Epidemiology 606

Department of Epidemiology

Professor James S. Koopman MD MPH

Susceptibility effects

The last exercise considered an exposure effect that increased the transmission probability from someone else to the exposed individual. We called that a "susceptibility" effect. Such an exposure might correspond to any deficiency in any physical or immune system barrier that allowed an agent in one's environment to enter a host and begin successfully multiplying. Examples include the presence of a genital ulcer in someone exposed to HIV, the lack of immunization in someone exposed to measles, the lack of prior infection in someone exposed to coxsackie 16, a deficiency in stomach acid in someone exposed to salmonella, etc.

We saw in the last exercise that such effects can be modeled with parameters that increase the transmission probability for the exposed people. We used a simple multiplicative parameter. Formulations other than multiplicative are possible. When we are dealing with only a single exposure, however, the mathematical details of how the parameter affects the transmission probability are not important. That is because it is always possible to transform a single parameter acting additively into one that acts multiplicatively. Since we only deal with single exposures in this course, we will not have to deal with the rationale for additive or multiplicative joint exposure effects for more than one exposure.

We now want to consider 3 additional exposure effects: effects on contagiousness, duration of infection, and contact rates. We will see that these exposure effects have somewhat different consequences for infection levels than susceptibility effects of exposure. One issue to be considered is which types of exposure effects will have the biggest effects on incidence and prevalence under different conditions.

The second effect we consider after a susceptibility effect is a "contagiousness" effect. Any factor that might increase the dissemination of a contagious organism from an infected individual would have a contagiousness effect. A genitourinary infection that draws T-cells to the genitourinary track would make an HIV infected person more contagious. A genetic or immune system alteration that makes it difficult for an individual to control infection could be a contagiousness factor. A behavior like wiping mucus from one's nose with a finger could increase one's contagiousness.

We will formulate contagiousness effects on the exposed in the same simple multiplicative way that we formulated susceptibility effects. The difference will be that we will perform the multiplication only when contact is made with an infected individual who is exposed.

The transmission system models in the first exercise had a contact rate parameter and a transmission probability given contact parameter. Contacts need to be discrete observable events in order to estimate these parameter values with data. If we define contact very broadly, such as being in the same room, we will have lower transmission probabilities than if we define contact to involve only skin to skin contact.

Exposures with contact effects are exposures that distinguish individuals who have a higher rate of contact from those who have a lower rate of contact. All contacts, of course, must be measured in the same way. In this course, we will model only symmetrical contacts. That is to say, the first person has just as much contact with the second as the second has with the first. When contact consists of breathing the same air as someone else in a room, the contact is likely to be quite symmetric. Many contacts in the real world are not symmetrical. Needle sharing contacts involved in HIV transmission can only go from the first person using a needle to the second person using a needle. We will not formally consider such asymmetric contacts in this exercise. We note, however, that it is often possible to model asymmetry with contagiousness or susceptibility effects.

The relationship between asymmetrical contact and contagiousness or susceptibility effects might be illustrated by considering a contact involving separate actions on the part of two individuals making contact. In such contacts, the overlap between contact effects and susceptibility or contagiousness effects can work as follows. Suppose the defined contact involves the infected person wiping their nose, then shaking hands with the susceptible individual, then the susceptible person wiping their nose. The hand shaking alone might be considered the contact. The infected person wiping their nose might be considered a contagiousness effect affecting the transmission probability during a hand shaking contact. The susceptible person wiping their nose might be considered a susceptibility effect on the transmission probability during the hand shake contact.

For a waterborne enteric infection, the infected individual has to do something that allows their feces to reach water and the susceptible person has to consume water. Now we are getting away from what might be readily conceptualized as a symmetric contact. When organisms have to survive or multiply in environments outside of the host in order to effect a transmission between hosts, we may want to include the environment, for example the water in a waterborne infection, explicitly in our model. Another example is that when contact occurs via mosquitoes. For mosquito borne transmission, we usually model the mosquito population separately. But we can capture something of the essence of this sort of transmission even without explicitly including the environment or the vector population in the model. For the objectives in this course, therefore, we will not model the organism in the environment. We will only use our very simple model of the organism in the host.

Contact effects can be modeled as an increase or decrease in the contact parameter attached to a segment of the population. We will again use a simple multiplicative formulation of contact effects so that our effect parameter multiplies the contact rate of the exposed group by a specified amount. Contact effects must be specified in terms of contact with whom. If you increase the contact rate of the exposed, you need to specify how much of the increase goes to other exposed individuals and how much goes to unexposed individuals. The pattern of who has contact with whom is a prime determinant of infection level in a population. One reason I separate contact parameters from transmission probability parameters in the models used in this course is that models that do not make this discrimination obscure important issues. Those issues include the things that generate specific patterns of contact and the ways that patterns of contact generate transmission dynamics.

In the model used in this exercise, it might seem that we just increase contact rates without specifying the pattern of contact with different segments of the population. In fact, we specify the contact pattern as being a proportionate mixing contact pattern. By constraining ourselves to this contact pattern, we do not need an additional parameter that specifies to whom additional contacts are directed. Contacts are directed to other individuals in proportion to the rates at which those individuals make contacts. In the next exercise, we add a parameter that specifies to what extent additional contacts are directed to groups like one's self or groups different from one's self. Because the contact effects we will model will increase contacts proportionately to available contacts, we will call this a proportionate contact effect.

The same factors that might affect the course of infection in a way that increases the contagiousness of an infection might affect the duration of that infection. A most important factor affecting the duration of infection is treatment. A genetic protective factor, or an induced immune response that shortens the duration of infection is also likely to decrease the contagiousness of infection. Treatment, on the other hand, is likely to shorten the duration of infection without affecting the contagiousness of infection before treatment is begun.

Consider a genetic factor or an immunization that enables the host to resist infection. In some cases, that factor might help the host to resist infection so effectively when the individual is exposed that the agent never even multiplies enough to reach the stage where it is recognized that infection occurred. In that case, the data would show this factor to be having a susceptibility effect. In other instances, the factor may not be enough to stop infection, but it might keep excretion levels low so that it has a contagiousness effect. When the level of agent is reduced, that may facilitate getting rid of the agent more quickly so that the exposure has a duration effect as well.

For the disease you will be discussing in your presentations at the end of the class, discuss with your group an important exposure that will increase transmission and try to specify how much that exposure affects contact, susceptibility, contagiousness, and duration. Specify as precisely as you can what you mean by contact.

Say an exposure has a three-fold effect. It may triple the transmission probability to an exposed individual who is susceptible -- that is to say, it might have a three-fold susceptibility effect. It might triple the transmission probability from an exposed individual who is infected -- that is to say, it might have a three-fold contagiousness effect. It might triple the contact rate of exposed individuals in a contact process that distributes contacts proportionate to their availability -- that is to say, it might have a three-fold proportionate contact effect. Or finally it might triple the duration of infection.

How will the population consequences of these effects differ? Which will have the biggest effect on infection prevalence, on total time spent with infection, on infection incidence, on the cyclical pattern of epidemics? A superficial consideration of this issue might lead one to conclude that any three-fold increase will have the same total effect. By examining this question using transmission system models, we hope you will gain some understanding as to why these different exposure effects have different consequences for the public health. Understanding these different effects is essential to setting good general priorities for using different approaches to discover controllable causes of infection and for organizing different types of infection control programs. In this exercise, we address these questions first in the context of short infections that induce immunity and that are characterized by epidemics that sweep through a population. That is to say, we address them for an SIR transmission system without vital dynamics. Next we consider the endemic situation of an SIR transmission system with vital dynamics.

We elaborate the model presented in the last chapter to add the three additional effects on contagiousness, contact rate, and duration. You can download the Stella IIÔ model from Dr. Koopman's ITD public space as ExpEff31.STM. The Stella IIÔ diagram of the model looks like the following:

The equations for this model are as follows:

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

INIT IE = .0001*(FractExposedAfterIntervention)

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

INIT IU = .0001*(1-FractExposedAfterIntervention)

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

INIT RE = 0

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

INIT RU = 0

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

INIT SE = .9999*(FractExposedAfterIntervention)

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

INIT SU = .9999*(1-FractExposedAfterIntervention)

ContRt = 5.2

Dur = 2

EffectOnContact = 1

EffectOnContagiousness = 3

EffectOnDuration = 1

EffectOnSuscept = 1

tpGvnCont = .1

FractExposedAfterIntervention = .1

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

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

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

TotalNewInfections = NewIE+NewIU

RiskDifference = FractEInf-FractTotInf

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

The risk factor effect on duration is a straightforward multiplication of the duration parameter. The three other risk factor effects in this model are all built into the derived variables that express the force of infection.

The susceptibility effect is only found in the force of infection for the exposed. You can see that it directly multiplies the product of the susceptible population segment times the transmission probability. This emphasizes that the direct effect of increased susceptibility is only on the exposed. The unexposed may have their risks indirectly affected because more exposed individuals will be around to infect them. But the primary effect is on the exposed. Both exposed and unexposed individuals will experience indirect effects of increased susceptibility in the exposed equally in this model. That is because contacts made by exposed individuals who are directly affected by the susceptibility effect are not biased toward the exposed or unexposed groups. Since the exposed have their infection risk raised by both the direct and indirect effects but the unexposed have their risks increased only by the indirect effects, the unexposed will always experience a lesser effect from a susceptibility effect than the exposed. When the combination of direct and indirect effects have saturated the exposed group with infection, there will still be individuals in the unexposed group who have been left uninfected because they did not experience the direct effects.

The contagiousness effect increases the force of infection equally in the exposed and unexposed population segments. Only exposed individuals experience this effect directly. But the direct experience of this effect does not increase the force of infection. In a sense, all contagiousness effects are indirect. They increase the risk of infection of other people, not the exposed people. Given the proportionate mixing formulation used in this model, both the exposed and unexposed individuals experience that indirect effect equally. That means that both the exposed and unexposed populations will be saturated with infection at the same time. Thus, when a contagiousness exposure effect is increased to the point where all the exposed are infected, all the unexposed are infected as well. A contagiousness effect in an otherwise homogeneous population can saturate the entire population with infection. A susceptibility exposure effect, we just observed couldn't be increased until the entire population is saturated with infection. Let us review that for a moment. The susceptibility effect increases the rate of infection faster in the exposed than in the unexposed population because direct susceptibility effects on the force of infection are experienced only by the exposed.

Contact rate effects influence the force of infection for both the exposed and unexposed population segments. But they do not do so equally. Let us bring these equations back down in our vision.

UForceInf = ContRt*tpGvnCont*{(IE*EffectOnContact*EffectOnContagiousness+IU)/ ((SE+IE+RE)*EffectOnContact+SU+IU+RU)}

The fractions in the curly brackets come from the proportionate mixing formula. In that formulation, each population segment is multiplied by its corresponding contact rate. Since all contact rates are the same in this model, they cancel out of the numerator and the denominator of the fraction in the curly brackets. The EffectOnContact multiplies every contact rate corresponding to exposed populations. As the rate of contact is increased in the exposed group, a higher fraction of the contacts for both the exposed and unexposed will be with exposed. Given the proportionate mixing formula used in this model, both the exposed and unexposed individuals will have the same fraction of their contacts with exposed individuals. As the contact rate is increased in the exposed, it will raise their risk and at the same time increase the contact rate of the unexposed with the now higher risk exposed group.

These descriptions of different exposure effects make it evident that we should see different effects on the population level of infection from exposure effects with the same multiplicative relationships on susceptibility, contagiousness, or contact rate.

Now consider exposure effects on the duration of infection. Just as with contagiousness effects, this exposure effect does not directly increase the force of infection in the exposed. It indirectly increases the force of infection in both the exposed and the unexposed. Given that mixing is proportionate, this indirect effect is experienced equally by the exposed and the unexposed. In this regard, duration effects are just like contagiousness effects. If a contagiousness effect triples the transmission probability, it will triple the number of individuals an exposed individual can infect just as a tripling of duration would do. The difference is that a contagiousness effect comes all within the original duration of infection period. The duration effect, on the other hand, is more spread out over time.

Is it the same thing to increase a transmission probability in a multiplicative manner with a contagiousness or susceptibility effect as it is to multiplicatively increase the contact rate or the duration of infection? If the transmission probability is already 0.5, the maximum multiplicative effect you could have would be 2. There is no such limitation on the contact rate or on the duration of infection. Thus the contagiousness and susceptibility effects of exposure are intrinsically more limited than the duration or contact rate effects.

It is possible to formulate contagiousness and susceptibility effects in a different way than multiplicatively. One bad thing about the way we have constructed the model we will examine is that it is possible to set parameter values for the transmission probability and the susceptibility and contagiousness effects that will raise the effective transmission probability above one. That of course is physically impossible. It is, however, mathematically the way we have formulated our contagiousness and susceptibility effects. Thus you must always check your self to see that the product of the transmission probability times the susceptibility times the contagiousness effect does not exceed more than one.

Susceptibility and contagiousness effects can be formulated such that they cannot raise the effective transmission probability above one. Such formulations may be more causally meaningful than the straightforward multiplicative formulation we have used here. The way that genetic host factors, acquired immunity, or environmental factors affect transmission probabilities goes beyond the topic I wish to address in this exercise. A major objective of this exercise is to compare different types of exposure effects with regards to their potential to affect population levels of infection. By using the straightforward multiplicative formulation, we make our effects more quantitatively comparable. We do so in a way that actually gives some advantage to the factors affecting transmission probabilities. We will see, however, that this advantage given to contagiousness and susceptibility effects is not enough to give them a greater impact than contact rate effects.

Let us set up a sensitivity comparison. We want to compare the situation where there are no exposure effects, to the situations where each of the four exposure effects we have examined has a threefold effect on its relevant parameter. Let us make the situation with no exposure effects be in the common situation where an R₀ is only slightly more than 1. With a contact rate of 5.5, a transmission probability of 0.1, and a duration of 2, The R₀ in the transmission system with no exposure effects is 1.1.

In the sensitivity setting window (found under the run menu), we can set five different runs. In the first, we set all four exposure effects to 1. In the next four, we set successively the susceptibility, contagiousness, contact rate, and duration effects to three. We expose only one tenth of the population. This might seem like a relatively small fraction of the population. But more than this one tenth of population is affected. The indirect effects on the other nine tenths of the population are quite important. The five epidemic curves generated by these settings appear in Graph 1.

We see that the contact rate effect created by far the more explosive epidemic. Tripling the contact rate of one tenth of the population turned a small, slow, low grade epidemic into a large explosive one. We also see that the contagiousness effect generated a bigger epidemic than the susceptibility effect. The duration effect generated a slower epidemic than the contagiousness or susceptibility effect. Let us examine these epidemics as the cumulative number of individuals infected in Graph 2

Here we see that increasing the exposures of only one tenth of the population had dramatic effects on the infection levels of all of the population. In every case, the increased exposure resulted in more than a doubling of the total number infected. Our observations on the last exercise that increasing R₀ had a much greater effect when R₀ was close to one than at other values explains in part why we got large effects.

Download this model (ExpEff31.STM) from Dr. Koopman's public space. Double the baseline R₀ by doubling the contact rate, the transmission probability, or the duration. Compare the effects of increased exposure under these conditions. (Before running make sure other parameters and initial values have the values outlined above.)

We also see that the epidemic under the influence of the duration effect infected the same total number as the epidemic under the influence of the contagiousness effect. It took longer to do so. But as commented earlier, the total transmissions from infected individuals will be the same under both effects.

Let us consider again why it is that the contact effect is the greatest of any of the exposure effects. The exposed individuals have their force of infection increased in two ways by the contact effect. First, they have more contacts and therefore proportionately more risk. But not only do their contacts increase, the fraction of their contacts that are with high-risk individuals increase as well. Both the exposed and unexposed population segments experience this effect.

Let us compare the influence of these exposure effects separately on the exposed and unexposed segments of the population. The unexposed populations can experience only indirect effects of exposure while the exposed segments experience both direct and indirect. Graph 3 presents the effects in the unexposed populations.

Here we see that the largest indirect effects are from the contact effect. The contagiousness and duration effects are two effects that do not have any direct effects on the exposed. They have less of an indirect effect in all, however, than the contact rate effect.

Using the model you downloaded for the last exercise, keep the R₀ for the baseline epidemic without exposure effects the same (1.1) but try to find some set of parameter values where the contact rate effect will be less than the other effects. What do you conclude about the generality of the phenomenon where contact rate effects are greater than contagiousness, susceptibility, or duration effects.

Now let us examine the effects in the exposed individuals.

Note that contact rate effects in the exposed population are particularly strong. Also, note that contagiousness and duration effects are the same in the exposed and unexposed populations. We commented before on why that should be.

We have seen that contact rate effects are the highest and that contagiousness and duration effects are bigger than susceptibility effects. The conclusion that equal multiplicative effects on contagiousness and susceptibility will result in greater population levels of infection for contagiousness effects is a general one. Do these conclusions mean that in a typical epidemiological study that compares the risk of infection in the exposed to risk of infection in the unexposed that the easiest risk factor effects to perceive will be contact rate effects followed by contagiousness and duration effects. Let us examine the pattern of risk differences over time for these four different exposure effects.

Graph 5

Risk differences given no exposure effects (1), three-fold susceptibility effects (2), three-fold contagiousness effects (3), three fold duration effects (4) and three fold contact rate effects (5) when the baseline R₀ in the absence of exposure is 1.1 and 10% of the population is exposed.

Note that contagiousness and duration effects both have greater population effects than susceptibility effects but they are not picked up at all by a standard epidemiological study that only compares risks in the exposed and unexposed. This is one of the most serious deficiencies in the standard approach to risk factor epidemiology.

Also note that although contact rate effects are much greater at the population level, they are not that much easier to pick up in a standard epidemiological study. The standard epidemiological study is strongly biased toward picking up susceptibility effects. If we want to maximize our effect on population levels of infection, we need to devise new methodologies to overcome this bias.

Explore the simulations to find conditions that increase or decrease the difference between susceptibility and contagiousness effects. You don't have to keep the R₀ constant in such an exploration. Look not only at baseline contact rates, transmission probabilities and durations. For example, examine the effects of a 10% increase in each of these. Also, look at the fraction of the population that is exposed and the size of the effects themselves. Describe as well as you can what baseline conditions affect the size of an exposure effect on the overall infection level in a population.