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Causal effect of covid vaccination on mortality in Europe"
André Redert, PhD"
Independent researcher"
Rodotti, Netherlands, 24 February 2023"
Abstract
This report investigates short-term causal vaccine-mortality interactions during booster
campaigns in 2022 in 30 European countries (population ~530M). An infection-vaccination-
mortality model is introduced with causal aspects of repeatability, random chance, temporal order
and confounding. The model is simple, has few or even zero prior model parameters and is
unbiased in causal mechanisms and strengths. Confounders are taken into account explicitly of
mortality-caused fear incentivizing vaccinations and four related to covid infections, and
generically for all long-term confounding. Bayesian probabilities quantify all interactions, and from
observed weekly administered vaccine doses and all-cause mortality, mortality on short-term
caused by a vaccination dose is estimated as Vaccine Fatality Ratio (VFR)."
#VFR results are 0.13% (0.05%-0.21%, 95% confidence interval) in The Netherlands and 0.35%
(0.15%-0.55%) in Europe, subtantially transcending covid IFR. Additionally, sewer-viral-particle
experiments suggested vaccination induces covid-infections and/or reactivates latent viral
reservoirs."
#The evidence of a causal relationship from vaccination to both infection and mortality is a very
strong alarm signal to immediately stop current mass vaccination programmes."
Statement of Interest
I declare that this work was done with an interest in science, and personal safety for myself, loved
ones, and humanity. Pro bono, independent, without payroll, not funded. The only competing
interest was time taken from my normal job (indy app developer in entertainment and music). If
you want to support my work, feel free to buymeacoffee.com/AndreRedert, or consider one of
the apps at rodotti.nl and qneo.net."
1. Introduction
Since the covid vaccination campaigns, high unexplained excess mortality rates have been
observed worldwide, starting in the second half of 2021. In The Netherlands (population ~18M),
excess mortality rates went up to ~80 people/day (excess ~20%) at end of 2022, before rising
even more due to influenza. Based on the sparse publicly available Dutch data on mortality and
vaccination, excess mortality was found to correlate positively with vaccination on long-term
[Re1], and short-term [Mee,Sch,Re2]. Figure 1 illustrates this correlation for weekly 4th/5th
vaccination campaign doses and all-cause mortality. Detailed case-based data has still not been
made publicly available, and it remains a scientific challenge to analyse excess mortality using the
sparse data that is both available, and reliable."
#This report has the same goal as my earlier work [Re2], estimating short-term Vaccine Fatality
Ratio (VFR) on the basis of weekly administered vaccine doses and all-cause mortality; two
integer, countable parameters that do not suffer from the subjectivity and ambiguities in PCR
testpolicies, covid diagnoses, cause of death atttributions, and modeling in mortality prognoses."
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"
Figure 1: Left) Weekly rates in the Netherlands for mortality and administered covid vaccine doses
[Src], scaled to illustrate temporal correlations. Right) Pearson correlations are very high. Different
regression factors may be related to the campaigns’ target ages (60+ and 12+).
New in this report is explicit causal modeling on the basis of repeatability, temporal order,
handling of infection-based and generic confounders in a combined infection-vaccine-mortality
(IVM) model, and the Bayesian framework to handle randomness and all statistic parameter
estimation. The extraction of short-term events introduced in [Re2] to remove effects and
confounding on long-term, see Figure 2, will be reused."
"
Figure 2: Same data as in Figure 1, short-term filtered to extract random weekly variations, easing
a single analysis with multiple campaigns combined."
Infection data will be used of PCR tests and viral particle presence in sewer wastewater [Med],
see Figure 3. As PCR tests are unreliable (false positives/negatives, test-willingness/policy-
dependence, arbitrary CT-values, etc), and sewer data is more objective but less widely available,
the relative performance of a simplified vaccine-mortality (VM) model without infections will be
evaluated to widen the applicability of the method. Experiments will be performed for booster
campaigns in 2022, in The Netherlands as well as 30 European countries (population ~530M)."
"
Figure 3: PCR+ tests and viral particle presence in sewer wastewater, The Netherlands. Sewer
data may be more objective, but lags PCR+ infections by a few weeks, and also lags vaccinations
and mortality as in Figure 1."
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2. Method
In the next sections I will present my method in many high-detailed steps. The final result,
however, is simple and contains a bare minimum, or even zero, model parameters."
2.1 Causality
The well-known “Correlation does not mean causation” is a vague statement that may lead to the
false believe that all correlation is insufficient as evidence of causation. However, correlation is all
one can observe, there is no alternative to determine causality. Pure causality cannot directly be
observed as it is an “abstract philosophical concept that indicates how the world progresses”
[Wik]. One can easily defend that:"
“The origin of all correlation is causality”
The well-known vague statement means that when A and B are correlating, causation may not be
between A and B (in either direction) but may involve a common cause known as a confounder.
Correlations may also occur by chance, when an “oracle” set of events in the universe conspires
to create the correlation; a confounder that cannot be known by definition, referred to by
“random”. Theoretically, such a random oracle could exist for every possible A and B, thereby
pulling any practical use of causality out of the scientific realm ."
1
#Requirements for establishing causality have been studied for epidemiology in particular, listing
consistency and strength of association, confounding, temporality, experiments and plausibility,
based on the famous Bradford Hill criteria [Io1, Shi]. Consistency and strength of association can
be rephrased as strong correlation, which can readily be observed in Figures 1 and 2.
Repeatability, or rechallengable, and random chance are already essential ingredients in
correlation analyses; e.g. covariances are characterizations of commonalities in many repeated
observations, and statistical techniques exist to obtain mean and deviation of such covariances.
Temporality, also known as temporal or causal ordering, and confounding are unique for causality,
the reason to include these in my prior work [Re2] via a causality test involving temporal
correlation and short-term-filtered observations to remove all long-term confounding."
#In controlled experiments, one can make event A occur at will, which is very effective for
excluding chance and confounders. Observational studies like in this report can, however, still
accomplish the same goal via additional requirements (see also e.g. [Gia]): natural repeated
occurrence of A, appropriately patterned in time to enable detection of the same pattern later in B
according to temporal order, plus more emphasis on excluding a confounder/chance origin of the
observed pattern. Finally, plausability requires a known mechanism that makes A cause B. For
current covid vaccines, there are plenty of plausible mechanisms that lead to mortality, e.g. acute
myo/pericarditis later followed by sudden cardiac arrest [Sun], and frail elderly for which
vaccination is the last push over the edge [Nor]."
#This report examines causality between observed vaccinations A and mortality B, incorporating
repeatability, temporal ordering, random chance and confounders via several known and unknown
mechanisms. These include a reverse mechanism from mortality to vaccinations and four
mechanisms via infections."
#One cannot keep adding possible confounders indefinitely, as that inevitably leads to
overmodeling. Every possible confounder exhibits random correlations with both A and B, even
unrelated ones such as the weekly number of planet-star eclipses visible from earth. A sufficient
number of confounders will swamp any causal effect between A and B at some point. My method
This seems to be the objective of the vague statement when it comes to vaccination and
1
mortality.
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includes short-term filtering of all observed data, eliminating all confounders that act on a term
longer than several weeks, even those unknown, while preventing overmodeling. As collateral
damage, the method can only measure causal effects that act on short term and is blind to
longer-term effects."
#The random weekly variations in infection, vaccination and mortality rates are used as
repeated, patterned ocurrences to detect temporal order in possible causal directions, while 40+
weeks of data in 2022 are used for statistical repeatability. The Bayesian probability framework is
used to handle random chance."
2.2 Infection-Vaccination-Mortality model
Figure 4 shows my causal infection-vaccination-mortality (IVM) model. Observables are infections,
vaccinations and mortality, all weekly absolute numbers in a population. Drivers of the
observables are viral waves, campaign dynamics and seasonal baselines. Natural immunity is
lifesaving in the real world but not modeled; it acts via a negative feedback loop on infection,
whose observations already include the full effect of natural immunity."
"
Figure 4: Causal infection-vaccination-mortality (IVM) model. The path of vaccines to mortality has
focus. The other five causal paths are confounders. Viral waves, campaign dynamics and seasonal
baseline drive the observables and are random sources. As infection data is less reliable, the
relative performance of a vaccination-mortality (VM) model without infections and only two causal
paths will be evaluated also.
Six directed paths, shown as arrows in Figure 4, model the causal interaction from some event in
one observable to events in another observable, either within the same week or in a few future
weeks, but not past weeks. Vaccine-induced immunity and damage are modeled together as a
single net interaction. Interaction strength is measured in units of a ratio such as mortality per
infection (the wellknown Infection Fatality Ratio or IFR), mortality per vaccination (Vaccine Fatality
Ratio or VFR), etc."
#The focus of this report is measuring VFR via the causal path of vaccination to mortality. The
backward confounding path models vaccinations incentivised by fear, caused by observed high
excess mortality. Such confounding may add to a positive correlation as in Figure 1, which can
easily, but falsely, be attributed to vaccine damage in an analysis that does not take temporal
order into account. The four paths connected to infections relate to other confounding
mechanisms, among which the depletion of the reservoir of vulnerable people."
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2.3 Vaccination-Mortality model without Infections
Infections are the least reliable in the IVM model, not only in terms of data reliability but also by
their origin and role in the model. Ideally, two types of infections are taken into account if reliable
data is available: biological infections which are caused by viral waves and lead to physical
illness/death, and reported infections that are published by by mass-media with main purpose to
induce psychological fear. Two different infection observables, however, strongly increases model
dependence on unreliable data, and complexity by number of causal interactions."
#I will simply keep one infection observable: the more objective sewer viral particles scaled to
equivalent PCR+ tests by calibration in the first part of 2022. The apparent delay in sewer-to-PCR
data will be investigated: possibly, compensating the delay may do more harm than good in a
causal context."
#The IVM model can be used when reliable sewer and/or PCR+ infection data are available. To
widen the method’s applicability, a simplified model without infections will be introduced: the
vaccine-mortality (VM) model, as was effectively used in my prior work [Re2]. The VM model does
not suffer from unreliable infection data, but cannot compensate explicitly for infection-based
confounders. Its performance will be evaluated relative to the IVM model."
2.4 Bicausal model for data delays
It may happen that observed data include delays due to a variety of reasons, e.g. test-delays until
symptoms occur and sewer viral particle data lagging PCR data, see Figure 3. Further, it is
common practice to delay vaccination status by several weeks after the “act of vaccination” to
2
account for immunity build-up. Delays are less expected for raw numbers of objectively-dated
events as vaccination doses or mortality, but they are still easily introduced e.g. by accident at
data transfer to 3rd party data aggregators;, or even purposefully for e.g. visualization or
integration of datasets."
#Due to unequal delays in observables, temporal ordering may get mixed up and effects may
migrate between forward and reverse causal paths in the model, see Figure 5 left. For example,
due to a one-week-delay in data, a several-weeks-enduring causal effect in one direction partly
migrates to the causal path in opposite direction. This may not be easily identifiable; the migrated
part will also change numerically, as paths in opposite directions have inverted units."
"
Figure 5: The bicausal model combines two oppositely-directed causal paths into a single
bidirectional (“bicausal”) path. All data is simulated for illustration purposes. The green zero means
a zero effect value.
The clarity of the word “vaccination” has eroded substantially in recent years.
2
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For this reason, I will model every causal path pair as a single bicausal path that interacts both
backwards and forward in time using one numerical unit, arbitrarily chosen from original forward
or backward path. Within a bicausal path, data delays manifest simply as a single shift, see Figure
5 at right. Whenever the forward or backward causal path dominates and activity persists for a
few weeks, any delays can be identified by visual interpretation, using common sense aided by
strength certainty intervals; I will not try to model/automate such an identification process. "
#It may appear that the bicausal model causes information loss: in Figure 5, the two causal
paths provide 2x5%=%10 measurements, while one bicausal path provides 9 measurements, one
less. The two instantaneous parts in the two causal paths at get combined into the single
instantaneous measurement of the bicausal path. This information loss, although real, is not due
to the bicausal model, as in the causal model the instantaneous parts get mixed up too, albeit
implicitly. Based on well-known onset latencies of the six causal paths’ mechanisms, however,
some interactions are zero in the first week (see Figure 5), and all measurements will be uniquely
assignable to the six causal mechanisms, both in causal and bicausal model."
2.5 Characteristics of drivers and causal paths
Table 1 shows onset latencies, temporal dynamics, and effect sizes of drivers and causal paths.
All effect sizes are very rough estimates by order of ten, and the expected sign of each of the six
paths is never used. Measurements will thus never be restricted by a priori sign expectations, but
a posteriori inspected for consistency. Free sign modeling is required for vaccine-induced
immunity and damage, as they have opposite signs but reside in the same path. Path interaction
strengths will be modeled both with prior expectations according to listed effect sizes, and
without, that is, completely free of any prior expectation."
#Numerically, covid IFR is in the order of 0.1% [Io2]. The VFR is below ~0.1% given that most of
the population was vaccinated yearly, baseline mortality is yearly ~1% and total observed excess
mortality is in the order of ~10%. Vaccine-Effectivity against infection (VE-I) and mortality (VE-M)
may start at ~100% but negative VE-I (positive damage) is known to possibly occur both in the
the first weeks during build-up of immunity, and after several months when immunity wears off.
The size of vaccine-caused immunity is not plain Vaccine-Effectivity against mortality (VE-M), but
scaled by IFR as the paths in my model relate to all-cause mortality instead of only infection-
attributed mortality, while assuming that ~100% of the population gets infected each year."
#Viral waves, campaign dynamics and seasonal baseline typically evolve on the longer term of
months/years, with random variations acting by definition on the shortest term of data resolution,
weeks in this report. Vaccine-caused effects have a fast, biological, single-persion underlying
mechanism, while mortality-induced fear has a slower, psychological, inter-personal nature, via
fear of death induced by observing deaths of others."
#For this report, the most essential property in Table 1 is the low onset latency of vaccine
damage: adverse systemic events can occur within minutes, with immunity build-up following only
in the next 2-4 weeks. The path from infection to death, illness, typically takes at least a week,
and slightly above 2 weeks on average [Mar]. The causal path of fear requires data collection,
aggregation, and mass-media reporting of high infection numbers, or high excess deaths that are
above expectation according to some prognosis. While in 2020 it was custom to broadcast fear-
inducing infection and mortality numbers daily, in most of 2022 the number of infections was
relatively low, and occurring high excess deaths were not attributable to covid and reported
seldom, late, or not at all."
#Finally, with depletion the reservoir of vulnerable people more susceptible to mortality declines
during periods of excess mortality. Although this effect has zero latency, it acts cumulative and is
thus extremely small on the short term, and dynamical, if at all, only on the long term.
Δt= 0
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Table 1: Time scales, onset latencies and rough order of effect size of drivers and causal paths.
2.6 Confounders
My IVM model handles 5 confounders explicitly, but clearly the list of actual confounders may be
longer and full of unknowns. As stated in the causality section, unlimited adding of possible
confounders leads to overmodeling. Instead, I will review a few known confounders that may have
effect in this model and generic classes of (unknown) confounders."
#First, when close-ones (family, friends, neighbours etc) die, shock and mourning may delay
planned vaccinations. This path is the same as the fear path, but with opposite/negative sign, not
delayed/amplified by mass-media and thus faster/smaller. In most cases, the vaccination will be
catched up shortly after, resulting in a slight delay in a few vaccinations. If present, this will be
measured as minor initial negative fear, with zero net effect after a few weeks."
#Secondly, during vaccination campaigns, typically the smaller group oldest and most
vulnerable get vaccinated first, and the broader group of younger and healthier people follow. If
vaccinations do cause mortality on the short term, concentrated in older/vulnerable people, the
order of vaccinations may create a bias towards “mortality first” followed by “later vaccinations”,
a temporal order and positive-signed effect equal to that of the fear path. This confounder makes
vaccine damage appear as mortality-induced vaccinations and thus leads to underestimation of
vaccine damage, an acceptable bias for measurements."
#Many confounders may additionally come into play, known or unknown. All that apply on the
long term will be irrelevant, as this study looks only at short-term events, as in Figure%2. Whenever
a confounder acts on the short-term, it may reveal itself in both causal directions with similar
Driver
Onset latency
Time scale
Effect size (10-order)
Viral infection waves
-
month
100% population/year
Vaccination campaigns
-
month
100% population/year
Seasonal mortality baseline
-
month
1% population/year
Random fluctations in each
immediate
week*
as observed from data
* zero, limited by source data resolution
Causal path
Onset latency
Time scale
Effect size (10-order)
Depletion M to I
>> week
months
Negligible < 0.1%
Illness I to M (IFR)
week or more
2 weeks
0.1%
Damage V to M (VFR)
immediate
months
0.1%
Immunity V to M (~VE-M)
2-4 weeks
months
0.1%
Damage V to I (~VE-I)
immediate
2-4 weeks
100%
Immunity V to I (~VE-I)
2-4 weeks
months
100%
Fear I to V
week or more
week
1
Fear M to V
week or more
week
100
IFR: Infection Fatality Ratio, VFR: Vaccine Fatality Ratio"
VE-I: Vaccine Effectivity against infection, VE-M: against mortality"
~ the modeled effects are proportional to VE, and involve additional products with IFR or driver effect size
VFR
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strength. In the bicausal measurements in Figure 5, this would appear as a signal without any
specific temporal order: evidence of confounding, and absence of causal evidence."
#If on the other hand a significant vaccine immunity/damage signal is measured, in absence of a
significant fear signal, this is evidence that the signal may not originate from a confounder, and
that the measured vaccine effect may be real and causal. Exactly such a signal was found in
[Re2]. Of course, it is possible that an unidentifed, short-term causally directed confounder exists.
Ignoring alarming evidence by assuming the existence of such a highly characterized but
unidentified confounder seems unwise."
2.7 Mathematical representation
Mathematically, I present my model first in full, long-term causal form, and reduce it subsequently
to the short-term bicausal model. To start off:"
#(1)"
#(2)"
# #Time, integer week number, from "
#Total number of analysed weeks in 2022 (typically for weeks 9-50)"
#Weekly number of infections, administered vaccine doses, and all-cause mortality"
#Long-term mortality baseline, captures slow variations over months/years"
#Short-term weekly variations, uncorrelated zero-mean random process"
#Same as mortality baseline, but for viral waves and vaccinations"
#Temporal convolution"
#Causal function from A to B, valued only for "
#Duration of short-term causal effect (typically weeks)"
#Some prognosis, expected mortality, slowly varying over time (seasonal)"
#Excess mortality with respect to the prognosis/expectation"
#Infection Fatality Ratio, total mortality caused by infection"
#Vaccine Fatality Ratio, total mortality caused by vaccination"
The and are the drivers. The “baselines” vary slowly on the longer-
term of a month/year. They capture viral presence waves, overall vaccination campaign dynamics,
and mortality of seasonal waves, other-cause waves (covid, NPIs). They also capture confounders
acting between them, on the long-term, such as vaccination campaigns planned during covid
waves. The random processes describe the spread around the baselines, or weekly
variability of infections, vaccination and mortality originating from chance. They are zero-mean
I(t) = bI(t) + rI(t) + {V*FV→I} (t) + {MΔ*FM→I}(t)
V(t) = bV(t) + rV(t) + {I*FI→V} (t) + {MΔ*FM→V}(t)
M(t) = bM(t) + rM(t) + {I*FI→M}(t) + {V*FV→M}(t)
MΔ(t) = M(t)−Mex pectat ion(t)
IFR =∑
0≤Δt<ΔT
FI→M(Δt)
VFR =∑
0≤Δt<ΔT
FV→M(Δt)
t
0≤t<T
T
T= 42
I,V,M
bM
rM
bI,rI,bV,rV
*
FA→B
0≤ Δt<ΔT
ΔT
ΔT= 2..5
Mexpectation
MΔ
IFR
VFR
bI,bV,bM
rI,rV,rM
bI,bV,bM
rI,rV,rM
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and uncorrelated over time and with respect to eachother, and non-stationary with variance
changing slowly over the longer-term together with . "
#The six causal functions model the six causal paths in Figure 4, with main focus on
. All operate by temporal convolution, a linear operator. The linearity of is
particularly justifed, as individual biological mechanisms within a population have no direct causal
interaction that could lead to substantial nonlinearities. For fear paths and , linearity
may be less defendable as the underlying psychological mechanism may involve nonlinear
effects, e.g. “100 deaths” causing double the fear of “99 deaths”. For immunity path , and
possibly , one can argue that these should include a product with viral presence/infections
, but this is not taken into account for mathematical simplicity. An argument that justifies this
choice partly is that covid has become endemic and viral waves are less extreme. Also, other
pathogens provide a permanent infectious background and it makes sense to evaluate vaccine
effectivity always and only against all-cause mortality [Ben]; ideally infections would represent all
pathogens, or even other mortal threats . Altogether, modeling of and may thus be
3
more accurate than the other paths, which is acceptable regarding the aim of this report."
#The mortality to vaccination and infection functions , have as argument; fear
and depletion are caused by excess mortality or above-expectation mortality .
There are other ways expectations may impact , e.g. when a newer expectation is published
that overshadows an older expectation, the newer creates additional fear and it may thus appear
with a positive sign: . For this analysis, it is irrelevant how and
when expectations are constructed, whether or not they include the effects of covid, NPIs,
whether calculated by a national institute or emerged within the minds of individuals, whether their
effect sign is positive or negative. What is important, is that expectations change slowly over time
t, and that the sign of actual, observed mortality is always positive in ."
2.8 Short-term filter
As in [Re2], I apply a linear temporal filter on source data to remove all long-term
events and extract short-term events in (with a hat), see also Figure 6:"
#(3)"
Applied to (1), this results in:"
#(4)"
bI,bV,bM
FA→B
FV→M
FA→B
FV→M
FM→V
FI→V
FV→I
FV→M
I
I
FV→M
VFR
FM→V
FM→I
MΔ
M−Mexpected
FM→V
Mnewer ex pectat ion −Molder ex pectat ion
M
MΔ
W(Δt)
I,V,M
I,
V,
M
W(Δt) = [−0.05 −0.25 + 0.6 −0.25 −0.05] for Δt∈[−2,2]
I(t) = {W*I}(t)
V(t) = {W*V}(t)
M(t) = {W*M}(t)
I(t) =
rI(t) + {
M*FM→I} (t) + {
V*FV→I} (t)
V(t) =
rV(t) + {
M*FM→V}(t) + {
I*FI→V} (t)
M(t) =
rM(t) + {
I*FI→M} (t) + {
V*FV→M}(t)
Mass-media have suggested vaccination protects against car crashes.
3
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"
Figure 6: The temporal filter W extracts events with a short temporal scale, and removes events
with longer temporal scale.
Importantly, all causal functions are unaffected by the short-term filter, and subsequently so
is in (2). The short-time versions of source data are zero-mean random
variables, illustrated by Figure 2. All long-term baselines and mortality expectation
have disappeared, also causing excess mortality in (1) to be replaced by plain
short-term filtered mortality in (4). The filtered versions of drivers have slightly
reduced variance compared to the unfiltered signals, and are still non-stationary."
#All of this is irrespective of the specific choice of filter , as long as it extracts short-term
events as in Figure 6. The filter choice (3) in this report is given by ,
where is a Dirac delta function, is a zero-mean, -deviation Gaussian, one of the smoothest
filters possible, and is chosen to ensure that is zero-mean (sum coefficients is zero). The filter
choice in [Re2], , was an intuitive approximation of the filter
in this report."
#As the same filter is applied to all observables, it does not bias the short-term measurements
towards any of the causal paths, or towards any direction within any path; the filter may even be
time-assymetric, e.g. a 3rd derivative filter [-1 +3 -3 +1], as possible time delays due to the filter
do not affect measurements of . The filter also does not bias measurements by sign. In
combination with zero-mean prior expectations on the values of , the filter may bias
measurements towards zero, with a relative strength growing with event time scale. Aggregates of
such as and will then be biased to zero, or equivalently underestimated in size."
#In the results section, I will illustrate a few alternative filters as well as absence of the filter, and
complete relaxation of the prior expectations. For readability: I will leave out all hats from this
point on. In all that follows, observables are short-term-filtered."
2.9 Short-term causal IVM model, prior variances and constraints
The short-term causal IVM model (4) is here denoted without hats:"
#(5)"
FA→B
VFR
I(t),
V(t),
M(t)
bI,bV,bM
Mexpectation
MΔ
M
rI(t),
rV(t),
rM(t)
W
W(Δt)≈δ(Δt)−KG1(Δt)
δ
Gσ
σ
K
W
[−0.1 −0.25 + 0.7 −0.25 −0.1]
FA→B
FA→B
FA→B
IFR
VFR
I,V,M
I(t) = rI(t) + {M*FM→I} (t) + {V*FV→I} (t)
V(t) = rV(t) + {M*FM→V}(t) + {I*FI→V} (t)
M(t) = rM(t) + {I*FI→M} (t) + {V*FV→M}(t)
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I model all and components of drivers and causal interactions as
independent, normal-distributed (Gaussian) stationary random variables. The original non-
stationarity of drivers is remodeled by a slightly higher overall variance, namely that of
the observed :"
#(6)"
For the variances one finds using the effect sizes in Table 1:"
#(7)"
These values are rough orders, which is accurate enough as they are only used as so-called
“priors” to slightly constrain the model; they bias measurements of to zero in the presence
of insufficient evidence in the observables. The priors’ influence decreases the more informative
observational data is available. I will alternatively allow for completely free causal
interactions, relaxing prior variance constraints (7), to prevent underestimation of interactions
:"
#(8)"
Using Table 1 one finds that fear and depletion causal paths have onset latencies beyond a week:"
#(9)"
Although the illness path, death caused by infection , also has an onset latency of a week, I
choose not to take it into account explicitly as a prior restriction. The three restrictions (9) already
suffice to disambiguate bicausal interactions at , and it is plausible that a highly vulnerable
person dies within a week after infection due to illness caused."
2.10 Time resolution and onset latency
Figure 7 illustrates how finite temporal data resolution affects observations and causal functions
. The underlying (inaccessible) time-continuous causal mechanism (with upper c) translates
to week resolution by a specific triangular-shaped time-symmetric aggregation filter :"
#(10)"
with continuous (real-valued) time in weeks. The triangular aggregation effect happens always,
no matter how fine the time resolution used. The week-resolution used in this report means that
any event pair separated by 3.5 days or more, is thus captured more by than ."
t
Δt
rI,rV,rM
FA→B
rI,rV,rM
I,V,M
σrI=σIσrV=σVσrM=σM
FA→B
σV→M= 10−3σM→I= 10−3σI→V= 1
σM→V= 102σI→M= 10−3σV→I= 1
FA→B
I,V,M
FA→B
σA→B=∞
FI→V(0) = FM→V(0) = FM→I(0) = 0
FI→M
Δt= 0
F
Fc
Λ
F(Δt) = {Fc*Λ}(Δt) = ∫Δtc
Fc(Δtc)Λ(Δt− Δtc)dΔtc
tc
F(1)
F(0)
of 11 22
"
Figure 7: Resolution effect example for vaccination to mortality. Event pairs separated by 2 days
are captured by and , randomly with probability 5/7 and 2/7 respectively,
depending on location of events within the week. The amounts are described by triangular
function of separation duration."
Most important for this report, any event pair separated by a week or more will under no
circumstance affect , justifying the onset latency constraints (9)."
2.11 Bicausal IVM and simplifed VM model
The bicausal IVM model combines the causal model’s six paths as in Figure 4 into three bicausal
paths, by new functions (note the double arrow):"
#(11)"
Each will be measured and is subsequently uniquely decomposable into and ,
as the latter are both only valued for . Ambiguities at are completely resolved by
onset latency constraints (9). The scale of has the same scale as namely B per A,
and thus ’s scale of A per B is corrected for in (11). Note that the in (11) models the
exact same physical event as in (5), with same accuracy, but with a slightly different mathematical
representation. Under expected circumstances where causal path effects are relatively small
compared to the drivers, the difference is negligibly small."
#The combined functions can be chosen as or for each of the three path pairs in
causal model (5). I choose them to match the biological causal paths, at the same time arriving at
only two equations:"
#(12)"
Variances (7) combine via (11) into:"
#(13)"
From here, the simplified bicausal VM model without infections is:"
#(14)"
FV→M(0)
FV→M(1)
Λ
F(0)
FA↔B
FA↔B(Δt) = FA→B(Δt) + σ2
B
σ2
A
FB→A(−Δt) , − ΔT<Δt<ΔT
FA↔B
FA→B
FB→A
Δt≥0
Δt= 0
FA↔B
FA→B
FB→A
FB→A
FA↔B
FB↔A
M(t) = rM(t) + {V*FV↔M}(t) + {I*FI↔M}(t)
I(t) = rI(t) + {V*FV↔I} (t)
σV↔M= 10−3σI↔M= 10−3σV↔I= 1
M(t) = rM(t) + {V*FV↔M}(t)
of 12 22
with only one prior from (13). In [Re2], of this bicausal VM model was estimated by a
Pearson correlation trend factor:"
#(15)"
This estimate was sub-optimal; next follows an optimal estimation procedure."
2.12 Bayesian probabilities and estimating parameters
The Bayesian probability framework is an ideal, systematic tool to obtain conditional probability
densities of model parameters given observed . From the relevant
statistics as mean and variance of can be extracted. Denoting all drivers by ,
one obtains for the causal IVM model :"
#(16)"
First row: the intermediate roles of drivers are taken into account via integrals, and observations
and model parameters are reversed bringing up additional priors, of which the model parameters
are independent. Second row: prior is expanded in 6 causal paths and all
components. Row 3: prior is split in individal drivers and time . Rows 4-6: causal IVM
model (5) enters via three Dirac functions plus so-called determinant of Jacobian that
describes the multiple occurences of in the arguments. #The Jacobian is a huge -by-
σV↔M
FV↔M
FV↔M(Δt)[Re2] ≈COV(V(t), M(t+Δt))
VA R(V(t))
PF|I,V,M
F
I,V,M
PF|I,V,M
F
r= {rI,rV,rM}
F= {FV→I,FI→V,FI→M,FM→I,FM→V,FV→M}
PF|I,V,M=∫r
Pr,F|I,V,M=∫r
PI,V,M|r,FPr,FP−1
I,V,M=P−1
I,V,MPF∫r
PrPI,V,M|r,F
=P−1
I,V,M∏
FA→B∈F
∏
Δt
GσA→B(FA→B(Δt))
∫rI,rV,rM
∏
t
GσI(rI(t))⋅GσV(rV(t))⋅GσM(rM(t))⋅
det J(F)⋅δ(I(t)−rI(t)−{V*FV→I}(t)−{M*FM→I}(t))⋅
δ(V(t)−rV(t)−{M*FM→V}(t)−{I*FI→V}(t))⋅
δ(M(t)−rM(t)−{I*FI→M}(t)−{V*FV→M}(t))⋅
=K∏
FA→B∈F
∏
Δt
GσA→B(FA→B(Δt))⋅
∏
t
GσI(I(t)−{V*FV→I}(t)−{M*FM→I}(t))⋅
GσV(V(t)−{M*FM→V}(t)−{I*FI→V}(t))⋅
GσM(M(t)−{V*FV→M}(t)−{I*FI→M}(t))
=Gμ,Σ(F)∼e−1
2{(F−μ)†Σ−1(F−μ)}
r
PF
PF
FA→B
Δt
Pr
0≤t<T
δ
J
I,V,M
δ
J
3T
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matrix full of zeros, ones, and (minus the) values of all 6 causal paths . Luckily, this
determinant can be computed analytically:"
#(17)"
where the 2nd equality is due to onset latency constraints (9). At the 3rd equality in (16), the
determinant is gone, prior ’s value is unknown and replaced by constant ; the prior is
constant as are fixed, observed source data. Also, the integrals transfer the causal model
from the functions to the priors of the drivers . As the final result contains only Gaussians and a
constant, it must be a simple multivariate Gaussian distribution with mean and
covariance matrix as in the final row."
#For the bicausal IVM and VM models, is also 1 and a similar multivariate Gaussian is
obtained. Mean and variance of ’s single components or linear combinations such as can
be obtained analytically. For example, one finds for the bicausal VM model a so-called least-
squares/minimum-norm solution (with for clarity):"
"
#(18)"
where is the identity matrix, is matrix transpose, and ± separates mean and standard
deviation. If the prior variance on is completely relaxed ( ), one gets the so-
called least-squares solution:"
#(19)"
3T
FA→B
det J(F) = {1 −FM→V(0)FV→M(0) −FI→V(0)FV→I(0) −FI→M(0)FM→I(0)
−FI→M(0)FM→V(0)FV→I(0) −FI→V(0)FV→M(0)FM→I(0) }T
= 1
P−1
I,V,M
K
I,V,M
δ
r
Gμ,Σ(F)
μ
Σ
det J(F)
F
VFR
ΔT= 3
V=
V(2) V(1) V(0) 0 0
V(3) V(2) V(1) V(0) 0
V(4) V(3) V(2) V(1) V(0)
.
.
.
V(T−2) V(T−3) V(T−4) V(T−5) V(T−6)
V(T−1) V(T−2) V(T−3) V(T−4) V(T−5)
0V(T−1) V(T−2) V(T−3) V(T−4)
0 0 V(T−1) V(T−2) V(T−3)
,O=
0
0
1
1
1
Σ= (σ−2
MV†V+σ−2
V↔MI)−1
μ=Σσ−2
MV†M
FV↔M=μ±diag(Σ)1
2
VFR =O†μ±(O†ΣO)1
2
I
†
FV↔M
σV↔M=∞
μ= (V†V)−1V†M
Σ= (V†V)−1σ2
M
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Bayesian approaches are ideal to obtain probabilistic answers to hard, binary questions such as
whether “vaccination has a net mortal effect in the first few weeks” (hypothesis ), or not
( ). With mean and variance from (18), one finds:"
#(20)"
This is a true probability, not a so-called likelihood ratio between two hypotheses: As the set of
hypotheses is complete in all possible outcomes, there are no other (unknown)
competing hypotheses possible."
#Any prior belief in any of the two hypotheses has already been accounted for in the calculation
of . If one has a strong prior belief in , one should incorporate this by a negative-
mean expectation of in the first few weeks, which is counter to commonly accepted
knowledge that protective effects of vaccines do not occur in the first few weeks. Logically, one
cannot first accept the computation of , and subsequently reject (20) on the basis that it
does not incorporate an additional explicit prior biasing towards reflecting one’s beliefs."
3 Results
I apply the bicausal model to booster campaigns during 2022 in The Netherlands (population
~18M). Infection, vaccination and mortality data are unstratified, weekly, total absolute numbers
from public national sources [Src]. With PCR+ and sewer viral-particle-based infection data
available, the IVM model is used to determine the usability of sewer data and relevance of
infection confounders, all compared to the relative performance of the infection-less VM model.
With the VM model, a few additional experiments are performed with different short-term filters
and periods, mortality age groups, and relaxation of prior variances on causal interactions ."
#Finally, I apply the bicausal VM model to 30 European countries (~530M people) with data from
aggregation sources [Eur, Owi] in 2022 during weeks 10-43 (limited by data availability)."
3.1 Sewer data for infections in the IVM model
Sewer viral particle data are publicly available for The Netherlands, see Figure 3. In weeks 1-20 of
2022, PCR tests were performed in higher volumes; I use this period to compute a scale constant
of PCR+ tests per sewer viral particle, and convert the more objective sewer data over entire 2022
to PCR-equivalent infections ."
#Figure 8 shows results of the bicausal IVM model, with PCR and sewer infection data, weeks
9-50 during the vaccination campaigns. Although few measurements are significant, a few
observations can be made. PCR-based infection-to-mortality suggests a peak at ,
matching the expected average time from infection to death. The sewer-based curve peaks just
significantly at , suggesting sewer-based measurements lag by a two-week time delay.
Applying a minus-two-week-shift in sewer data to compensate the delay brings the sewer-based
infection-to-mortality peak also at ."
#The individual PCR-based vaccination-to-mortality measurements of and 5-week-net-
result -9±7% after vaccination are just significant, and the probability that the vaccine protects
against infection (hypothesis ) is 91%. The original unshifted sewer-based measurements
suggest that vaccination causes infections with a probability 83%. The 2-weeks shifted sewer
Hmortal
¯
Hmor ta l
VFR
Pr{Hmortal} = Pr{V FR > 0} = 1
2+1
2erf(μVFR
2σVFR
)
{Hmortal,¯
Hmor ta l}
VFR
¯
Hmor ta l
FV→M
VFR
¯
Hmor ta l
F
I
FI→M(2)
FI→M(0)
FI→M(2)
FV→I
H−
of 15 22
measurements solve this uncomfortable finding, but bring up a logical, causal issue: the shift
brings the peak towards , infection-induced-vaccinations via fear. This is, however, not
causally possible as the original unshifted sewer data cannot cause fear before being measured.
As can be seen in Figure 3, PCR+ testing was quite low for most of 2022, and its media-reporting
seem unable to have simultaneously produced the same required fear, as is also visible in Figure 8
by with 5-week-net-result -0.53±2.67."
"
Figure 8: Results for bicausal IVM model, infection-mortality and vaccination-mortality interactions,
with PCR+ and sewer measurements, weeks 9-50 of 2022 in The Netherlands. Black dotted lines
indicate week is assigned to right, biological causal path according to onset latencies. Y-
axis has biological path scale. Grey areas"indicate"±1 sigma interval. Pr indicates Bayesian
probability of the biological path to be net positive or negative (hypotheses and ).
FI→V(1)
FI→V
Δt= 0
H+
H−
of 16 22
These results suggest that sewer data should not be back-shifted in time, and that the viral
particles in the sewer are caused by vaccination. The possibility that sewer data is measuring
spike (S) particles created directly by vaccinations can be excluded, as there is no overlap
between viral genes used for sewer measurements (N/E) and vaccination (S) [Med]. Recent
research did find evidence that infections correlate strongly with vaccinations [Shr]."
#Using , a PCR-equivalent ratio of infections/reactivations per vaccination is found as
just-not-significant 0.35±0.37, see Figure 8 right-center. Figure 9 shows the same result with
: with more than 99% probability, vaccination causes viral particles to increase ( ) in
the first three weeks since vaccination. The amount of equivalent-PCR-infections per vaccination
is 0.50±0.21 (95%%CI%0.08-0.92). This number is very high; possibly the equivalent PCR+/sewer-
particle scale factor is not well estimated or applicable. An additional explanation is that besides
triggering infections, vaccinations may reactivate latent viral (particle) reservoirs."
"
Figure 9: The causal effect of vaccinations on infections as measured effectively by sewer viral
particles."
3.2 IVM versus VM model
Figure 10 shows and obtained via the IVM model via PCR+, plain and 2-weeks-
backshifted sewer-based infections, and the VM model without infections, in weeks 9-50 in The
Netherlands. In all four cases a causal temporal order effect can clearly be seen, with mortality
insignificant before vaccination and significant after. Also in all four cases, equals
0.09±0.03%, that is, vaccine-induced mortality in the same week of vaccination is near 0.1%, with
3-sigma confidence. Finally, again in all 4 cases, over 5 weeks is just insignificant at ca
0.07±0.09%."
#Apparently, infections did not confound mortality and vaccination in The Netherlands in 2022,
in a significant way measurable by the IVM model. Based on these results, I conclude that the IVM
model does not offer substantially different or better results compared to the simpler VM model
without infection confounders."
#The insignificance of appears caused by a negative after two weeks and an
increasing sigma when aggregating over weeks. The positive-negative-dynamics
of are consistent with vaccination being the last push to mortality for highly vulnerable
people. The probability that vaccination causes net mortality over the entire 5 weeks after
vaccination ( ) remains however ca 75%-80% in all cases. The found is lower but order-
comparable to that of my prior work [Re2], which found ~0.18% in weeks 9-34 of 2022, using
age-correction in vaccination-mortality data, and a suboptimal estimation procedure."
ΔT= 5
ΔT= 3
H+
FV→M
VFR
FV→M(0)
VFR
VFR
FV→M
FV→M
ΔT= 5
FV→M
H+
VFR
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"
Figure 10: Results for and with bicausal IVM and VM models, with and without
infection confounder, weeks 9-50, The Netherlands."
3.3 Short-term filter, period, relaxation of prior, age groups
Table 2 shows using the VM model, for different short-term periods , filters and
relaxation of prior variance . It shows that shorter periods lead to better/lower
result variances, and that varying filter has marginal influence. Most importantly, relaxing the
prior from zero-mean Gaussian with to a uniform unbiased prior with
has no significant influence."
#Figure 11 illustrates a few results: the significant result with without prior, an isolated
central peak in with , and the erratic effect of not using any short-term filter.
Figure 12 shows a result for age groups: significant is concentrated at ages 65+."
Table 2: and ( ) with bicausal VM model for several different short-term
periods, filters, and relaxation of prior variance.
FV→M
VFR
VFR
ΔT
W
σV→M=∞
ΔT
W
σV→M= 10−3
σV→M=∞
ΔT= 2
FV→M
ΔT= 16
FV→M
default VM model
0.12±0.04% (99.9%)
0.09±0.05% (97%)
0.07±0.08% (80%)
0.13±0.04% (99.9%)
0.10±0.05% (98%)
0.07±0.09% (79%)
0.14±0.04% (99.9%)
0.11±0.05% (97%)
0.07±0.09% (80%)
0.14±0.07% (97.7%)
0.13±0.10% (89%)
0.06±0.16% (66%)
ΔT= 5
ΔT= 3
ΔT= 2
= [-1 +3 -3 +1]
W
Relaxed prior
σV→M=∞
= [-0.25 +0.5 -0.25]
W
VFR
Pr{V FR ≥0}
of 18 22
"
Figure 11: A few results with short-term periods , and without short-term filter.
"
Figure 12: Results with bicausal VM model for several age groups (mortality data age-stratified,
vaccination data is all-ages). is concentrated at ages 65+."
3.4 Europe
I applied the bicausal VM model on 30 European countries using data from aggregators [Eur,Owi]
in weeks 10-43. Countries include Austria, Belgium, Bulgaria, Croatia, Czechia, Denmark, Estonia,
Finland, France, Germany, Greece, Hungary, Iceland, Italy, Latvia, Liechtenstein, Lithuania,
Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain,
Sweden, Switzerland and United Kingdom."
#Figure 13 shows results for The Netherlands, with data from the same national source used in
previous sections and the aggregation sources. Clearly, aggregator data has been processed.
This may involve smoothing or small accidental/deliberate data shifts in time, e.g. to align data
sets or optimize visualization for their online services. In this study, however, it negatively affects
the significance of measurements."
"
Figure 13: Results with bicausal VM model for The Netherlands, weeks 10-43, data from three
different sources [Src,Eur,Owi].
Figure 14 shows results for a selection of European countries. Clearly, the individual results lack
significance.
ΔT= 2
ΔT= 16
VFR
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"
Figure 14: Results for several individual countries, weeks 10-43 (data from [Eur], UK data was
available only at [Owi]).
Significant results are obtained when data from all 30 countries are combined. Table 3 shows
results for , with/without relaxation of prior expectations. Figure 15 shows a result with
: all are significantly different from zero, and all are not, evidence for a
significant mortality effect caused by vaccination, not involving a confounder. In absence of any
prior expectation, a of 0.35% ±0.10% is obtained according to Table 3 in the first 3 weeks."
Table 3: and ( ) of all countries combined, at , per data source, with
and without prior model."
ΔT= 3
ΔT= 5
FV→M
FM→V
VFR
Data source [Eur]
Data source [Owi]
0.26±0.08% (99.9%)
0.01±0.03% (59%)
0.35±0.10% (99.98%)
0.01±0.03% (60%)
Prior model variance
σV→M
10−3
∞
VFR
Pr{V FR ≥0}
ΔT= 3
of 20 22
"
Figure 15: Results for 30 European countries combined, weeks 10-43.
4 Conclusions and discussion
This report presents a causal model of infections, vaccinations and mortality (IVM), with main goal
to estimate Vaccine Fatality Ratio on the short-term of a few weeks. The model has few to
zero prior model parameters, which are unbiased in terms of causal effect direction (from A to B or
vice versa), sign (enforcing or supression, protection or damage), and even strength (strong or
weak). Bayesian probabilities are used to quantify all interactions. Five confounders are explicitly
taken into account, plus all long-term confounding using a filter that extracts short-term events
only. A simpeler VM model without infections and only one “bicausal” interaction is shown to
provide essentially the same results, indicating that during the analysis period, infections did not
play a significant confounding role. "
#Evidence was found of a causal effect from vaccination to mortality during booster campaigns
in the Netherlands (2022 weeks 9-50) and Europe (weeks 10-43 due to data limtations). A positive
Vaccine Fatality Ratio was found within 2-3 weeks after vaccination of 0.13%
(0.05%-0.21%, 95% CI) for The Netherlands, and 0.35% (0.05%-0.55%) for Europe. These s
transcend the of covid substantially [Io2]."
#The high on the short-term was found to be partially compensated a few weeks later. A
single, partially age-stratified experiment did indicate that vaccine-induced mortality focuses on
the 65+ age group. This supports the mechanism of very frail elderly whose death is accelerated
1-2 weeks due to vaccination, associated with a low loss of QALYs (Quality Adjusted Life Years). If
present, this mechanism is only partial; over 5 weeks still has 68-80% probability of being
net positive."
#Additionally, experiments using the IVM model with sewer-viral-particle data in The Netherlands
suggested vaccination induces covid-infections and/or reactivates latent viral reservoirs, at a rate
scaled to equivalent-PCR-infections per vaccination of 0.50 (0.08-0.92). Recent research reported
that infections strongly correlate with vaccinations [Shr]."
#This study was limited in many ways. The available source data was not case-based but
national weekly overall rates. The short-term filter approach is insensitive to all long-term effects.
Significance levels were very low for many measurements. The method used is very sensitive for
preprocessing that data aggregators may apply, e.g. smoothing or small accidental/deliberate
data shifts in time, e.g. to align data sets or optimize visualization. Infection data was of low
reliability, by nature. Finally, my method has all kinds of flaws unknown to me, to all [Bre]. Despite
all this:"
#The evidence of a causal relationship from vaccination to infections and mortality is a very
strong alarm signal to stop the current mass vaccination programs."
VFR
VFR
VFR
IFR
VFR
VFR
of 21 22
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