Epidemiology 606

Department of Epidemiology

Professor James S. Koopman MD MPH

Contact patterns determine both the direct and indirect effects of treatment, vaccination, or risk factor elimination. Contact patterns can cause the effects of risk factors on the individual exposed to the risk factors to be attenuated or amplified by indirect effects. They determine the size of indirect effects in the population of unexposed individuals and often can account for a greater number of indirectly caused infections in the unexposed population than directly caused infections in the exposed population. Thus, risk factor effects on both the populations exposed to risk factors or unexposed to risk factors have to be attributed to the contact patterns that permit those effects as well as to the risk factors themselves. This exercise demonstrates these risk factor effects and provides insight into what contact patterns will have what effects given different transmission probabilities, overall contact rates, durations of infections, and nature of the risk factor effects.

The number of dimensions in which contact patterns can be assessed is almost limitless. There are dimensions that relate to the overall population such as how extensively risk factor exposed individuals interact with risk factor unexposed individuals or the extent to which high contact rate individuals make contact with low contact rate individuals. For this type of contact pattern description, the possible number of dimensions grows in an explosive manner as the number of categories increase. Additional dimensions relate to patterns of closure in connection between individuals. For example one measure might assess the chance that an individual will contact another contact of one of their other contacts. The number of dimensions here is higher than N^N where N is the population size. Additional dimensions relate to the timing of contacts. Whether people have simultaneous or sequential contacts, for example, can be important for the spread of infectious disease.

A most important dimension relates to the degree of symmetry in a contact. It is possible that in a contact between two groups, the potential for transmission will be stronger in one direction than in the other. For example, contact may be mediated by both individuals touching the same spot. Transmission could only take place from the first person touching the spot to the second. We, however, will only examine symmetric contacts.

The mathematicians working on continuous compartmental transmission system models only began attending to contact patterns about 20 years ago. They addressed only the first class of dimensions of contact patterns mentioned above and the number of dimensions for which useful measures have been developed is limited. This work has only made a significant impact on epidemiological literature in this decade. Sociologists have a somewhat longer tradition of examining contact patterns buts only a minority of sociologists work in this tradition. This tradition has developed measures along the second type of dimensions measured above. Epidemiologists could benefit from collaboration with sociologists in this type of work. There is not very much work developing measures along the time dimension. What work there is has been the product of epidemiologists or sociologists working on epidemiology problems of infection transmission. The timing of contacts is important in establishing potential chains of transmission.

Only the first type of dimension mentioned above can be examined with the continuous compartmental models we use in this course. In this exercise, we examine only the simplest dimension based on only two categories of individuals and using one of the simplest assortative mixing formulations called "preferred mixing". This limits the lessons we can learn about how mixing patterns determine risk factor effects. But it allows us to illustrate a couple points about assortative (like individuals are more likely to encounter each other) mixing that is important to anyone making infection control decisions. One point is that aspects of contact patterns that raise the R₀ in a subgroup above 1 can have dramatic effects on overall population dynamics. Another is that when one subgroup is above threshold and the other is below threshold, the population level of infection can be either raised or lowered by increasing contact between groups, depending on the state of the system. A third is that measurement of the direct effects of exposure can be dramatically distorted by non-random contact patterns and that standard means for controlling for confounding cannot properly adjust for this distortion.

To examine these issues, we return to a model with all four risk factor exposure effects that we included in the model of Exercise 3. We add derived variables for effect measures to this model like those in the Halloran article. We examine only those measures at the cumulative risk level. Instead of making these effect measures one minus the risk ratios, however, we make these merely the risk differences. Remember that the 1 minus the risk or rate ratio measures are the risk difference divided by the risk in the high-risk group. Such measures are appropriate only for protective effects given risk factor exposure. We will examine risk factor exposures that increase risks. The contrasts in the Halloran article were:

1. I: Contrasts between exposed and unexposed populations in the intervention population
2. IIA: Contrasts between the unexposed population in the intervention population and the unexposed population in the control population without the exposure.
3. IIB: Contrast between the exposed population in the intervention population and the unexposed control population.
4. III: Contrasts between the intervention population as a whole and the control population.

A most important modification is to allow for assortative mixing using the preferred mixing model.

The preferred mixing formulation

The mathematics of the preferred mixing formulation do not parallel the physical process of mixing. In other words, they do not provide a realistic parameterization of what goes on in a mixing process. We are not going to examine the mixing process, however. We are just going to examine the consequences of contact patterns generated by a mixing process. Given only two categories of individuals in the population, this formulation can reproduce any assortative mixing pattern desired. Since we are going to stick with only two groups, we will use preferred mixing because it has nice mathematical properties that make it easy to handle when analyzing models.

The contact process in the preferred mixing formulation is a symmetric one. If group A contacts group B in a way that has potential to transmit infection, group B contacts group A with the same transmission potential.

In the preferred mixing formulation, besides the general mixing in the population which is formulated as proportionate mixing, there is a fraction of contacts which are made exclusively within one's own contact group. We usually designate the fraction reserved as r (rho). The general formulation of reserved mixing has separate reserved fractions for each category of individuals in a population. For our purposes, however, we need only examine models where the reserved fraction is the same in the segment of our population which is exposed to the risk factor and in the unexposed segment.

When we have our population divided into two categories and contact is symmetric, there are three contact rates of interest. These are exposed with exposed, unexposed with unexposed, and exposed with unexposed. The symmetry of transmission makes exposed with unexposed the same as unexposed with exposed.

Within the reserved fraction, one makes contact only with one's own group. The unreserved fraction of contacts is distributed proportionately with all other individuals in the population. Thus, some of the unreserved fraction of contacts is made with one's own group as well. The total number of contacts within one's own group for the exposed would be:

Where N_E is the total number of exposed individuals, N_U is the number of unexposed individuals, c_U is the rate at which unexposed individuals make contact overall, c_E is the corresponding rate for exposed individuals, and r is the reserved fraction. The contact between different categories only occurs in the unreserved fraction so we have

Note that the sum of C_EE and C_EU is just N_Ec_E, the total rate at which all exposed individuals make contact with anyone.

This formulation is used to calculate the forces of infection in our model. In the model formulation, c_E is entered as c_U times the contact rate effect of exposure.

The model we will examine is as follows:

IE(t) = IE(t - dt) + (NewIE - NewRE) * dt

……..INIT IE = InfectSeed*(FractExpRiskFact)

……..NewIE = SE*EForceInf

……..NewRE = IE/(Dur*EffOnDurat)

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

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

……..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

RE(t) = RE(t - dt) + (NewRE) * dt

……..INIT RE = 0

……..NewRE = IE/(Dur*EffOnDurat)

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

SE(t) = SE(t - dt) + (- NewIE) * dt

……..INIT SE = (1-InfectSeed)*(FractExpRiskFact)

……..NewIE = SE*EForceInf

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

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

……..NewIU = SU*UForceInf

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))

ContRt = 4

tpGvnCont = 1/32

Dur = 4

EffOnContag = 1

EffOnContRt = 1

EffOnDurat = 1

EffOnSuscept = 1

FractExpRiskFact = .1

InfectSeed = .00001

ResFract = 0

EForceInf = ResFract*ContRt*tpGvnCont*EffOnSuscept*EffOnContRt*(IE*EffOnContag)/(SE+IE+RE)+(1-ResFract)*ContRt*tpGvnCont*EffOnSuscept*(IE*EffOnContag*EffOnContRt+IU)/((SE+IE+RE)*EffOnContRt+SU+IU+RU)

UForceInf = ResFract*ContRt*tpGvnCont*(IU/(SU+IU+RU))+(1-ResFract)*ContRt*tpGvnCont*(IE*EffOnContag*EffOnContRt+IU)/((SE+IE+RE)*EffOnContRt+SU+IU+RU)

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

FractEInf = If FractExpRiskFact = 0 then 0 else (IE+RE)/(SE+IE+RE)

FractTotInf = (IE+IU+RE+RU)/(IE+IU+RE+RU+SE+SU)

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

NumCasesAttribExpos = (FractEInf-FractUInf)*(SE+IE+RE)

TotNewInfectFlo = NewIE+NewIU

RD_I = FractEInf-FractUInf

RD_IIa = (FractUInf-FractContInf)

RD_IIb = (FractEInf-FractContInf)

RD_III = FractTotInf-FractContInf

Our model is again an SIR model without vital dynamics. We first focus on the situation of strong risk factor effects that are able to raise the internal R₀ in the exposed group above one if enough of the contacts made by exposed individuals are with other exposed individuals. Note that the model parameters include a contact rate = 4 per time unit, a duration of four time units, and a transmission probability per contact of 1/32. If there are no exposed individuals in the population, given these parameters the population will not be above the transmission threshold and the epidemic will not take off.

In the model we will examine now, we will expose 10% of the population to the risk factor effect and make that effect be a seven-fold increase. We will examine effects first on duration, then on susceptibility, then on contagiousness, then on contact rate. We will examine how the size of the epidemic and each of the effect measures using the Halloran et al. comparisons. We will vary the reserved fractions from zero (random mixing) to one (complete separation of exposed from unexposed) with intermediate values of 0.25, 0.5, and 0.75. First let us look at the size of the epidemics at these five reserved fractions when the seven-fold risk factor effect is on duration of infection.

We note that there is no epidemic when mixing is proportionate (random). Even though the exposed individuals generate 3.5 infections over the course of their infection, they generate 90% of those in unexposed individuals who generate on average only one fourth of an infection. Consequently, the subsequent generations of transmission from the exposed individuals decrease to a value of less than one. Given a reserved fraction of 0.25, the exposed individuals generate enough of their transmissions to other exposed individuals so that an epidemic takes off. The generate 25% in their own exposure group due to the reserved fraction parameter and they generate another 0.1 times 0.75 of their contacts in their own exposure group due to proportionate mixing among the unreserved fractions. Since every infected individual makes 0.9 times 0.75 of their contacts with unexposed individuals, they spread quite a bit of infection to that unexposed population.

As the reserved fraction goes up to 0.5, the epidemic in the exposed population rises to a higher level. While each infected individual makes fewer contacts with the unexposed individuals at these reserved fraction settings, the increase in the size of the epidemic is enough to make the overall size of the epidemic continue to increase.

Let us examine the infection prevalence in the exposed and unexposed populations separately given reserved fractions of 0.25 and 0.5

At any one point in time, there are more infected individuals in the exposed than in the unexposed population. That is because infections in this population last longer. The total number of new infections is greater in the unexposed population than in the exposed population although most of the infections in the unexposed individuals arose from exposed individuals. We see the total number of infections by examining the immune compartments. In the model without vital dynamics, these compartments accumulate all infections.

Let us now see what happens when we increase the reserved fraction to 0.5.

The epidemic rises a lot faster as exposed individuals transmit more often to others who will carry on the chain of infection. It also rises to a higher level because the fraction of both the exposed and the unexposed population that is infected increases. Note, however, in this situation where only the exposed are intrinsically above threshold, the ratio of total infections in the exposed and unexposed populations has to go down as the reserved fraction increases.

Let us examine the immune compartment levels in the exposed and unexposed at each of the five levels of reserved fractions.

The fraction of the exposed population that is infected has to continue rising as the reserved fraction increases.

The fraction of the unexposed population that is infected at first rises as the reserved fraction increases. After 50% however, there are not that many more exposed individuals who get infected and those exposed individuals are making fewer contacts with unexposed individuals so that the overall infection rate in the unexposed population goes down. When the unexposed population makes no contact with the exposed population (reserved fraction = 1) there is no epidemic at all in the unexposed population.

Now let use examine the contrast that Halloran calls the "direct" effect. This is just the classic risk difference. It goes up with increasing reserved fraction because the risk in the exposed has to go up faster than the risk in the unexposed as the reserved fraction increases. The fact that only the exposed population is intrinsically above threshold accounts for this. If the unexposed population could also be above threshold, this continuous increase is not a necessity.

We see in the above graph that what Halloran calls "direct" effects are markedly influenced by contact patterns and thus cannot truly all be "direct" effects.

Next we examine the contrast that Halloran calls the "indirect" effect. Since no epidemic ever takes off in the control population, this is just the same as the pattern see for infection in the unexposed individuals who are in the same population with the exposed individuals.

Likewise the contrast IIb of Halloran reflects the pattern in the exposed population.

Similarly Halloran's contrast III just corresponds to the overall infection rate in the population since the control population has no infection at all.

We note that the total effects of a risk factor that increases contagiousness seven-fold are the same as the effects of a risk factor that increases duration of infection seven-fold. The difference is that the contagiousness effects are felt much more rapidly.

We now examine the same parameter of total population effects but where the risk factor affects contagiousness rather than duration.

As we saw in exercise 3, contagiousness and duration risk factors have the same effects on the final infection rate in an SIR epidemic without vital dynamics. The contagiousness effects are just achieved very much more quickly.

Now we examine the same series of simulations but this time we set the all exposure effects to one except for susceptibility to infection. We again set this value to 7. The big lesson here is that indirect effects of susceptibility risk factors are not nearly as large as indirect effects of duration or contagiousness risk factors. We saw this in exercise 3 but here we see that realistic contact patterns where individuals are more likely to mix with their own kind considerably amplify the relative importance of contagiousness or duration effects.

We see below that there is not that much difference in the exposed population that is infected. There is, however, a big difference in the unexposed population.

The total population effects are correspondingly considerably less given that a risk factor affects susceptibility than if it affects contagiousness or duration.

In exercise 3, we saw that given proportionate mixing, the order of effects on the final size of an SIR epidemic with out vital dynamics were contact rate effects > contagiousness effects = duration effects > susceptibility effects. We have seen that assortative mixing can expand the difference between contagiousness or duration effects and susceptibility effects. Now let us examine contact rate effects. Let us begin by examining the fraction of the total population that is infected as we go from a reserved fraction of zero to one.

We see that the maximum fraction of the population infected is somewhat less than for the case with contagiousness or duration effects. The same phenomenon of rising then falling risks as the reserved fraction is increased as was for the other type of risk factors is again observed with contact rate effects. Let us break this down into effects in the exposed and unexposed populations.

These rates are almost the same as the duration effects but at the 25% and 50% reserved fractions are just perceptibly less. Now let us look at the rates in the unexposed.

These infection levels are higher than for susceptibility effects but quite considerably less than for duration or contagiousness effects. At the particular parameter settings which we have used here, contagiousness effects can be greater than contact rate effects because more of the contact is concentrated in the high-risk group given contact rate effects than is the case for contagiousness effects. The pertinent parameter settings are that only 10% of the population is exposed, the population is below threshold without exposure effects, and the exposure effects are relatively strong. At other parameter settings, especially where the unexposed population is above threshold, we can see that contact rate effects might be greater than contagiousness effects.

Susceptibility effects always have the lowest influence at the population level. Contact rate, contagiousness, or duration effects are most often greater. Whether contagiousness or duration effects will be greater than contact rate effects depends upon the parameter settings. At the parameter settings seen here, changing the contact pattern can generate a large increase in the total effects of exposure. A general and important phenomenon is seen when the unexposed segment of the population is below threshold. It is the following. Restricting contact between exposed and unexposed individuals can raise the total rate of infection in the population by increasing the size of the epidemic in the exposed population. The larger size of the epidemic in the exposed generates more contacts of the unexposed individuals with infected exposed individuals although their total number of contacts with all risk factor exposed individuals goes down.