Chap8.htmTEXTMSWDU²°†b°††n Chapter 9

Chapter 8

Partial Vaccine Effects

Epid 802 Lecture Notes

Jim Koopman

8.0 Purpose of this chapter

This chapter presents a "partial effects" model of vaccine effects which is more realistic for most vaccines than the "all or none" model. In its simplest form this model may have only one parameter which alters the transmission probability to the vaccinated person given contact with an infected person. Parameters reflecting an altered course of infection in vaccinated individuals who do become infected are another part of the model.

Having a more realistic model and a standard statistic that estimates a parameter in a less realistic model provides an opportunity to experience the errors that can arise from using statistics that estimate the wrong parameters. An example presented relates to the estimation of an effect for a vaccine in a low transmission environment and then the generalization of that effect to a high transmission environment. This is what happened with the polio vaccine. Using the right model, one predicts that the standard vaccine efficacy statistic should have markedly lower values in high transmission countries. Since the wrong model of vaccine effects held sway, however, when low values of the standard vaccine efficacy statistic were encountered in high transmission countries, the hypothesis of viral interference was generated to explain this observation. It seems to me that the difference in VE estimates using the standard statistic is better explained by the partial effects model than by viral interferrence.

The moral of this story is that a scientifically correct model is needed to make scientific generalizations and that failure to examine carefully the model on which the statistics we use are based can lead to serious misperception of disease causation processes. This same moral will be emphasized later in the case of multivariate models for chronic diseases. (If we ever get that far this semester!)

8.1 The need for more realistic models of vaccine effects:

The "all or none" model would serve to provide predictive and generalizable estimates of vaccine effects if it corresponded to the way the real world works. But it does not take a great deal of insight into the role of vaccine stimulated immunity to realize that for most vaccines the "all or none" model is unrealistic and therefore estimates of its parameters will not be generalizable across different populations or predictive of vaccine effects. If we seek predictiveness and generalizability of estimated vaccine effects, there is a need for more realistic models of vaccine effects. Even if we don't seek predictiveness and generalizability of estimates, it is possible that our qualitiative understanding of how vaccines affect transmission dynamics might be seriously flawed if they are based upon unrealistic models. Thus we should seek to explore how vaccines under more realistic models affect transmission dynamics. We should see if the insights about critical vaccination levels and indirect effects are robust to model changes that make our model more realistic.

There are many different ways that models of vaccine effects can be made more realistic. The most productive modeling efforts will define a range of different ways and then determine which aspects of realism are necessary for different specific purposes. To maximally increase our understanding of the phenomenon being modeled, the art of modeling must be practiced in a way that selects the simplest model that captures the essence of the real world which is necessary to address a specific problem. That entails a process of considering various alternatives and deciding which alternatives capture aspects which are essential to the issues being addressed. The issue which I wish to address in the rest of this chapter is the following:

"When will estimates of vaccine effects based upon erroneous models lead to serious errors in Public Health practice."

All models are erroneous in the sense that they are simplifications and abstractions that ignore parts of reality. But what are the errors which might lead us to make serious errors in public health practice. There is no simple way to devise all possible models and test them in all possible situations in order to make this determination. We must be guided by judgements which lead us to pursue some paths and ignore others.

Realistic vaccine effect models might specify how vaccination affects immune responses and how these altered immune responses in turn affect the course of infection. They might specify how vaccination affects the risk of infection and the course of infection. To be complete they might do so specifying vaccine effects for different doses of exposure and different times between exposures. To develop models with such specificity we would need to connect models of how the immune response affects the course of infection in individuals to models of vaccine effects in the population.

8.2 The "partial effects" model of vaccine effects

In keeping with the idea that we should examine simpler models always with an eye toward their limitations and how we might need to modify them in the future, we will begin with a simple model alternative to the "all or none" effects model. The model we will consider is called the "partial effects" model. In this model the vaccine acts to reduce the risks of infection in everyone who receives it. No vaccine recipient, however, is absolutely protected against infection. In its simplest form which we examine first, the partial effects model has only a single vaccine effect. It reduces the susceptibility of individuals to infection. The assumption is made that the course of infection in vaccinated individuals is unaffected. This is undoubtedly an unrealistic assumption which we will need to relax later when we use our model for a full examination of how using the wrong model of vaccine effects can affect Public Health vaccine program decisions. Right now, however, our purpose in using this model is examine the consequences of putting the vaccine effect of reducing susceptibility on individuals or on the interactions between individuals when transmission can take place. For this purpose a model of vaccine effects only on susceptibility to an SIR infection in a randomly mixing population will do. We will call this the "partial effects on susceptibility" model of vaccine effects in comparison with the "partial effectson on infection" model which will comprehend vaccine effects on contagiousness and on duration of infection as well.

In our simplified version of the partial effects model, vaccination reduces the probability of infection given exposure by the same relative amount in all individuals irregardless of the dose of exposure or time course of exposure that they had. Vaccination will not alter the course of infection in vaccinated individuals who become infected. The transmission probability to a vaccinated individual will be reduced by a fixed amount. We will formulate this effect in a multiplicative manner. We assign a fraction less than one such that the transmission probability from an infected person to a vaccinated person is this fraction times the transmission probability to an unvaccinated person. We label this fraction _.

ß_V = _ß_UnV..................................(Equation 1)

The parameter _ will have a low value if the vaccine is very effective and a value closer to one if the vaccine is less effective. We usually express vaccine effects as the complement of these relationships. For that reason we create a parameter for the Partial Vaccine Effect on Susceptibility which we label PVES

PVES = 1-_.................................(Equation 2)

The statistic to estimate this parameter we will call PVEStat.

We remind you that the statistic to estimate Standard Vaccine Effect on Susceptibility is

An important difference between this model and the "all or none" model is when chance acts. Chance will not act at the time of vaccine administration in this model. Chance will act at the time that a vaccinated individual is exposed to infection. The benefit of vaccination will be manifest at this time. The benefit will be to reduce the chances of infection given a specific exposure. At the time of exposure, nature will roll the dice to determine if an individual becomes infected. Having received a vaccination will weight the dice in the vacinee's favor.

Note that in our deterministic, compartmental, simulation models, we deal only with populations and not with individuals. Likewise we deal only with deterministic processes and not chance events. Thus some of the language in the previous paragraph might seem inappropriate. We do, however, have model parameters that correspond to transmission probabilities. Unlike the "all or none" model, vaccination will change these transmission probabilities. Thus we can speak of the chance effect of the vaccine acting at the time of exposure when the transmission probabilities are changed. The population effect will be fixed at the expected proportion of vaccinated individuals that would be protected against otherwise inevitable infection had they not been vaccinated.

8.2.1 Estimating the partial effect on transmission probabilities of vaccines

In the continuous population and instantaneous contact process framework of the infection transmission models that we have built with STELLA™, the effect of a vaccine fitting the partial effects model can be thought of as an effect upon the rate of transmission per contact. Label the rate of transmission per contact in vaccinated as lv. Label the rate of transmission per contact in unvaccinated individuals as lu. We could label the vaccine efficacy in the partial effect model as PVE and define it as . Over a defined period of risk in which vaccine efficacy is being assessed, we assume that the number of contacts made by vaccinated and unvaccinated individuals with infected individuals is the same. Label this number of contacts "i". Then the observed risk of infection in the vaccinated, Rv, over the period of observation is 1 - e-ilv. (Remember from Chapter 2 that this is the closed form solution. "i" in this case is our marker of constant accumalation of risk which can be measured by time.) The observed risk of infection in the unvaccinated, Ru, over the period of observation is 1 - e-ilu. This should be evident from the exercize in chapter 2. Subtracting one from each side of these equations and multiplying through by minus one we have:

1 - Rv = e-ilv

1 - Ru = e-ilu

Taking the log of each side, dividing the first equation by the second and subtracting this from one we have

..........................(equation 4)

Note that our statistic PVEStat has the degree of contact intrinsically specified because lv and lu are expressed in per contact units. But we do not need to measure the number of contacts to estimate this measure of vaccine efficacy because the number of contacts in the numerator cancels out the number of contacts in the denominator (if the assumptions of equal contact rates in the vaccinated and unvaccinated and of random mixing hold). If we did not have randomized vaccine administration and there was non-random mixing, we could not make this cancelation.

8.2.2 Stella™ Implementation of the "partial effects on susceptibility" model

In the file PEffSusc I have constructed a simple model of this type. It can be appreciated in Diagram 8.1. Both the standard vaccine efficacy statistic (SVEStat) and the partial vaccine efficacy statistic PVEStat are calculated as derived variables

Diagram 8.1

A Stella™ model of partial vaccine effects on susceptibility

DIFFERENCE EQUATIONS

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IUnV(t) = IUnV(t - dt) + (NewIUnV - NewRUnV) * dt

INIT IUnV = (1-FractVacc)*InfectSeed

NewIUnV = SUnV*UnVForceInf

NewRUnV = IUnV/Dur

IV(t) = IV(t - dt) + (NewIV - NewRV) * dt

INIT IV = FractVacc*InfectSeed

NewIV = SV*VForceInf

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-FractVacc)*(1-InfectSeed)

NewIUnV = SUnV*UnVForceInf

SV(t) = SV(t - dt) + (- NewIV) * dt

INIT SV = FractVacc*(1-InfectSeed)

NewIV = SV*VForceInf

MODEL PARAMETERS

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beta = .5

c = 4

delta = .2

Dur = 2

FractVacc = .5

InfectSeed = .00001

DERIVED VARIABLES

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UnVForceInf = c*beta*((IUnV+IV)/(IUnV+IV+RUnV+RV+SUnV+SV))

VForceInf = c*beta*delta*((IUnV+IV)/(IUnV+IV+RUnV+RV+SUnV+SV))

PVEStat = 1-(LOGN(1-RiskV)/LOGN(1-RiskUnV))

SVEStat = 1-RiskV/RiskUnV

RiskUnV = (IUnV+RUnV)/(IUnV+RUnV+SUnV)

RiskV = (IV+RV)/(IV+RV+SV)

RiskTot = (IUnV+IV+RUnV+RV)/(IUnV+IV+RUnV+RV+SUnV+SV)

At the above parameter values we get the output in graph 8.1:

Graph 8.1

Estimated vaccine efficacy statistics over the course of an epidemic

Here are some things to note about this output. At the very beginning of our simulation the risks in our estimated statistics are dominated by the seeding population rather than by the transmission dynamics. Since we seeded both the vaccinated and unvaccinated populations equally, the efficacy statistics start out at zero. By the time there is a perceptible level of infection in the population, however, the vaccine efficacy statistics are both equally at 1 - _. The equality of these two statistics at low risks of infection and their correspondence to the causal model parameters is important to note since many times vaccine trials may be conducted in low risk settings. We see, however, that as the overall infection risk in the population rises, these two statistics begin diverge. In some cases, such as enteric infections in developing countries, the force of infection can be very high and divergence of these two statistics could be quite large.

Note that if the force of infection is high enough, vaccine efficacy calculated using the standard statistic will be quite close to zero even when a vaccine reduces the risk of transmission in a contact by 90% or more. That is because almost all vaccinated individuals will become infected despite being vaccinated if they are exposed frequently enough.

Homework 8.1

In developing countries, it is possible for almost all infants to become infected with polio virus by 18 months of age. Given the considerations of how to measure R₀ in chapter 5, that implies an R₀ of about 30. The R₀ of polio infection in developed countries might be 1.5. Assume that you are evaluating a polio vaccine in a developed country and a developing country. You observe the infection status of your study population serologically at the end of your study. Suppose you have a serology that can distinguish a vaccine induced immune response from a natural infection induced immune response. You find that 5% of your control population has been infected in the developed country setting and your standard vaccine efficacy statistic is calculated to be 90%. In the absence of interference from other viruses, determine the value of the standard vaccine efficacy statistic you would expect in the developing country study given that the model of partial vaccine efficacy on susceptibility holds.

8.3 Vaccine effects on the course of infection:

The virus watch studies of John Fox in New Orleans were conducted at the time that polio immunization with the Salk or killed polio vaccine. These studies demonstrated that immunization did not reduce the risk of polio infection at all. The benefit of the vaccine was that it prevented disease even though it did not prevent infection. It prevented disease by keeping the enteric infection with polio virus under sufficient control that the virus did not reach the spinal cord.

Despite this lack of protection against infection, Salk vaccine administration in the Scandinavian countries has completely eliminated polio virus circulation. How is this possible? How can a vaccine that does not prevent infection stop the circulation of a virus? The answer to that rhetorical question is simple. The vaccine can lower the R₀ below one by reducing the contagiousness of vaccinated individuals who become infected or by reducing the duration of infection. Most likely both effects will occur. The excretion rate of polio virus from vaccinated individuals is on average a couple logs less than the excretion rate of virus from infected individuals who were never vaccinated. Thus one might expect that vaccinated individuals who become infected would not be as contagious as unvaccinated individuals who become infected.

The conjugated Haemophilus Influenza B vaccine recently surprised almost everyone with its ability to almost completely eradicate HIB infection. This was quite an unexpected result given the vaccine trial results which only measured effects on susceptibility. It was thought that the vaccine had only a limited effect on preventing colonization with HIB. The major effect of the vaccine was thought to be the prevention of invasive infection. But clearly the vaccine must have had some effect on the contagiousness of infected individuals in order to stop transmission.

We surely should be interested in such vaccine effects which could stop the circulation of an infection. With HIV vaccines, we have argued that effects on contagiousness are the only effects that we can expect from a vaccine but that these will be quite sufficient to stop the epidemic. But now we have three vaccine effects to measure and not just one. To unify notation we will call these VES for the vaccine effect on susceptibility which we just estimated using PVEStat. We have VEC for the vaccine effect on contagiousness. We have not shown how to estimate this and will not do so in this exercize since discrete individual models with discrete contacts are necessary to do this. You can see our article in the Amer. J. Epidemiol. on "Assessing HIV Vaccine Effects" to see how to do this. We also VED which is the vaccine effect on the duration of infection.

Define ß_UU as the transmission probability given a contact between a susceptible individual who has not been vaccinated and an infected individual who has not been vaccinated. Define ß_VU as the transmission probability from an unvaccinated infected individual to a vaccinated individual who still has some susceptibility. Define ß_UV as the transmission probability from an vaccinated person who became infected after vaccination to an unvaccinated person. Define ß_VV as the transmission probability given a contact between an infected and susceptible individual who have been vaccinated.

Assume that and that so that we can define

Homework 8.2

Modify the PEffVacc model to include not only the VES effects it already has but the VEC and VED effects defined above as well. Set transmission parameters in the model so that without any vaccination the R₀ will be 4. Set VES at values of 0. 0.4 and 0.8. Then set VED and VEC at values that will make the critical vaccination level in the population to stop transmission equal to 80%. Demonstrate that in fact this is a critical vaccination level in your model.

Note that you could use the model from Chapter 6 with appropriate parameter settings if you did not want to have to modify this model. If you change the exposed to vaccinated in the Chapter 6 model, you can explore a whole series of issues in vaccination. This might serve as the basis for a term paper.