Epidemiology 606

Department of Epidemiology

Professor James S. Koopman MD MPH

We consider in this exercise a vaccine whose effect is to accelerate and boost the immune response to infection so that infection is controlled more quickly. At some point, the vaccine-boosted immune response will be so accelerated and strong that it will appear as if infection was completely prevented. For our purposes here, even if there is some detectable immune response, if infection never becomes contagious we can talk about prevention of infection in these cases. Only when infection is prevented will we see VES effects. This is true whether we calculate VESrisk or VESrate statistics from the previous exercise.

Given human variability in the response to the vaccine, some individuals will have a response that for practical purposes prevents infection while other individuals will have detectable infection whose course is modified to different degrees. Likewise, there will be considerable variation in the doses of agent transmitted during different contacts even for the same individual. The continuous compartmental models of transmission used in this course are clumsy and impractical for dealing with a lot of variation in individual characteristics or in contact characteristics. Still we can gain insight into vaccine behavior as a function of the immune response using this type of model.

We can simplify vaccine effects by formulating them precisely the way that we formulated risk factor exposure effects in the model in exercise. I have modified that model for use in this exercise. I have eliminated the contact rate effects from that model since the effect of vaccines on contact rates is not a very direct effect. Halloran has argued, however, that this effect can be quite important for sexually transmitted diseases when the vaccinated individuals think that their level of protection gives them more freedom to engage in risky sex. None-the-less, for the sake of simplification, we ignore that effect here.

I have labeled the exposed group as vaccinated. I have introduced a control population in the same manner as for exercises 4 and 5. I derived VESrisk and VESrate statistics in the same manner as exercises 4 and 5. Note that VESrisk and VESrate correspond to the cumulative incidence (row IV) and incidence rate (row II) in the first column for design I in table 1 of the Halloran et al. article that was assigned.

And I introduced new vaccine effect parameters. These correspond to the unconditional measures relating cumulative incidence in row IV of the table. They are

1. The measure in column IIA comparing unvaccinated individuals in the control and vaccine populations,
2. The measure in column IIB comparing vaccinated individuals in the vaccinated population to unvaccinated individuals in the control population.
3. The measure in column III comparing the overall infection level in the vaccinated and unvaccinated populations.

The model can be obtained for your education and entertainment in the file VaccComp.stm file in my public IFS space. The diagram of that model is as follows:

The equations for this model are:

IU(t) = IU(t - dt) + (NewIU - NewRU) * dt

……..INIT IU = InfectSeed *(1-FractVaccinated)

……..NewIU = SU*UForceInf

……..NewRU = IU/Dur

IU_2(t) = IU_2(t - dt) + (NewIU_2 - NewRU_2) * dt

……..INIT IU_2 = .InfectSeed

……..NewIU_2 = SU_2*ContRt*tpGvnCont*(IU_2/(IU_2+RU_2+SU_2))

……..NewRU_2 = IU_2/Dur

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

……..INIT IV = InfectSeed *(FractVaccinated)

……..NewIV = SV*VForceInf

……..NewRV = IV/(Dur*EffectOnDuration)

RU(t) = RU(t - dt) + (NewRU) * dt

……..INIT RU = 0

……..NewRU = IU/Dur

RU_2(t) = RU_2(t - dt) + (NewRU_2) * dt

……..INIT RU_2 = 0

……..NewRU_2 = IU_2/Dur

RV(t) = RV(t - dt) + (NewRV) * dt

……..INIT RV = 0

……..NewRV = IV/(Dur*EffectOnDuration)

SU(t) = SU(t - dt) + (- NewIU) * dt

……..INIT SU = (1-InfectSeed)*(1-FractVaccinated)

……..

SU_2(t) = SU_2(t - dt) + (- NewIU_2) * dt

……..INIT SU_2 = (1-InfectSeed)

……..NewIU_2 = SU_2*ContRt*tpGvnCont*(IU_2/(IU_2+RU_2+SU_2))

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

……..INIT SV = (1-InfectSeed)*(FractVaccinated)

……..NewIV = SV*VForceInf

The parameters are

ContRt = 4

Dur = 4

EffectOnContagiousness = .25

EffectOnDuration = .25

EffectOnSuscept = .75

FractVaccinated = .5

tpGvnCont = 1/8

Derived variables are

FractContInf = (IU_2+RU_2)/(IU_2+RU_2+SU_2)

FractTotInf = (IV+IU+RV+RU)/(IV+IU+RV+RU+SV+SU)

FractUInf = If FractVaccinated = 1 then 0 else (IU+RU)/(SU+IU+RU)

FractVInf = If FractVaccinated = 0 then 0 else (IV+RV)/(SV+IV+RV)

NumCasesAttribExpos = (FractVInf-FractUInf)*(SV+IV+RV)

TotalNewInfections = NewIV+NewIU

VEIIa = 1-( FractUInf / FractContInf)

VEIIb = 1-(FractVInf/FractContInf)

VEPopIII = 1-FractTotInf/FractContInf

VESrate = 1-(logn(1-FractVInf))/(logn(1-FractUInf))

VESriskI = 1-FractVInf/FractUInf

VForceInf = ContRt*tpGvnCont*EffectOnSuscept*(IV*EffectOnContagiousness+IU)/(SV+IV+RV+SU+IU+RU)

UForceInf = ContRt*tpGvnCont*(IV*EffectOnContagiousness+IU)/(SV+IV+RV+SU+IU+RU)

Our model examines only effects on the course of epidemics when everyone is vaccinated before the epidemic begins.

Our vaccine effect parameter settings reduce susceptibility to 75% of the value without vaccination and they reduce contagiousness and duration to 25% of their values. The total number of individuals that a vaccinated individual who gets infected will infect is thus only one sixteenth of the number that an unvaccinated individual will infect.

As we increase the potency of vaccines, I suspect that we will in most cases reduce transmission to below one sixteenth even before we see any susceptibility effect at all. How this works out in reality, however, is still unknown.

We see in the following graph of vaccine effect statistics that at these parameter settings, where 50% of the population is vaccinated in the intervention population, indirect vaccine effects dominate over direct susceptibility effects. Even though the direct effects are 25%, there is little difference between the pure indirect effects of design IIA and the combined effects of design IIB.

We also note in the following graph that vaccine effect statistics using comparisons to the control population rise and then fall. That is because the epidemic in the vaccinated population takes off quite a bit after the epidemic in the intervention population. In practicality, infection levels at the end of the epidemic are likely to be compared. But it is worthwhile to note that it is possible for vaccination only to delay an epidemic and if you take your measures too early, you may overestimate vaccine effects.

Let us examine the same graph but when R₀ has a value of 4 because we doubled the contact rate.

At higher R₀ values, the indirect vaccine effects are less. That is because if a person who might have been a source of infection has their contagiousness reduced or their infection entirely prevented, they are still likely to be other infected individuals around who will negate that indirect effect by causing infection in the individuals receiving indirect effects.

Note that for the reasons discussed in the last exercise, VESrisk diminishes when the R₀ increases and the true vaccine effect model is one of partial vaccine effects.

We now vary the R₀ by varying the contact rate with the values 4, 8, 16, and 32 giving R₀ values of 2, 4, 8, and 16. The last exercise dealt with comparing VESrisk and VESrate under these conditions. We do not examine the rate versus risk dimension of Halloran et al's table 1 so here we just use the risk measures. First, we see what happens to VESrisk under these conditions by observing the value of the VESrisk parameter from the beginning of transmission to the end of the epidemic. We are likely only to observe the value at the end of the epidemic in an actual study of design I, but we may stop at other points.

The reason for the decreasing value as R₀ increases was explained in exercise 5. The values early on are higher than the values later. That is because the epidemic in the vaccinated population takes longer to take off.

Next we observe what Halloran et al call the indirect effect as reflected by their design IIA. This is not the total indirect effect. It is only the indirect effect in the unvaccinated population as the measure only compares cumulative risks in the unvaccinated individuals in the vaccine population and the control population. The indirect effect here includes effects on contagiousness and duration of infection in individuals who are a source of infection to others as well as effects due to the direct protection from infection of individuals who might have been a source of infection.

Note that these indirect effects are considerably greater than the direct effects even when we have only 50% of the population vaccinated. If we vaccinate higher percentages of the population we of course get even greater indirect effects.

Next we examine what Halloran et al. call the combined direct and indirect effects with design IIB.

The combined direct and indirect effects with measure IIB are only slightly greater than the indirect effects alone with measure IIA. Most of our effects here are indirect. The fact that we cut the total contagiousness of vaccinated individuals 4 fold by a contagiousness effect and fourfold by a duration effect giving a total effect of 16 fold is the major explanation for this. Note that at intermediate values of R₀ the combined effects are meaningfully greater than the indirect effects alone.

Finally, we examine the pattern of total effects.

The pattern is very similar as the indirect effects again dominate it.

Let us vary the fraction of the intervention population that is vaccinated using the values 25%, 50%, 75%, 90% and 99%. We compare the different vaccine effect measures in a model where the R₀ has a value of 8 with a contact rate of 16. First let us look at the pure indirect measure comparing the unvaccinated individuals in the intervention and control populations.

In the above graph we see that vaccinating 90 or 99% of the population gets us below threshold. We also note that as we get closer to the threshold, the vaccine effects increase dramatically. This is the same phenomenon as we saw for R₀ and the final fraction of the population that is infected. As we approach threshold, effects increase dramatically. This is one of the most important lessons that you can get out of all of these modeling exercises.

Now let us look at the same results but using the total effect measure from design III.

We observe basically the same phenomenon.

To see this phenomenon once again, let us raise the R₀ to 16 by putting the contact rate up to 32.

Here we see almost no effect at 25% and almost no improvement by doubling the coverage to 50%. As we reach 75% we barely begin to see an effect. At 90% the effect is quite notable but certainly not dramatic. At 99%, however, we have crossed the threshold. This is the same phenomenon that at high R₀ we may get minimal effect with interventions until those interventions are able to lower R₀ close to threshold.

This exercise emphasizes the importance of getting coverage to high levels when R₀ is high and a realistic combined effect model of vaccine effects holds. But overall levels are not the only thing that are important. Leaving pockets of susceptible individuals after a vaccination program can negate high vaccination levels. To learn that lesson, we can't use models where everyone mixes randomly. The key to understanding transmission dynamics is realizing that overall the largest determinant of infection levels in any population is the contact pattern. That is the lesson of the next exercise.