Exercise 4
The "All or None" model of Vaccine Efficacy
Professor
James S. Koopman MD MPH
This exercise presents the standard vaccine efficacy statistic calculated from risk of infection data. The major objective of the exercise is to present and explore the model upon which the proper use of that statistic depends. We will call this statistic VES(risk) for "Vaccine Effect on Susceptibility (estimated from risk data).
where Ru is the risk in the unvaccinated population and Rv is the risk in the vaccinated population.
Until recently, this statistic was viewed as "the" vaccine efficacy statistic. It was used without an appropriate understanding of the causal model to which it was relevant. Many epidemiologists mistakenly believe that this statistic just "describes what is observed in vaccine trials" and that its use does not depend upon any model. They would argue that it is just a descriptive statistic. We will see in this exercise and the next that under this philosophy, descriptions can be quite distorted and the use of VES(risk) can lead to important errors.
One can predict the population effects of vaccines on the basis of vaccine trial results only if one has an appropriate causal model of vaccine effects for the task and can estimate the parameters of that model on the basis of the vaccine trial results. In this exercise, we present the model for which VES(risk) provides a meaningful parameter estimate. We then explore some of the behavior of this model. To understand the consequences of misusing the standard vaccine efficacy statistic, we need to examine a more realistic model of vaccine effects. Thus we leave the illustration of the deficiencies of this statistic to the next exercise. We do, however, present some initial discussion of these deficiencies as background to the general issue of vaccine effect estimation.
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 being made more potent through a better understanding of adjuvant effects that increase the immune response to specific agents. Vaccines 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, the Hepatitis A vaccine, the new vaccines against otitis media caused by Streptococcus pneumoniae or Moraxella catterhalis, and rotavirus vaccines. 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 vaccines being developed, we need a new population science of vaccine effects. That science should link the effects of vaccines on the immune responses of vaccinated individuals to the population effects of vaccination on the circulation of infectious agents in populations. I find that the transmission models and vaccine effect statistics most commonly examined by epidemiologists are not up to this task. The past several years, however, have seen a proliferation of vaccine effect models. The progress toward developing the models needed to advance the science of vaccinology is notable. But more progress is needed. This chapter and the next two are designed to help position you to contribute to that progress.
An alternative position to the idea that the major deficiencies in vaccinology are the models used might be that better data alone can advance the use of vaccines. Over the course of this exercise and the next two 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.
Different models of the same reality are needed for different purposes. The currently dominant vaccine effect models and vaccine effect statistics were developed to test the hypothesis that a vaccine has an effect on the susceptibility of vaccinated individuals to either infection or to disease. The same VES(risk) statistic is used for this hypothesis testing when the outcome variable is infection as when it is disease. This statistic is fine for this purpose. The problem is that the hypothesis testing purpose is not what we need to develop, evaluate, and optimally employ vaccines.
The current needs of vaccine studies go considerably beyond the issue of hypothesis testing. This is especially true when we are developing new vaccines to replace old vaccines. We need studies that compare the population effects of different vaccines and different vaccine administration strategies. That means that we need studies that estimate causally appropriate parameters rather than just test hypotheses. It also means that studies must capture vaccine effects on the spread of infection from vaccinated individuals who become infected despite vaccination as well as the susceptibility to infection of vaccinated individuals. VES(risk) was never intended to do that. For some vaccines, such as the conjugated Haemophilus influenzae B vaccines, the effects on transmission have proven to be far more important than the effects on susceptibility to disease.
For these objectives, the vaccine effect models with parameters estimated by VES(risk) are inappropriate. It is the task of this exercise just to get you to understand the model implied when one uses this statistic. The next exercise will take one step in the direction of developing models that are more appropriate and will demonstrate how the inappropriate use of the standard vaccine efficacy statistic could lead to quite bad decisions.
We begin by examining the old standard statistic, VES(risk), used to assess vaccine efficacy in epidemiological studies and vaccine trials. 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 Exercise 3 that infectious disease transmission parameters reflect interactions between individuals in the social plane. In the next chapter we will examine 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 independent of the level of circulation of infectious agent in the population. Of course, vaccines alter the circulation of infectious agent. But because this was a relative measure where the absolute effect of vaccination was divided by the risk in the vaccinated, the hope was that this measure would be independent of the level of infection in a population. That is to say, the hope was that this measure would reflect the effect of the vaccine no matter what transmission risk factors existed in the population and no matter what the level of infection circulation was in the population. In this exercise, we will examine the model under which that hope is realized. In the next, we will examine a somewhat more realistic model and we will see that in reality this hope for stability of VES(risk) values is not realized.
The risk assessed by the VES(risk) statistic is either infection or disease. The outcome of disease given infection seems 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 VES(risk) 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 is 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. We will present that shortly. The parameter that VES(risk) estimates is indeed at the individual level. VES(risk) 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. The problem is, a model with this sort of parameter is unrealistic for almost all vaccines. Let us now examine this statistic and consider its interpretation in the context of the individual effects models we examined in Chapter 4.
can be divided by Ru to yield
VES(risk) = 1 - RRv
where RRv is the relative risk in the vaccinated as compared to the unvaccinated. 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. Only the risk of infection has meaning when this statistic is used as an estimate of a parameter in a transmission model. Whether the outcome is infection or disease makes little difference, however, when the VES(risk) is interpreted using standard attributable risk theory which has the hidden assumption we saw in previous exercises that no transmission is generating dependencies in the outcomes of different individuals.
In terms of standard attributable risk theory as presented in epidemiology 655, if the assumptions about independence between individuals held, VES(risk) would reflect the fraction of unvaccinated and infected individuals whose infection would be attributable to being unvaccinated. In other words, if the independence assumptions of attributable risk theory hold, VES(risk) estimates the fraction of infections in unvaccinated individuals that would be prevented by vaccination.
We can see this if we start by noting that if we treat the unvaccinated as the high-risk or risk factor exposed population, then VES(risk) is a risk difference divided by the risk in the exposed. (I will try to use "risk factor exposed" when the standard use of "exposed" in non-infectious disease epidemiology is used and "agent exposed" when contact has been made with an infectious individual.)
This is a dimensionless because the units of the numerator and denominator cancel out. It is a proportion as long the risk in the exposed is higher than the risk in the unexposed.
You have learned in 601 and 655 how the risk difference under standard attributable risk theory is the fraction of the exposed population with disease attributable to exposure. We designate the exposed population with cases attributable to exposure as "AC" for "attributable cases", the entire exposed population as E, and exposed individuals with disease whether or not that disease is attributable to exposure as ED. Then
and
So 
.
The numerator is the number of cases attributable to being unvaccinated and the denominator is the total number of cases in the unvaccinated.
As we saw in exercise 2, however, this interpretation is quite meaningless for risk factors for infection where the assumptions of independent attributable risk theory do not hold. That is because there are infections that are indirectly attributable to risk factor exposure as well as infections that are directly attributable to risk factor exposure. We saw in exercises 2 and 3 that even infections where the risk factor acted in their causal pathway and are therefore directly attributable to the risk factor may also be indirectly attributable to risk factor exposure. That is because they depend upon the action of risk factor exposure in other individuals in order to have had contact with an infectious individual.
The original concept behind VES(risk) was to have a measure of risk reduction attributable to vaccination that was at the individual level. VES(risk) was intended to capture the frequency of vaccine effects in the individual and not vaccine effects on the outcomes of interactions between individuals. We will now examine the transmission system model where this objective is achieved.
The transmission model where an individually based vaccine effect model is meaningful is called the "all or none" model. In this model, at the time of vaccination, an individual is assumed to get either an "all" response or a "none" response. The "all" immune response to the vaccine is assumed to always be so effective that it will prevent all transmissions when a vaccinated individual contacts an infectious individual. On the other hand, the "none" response is assumed to be so ineffective that it will under no conditions prevent transmissions that would have occurred in the absence of vaccination. This excludes, for example, vaccine effects such as increasing the effective number of agents that must be inoculated to cause infection in vaccinated individuals. The "none" response also assumes that vaccination will not affect the course of infection after transmission occurs to a vaccinated individual.
Let us restate the assumptions of the "all or none" model. 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 individual's interactions with other individuals. Once the individual has benefited, there is no chance acting to determine whether that individual will be protected from a particular exposure. If an individual has benefited 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 vaccinated individual. 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 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 the 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. Only during recent times has the circulation of measles virus been so low that low level and largely undetectable infections in vaccinated individuals failed to keep stimulating immunity and make that immunity relatively permanent. In the current era, both naturally acquired and vaccine acquired immunity can wane over time so that a significant chance of infection can reemerge.
Under the all or none effects model, we assume that we have no nefarious situations where vaccination might have caused infection. Given this assumption, a fixed proportion of the vaccinated will be protected no matter what the dose of exposure. Label 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).
This holds because 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. If there is random vaccination and mixing, then vaccinated and unvaccinated individuals should have the same frequency of agent exposure. In that case
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 Exercise 2.
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. This exercise will demonstrate that to you.
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.
For purposes of comparison with a population that receives no vaccination, a completely separate second transmission system is included where no one is vaccinated.
Model Diagram 4.1
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))
Note from the equations for NewIUnV and NewIV that the vaccinated and unvaccinated susceptible individuals experience exactly the same force of infection. The total rate of infection in the vaccinated population is lower than the rate in the unvaccinated because the vaccinated have an additional population, namely the VaccImmune, that experiences a zero level force of infection.
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 = 1/8
c = 4
Dur = 4
InitialInfectionSeed = .0001
p = .8
PropVaccProg = .5
Derived variables
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VES(risk) = 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 4.1
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. The concept of critical vaccination levels depends upon population mixing pattern assumptions as well as the assumptions 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(R). 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(R) above which the reproduction number falls below one as Pc(R).
therefore
so
Since the critical level of immunity is the value that causes the reproduction number to equal one,
If one knows the basic reproduction number, that is ßcD in a homogeneous population, then one can calculate Pc(R) as:
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 R0 affects it.
Graph 7.2
Proportion of a population that must be effectively immunized (fraction immunized times p) to prevent all infections through herd immunity as a function of the basic reproduction number given that mixing and immunization are random
{[ Pc(I) in the above figure is meant to be Pc(R) ]}
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 R0 increases, the effects of herd immunity from a given level of immunization decreases, is robust to most model modifications that make the underlying model more realistic. We will not demonstrate that robustness in this exercise because we need models where the assumptions of this model do not hold. Latter models will be used for this purpose.
Homework 4.2
a) Calculate the critical vaccination value for three different widely divergent sets of all or none model parameters and then demonstrate empirically using Stella II
Ô that those critical vaccination values in fact correspond to thresholds where there will or will not be epidemic transmission.