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Redesign of the didactics of S(E)IR(D) -> SI(EY)A(CD) models of infectious epidemics

Thomas Colignatus

This notebook and package redesign the didactics of the classic SIR, SIRD and SEIR(D) models, using the example of the SARS-CoV-2 pandemic. (Issue 1) 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. In SIA(CD) we have A' = γ I, with γ the acquittal rate from infectiousness, with I' incidence, I prevalence and A cumulated prevalence. The format of ordinary differential equations (ODE) should not distract. The basic structure is given by the Euler-Lotka renewal equation. The deceased are a fraction of the acquitted, D = φ A, with φ the infection fatality factor (IFF). The ODE format D' = μ I or D' = μ Y can be rejected since it turns the model into a course in differential equations, with the need to prove D = φ A which can already be stated from the start. The ODE format also causes distracting questions what μ might be and whether there is a difference between a lethal acquittal period and a clearing acquittal period, and how parameters values must be adapted when the acquittal rate γ changes. (Issue 2) The term "herd resistance" is much more preferred above the term "herd immunity". The notion means that the herd survives and doesn't become extinct, even when the infection is lethal to perhaps most members. Policy makers and the general public associate "herd immunity" with protection for all members, which is not what the notion means (and which special outcome may only occur with vaccination in a well-mixed population before the onset of infection). This source of confusion between experts and non-experts can be prevented by using the term "herd resistance". The common formula on herd resistance 1 - 1 / R0 uses (i) a steady state or (ii) vaccination before the start of an epidemic. (ad i) SI(EY)A(CD) has only an asymptotic steady state, for which the formula does not apply. (ad ii) Vaccination during an ongoing epidemic means that there are still large numbers of (asymptomatic) infected persons, so that the infection continues, called "overshoot", which does not give the protection that is suggested by the term "herd immunity": there is only a herd resistance. 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 R0 = 2.5 and "herd immunity" (i.e. herd resistance) of 60%, presumably using another type of model with a proper steady state. In SI(EY)A(CD) an infection with R0 = 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. (Issue 3) The objective of Public Health is 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. If this is rejected because the virus is so nasty then this is an argument for eradication. 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 - lock-off approach, that is risky and prolongs the economic crisis. The SI(EY)A(CD) model uses lives saved (lives extended) but other life gain measures are the (quality adjusted) life-years gained, fair innings, proportional shortfall and UnitSqrt. A life table computation that assumes annual loss of immunity shows that a 10% rise in annual cumulated prevalence implies about a 0.5 year drop in life expectancy. The effect is relatively small since the fatalities are mostly in the higher age groups. If cumulated prevalence would be 60% as RIVM suggested, then life expectancy would reduce with 3 years, but the overshoot to 90% would reduce life expectancy by some 4.5 years. The annual death toll is an acceleration of the mortality by comorbidity. (Issue 4) Public Health and epidemiology exist 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.

The PDF is available at 10.5281/zenodo.4007638. This 2nd version includes vaccination, life gain measures and life table.
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