Exercise 2
Assessing Exposure Effects That Increase Transmission Probabilities To Exposed Individuals In SIR Transmission Systems
Professor
James S. Koopman MD MPHOne goal of epidemiology is to identify risk factors. Once risk factors have been identified, a second goal is to determine how much disease will be prevented by controlling those risk factors. Risk factor identification and quantitative risk assessment are different for non-transmissible and transmissible diseases. Use of any of the risk factor assessment parameters studied in Epidemiology 601 or 655, such as the risk difference, risk ratio, odds ratio, assumes that the outcome in one individual is independent of the outcome in other individuals. That is to say that they assume that the important events in disease causation all occur within the individual and are not a function of an individual's contacts with other individuals. Since a contact between individuals is the basis of transmission, and since the outcome of that contact depends upon the outcome of prior exposure in the individual contacted, the independent outcome assumption intrinsic to standard methods in epidemiology does not hold for infectious diseases. Consequently, the standard methodologies can be unproductive or deceptive when applied to infectious diseases.
We must distinguish the goal of risk factor identification from the goal of quantifying risk factor effects. Methods assuming that outcomes are independent between different subjects can identify many risk factors even if they cannot quantify their effects accurately. But methods based on the erroneous assumption of independence cannot identify all risk factors. For that reason it is useful to distinguish the principal risk factors for transmission from secondary risk factors. Analytic methods assuming independence of outcomes often work well to identify the principal risk factor for a transmissible disease. They often fail, however, to identify secondary risk factors. What we will demonstrate in this exercise is that statistics that quantify risk factor effects based on the assumption of independence cannot quantify risk factor effects. For further discussion of this issue we refer you to two articles: 1) Koopman JS, Longini IM, Jacquez JA, Simon CP, Martin W, and Woodcock D. Assessing risk factors for transmission. Am J Epidemiol. 1991; 133(12). 2) Koopman JS and Longini IM. Ecological effects of individual exposures and non-linear disease dynamics in populations. Amer J Pub Hlth 1994; 84(5):836-842.
Quantitative risk assessment involves determining how much an exposure or risk factor affects the risk of a disease. The most common exposures examined are personal behaviors, environmental contaminations, and biological conditions. Risk assessment can be performed on an individual or a population level. At the individual level we attempt to determine the chances that an individual will develop disease as the result of the exposure. At a population level we try to determine how much disease is (or would be) caused in a population by an exposure.
For non-transmissible diseases, the individual and the population levels are rarely distinguished. To quantify the risk attributable to an exposure at the individual level, group rates in the exosed and unexposed always used. Usually these are determined by observations made directly on individuals but sometimes they are determine by observations made at the aggregate level. The true risk of an exposure cannot be determined for an individual. That would require observing the individual outcomes in the same individual and the same conditions with and without the exposure. Since a first exposure will change the immune response of an individual, this can never be done. Instead, individuals are classified into exposed and unexposed groups and their average risk difference at the group level is taken as representative of each individual. In fact, every individual has their unique level of risk. If specified accurately enough for all the conditions affecting risk, each individual's risk is truly either 1 or 0. But we can only calculate risk on the basis of characteristics that put individuals into groups.
At the crudest level, where confounders are not considered, the groups are just exposed and unexposed. To calculate the risk difference one determines the risk over a defined period in the exposed group and in the unexposed group. Then one takes the difference. This risk difference is often adjusted for confounding variables with multivariate models. For non-transmissible diseases this risk is then taken to apply to all the individuals in the group. The other side of this coin is that it is assumed that you can sum up the risks due to an exposure at an individual level across all individuals in a population and you will then have the level of disease in the population caused by the exposure. We will show in this exercise that this is not the case for transmissible infections.
The reason you cannot sum up individual risks to get the population risk for transmissible infections is that the basic risk event, transmission of infection, does not involve just one individual. It involves a contact between two individuals, one infected and one susceptible. Each of those individuals are in turn connected to other individuals in ways that are capable of establishing chains of transmission. If someone anywhere down a chain of transmission changes their exposure category, it will affect the risk of infection in everyone further down that chain of transmission. Thus an individual's risk is not just determined by their exposures, but by everyone else's exposures as well.
Even if we determine the exposure status of everyone in a population, we do not have enough information to determine the risk status of an individual. Even though each individual in one population may correspond exactly to an individual in another population in terms of their exposures, if the individuals in two populations are connected differently into potential chains of transmission, the risks of the individuals in the two populations will be different. To assess the risk of infection in individuals, one must assess the shape of the chains of transmission at the population level. The methods to assess the shape of potential chains of transmission and translate those measurements to risk assessments are in quite preliminary stages of development.
The term "indirect effects" has long been used by infectious disease epidemiologists to refer to protection or causation of infection in some individuals through preventive or causal actions on other individuals. Recently the term has suffered some lack of specificity as non-infectious disease epidemiologists have adopted it for cases where one variable causes another variable which directly affects an individual. We use the term here only in the classical infectious disease sense. Indirect effects explain why risks to individuals do not sum up to risks in populations for transmissible infections
There are two major sources of indirect effects: transmission and immunity.
In many cases the indirect effects of risk factors or public health interventions may be greater than the direct effects. Classic epidemiological measures like risk ratios, risk differences, and etiologic fractions only assess direct effects. To understand how indirect effects from transmission and immunity can determine the level of infection in a population that does not correspond to the sum of individual risks, we will consider some idealized examples. The points to be presented are usually presented in a very mathematical form. This is a non-mathematical presentation.
The person directly vaccinated may not be the only one to benefit from that vaccination. There are indirect benefits of the vaccination to the people on the potential chain of transmission that involved the vaccinated individual. When vaccination breaks that chain, many people further down the chain who did not get vaccinated may benefit. We can protect some people indirectly by vaccinating others. Vaccination of others may protect one indirectly in two ways. First vaccination may prevent infection in the person who would have been a source of infection either because that person was directly or indirectly protected by vaccination. Second, vaccination may not prevent infection in the person who is the source case of infection. Vaccination of that person before they become infected may merely reduce the contagiousness of that person.
How much indirect benefit there is from vaccination will depend upon the transmission routes available which infection might take to reach the individuals further down potential chains of transmission from a vaccinated person. If there are so many routes that all individuals further down a potential chain of transmission that was cut by immunization will be infected anyway, then there will be little indirect effect. If vaccinating a few individuals cuts off long and highly branched chains of transmission at the trunk, then there will be large indirect effects. Thus the extent of indirect effects from vaccination depends upon the extent of contact that can transmit infection in a population, the pattern of those contacts, and the pattern of immunizations. Many administrative decisions in vaccination programs affect the pattern of who will and who will not get vaccinated. Some people may cost 10 times as much to reach with vaccination. If they generate more than10 times as many indirect effects, however, it will be cost effective to spend 10 times as much to reach those individuals. One of the most important aspects of administering immunization programs is knowing how to make decisions which will maximize the indirect effects of vaccination.
The indirect effects of immunization are called herd immunity. Later we will try to clarify what determines herd immunity and we will try to clear up some misconceptions about it.
Let us consider indirect effects from STD risk factors on an individual level first. Then we will consider them on a population level. Say that the boyfriend of a man's wife starts using a condom when he sees a prostitute. That is going to reduce that husband's risk of venereal infection even if the husband doesn't change his risk behaviors or with whom he has sex.
On a population level, consider a situation where only 1% of all sex in a society is with prostitutes and only 5% of all gonorrhea is in prostitutes, but these prostitutes form key links in the chain of transmission that keeps an agent like gonorrhea circulating. Say that each infected prostitute infects 10 other individuals. Only half of these individuals may in turn infect another individual. If the chain of transmission that these individuals start never gets back to the prostitute population, the chain of transmission will eventually end. If it does get back to the prostitute population, it will start 10 new chains. If this idealized example were really the case, it might be possible to completely eliminate an agent like gonorrhea from that population just by affecting the 1% of all risk behavior that involves prostitutes. Thus, an intervention that directly reduced the risk of only 1% of the population that experiences only five percent of the gonorrhea would reduce gonorrhea at the population level not by 5% but by 100%.
Consider an enteric infection like Shigella flexneri. Flexneri used to be the most common Shigella in this country as it still is in most developing countries. In the middle of the 20th century, however, this country underwent a change in enteric agent transmission dynamics that considerably reduced flexneri while it had a much smaller effect on sonnei. Let us consider why this shift took place.
Most Shigella are highly transmissible via direct contact. In one outbreak investigation I conducted in Cali, 20 children who were infected from contaminated food in a school slept in the same beds with 23 other children in their homes. All of those children not only became infected, but they became ill with infection. It takes a higher dose of organisms to transmit flexneri than it does to transmit sonnei. Once infected, however, individuals with flexneri might produce a somewhat higher number of organisms and once flexneri contaminates food, it might reach a somewhat higher level. Growth in food, however, allows the infectious dose to be reached rather readily for either flexneri or sonnei. Individuals infected with flexneri will require more intimate contact to spread infection directly because they have to transmit a higher number of organisms. Thus the direct effects of improved food hygiene in the middle of this century may have been the same for both species. If both species depended upon food contamination to eventually sustain all chains of transmission, then improved food hygiene would have had greater indirect effects on sonnei than on flexneri. Sonnei, however, can be sustained through chains of transmission that occur in nursery schools while such sustained transmission is rare for flexneri. Thus eliminating food contamination had greater indirect effects for flexneri than it did for sonnei.
Dengue is a virus that probably hasn't changed in centuries. There are four different variants of the dengue virus. Over time something remarkable has happened to make dengue type 2 emerge as a severe threat to child and even adult health. Although all virus types can cause severe dengue, the type two virus more often causes severe dengue than the other types. The first type two viruses isolated in the mid 40s in Thailand have remarkably similar nucleotide patterns to the type two viruses isolated there today. Severe hemorrhagic dengue has emerged not because the virus has changed, but because the old pattern of intermittent epidemics changed.
Before the late 1940s in Southeast Asia, dengue epidemics were sporadic events usually separated by many years. Then more frequent epidemics began to appear and finally the situation of today has emerged where there are annual epidemics with multiple serotypes. Mexico and tropical Central and South America seem to be following a similar course but about 20 years behind Southeast Asia.
Type two viruses do not cause severe dengue when they are the first virus to infect an individual. Neither do they cause severe hemorrhagic dengue when they are the third virus to infect an individual or when five years has passed between the first and the second dengue virus infection. But they do cause severe dengue when they are the second virus to infect an individual within a five year period. When multiple agents circulate each year, the chance of a sequence where a second infection with a type 2 virus follows a first infection with another strain is increased.
The exposure of people to the different Aedes species that can transmit dengue probably did not change much over the interval when annual epidemics with multiple strains emerged. Indeed, electric fans and mosquito repellents may even have reduced individual exposures. What changed was the movement of people that made it more possible for small epidemics among localized villages or neighborhoods to spread beyond their borders. Mobile infected humans carry the infection from one population of mosquitoes to another and thus sustain transmission across large segments of population.
What has changed with dengue is not the level of exposure of humans to mosquitoes, but the pattern of contact between mosquitoes and humans. In the past, infected humans were less likely to expose mosquitoes in distant locations to their infection. With modern travel methods, they are now much more likely to expose mosquitoes in distant locations.
Dengue is just one of many examples where human population patterns are building a critical mass needed to sustain transmission. A most important example of this phenomenon nowadays is AIDS.
Most infections can be transmitted by several different routes. Hepatitis B and HIV can both be transmitted either sexually or percutaneously. HIV in heterosexual populations may have a reasonable probability of being transmitted through two or three generations, but the chances of maintaining continuing lines of transmission beyond 7 or 8 generations might be slim. Thus most introduction of HIV into the heterosexual population might die out. For each dying chain of transmission along the heterosexual route, what if there were a new chain started by percutaneous transmission. It might be that only 1 in four transmissions in the heterosexual population were from percutaneous contact. But if you could eliminate that contact, infection would die out of the heterosexual population. Thus affecting one fourth of the exposures could prevent all of the infections so that the sum of the risks on the individual level do not equal the risks on the population level. The indirect effects from controlling this risk factor would be great.
Transmission system analysis becomes especially important when there are multiple modes of transmission and/or multiple different points in the transmission system where one might decide to concentrate ones control resources. We are increasingly addressing the control of infections that are less uniform, less likely to cause a distinctive syndrome in a majority of cases, and more likely to have complex transmission systems which have not yielded to the single focus control programs of the past. That is to say, we are increasingly addressing agents that have adapted in a multitude of ways to sustain their circulation. Thus, the frequency with which we face choices regarding control options directed to different modes of transmission is increasing. It is important that we develop ways to determine which control actions will have the most indirect effects. The first step is to develop a transmission model where we have exposed and unexposed individuals that enables us to appreciate the nature of indirect effects.
Indirect effects arise because individuals are connected to each other. The connections between individuals provide a social dimension to epidemiological data that is not often taken into account.
Epidemiological data is usually structured such that individuals studied are arranged in rows and variables measured on those individuals are in columns. Some columns represent dependent variables like disease and some represent independent variables like exposure status. All of the statistics University of Michigan epidemiology students learn during their master's training assume that this data structure is valid for assessing risk factor effects and controlling such effects for confounding. The data structure may be elaborated with repeated observations at different times, or by clustered sampling, but the analysis proceeds by using models with parameters that relate variables to each other rather than by using models with parameters that relate individuals to each other. We will see that this is an essential difference in infectious disease models. To see this difference, let us present the standard data form first in the following table. When standard analytic methods use this data form, the order of individuals in different rows makes no difference.
Table 1: Standard Epidemiological Analysis Data Structure
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Outcome 1 |
Outcome 2 |
Exposure 1 |
Exposure 2 |
Exposure 3 |
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Individual 1 |
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Individual 2 |
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Individual 3 |
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Individual 4 |
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Individual 5 |
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In reality each individual is connected to other individuals. It is these connections through which infection gets transmitted. Social Network Analyses, as described by Wasserman and Faust in their 1996 Oxford University Press text titled Social Network Analysis, is concerned with describing the patterns by which individuals are connected to each other. The data structure for social network analysis has individuals in both rows and columns and the table entries describe something relevant to the connections between the individuals in the rows and in the columns.
Table 2: Social Network Analysis Data Structure
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Individual 1 |
Individual 2 |
Individual 3 |
Individual 4 |
Individual 5 |
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Individual 1 |
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Individual 2 |
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Individual 3 |
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Individual 4 |
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Individual 5 |
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Data that might be entered into such a table include the rates at which individuals contact each other or their probabilities of transmission should they make contact. These are the basic parameters of the infection transmission models studied in this course. The basic parameters of infection transmission models lie in this social dimension, not in the dimension connecting exposure to disease. In this type of data structure, the data values depend upon the arrangement of individuals. The arrangement of individuals is the central aspect of the data structure.
Let us now consider how data in a social network data structure generates networks. To simplify things we consider that the only data in the table is a zero or a 1 where 1 indicates a contact and zero indicates no contact. The direction of the contact is from the row individual to the column individual. The data might be that of Table 3.
Table 3: Social Network Data Indicating the Presence or Absence of a Connection
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Individual 1 |
Individual 2 |
Individual 3 |
Individual 4 |
Individual 5 |
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Individual 1 |
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1 |
1 |
0 |
1 |
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Individual 2 |
1 |
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0 |
0 |
0 |
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Individual 3 |
0 |
0 |
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1 |
0 |
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Individual 4 |
1 |
1 |
0 |
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1 |
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Individual 5 |
0 |
0 |
0 |
0 |
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Such data would be describing the following connections between individuals.
Figure 1: Connections between individuals described by social network data
The complete structure of epidemiological data is a three dimensional structure that has each of the data structures just presented as particular planes in the three dimensional data. Such a data structure is needed to assess any disease where the connections between individuals matter, that is to say where connections between individuals play a role in disease processes,. This is seen in figure 2.
Figure 2: The three dimensional structure of epidemiological data
The analytic methods you have been taught in your epidemiology and biostatistics courses use data only from the individual effects plane that crosses individuals with the outcome and exposure variables measured on those individuals. The effects of exposures cannot be assessed just in the individual effect plane because those exposure effects alter the outcomes of interactions between individuals.
In a three dimensional data structure such as this, the combination of infection status and exposure status might be used to determine if there are connections between individuals. For example, Figure 1 might only have lines where the source individual is infected. This then would describe a potential chain of transmission. There might be different types of connections depending upon exposure status. The connections might also have different strengths depending upon exposure status. For example, immunity status might be one exposure variable and the connections to someone with immunity might be very week or non-existent. Another example might be whether an individual washes their hands after defecation. Such washing might diminish the value of connections from the source individual.
Because the effect of each interaction depends upon whether the individuals in the interaction have been affected by infection and immunity processes in the past, assessing the effects of exposure variables in the population depends upon formulating how infection is spread through the population. A model that specifies the patterns of contact is required. The transmission models we consider in this course capture an essential aspect of the pattern of connections just discussed despite the fact that those models have no individuals and therefore do not generate data structures like those we have just considered.
Transmission models take this three-dimensional structure dynamically through time. The models in the first Epid 606 exercise did not make any distinctions between individuals that affected whether or not they were connected to each other. Those models just assumed that all segments of the population were equally connected to all other segments of the population. Although individuals were distinguished by whether they were in the S, I, or R categories, those models assumed that these distinctions did not affect contacts between these population segments.
The models in this chapter add another distinction between individuals. We distinguish individuals as being exposed or unexposed. In the model to be presented in this chapter, the exposed and unexposed individuals have the same contact rates. Given this fact and the fact that exposure status will not influence the chances that two individuals make contact, one might think that the social dimension will not make a difference in risk assessment in such a model. We will see that this is wrong. We will see that because the parameters of contact and transmission lie in the social dimension, exposure risks are strongly influenced by this dimension despite the fact that contact processes don't distinguish individuals by exposure status.
Very few epidemiological data sets have variables that express the degree of connection between individuals or between broad classes of individuals. Data from STD studies do tend to collect more of such data. For example, it is very common to ask about specific partnerships and specific activities in those partnerships. For airborne or enteric infections, relational questions are also used. For example, questions on crowding, on number of people in the house, on how often one is in crowds, etc. are often used. But even when epidemiologists expressly state the need for variables that relate individuals to each other, they most often analyze the data with methods that assume individuals are not related to each other.
One reason for that deficiency may be that very few data analytic methods have been developed for data on relationships between individuals. There is a chicken and egg problem here. Biostatisticians tend to develop methods to analyze the data that epidemiologists collect and find important. A case in point is the sexually transmitted disease variable of time between meeting a partner and having sex with them. This is data that clearly lies in the social plane and which Sevgi Aral at CDC has long emphasized as being of great importance as a determinant of transmission potential. But when this data gets analyzed as if it existed only in the individual plane, it loses most of its value. This then tends to lessen the importance of collecting this sort of data. So epidemiologists don't collect the data and biostatisticians don't develop the most appropriate methods to analyze the data.
It is my hope that by demonstrating the importance of the social dimension through the exercises in Epid 606, that a new stimulus to collect and to develop analytic methods for such data may be stimulated.
The models we present here get us out of the standard approach in epidemiology which has recently come to be known somewhat disparagingly as "risk factor epidemiology". I do not want to be counted among those who are disparaging risk factor epidemiology. Some of our most productive tools for identifying controllable causes of disease lie in the risk factor epidemiology dimension. Until we develop better methods to gather and use data from the social dimension, I think we should seek maximum value from methods that do not. The important message of this exercise is that we need to apply good judgement as to when our failure to use data from the social dimension in our analysis to estimate parameters in the social dimension will cause us problems and might lead to bad infection control policy decisions.
Despite the fact that the analytic methods you have learned in your training so far make clearly erroneous assumptions when applied to infectious diseases, those methods have proven utility. There is still a lack of methodology which takes the social dimension into account. Rather than disparage risk factor epidemiology, I think our task should be to define methodologies for a more comprehensive epidemiology which takes the social dimension and the cellular and molecular dimensions into account.
A first step toward developing new methodologies that integrate the social dimension into epidemiological analysis is to demonstrate that errors arise when the social dimension is not taken into account. Demonstrating errors, however, is insufficient motivation to change epidemiological practice. Historical and philosophical analyses of the scientific process have clearly demonstrated that science must proceed by working with theories and methods that are not wholly correct and are error prone. Those historical and philosophical analyses demonstrate that the way to move science in new directions is not to demonstrate errors, but to demonstrate the utility of alternative approaches. To develop the new methods, however, we must make the nature of the errors clear. We do that in this chapter by constructing theory and models that incorporate the social dimension and then showing the error of the inferences made by standard analyses that do not take that dimension into account. Once useful theory and models are available, then the path to developing methods that are based on assumptions consistent with the reality of transmission should become clearer.
Because the task of developing study design and analytic methods must deal with discrete individuals, the compartmental models we study in this course do not provide a sufficient basis for such development. Discrete individual models are required for this task. Discrete models, however, are less immediately tractable than compartmental models. The understanding of system phenomenon in epidemiology that one gains from working with compartmental models is an essential first step to working effectively with discrete individual models.
(This section is an advanced topic that can be skipped by most 606 students. I include it because I know of at least one student in this course who is interested in this topic.) There has been much discussion in the epidemiological community lately about the need to go beyond risk factor epidemiology. A need to integrate analysis of hierarchical systems has been clearly expressed by Mervyn and Ezra Susser who propose that the metaphor of Chinese boxes which all fit one inside the other should guide the formulation of epidemiological theory and methods. In a similar fashion social epidemiologists like as Steve Wing and Nancy Krieger have emphasized the need to take social and political dimensions into account in epidemiological analyses.
One approach to developing methods appropriate to these new metaphors has been the development of "hierarchical analytical models". In these models, all final causal actions producing disease take place at the individual level. Social and ecological variables that cannot be measured at the individual level are, however, integrated into the analysis. The way they are integrated is by assuming that descriptive rather than dynamic characterization of the social dimension effects is adequate and that descriptions of social environments are meaningful determinants of individual risks.
The use of hierarchical models is an improvement over staying wholly in the individual risk factor dimension of epidemiological analysis. For infectious disease epidemiology, however, it is an inadequate and inefficient solution. In order to relate individual and population risks for infectious disease phenomenon, we need to define the causal processes which generate potentially infection transmitting interactions between individuals and which determine the outcomes of those interactions. Static descriptions at the ecological level will not do because the nature of immune processes continually change the ecological setting of infection transmission as infection spreads through populations. A description of infection patterns or immunity patterns at one point in time is not adequate because those patterns are in a continual state of flux. The direction of the flux is very difficult to predict because contact and transmission systems are so highly non-linear. Likewise, static descriptions of the contact patterns through which infection flows are inadequate for predicting future risks. That is because the future of infection transmission through any system is very highly dependent upon the history of that transmission and the patterns of immunity and sources of contagion that history has left.
To effectively integrate the social dimension into epidemiological analysis, we need dynamic models of contact patterns, infection, and immunity. Let us develop a very simple model of this type.
In the first exercise we did not distinguish exposed and unexposed classes of individuals. Here we do. When exposure status affects the rate at which people make contacts with other people, the rates at which people in different classes contact each other must be formally specified. When the rates of contact for exposed and unexposed categories are the same, we could do just as we did in the first exercise and say that our classification of individuals makes no difference for contact. The model in this exercise will in fact have exposed and unexposed individuals making contact at the same rate. But we will formulate contact in a more general manner that is capable of defining contact rates between groups that have different overall contact rates. That will best serve our purposes for the next exercise.
There are many ways to formulate contact between different segments of a population. Our purpose in this chapter and the next couple of chapters is simply to show the importance of contact patterns. For this purpose, we do not have to employ formulations which correspond to the actual process of making contacts. We only need a formulation which allows us to specify different degrees of interaction between exposed and unexposed population segments. For that purpose we can use two of the most commonly employed mixing formulations. These are the "proportionate mixing" formulation and the "preferred mixing" formulation. This exercise will use only the proportionate mixing formulation. In the fourth exercise we will use the preferred mixing formulation that builds upon the proportionate mixing formulation.
The proportionate mixing formulation in an unbiased mixing pattern in which individuals encounter other individuals in proportion to the total contacts made by those other individuals. The advantage of the proportionate mixing formulation is not that it is realistic. It is not realistic to think that people make encounters randomly in proportion to the rate that other people are making encounters. The great virtue of the proportionate mixing formulaiton is that it is very tractable mathematically. The same can be said for the preferred mixing formulation. It is widely used by mathematical modelers of infectious diseases not because it is realistic, but because it has mathematical advantages.
Proportionate mixing assumes that all contacts occur at random. This is not the same as saying that individuals mix randomly. If individuals mix randomly, then each individual would have an equal probability of making contact with each other individual. But some individuals may be making more contacts than other individuals. The proportionate mixing formulation assumes that ones chances of making contact with specified other individuals are proportionate to the rate at which those other individuals are making contacts. For example, you have 10 times the chance of making contact with someone who makes 10 contacts a month as you do with someone who makes 1 contact per month. In the formulation presented below, it is assumed that contact events have no directionality. That means that the contacts are not distinguished by who initiated the contact nor by who has the potential to transmit to whom within the contact. We assume that transmission potential is symmetric during a contact. Formulations with directionality are quite important for many infectious disease system analyses. They are not, however, necessary for the purposes of this exercise.
In the very first model we examine, we have exposed and unexposed individuals making contact at the same rate. That means the contact rates in expose and unexposed individuals would cancel out of the formulas below. In the next model, however, we will model exposed populations that are making contacts at a higher rate than unexposed populations. In that case, we formulate proportionate mixing more generally as follows:
Define
Ne = the number of exposed individuals (or better said the size of the exposed population segment) (Note that Ne is the sum of Se, Ie, and Re)
ce = the rate at which exposed individuals make contact
Nu = the number of unexposed individuals
cu = the rate at which unexposed individuals make contact
Ceu = Cue = the overall rate of contacts in the population which are between individuals where one is exposed and one is unexposed.
Cee = the overall rate of contacts in the population where both individuals are exposed
Cuu= the overall rate of contacts in the population where both individuals are unexposed
The total number of contacts per unit time made by exposed individuals will be Nece. These will be distributed proportionately between exposed and unexposed individuals according to the total number of contacts made by exposed and unexposed individuals. The fraction of contacts by the "e" population that will go to other individuals in the "e" population will be 
. That means that Cee, the overall rate of contact between "e" individuals, will be 
. The fraction of contacts by the "e" population that will go to other individuals in the "u" population will be 
so that 
. Note that we get the same rate of contact between exposed and unexposed segments of the population whether we start by calculating the rate of contacts made by exposed individuals with unexposed individuals or the rate of contacts made by unexposed individuals with exposed individuals. This necessary condition of symmetry is a convenient aspect of the proportionate mixing formulation.
We now present a standard SIR model without vital dynamics similar to the one presented in exercise 1 but with division into exposed and unexposed populations. This model assumes that everyone mixes proportionately to his or her contact rates. We divide the Ne exposed individuals into Se, Ie, and Re in the fashion of the SIR models presented in the first exercise. We will make a similar division of the unexposed. Infection status is not a determinant of contact patterns, however, so that the formulas we will present could be collapsed into those just presented for proportionate mixing.
In this model, the only effect of exposure is to increase the transmission probability to an exposed person when that person makes a contact with an infected individual. This exposure effect can be viewed as an increase in susceptibility of the exposed individual that increases their risk of infection but does not increase the severity of infection once it occurs. An example might be the presence of a genital ulcer that can allow access by the HIV virus to the host. The model is presented in the ExpEff1 Model 1 Diagram and equations which follow. It is available in the Public IFS space of Dr. Koopman as ExpEff1.STM.
ExpEff1 Model Diagram
If you click on the down arrow on the left hand border of the diagram window, you will get a Stella IIÔ equation window. From this window you can see exactly what equations have been entered into either the flow regulators or the converters where derived variables like the forces of infection (UforceInf and EforceInf which are the forces of infection experienced by unexposed and exposed individuals respectively). Of course, if you want you can also see these formulations by double clicking on the flow regulators or converters.
DIFFERENCE EQUATIONS FOR STOCKS
IE(t) = IE(t - dt) + (NewIE - NewRE) * dt ::
INIT IE = .0001*( FractionExposed)
INFLOW: NewIE = SE*EForceInf
OUTFLOW: NewRE = IE/Dur
IU(t) = IU(t - dt) + (NewIU - NewRU) * dt ::
INIT IU = .0001*(1- FractionExposed)
INFLOW: NewIU = SU*UForceInf
OUTFLOW: NewRU = IU/Dur
RE(t) = RE(t - dt) + (NewRE) * dt ::
INIT RE = 0
INFLOW: NewRE = IE/Dur
RU(t) = RU(t - dt) + (NewRU) * dt ::
INIT RU = 0
INFLOW: NewRU = IU/Dur
SE(t) = SE(t - dt) + (- NewIE) * dt ::
INIT SE = .9999*( FractionExposed)
OUTFLOW: NewIE = SE*EForceInf
SU(t) = SU(t - dt) + (- NewIU) * dt ::
INIT SU = .9999*(1- FractionExposed)
OUTFLOW: NewIU = SU*UForceInf
PARAMETERS
ContRt = 1.5
tpGvnCont = .25
Dur = 2
EefctOnSuscept = 3
FractionExposed affects initial division of population into exposed and unexposed
In the diagram of baseline conditions, (which is further down in the same model as the one presented in the above diagram) the fraction exposed remains fixed at baseline conditions set in the parameter labeled "BaselineFractionExposed".
DERIVED VARIABLES
UForceInf = ContRt*tpGvnCont*{(IE+IU)/(SE+IE+RE+SU+IU+RU)} This is the force of infection experienced by unexposed individuals. Note that this uses the proportionate mixing formulation. Since ce = cu however, the c's cancel out of the numerator and the denominator.
EForceInf = ContRt*tpGvnCont*EefctOnSuscept*(IE+IU)/(SE+IE+RE+SU+IU+RU) This is the force of infection experienced by exposed individuals.
FractUInf = (IU+RU)/(SU+IU+RU) The fraction of unexposed individuals that are infected or have been in the past infected at any point in time. In the Stella model, if there are no unexposed individuals we set this fraction to zero using an "if … then … else …" statement. This avoided a division by zero problem.
FractEInf = (IE+RE)/(SE+IE+RE) The fraction of exposed individuals that are or have been in the past infected at any point in time.
FractTotInf = (IE+IU+RE+RU)/(IE+IU+RE+RU+SE+SU) The fraction of the total population that are or have been infected at any time in the past.
NumCasesAttribExpos = (FractEInf-FractUInf)*(SE+IE+RE) (This number of attributable cases is the risk difference times the number exposed)
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Note that we treat both I and R compartments as infected for the purpose of calculating the attack rates FractUInf and FractEInf. One would observe these attack rates if the entire population was studied at any time during the epidemic and cases that occurred at any previous time were counted as infected. One would need a combination of agent detection methods and serology to detect all such cases.
The model I have constructed for your use contains a duplicate model that is unaffected by FractionExposed. It has a BaselineFractionExposed. This enables one to compare SIR epidemics with and without the elimination of exposure. In Epid 601 and 655 you learned that if there is no confounding and if the independent outcome assumption holds, you can use the risk difference to predict population level effects of eliminating exposure. In the model we have constructed here, there are no third variables to confound exposure-disease associations. The entire population is examined to measure exposure effects so there can be no sampling biases. Measurement is 100% accurate so there are no information biases. Thus, the risk difference is an unbiased measure. We will see, however, that that the risk difference is not a faithful measure of risk factor effects. That is because the risk difference estimates the causal effect parameter in a model of independent individuals. It does not estimate any causal parameter in a transmission system model. It estimates a parameter in the standard risk factor data plane. In our model, the true risk factor effect affects the transmission probability when individuals make contact. That is to say, the true parameter lies in the social dimension. There are no stable causal parameters in the standard data plane.
In the next chapter, we will examine other risk factor effects in transmission systems. In all of those models, in all transmission system models, in all real world situations where transmission is a fact of life, the causal parameters will not be in the plane where standard epidemiological data is found. The causal parameters that we should pursue lie in the social dimension. At a minimum, we must begin to understand the errors that will arise if we try to use risk differences as causal parameters for transmission systems.
Here we see that the risk difference is not useful for predicting the effects of eliminating infectious disease exposures. In the following graph, we compare the attack rates in the population where half of the individuals are exposed (a population where BaselineFractionExposed = .5) and in the population where all of the individuals are unexposed (a population where FractionExposed = 0). If the assumptions of attributable risk calculation hold, eliminating exposure from the population that was exposed would have caused all of the population to have the disease rates in the unexposed. In other words, in the figure below, it would have caused line 3 to coincide with line 2 and this would have made line 1, the risk in the total population, to correspond to the coincident lines 2 and 3. The assumptions of attributable risk calculation are that all effects are direct effects. In Figure 2, however, we see that eliminating exposure from the population that was exposed resulted in almost complete elimination of infection. In this case, and in many cases in the real world, the indirect effects are greater than the direct effects. Instead of the direct effects only lowering the infection levels in the exposed to those of the unexposed when their exposures were eliminated, they lowered the infection levels of both the exposed and the unexposed to zero. All of the reduction in the unexposed is an indirect effect. The lowering in the exposed to levels below the original unexposed levels can also be considered to be indirect effects.
Figure 2
Model 1 cumulative infection patterns when half of the population is exposed (2:FractUInf 2 and 3:FractEInf 2) and when almost no one is exposed (1:FractUInf)
Explain why eliminating exposure from the half of the population that was exposed caused the elimination of infection in both the exposed and unexposed populations. Don't just say this was an indirect effect. Try to think about what is happening to the forces of infection in the exposed and the unexposed and explain why those forces of infection cannot generate an epidemic. In your explanation, you may want to compare the forces building up the infected compartment with the forces draining it. While it is not mandatory, I encourage you to use the model provided to explore model behavior in ways that enhance your explanations. For example, you might want to eliminate exposure from different fractions of the exposed populations and observe the different sizes of effects you get in the unexposed populations. Another thing you might do to help your explanation is to make derived variables that express how much of the risk of infection in the unexposed derives from the exposed. To do this you would divide the force of infection into two components. One would be the IU component and the other would be the IE component. You can see that IU and IE are multplied in the force of infection equations.
Describe the relationship between the final size of the epidemic and the fraction of the population that is infected over the course of the epidemic as the fraction of the population that is exposed decreases. What you need to do here is run the model with different fractions exposed and record the fraction of the population that is infected at the end of the epidemic. Then you will need to take those separate fractions infected and put them in some graphics program to plot out the relationship between fraction exposed and final fraction of the population infected. Remember that you can set up a sensitivity run to get all of your final fractions infected with a single run command. You do this under the run menu at sensispecs. Your sensitivity variable will be FractionExposed. Then you gather the final fractions from each setting and plot the final size of the epidemic versus the fraction of the population that is exposed. Use 11 settings at 10% intervals from 100% to 0% exposed . Before you run any models, draw out a curve by hand of the results that you expect to get. Remember your axes are fraction of the population that is infected at the end of the epidemic and the fraction of the population that is exposed.
Explain why the curve you generate has the shape that it does. If it differed quantitatively or qualitatively from the curve you drew before you ran the simulation, explain why you missed the prediction of what to expect. Note that this question is just an extension of the previous question from explaining complete elimination to explaining the pattern of partial effects as well.
Self evaluated homework:
Double the exposure effect in the model so that transmission probabilities to exposed individuals will be 6 times the exposure probabilities to unexposed individuals. Now predict the shape of the same curve you just generated and check your predictions by running the model.