Epidemiology 606

Partial Vaccine Effects On Susceptibility

Department of Epidemiology

Professor James S. Koopman MD MPH

5.0 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 interference.

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.

5.1 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 predictive power and generalizability of estimated vaccine effects, there is a need for more realistic models of vaccine effects. Even if we don't seek predictive power and generalizability of estimates, it is possible that our qualitative 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 affect transmission dynamics assuming more realistic models. 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 I will address in the rest of this chapter is the following:

"When will estimates of vaccine effects based upon an inaccurate 'all or none' model when reality is consistent with partial effects 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 that lead us to pursue some paths and ignore others.

Realistic vaccine effect models might specify how vaccination alters immune responses and how these altered immune responses in turn alter 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 models of the immune response in individuals should be integrated into models of transmission dynamics in populations.

5.2 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 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 an unrealistic assumption. We make it only so that we can see the isolated conceptual effect of using an "all or none" model parameter estimate when a partial effects model holds.

In our simplified version of the partial effects model, vaccination reduces the probability of infection given agent exposure during a contact by the same relative amount in all individuals irregardless of the dose of agent in the exposure or the time course of exposure. 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 d.

ß_V = dß_UnV..................................(Equation 1)

The parameter a 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 VES(rate).

VES(rate) = 1-d.................................(Equation 2)

To distinguish the statistic to estimate this parameter from the parameter itself, the usual practice would be to give it a "hat". I don't have time to figure out how to make this work in html so I will use VES(rate) to refer to the statistic and 1-d to refer to the parameter.

We remind you that the statistic to estimate the parameter for the "all or none" model is often used without consideration of what model applies. That statistic 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 vaccinated individual's favor.

Note that in our deterministic, compartmental, 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.

5.2.1 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 l_v. Label the rate of transmission per contact in unvaccinated individuals as l_u. Then .

In epidemiological studies, we never measure instantaneous rates. We sometimes measure rates across intervals assuming that rates stay constant. More commonly, we just measure risks across defined time intervals. Since VES(risk) was measured using risk data and we now want to examine VES(rate) using the same sort of data, we need to convert our rate parameter into a form that can be estimated using risks. When a rate is constant, we remind you of the relationship between risks and rates.

If you take epidemiology 802 you will understand exactly where that equation comes from and what it means. But for now just note that the longer the rate is experienced, the higher the risk goes. It cannot, however, go above 1, even when the rate is above one.

The rate at which vaccinated individuals get infected in an SIR model is

where c is the contact rate and b is the transmission probability. Likewise the rate at which unvaccinated individuals become infected is

.

We want a statistic that estimates the ratio of the l_v to l_u using rates. That is easily found by taking the log of

as

ln(1-R_v) = -cl_v{S/(S+I+R)}.

The same can be done with the unvaccinated yielding

ln(1-R_u) = -cl_u{S/(S+I+R)}.

The second equation can be divided into the first giving

so that

.

This is the statistic that you should use to estimate vaccine effects on susceptibility of vaccinated individuals if you think that the partial effects model holds rather than the all or none model. Note that the fraction susceptible and the contact rates cancel out only 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 cancellation.

5.2.2 Stella IIÔ 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. The standard vaccine efficacy statistic, VES(risk), is labled SVEStat because I did not have time to change it. Likewis the partial vaccine efficacy statistic, VES(rate) is labeled PVEStat. Both of these statistics are calculated as derived variables. We could have used the model from exercise 3 for this purpose but it has extraneous effect parameters that we do not need right now such as effects on contact rates, durations, and contagiousness. This model is thus a little cleaner. I would hope, however, that you note how this model is just a reduced version of that model with a couple of extra derived variables.

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 5.1:

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-d. 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 vaccine trials may be conducted in low risk settings. In such settings, VES(risk) and VES(rate) have almost identical values. The observation has been made that in some measles outbreaks in the post-vaccine era, it makes little difference which statistic is used. This observation should not be interpreted as meaning that the distinction between these statistics is never important.

Another thing to note is that VES(rate) faithfully estimates 1-d once the indigenously infected population contributes significantly more to the infection prevalence than the seeded population. This statistic can thus be used to project vaccine effects from one population to another when the partial effects model holds.

We see that the distinction between the two statistics becomes important later in the epidemic. As the overall infection risk in the population rises, these two statistics 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.

Demonstrate to yourself that VES(rate) estimates 1-d under a wide range of parameter setting for this model.