The standard statistic used to assess vaccine efficacy was used
until relatively recently without an understanding of the causal
model to which it was relevant. The mistaken belief that this
statistic was just describing what was observed in vaccine trials
and did not depend upon any model for its use has led to many
errors in the study and use of vaccines. This chapter introduces
this standard statistic used to measure vaccine efficacy and the
model parameter for which that statistic is an estimate. The
relevant model is not a realistic one for most vaccines. It serves,
however, as a basis for exploring some of the indirect effects
of vaccines.
The concept of a critical vaccination level has been built upon
this rather unrealistic model of vaccine effects. We examine
the dependence of this critical vaccination level concept upon
the assumption of random mixing and demonstrate how severely limited
this concept is by this most unrealistic of assumptions. The
estimation of the vaccine effect in this model is also shown to
be highly dependent upon the assumption of random mixing.
The problems in using this statistic cannot be fully explored
within the context of the model in which it is based.. The behavior
of the statistic in more realistic models must be explored to
accomplish that purpose. Therefore a full exploration of the
problems with this statistic are left until the next chapter when
alternative models of vaccine effects are explored.
Immunization is one of the basic preventive activities in Public
Health. Its importance is likely to grow rapidly in the next
couple of decades. During the last decade we saw only a few new
vaccines come into wide use and we saw many vaccine makers abandon
production because they became afraid of law suits. But biotechnology
is now providing an explosion of new opportunities for effective
infection control through vaccination. New vaccines are becoming
more potent as we begin to understand the nature of adjuvant reactions
that increase the immune response to specific agents. They are
becoming more targeted and are generating fewer side effects as
we identify specific epitopes (or antigenic sites) that are crucial
to infection control. Vaccines targeted toward common infections
are promising to significantly lower total health care costs in
an era when the switch in health care funding should lead to new
sources of support for immunization. The newer vaccines on the
drawing boards do not depend upon agent replication in the host.
Thus they generate little if any interference so that multiple
vaccines can be administered simultaneously and the cost per vaccine
administered is lowered correspondingly.
Examples of new or newly altered vaccines that have recently come
into use are the Haemophilus influenza B vaccine, the new modifications
of the Hepatitis B vaccine, and the Hepatitis A vaccine. Many
vaccines are under development for enteric infections and many
common respiratory infections are susceptible to control through
vaccination. In the future we can envision vaccines for many
sexually transmitted infections. HIV vaccines could play an especially
crucial role along with risk behavior reduction in an integrated
program designed to stop the spread of infection.
I maintain that to cost-effectively and safely use the new biological
vaccines being developed, we need a new population science of
vaccine effects. That science should link the mathematics used
in epidemiological study design and analysis more closely to the
biological effects of vaccine induced immunity. In other words,
we need better models of vaccine effect upon which to base epidemiological
studies of vaccines. An alternative position might be that the
models we have are fine and that better data alone can advance
epidemiological studies of vaccines so that they can optimize
the choice of vaccines. Over the course of this chapter and the
next I will attempt to demonstrate that it is not only better
data that is needed. Better models and a better understanding
of when different models are applicable are needed even more urgently.
There are several major deficiencies in the way epidemiologists
have approached vaccine effects. First they seem almost never
to formulate a model about how the vaccine is producing its effect
in the individual so that this model can be integrated into a
population model. They have calculated the statistic called vaccine
efficacy but they have been unclear what model parameter this
statistic is meant to estimate. They often act as if the statistic
itself is of interest no matter whether or not it has any meaning
in terms of giving an indication about the value of a parameter
in some model. They then proceed to use their statistic, however,
as a parameter estimate without apparently being aware that they
are doing so. This represents a fundamental deficiency in the
thinking of epidemiologists with regard to the relationship between
statistics and parameters and their use. I hope that the experiences
of this chapter and the next will make you more aware of these
relationships.
Second, epidemiologists seem only rarely to evaluate the consequences
of vaccine adminstration on transmission system dynamics. When
they do so, they usually only do so in a very qualitative fashion.
The reason for this deficiency is that the analytic traditions
of epidemiology have been developed for non-transmissible diseases
and there is no tradition of transmission system analysis using
transmission models like those presented in the last two chapters.
In the non-infectious disease tradition all of the effects of
risk factors and preventive agents act directly upon the individual
exposed to the risk factor or the preventive agent. When a non-infectious
risk factor or a preventive agents acts upon one individual, it
is assummed that the risks of only that individual are changed.
When a risk factor causes infection in an individual, however,
or when a vaccine prevents an infection, a new source of infection
is created or prevented for other individuals.
There is a set of analytic traditions that has been developed
over almost a century that are intended to address some aspects
this situation. But these traditions have been developed by mathematicians
rather than epideminologists and they are almost wholly unused
by epidemiologists. Because epidemiologists have not been very
actively involved in creating these analytic traditions, they
have for the most part been developed along lines that epidemiologists
either do not understand or see as irrelevant to the issues that
they face. Epidemiologists have used the concept of herd immunity
to express the indirect effects of vaccination on unvaccinated
as well as vaccinated individuals. This concept, however, has
rarely been expressed in transmission system terms. The effect
of vaccination on the R0 or upon the force of infection
is rarely conceptualized.
A third deficiency in the way epidemiologists approach vaccines
is that they do not integrate an accurate biological conceptualization
of vaccine effects into the way they analyze those effects in
populations. That is to say they do not use population models
of vaccine effects that correspond to the vaccine actions that
they believe are taking place at the individual level. Consequently
they most often disregard the effects of vaccines on the course
of infection in vaccinated individuals which reduce the total
contagiousness of vaccinated individuals when these individuals
become infected. These contagiousness effects may often be the
most important effects of vaccines upon transmission dynamics.
Transmission can be equally stopped by decreasing either susceptibility
or contagiousness. To see this consider the case when everyone
in a population has been vaccinated. In Chapter 5 we saw that
R0 = cßD. If everyone is vaccinated R0
will be equally reduced to the extent that ß is reduced
by reducing the susceptibility of vaccinated individuals or by
reducing the contagiousness of individuals. If R0
is the same after either reduction, the effect of vaccination
on infection level in the population will be the same after either
effect.
Epidemiologists often conceptualize vaccine effects as preventing
infection. Some vaccines are conceptualized as preventing disease
rather than infection. But when vaccines are conceptualized as
affect transmission dynamics, they are seen as doing so by preventing
infection. This is an abstraction which may not be ideal.
Vaccines never eliminate susceptibility completely. They induce
a memory immune response that, once an infectious process begins,
can be brought into action to control that infection. No vaccine
induced immune response is likely to completely prevent the uptake
of an infectious agent and thus completely prevent the initiation
of some infectious process. Sometimes the vaccine induced immune
response might come into play so quickly and effectively that
only a couple the originally invading infectious agents succeed
in replicating in a host. In that case, the effect of the immune
response to vaccination can justifiably be abstracted as a prevention
of infection. But there may be very few situations where this
abstraction is justified. There are probably no vaccines in use
today that will consistently over time keep an initial bolus of
infectious agent from reproducing in the host to several times
the number of agents involved in the original invasion. The majority
of immune responses stimulated by vaccines will, however, accelerate
the immune response to an invasion and slow the initial reproduction
of infectious agent so that extremely high levels of infectious
agent, and high contagiousness of the infected person, are not
achieved.
Another deficiency in the modeling of vaccine induced control
of infection is the temporal limitations of protection. All known
immune responses by vaccines can be expected to wane over time
if there is no boosting by natural infection. That waning will
in most cases result in some cases where the agent will reproduce
enough in the host to cause disease or to have some degree of
contagiousness. We have in the past thought of immunity as being
lifelong for some agents like measles. But boosting by aymptomatic
natural infection was very likely the reason we saw lifelong immunity.
This issue of waning is not addressed in this chapter or the
next but I hope that someone might see that the models presented
in the next chapter could be readily modified to address this
issue and that it could make a good topic for a term paper.
New logic, new mathematics, and new study designs are needed to
address the extent to which reduction of infectious agent replication
in a host by a vaccine augmented immune response will reduce the
level of agent replication in the host and slow the spread of
agent through a population. For these to be pursued, much less
adopted, there will have to be considerable cultural change in
the way epidemiologists conceptualize their work and their roles.
There will have to be much more focus on the patterns of infection
spreading contact in populations. The unit of interest in epidemiological
inquiry will have to be moved from the individual at risk from
their environment to the ecological system that is affected by
the events experienced or the actions taken by individuals.
The nature of cause and how one makes inductions and deductions
about cause from available evidence will also have to be changed.
Historical and observational evidence involving unique events
at the population level will have to assume a new importance.
Epidemiologists usually conceptualize causal events as potentially
reproducible micro-events in a sample selected from huge numbers
of individuals experiencing those events. The assumption that
entire populations of individuals can be conceptualized as having
the essential elements to experience the same causal effects as
a studied sample of individuals is the current basis for making
causal conclusions in epidemiology. But the emergence of infectious
agents in an epidemic or an endemic pattern for the most part
are unique events in the history of our world. Likewise the optimal
vaccination and behavior change strategy for the control of infection
in any particular population are likely to be unique. Complexity
science is addressing the issue of how one can make inferences
from unique system behaviors. In order for epidemiologists to
address the assessment of vaccines completely, they will have
to learn the lessons that complexity science is learning in this
regard.
We begin by examining the old standard statistic used to assess
vaccine efficacy in epidemiological studies and vaccine trials.
This statistic was thought to describe vaccine effects sufficiently
well to provide a basis for choosing between different vaccines
and for deciding whether it was worthwhile to adopt a new vaccine
for widespread use. It was not designed to estimate the parameter
of any particular causal model.
This statistic was intended to reflect vaccine effects at an individual
level. That is to say, it was designed to assess the effects
of vaccines upon the risks of infection experienced by individuals
rather than the level of infection experienced by a population.
We saw in Chapter 6 that infectious disease transmission parameters
reflect interactions between individuals in the social plane.
In the next chapter we will examin vaccine effect statistics
that act in this social plane as well. But the old standard vaccine
efficacy statistic was not designed to reflect effects upon transmission
during interactions between individuals. It was designed to assess
individual risks and it required a peculiar sort of logic to sustain
the use of this statistic when it was clear that one of the vaccine
effects of primary interest was the effect of reducing the risk
that infection would be transmitted to the vaccinated person.
The outcome which the standard vaccine efficacy statistic assessed
could be either infection or disease. The issue of whether different
statistics would be appropriate for different outcomes was not
one that seemed to concern epidemiologists using this statistic.
The outcome of disease given infection seems clearly to be an
individual level outcome for which an individual based statistic
would be appropriate. When disease was used as an outcome, however,
it was exceedingly rare for that outcome to be conditioned on
the infection status. Thus when disease was the outcome, the
effects being estimated were a combination of transmission of
infection and development of disease given transmission. Many
of the uses to which this statistic was put reflected a desire
to assess the effects of vaccination upon transmission. Indeed
many transmission modelers were led to use this statistic in the
context of transmission models in order to assess issues like
what the critical level of vaccination ise in a population that
will result in the elimination of transmission.
The use of this statistic in transmission models eventually led
to a clarification of what statistic in what sort of transmission
model this statistic estimated. The exercize you will perform
in this chapter should make that clear to you. The parameter
that this statistic estimates is indeed at the individual level.
The statistic does not estimate a parameter acting during the
interactions between individuals where transmission might take
place. It estimates a parameter that puts individuals in one
category or another. Let us now take a look at this statistic
and consider its interpretation in the context of the individual
effects models we examined in Chapter 4.
VE (vaccine efficacy) =
VE =
.................... (equation 1)
The Risk might be risk of disease or risk of infection. This
distinction makes a huge difference when we consider the transmission
dynamics implications of the statistic but it makes little difference
when we consider its interpretation at the individual level.
In terms of chapter four, if the assumptions about independence
between individuals held, that statistic would reflect the fraction
of unvaccinated and infected individuals whose infection would
be attributable to being unvaccinated. In other words, the fraction
of infections in unvaccinated individuals which would be prevented
by vaccination. As we saw in chapter 6, however, this interpretation
is quite meaningless for the type of risk factor effects that
we examined in that chapter. It would seem that vaccine efficacy
would reflect an effect on the susceptibility of individuals and
we saw in Chapter 6 that such effects are highly dependent upon
the transmission context in which they occur. This is true both
for their population effects and their individual risk effects.
It turns out, however, that there is a particular type of parameter
which is wholly at the individual level but that can be part of
a transmission model which the standard VE statistic can estimate
under certain conditions. This parameter, however, implies a
biological mechanism that is very rare for vaccines. You will
see this in an exercize which demonstrates the causal model for
the parameter which this statistic estimates.
The vaccine mechanism implied by the parameter which the standard
VE statistic estimates is as follows: A vaccine induced immune
response is able to absolutely prevent infection in successfully
vaccinated individuals. In unsuccessfully vaccinated individuals
in whom it fails to prevent infection, vaccination has no effect
whatsoever on the course of infection. Thus this standard vaccine
efficacy statistic corresponds neither to what we know of the
biology of vaccine induced immunity nor the population dynamics
of infectious diseases. That does not mean that calculation of
this statistic is not useful. It just means that we need to understand
better how to use this statistic and how to avoid its misuse.
Let us explore this vaccine efficacy statistic more thoroughly.
As is the case with the epidemiologic statistics reflecting exposure
effects presented in Chapter 4, vaccine efficacy is usually calculated
as a dimensionless proportion expressing a risk rather than a
rate. We saw that the risk difference estimates the proportion
of the exposed population that gets disease attributable to exposure.
We saw that the etiologic fraction is the proportion of the diseased
population that gets disease attributable to exposure. The classical
measure of vaccine efficacy can be viewed as the proportion of
the exposed population with disease that has that disease attributable
to the exposure. Exposed in this case means the unvaccinated.
With regard to the risk difference and the etiologic fraction,
we noted that our interpretations of these measures in the above
terms depended upon the assumption that the exposed and the unexposed
were just alike with regard to their risks of disease except for
the exposure. Another assumption necessary for the above interpretation
of these measures is that risks to individuals will not change
as other individuals change their exposures or develop disease
as a result of their exposures. Both of these assumptions are
equally important to the standard interpretation of the meaning
of vaccine efficacy. For most infectious diseases in most populations,
however, these assumptions are quite erroneous. But all models
are wrong in the sense that they are simplifications that fail
to take into account some aspect of reality. That doesn't mean
that the models are useless.
Let us consider again the elements of this statistic.
The unimmunized are the higher risk group so we can think of them
as the exposed. The immunized are the lower risk group and we
can think of them as the unexposed. Then the above formula becomes
the risk difference divided by the risk in the exposed.
The 
is called the attributable risk percent
in Hennekens textbook. It is called the attributable proportion
by Rothman. It is not dealt with in Kramer's text and in various
other places is called the relative attributable risk, the population
attributable risk percent, and even the attributable risk and
the etiologic fraction. Note that in Miettinen's 1974 paper defining
the etiologic fraction, he also presented this statistic and called
it the etiologic fraction in the exposed and then Rothman and
Hennekins cite the original paper as calling this fraction the
etiologic fraction. The terminology has been hopelessly misused
because of inadequate understanding of the meaning of the interpretation
of the absolute risk effect measures.
Given the interpretation of the risk difference which we developed
in Chapter 4, we can interpret the above formula as:
................................(equation
2)
Putting this back in terms of immunized and unimmunized, this
is the fraction of cases in unimmunized individuals which are
attributable to being unimmunized. Another way of saying this
would be the fraction of cases in the unimmunized that would have
been prevented by immunization. But remember, this interpretation
depends upon assumptions which are just plain untenable for most
infectious diseases.
Most people presume that this measure averages in some way across
all the different individuals in a population whose personal measures
of vaccine efficacy could be different. They presume that the
measure can thus be used as a population measure. But individual
effects do not sum up to population effects for infectious diseases.
There are three reasons for this. 1) There are always indirect
effects that go beyond the individuals directly affected by an
exposure. 2) Escaping a causal exposure may not result in an
escape from infection because there are so many other sources.
3) Infection may generate immunity which protects one against
the effects of other sources of infection.
Just because individual effects will not add up to population
effects does not mean that there is not a transmission model in
which an individual based parameter is not meaningful. Let us
examine a transmission model where this parameter is in fact meaningful.
The transmission model where an individually based vaccine effect
model is meaningful is called the "all or none" model.
In this model, the immune response to the vaccine is assummed
to always be so effective upon exposure to the infectious agent
that no symptoms of infection will develop and in fact so little
of the agent will be reproduced in the vaccinated host that the
vaccinated host will never become contagious. It is also assummed
that if the vaccine fails to protect an individual against one
exposure to an infection, it will fail to protect them against
all other exposures and that if it protects them against one dose
of exposure to the virus, it will protect them against all exposure
doses.
In this model chance acts at the time of immunization to determine
whether or not an individual benefits from a vaccine. The vaccine
acts on the individual, not on that individuals interactions with
others. Once the individual has benefitted, there is no chance
acting to determine whether that individual will be protected
from a particular exposure. If an individual has benefitted from
the vaccine, that benefit is absolute protection against all exposures.
If an individual did not benefit from a vaccine effect at the
time of vaccine administration, then at the time of exposure to
an infectious agent the vaccinated individual's chances of getting
infected will be the same as if the individual had never received
any vaccination. Also, when vaccinated individuals become infected,
their infection will last just as long as they would if they had
not been vaccinated and they will be just as contagious as they
would have been if they had never received any vaccination.
The vaccine to which this model is probably most applicable is
the measles vaccine. With this live virus vaccine, the chance
event at the time of vaccination which determines whether or not
the vaccine will be effective is whether or not the vaccine infection
takes off in the host. If the vaccine strain has been killed
because the practitioner administering it left their refrigerator
door open, then there will be only a negligible amount of measles
antigen that stimulates an immune response in the vaccinee. This
response might not affect the risk of infection nor the course
of infection at all. On the other hand, if the vaccine infection
does take off, a much broader set of immune responses will be
stimulated than would be the case if the vaccine antigens were
not part of a replicating virus. This broader response might
be highly effective if not absolutely effective against the risk
of subsequent infection.
In summary, if upon vaccination the dice fall in favor of one
getting the "all" response to vaccination, then all
subsequent infections which that vaccinated individual might get
can be disregarded because the immune response neutralizes them
so quickly and effectively. On the other hand if athe dice gave
the "none" response to vaccination, then one's risks
of infection and course of infection would be no different than
those of an unvaccinated individual.
This model probably never applies in reality because the protection
against infection of every vaccine turns out to be relative to
some degree. We know now that in fact low level measles infections
that produce no symptoms are common in both individuals who have
had natural infection and in vaccine recipients. Even right after
immunization, some measles virus replication takes place upon
exposure to measles virus. Moreover, over time that amount of
virus replication can increase significantly. But if the infection
that is allowed is almost always non-contagious and does little
more than give an individual a boost in their immunity to an agent,
then for many purposes it can be disregarded in a transmission
model.
Under the all or none effects model, we assume that we have no
nefarious situations where vaccination might have caused infection.
Then under this model a fixed proportion of the immunized will
be protected no matter what the dose of exposure. Lable that
proportion as "p" and the proportion that is not protected
as "1-p". Then the total risk in the vaccinated is
the average risk between the truly protected fraction and the
unprotected fraction or
Rv = 0*p + (1-p)*Ru =
Ru(1-p)
since the risk in that fraction "p" who got the protective
effect is zero and the risk in that fraction "1-p" who
did not get it is the same as the risk in the unvaccinated population
when there is random mixing. The Standard Vaccine Efficacy Statistic
is then
........................(equation
3)
With this equation, we see that under the all or none effects
model given random mixing, the standard vaccine efficacy statistic
is independent of the level of infection. We also see that the
standard vaccine efficacy statistic reflects the basic parameter
of vaccine efficacy in the all or none effects model.
We consider first the simplest SIR model. Then we integrate the
insights from this model into the general exposure effects model
that we developed for Chapter 6.
In the "all or none" model of vaccine action, we could
theoretically classify everyone who has been administered a vaccine
as being protected or not being protected. When a vaccine "takes"
in this model, that is to say when an administered vaccine proliferates
enough to stimulate an immune response, the effect in the context
of an SIR model is to move individuals from the "S"
state to the "R" state. Infection with the vaccine
virus is presumed to have the same effect on immunity as infection
with wild virus. If we are considering an illness like influenza
and we do all of our immunizing before the seasonal influenza
epidemic begins, then the number of individuals moved from the
S to the R state would be the number of S individuals immunized
times the proportion of the immunized who are effectively immunized.
We will label this parameter "p" as we have done above.
This "p" is equal to the vaccine efficacy statistic
if the all or none effect model holds and mixing is random.
Thus the two basic parameters of immunization in the population,
namely the proportion of the population immunized and the proportion
of the immunized who are effectively immunized do not connect
to any flows in the model of the transmission system in which
we are modeling vaccine efficacy. They merely determine the starting
numbers in the S and R states for the epidemic simulation. I
have constructed a model of a randomly mixing population where
vaccination affects only the starting conditions according to
the model of all or none effects. You can find this in the Public
folder of my IFS space as "AllNone". It has the population
first divided in vaccinated and unvaccinated segments, each of
which is then divided into S, I, and R segments. There are two
kinds of vaccinated individuals who are immune individuals: 1)
individuals in whom the vaccination "took" so that they
were completely protected and 2) individuals in which the vaccine
did not take and who entered this segment after they later became
infected with the wild virus. The first class of individuals
cannot move to any other state in this model and there is no way
to get there except by vaccination before the epidemic is run
so they have no flows in or out.
Difference equations for the population having some vaccinated individuals:
------------------------------------------
SUnV(t) = SUnV(t - dt) + (- NewIUnV) * dt
INIT SUnV = (1-PropVaccProg)*(1-InitialInfectionSeed)
NewIUnV =
SUnV*c*beta*((IUnV+IV)/(IUnV+IV+RUnV+RV+SUnV+SV+VaccImmune))
SV(t) = SV(t - dt) + (- NewIV) * dt
INIT SV = (PropVaccProg)*(1-p)*(1-InitialInfectionSeed)
NewIV = SV*c*beta*((IUnV+IV)/(IUnV+IV+RUnV+RV+SUnV+SV+VaccImmune))
IUnV(t) = IUnV(t - dt) + (NewIUnV - NewRUnV) * dt
INIT IUnV = (1-PropVaccProg)*InitialInfectionSeed
NewIUnV =
SUnV*c*beta*((IUnV+IV)/(IUnV+IV+RUnV+RV+SUnV+SV+VaccImmune))
NewRUnV = IUnV/Dur
IV(t) = IV(t - dt) + (NewIV - NewRV) * dt
INIT IV = (PropVaccProg)*(1-p)*InitialInfectionSeed
NewIV = SV*c*beta*((IUnV+IV)/(IUnV+IV+RUnV+RV+SUnV+SV+VaccImmune))
NewRV = IV/Dur
RUnV(t) = RUnV(t - dt) + (NewRUnV) * dt
INIT RUnV = 0
NewRUnV = IUnV/Dur
RV(t) = RV(t - dt) + (NewRV) * dt
INIT RV = 0
NewRV = IV/Dur
VaccImmune(t) = VaccImmune(t - dt)
INIT VaccImmune = (PropVacc)*p
Difference equations for the control population that had no vaccination
--------------------------------------------------------
SUnV_2(t) = SUnV_2(t - dt) + (- NewIUnV_2) * dt
INIT SUnV_2 = (1-InitialInfectionSeed)
NewIUnV_2 = SUnV_2*c*beta*((IUnV_2)/(IUnV_2+RUnV_2+SUnV_2))
IUnV_2(t) = IUnV_2(t - dt) + (NewIUnV_2 - NewRUnV_2) * dt
INIT IUnV_2 = InitialInfectionSeed
NewIUnV_2 = SUnV_2*c*beta*((IUnV_2)/(IUnV_2+RUnV_2+SUnV_2))
NewRUnV_2 = IUnV_2/Dur
RUnV_2(t) = RUnV_2(t - dt) + (NewRUnV_2) * dt
INIT RUnV_2 = 0
NewRUnV_2 = IUnV_2/Dur
Parameter values including initial conditions
------------------------------------------------------
beta = .5
c = 2
Dur = 2
InitialInfectionSeed = .0001
p = .8
PropVaccProg = .5
Derived variables
-------------------------------------------------------------------
VEstat = 1-((IV+RV)/(IV+RV+SV))/((IUnV+RUnV)/(IUnV+RUnV+SUnV))
TotInfVaccPop = IUnV+IV+RUnV+RV
TotInfContPop = IUnV_2+RUnV_2
TotInfPrev = TotInfContPop-TotInfVaccPop
Homework 7.1
a) Describe how the vaccine efficacy statistic relates to the all or none vaccine effect parameter at different values of the parameters c, beta, dur, and p, and given different proportions of the population which are vaccinated.
b) Explain why the relationships in a) were observed.
c) List the assumptions in this model on which the relationships in a) depend.
d) Describe how the total number of infections prevented relates to the all or none vaccine effect parameter at different values of the parameters c, beta, dur, and p, and given different proportions of the population which are vaccinated. When "p" is low, determine at which other parameter settings the herd immunity effects are the greatest and the least. Do the same for high values of "p"
e) Suppose one has a VEstat estimate from
another population where the all or none mode holds. Explain
what one has to do to predict what fraction of a randomly mixing
population will have infection prevented given vaccination in
a specific fraction of the population.
Many modelers using models precisely like those we have worked
with above have examined the proportion of the population that
needs to be vaccinated in order to stop transmission. Transmission
can be stopped by getting the reproduction number below 1. The
reproduction number (not the basic reproduction number) is lowered
by getting a higher fraction of the popultion into the VaccImmune
category in which vaccination has successfully taken to provide
complete protection. This could be called the proportion of the
population that needs to be effectively immunized in order to
protect everyone through herd immunity effects. More commonly
it is just referred to as "the critical vaccination level".
The concept of critical vaccination levels has been developed
almost wholly using the assumptions of the quite unrealistic "all
or none" model. But the concept of critical vaccination
levels depends upon many other assumptions besides those of the
"all or none" model. The only models with a single
critical immunization level are models which assume that vaccination
has been randomly administered and that everyone mixes homogeneously.
These models are very far from any feasible reality for any infection
or for any realistic immunization program. I think that promoting
the idea of a single critical immunization level does a disservice
to the practice of Public Health and that there should never be
any occasion to calculate a single percentage of a population
that needs to be immunized in order to stop transmission. What
should orient our immunization programs should be the need to
eliminate foci of individuals in contact with each other who can
sustain chains of transmission. Programs with such an orientation
will not judge their success by what fraction of their population
has been vaccinated but rather by whether they have reached key
populations they need to reach in order to stop transmission.
But the calculation of critical immunization levels is widespread
in the literature and there seem to be a great many people who
are willing to accept conclusions drawn from models which assume
homogeneous mixing. As a consequence, we should understand the
logic of such calculations.
In the all or none model of vaccine effect, the reproduction number
at the start of the epidemic will be 
.
This is not the initial reproduction number because not everyone
is susceptible. But if this number is less than one at the beginning
of the epidemic, each case will generate less than one other case
and there will be negative exponential growth in the number of
cases rather than positive exponential growth.
The fraction 
in the above formula for the
reproduction number is one minus the proportion of the population
that is immune after an immunization program. Let us denote the
proportion of the population that is immune after an immunization
program as P(I). This might be called the effective vaccination
level. It is the fraction of the population which is vaccinated
times the fraction of the immunized population that got the "all"
response to vaccination rather than the "none" response.
We label the critical P(I) above which the reproduction number
falls below one as Pc(I).
P(I) = 1 - 
so 
= 1 - P(I)
If one knows the basic reproduction number, that is ßcD, then one can calculate Pc(I) as:
Pc(I) = 1 - 
or
1 - 
.....................(equation
3).
Thus if you are willing to accept a model with homogeneous susceptibilities
and exposure and homogeneous mixing and if you are willing to
accept the all or none model of vaccine effect and if you have
calculated the initial reproduction number, you can calculate
the critical level at which effective immunization will stop an
epidemic. It is worthwhile just plotting out the function for
Pc(I) to see how ¬o
affects it.
Beyond a basic reproduction number of one, the proportion of the
population that must be immunized to stop transmission given our
rather unrealistic model rises very quickly. At a basic reproduction
number of 20, 95% of the population must be effectively immunized.
That means that given a vaccine efficacy of 94% assuming the
"all or none" model, you could never get above the critical
level. Although reality might not conform to the model underlying
the above graph, the general conclusion that as ¬o
increases the effects of herd immunity from a given level of immunization
decreases appears to be quite robust to model modifications that
make the underlying model more realistic.
Homework 7.2
a) Calculate the critical vaccination value
for three different widely divergent sets of all or none model
parameters and then demonstrate empirically using Stella that
those critical vaccination values in fact correspond to thresholds
where there will or will not be epidemic transmission.
We will now demonstrate how critically dependent the concept of
critical vaccination levels is to assumptions about mixing patterns.
In chapter 6 we introduced the preferred mixing formulation as
an easy way to bias mixing between different exposure groups.
Just as we did there, we will use a "reserved fraction"
that is the same for all subgroups. That then gives us the following
model which you can find as AlNoPref in my Public file.
DIFFERENCE EQUATIONS
----------------------------------------
IUnV(t) = IUnV(t - dt) + (NewIUnV - NewRUnV) * dt
INIT IUnV = (1-PropVacc)*InfectionSeed
NewIUnV = SUnV*c*beta*((rho*IUnV)/(IUnV+RUnV+SUnV)+((1-
rho)*(IUnV+IV)/(IUnV+IV+RUnV+RV+SUnV+SV+VaccImmune)))
NewRUnV = IUnV/Dur
IV(t) = IV(t - dt) + (NewIV - NewRV) * dt
INIT IV = (PropVacc)*(1-p)*InfectionSeed
NewIV = SV*c*beta*((rho*IV)/(IV+RV+SV+VaccImmune)+((1-
rho)*(IUnV+IV)/(IUnV+IV+RUnV+RV+SUnV+SV+VaccImmune)))
NewRV = IV/Dur
RUnV(t) = RUnV(t - dt) + (NewRUnV) * dt
INIT RUnV = 0
NewRUnV = IUnV/Dur
RV(t) = RV(t - dt) + (NewRV) * dt
INIT RV = 0
NewRV = IV/Dur
SUnV(t) = SUnV(t - dt) + (- NewIUnV) * dt
INIT SUnV = (1-PropVacc)*(1-InfectionSeed)
NewIUnV = SUnV*c*beta*((rho*IUnV)/(IUnV+RUnV+SUnV)+((1-
rho)*(IUnV+IV)/(IUnV+IV+RUnV+RV+SUnV+SV+VaccImmune)))
SV(t) = SV(t - dt) + (- NewIV) * dt
INIT SV = (PropVacc)*(1-p)*(1-InfectionSeed)
NewIV = SV*c*beta*((rho*IV)/(IV+RV+SV+VaccImmune)+((1-
rho)*(IUnV+IV)/(IUnV+IV+RUnV+RV+SUnV+SV+VaccImmune)))
VaccImmune(t) = VaccImmune(t - dt)
INIT VaccImmune = (PropVacc)*p
MODEL PARAMETERS
------------------------------
beta = .5
c = 2
Dur = 2
InfectionSeed = .0001
p = .8
PropVacc = .625
rho = .8
DERIVED VARIABLES
-------------------------------
TotInfVaccPop = IUnV+IV+RUnV+RV
VEstat = 1-((IV+RV)/(IV+RV+SV+VaccImmune))/((IUnV+RUnV)/(IUnV+RUnV+SUnV))
Homework 7.3
Examine the different parameter sets you
presented in homework 7.2 but now with values for rho of 0.0,
0.4, and 0.8.
a
Determine how non-random mixing affects
the critical vaccination level in each of your three parameter
sets and describe these effects of non-random mixing as concisely
as you can. Explain why the relationships you observe occur.
b
Determine how estimation of "p" by the vaccine effect statistic is affected by non-random mixing in each of the different parameter sets. Explain the reasons for the relationships that you observed.