomatic disease (e.g. the Goh (2020) “epidemic calculator”). For SARS-CoV-2, the crucial phenomenon is asymptomatic infectiousness.

3.2.7. Models SEYCDT and SEYCDB for estimation

Chapter 12 discusses two methods to present an intervention.
The first method has a Base SEYCD model alongside an Alternative SEYCD model, and the models differ in parameters and time of intervention.
Having an intervention and estimation may require an administration of the N[0] that is used as the denominator of the proportions. This is not required for the latter SEYCDT and SEYCDB with the WhenEvent approach,

3.2.8. Modularity

A fundamental programming principle is to build with blocks that are fully functional and well-tested. The user often knows little about the building blocks and only requires that everything works as stated in the manual.
It is tempting to create flashy results (e.g. some plots below) using quick and dirty routines. In such cases, it takes only a small glitch and the whole edifice doesn't work anymore. Even worse: if programming isn’t well-structured, then it is often impossible to trace its cause and repair the issue.
The present setup provides both the building blocks and the higher-up routines. The building block have been tested (by writing this notebook, also in steps) and are reliable. If there would occur glitches in the higher-up routines (e.g. entering 1/0) then it remains possible to redo the process in steps by using the underlying building blocks.
For example, in steps (available building blocks) we define the equations, assign parameter values, do the NDSolve for the path, set the level variables Sq, Iq, Aq, ... to the interpolated functions, set the proportion variables Sp, Ip, Ap, .... Normally all steps have a standard sequence, so there is also a routine to do the steps for us. The higher-up routines are less-tested than the building blocks, so if something happens, like with a parameter out of range, then it still is possible to do the steps. For example, while the package presents a main plot format with a particular choice of plotting options, and there would be a glitch, say by a newer version of Mathematica that changes something in plotting, it always remains possible to use the Plot routine that is standardly available within Mathematica.

3.2.9. Administration of scenarios

Apart from this discussion of what the models and package do, there is also the application that requires record keeping of assumptions, parameters and outcomes. This is handled in another package Survival`ApplySIA`, not shown or discussed here. My administration is organised but it has not been documented for use by others.

3.3. General setting on programming

3.3.1. Reasons for choosing Mathematica

Reasons for choosing Mathematica are the same as for writing packages in Mathematica in general:
- Use of a general environment, so that code isn’t locked-up within the confines of a particular application, but can be used in direct combination with other computable aspects, like estimation and economic implications. There is a price to pay though: this notebook and package come with the confinement of the Mathematica environment.
- While the language of mathematics is universal, also on the computer, Stephen Wolfram had a talent to bring it to the computer. Mathematics has been developing over millennia, and my impression is that Mathematica has caught this development in the best manner for application on the computer. Other applications obviously must satisfy mathematics if they are to work, and mathematics can have many dialects, so this reason need not be an universal truth itself, and likely it is only a matter of (acquired) taste.
- All this is embedded within my proposal for redesign of mathematics education, see https://zenodo.org/communities/re-engineering-math-ed/about.
I made a choice for Mathematica in 1993 and have much benefitted from it. Over the years, other computer algebra packages have not developed as much. New options like R and Python do not have the appeal of the language.

3.3.2. The Economics Pack

This notebook and its discussion are open access. At first my objective was to make the package open access too, but soon I found it more efficient to use features from The Economics Pack, which features had been developed in the past for good reason. The pack is available at a cost, see Colignatus (1995, 2020e). Considering that others can use notebooks in Mathematica only by also acquiring Mathematica anyhow (except for the MathReader, now WolframPlayer), I dropped the notion of an open access package. The new package is basically for convenience. The Economics Pack already contained a section on the life sciences, and Survival`SIA` is included there. When the Pack has been installed on the computer, use the file explorer to find the notebooks. A complication is that the Help function of Mathematica has abolished the HelpBrowser, and switched to a paclet structure. The Pack hasn’t adapted to this yet. However, the user can find the explanatory files by use of the file explorer.

3.4. The SIA package

4. Compartments

4.1. Principle

Principles for this chapter are:
(1) (Public health) compartments appear to be relevant for quarantine controls. The following is a suggestion for such compartments.
(2) The SARS-CoV-2 pandemic requires clear communication in and over a variety of cultures in the world. We should use as much common ground as possible.
(3) One common ground is the human sensitivity to colours. Colour coding seems rather important for effective communication. Humans are sensitive for very subtle differences in colours, but, while the rainbow shows all frequencies, or colours ordered by hues, there is a tendency to see only some 10 hues, differing by about 0.1 degrees.

4.2. The meaning of red for regulation

Like with traffic lights, we need colours to manage not only people but also locations. It is insufficient to paint “dangerous cars” red and “safe cars” green. Red is the general colour for alert status (likely because blood is red), and it is best to use red for the barrier between any quarantine group rather than for a particular group.
The basic idea is that red identifies a barrier.
The following suggested colour scheme uses identifiable patterns in the hue or RGB schemes.
In the RGB rainbow:
- B goes from 0 (uninfected) to 1 (infected) (uphill only).
- R has the path 1-0-1 (valley).
- G has the path 0-1-0 (up and down a hill).
However, the rainbow is subsequently rearranged, namely to put red between the vulnerable and less vulnerable quarantine classes.
The interpretation is: the further a quarantine class is away from the barrier, the more extreme the position will be: either uninfected (B is 0) and more vulnerable, or infected (B is 1) and getting sicker. PM. The colour of DeeperPink for the ICU has an infringement upon the B = 0 or B = 1 dichotomy.

4.3. Suggested colour scheme

The colours might also be used for SI(EY)A(CD)-type of modeling with more compartments. The scheme allows that SI(EY)A(CD) is used for both infections and symptomatic disease. The “bluepurple” group was included for symptomatic disease. For quarantine it would likely be joined with “freshpurple”.
For SI(EY)A(CD) modeling, mortality can be indicated by dashed Black / White lines, which fits European / Chinese conventions on these “colours”.
We can put the RGB combinations in the familiar RGB-plot reminding of a Venn-diagram. (PM. Deeper pink is not shown here.)
(a) The larger disks give a Venn diagram indeed. For example, the yellow group is safe because they are protected by a quarantine barrier. For example, the blue group is infected but it is unknown whether they will become immune (green) or have to be hospitalised (magenta).
(b) The smaller disks do not form a Venn diagram. They highlight special conditions. For example, for “fresh purple”, the barrier is breached, and those of the vulnerable group must be hospitalised. (Hospitalisation itself shows that the barrier was crucial for them.)

4.4. Use of these colours in the SI(EY)A(CD) plots

These colours will also be used for plotting the SI(EY)A(CD) graphs. For the Acquitted, the combination of Cleared (Green) and Deceased (Black and White) results into some brownish colour.
The SI(EY)A(CD) model, while officially named for infections, is also used by researchers for symptomatic disease. Instead of the “exposed” there is “incubation”, and instead of “infectious” there is “symptomatic”. In itself this is not so strange, see Chapter 10 about the distinction between the “generation interval” and the “serial interval”. For the SARS-CoV-2 virus, it appears that both such applications are used indeed. These models however have different assumptions, and there is some risk of confusion. Thus it is necessary to have legends that fit the type of modeling assumptions. These are the two sets of categories, and the symptomatic case tends to be a bit darker or up in the rainbow than the infections.
The command PlotLegendsSIA[SEYCD] sets the legends and colour schemes for the standard interpretation of SIA, SICD and SEYCD for infections. PlotLegendsSIA[“Symptoms”] sets those for symptomatic disease.

4.5. When in lack of colours

These ten or eleven “identified possible quarantine groups” and their colours will not be enough to serve the large variety of cases. However, it is always possible to combine codes. We could show colours in bands, and not blend the hues.
For example, an icon that shows both magenta and freshpurple indicates that a hospitalised person belongs to the vulnerable group, and if this is clear, then we might add a little bit of yellow to indicate that the origin is from the elderly and comorbidity group or a little bit of orange for the younger group of vulnerables. In a graphical plot we could have a line dashed with those colours (except that this hasn’t been implemented yet).
PM 1. There is a curious effect. In everyday life, when we use colors red and green, we do not necessarily think of traffic lights. With above colour scheme for infection status, the colours in a graph on another subject may cause the question whether that other subject concerns infection status or not ... It is something to get used to.
PM 2. It would be fortunate if Mathematica adapts a notion of "generalised colour" that includes such band-dashing in different normal colours, so that the options for Mesh and Legend can be combined. Now a trick is to plot a function twice, in both a base colour and a dashed other, but this means that one also gets a double legend.