It is useful to separate the goal of risk factor identification from the goal of risk factor quantification. Similarly it is useful to distinguish the dominant risk factors for transmission from secondary risk factors. By dominant risk factors we mean those which directly cause the majority of infections. By secondary, we mean all others. The methodology used in non-transmissible diseases often works well for the identification of the principle risk factor for a transmissible disease. It does not work well for the identification of secondary risk factors. Neither does it work well for the quantitatively predicting the effects of risk factors for infectious diseases -- be they dominant or secondary risk factors. Even for dominant risk factors, some standard analytic procedures, such as stratifying by geographic locality, can completely obscure their detection. This is demonstrated in a May, 1994 article in the American Journal of Public Health by Koopman and Longini entitled "Ecological effects of individual exposures and non-linear disease dynamics in populations".

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 assess risk at the individual level, it is often assumed that the average risk of an exposure to an exposed individual equals the risk difference. Every individual has their own unique level of risk. But we can only calculate risk on the basis of characteristics that put individuals into groups. At the crudest level the groups are just exposed and unexposed. To calculate the risk difference one determines the risk over a defined period of time in the exposed group and in the unexposed group and takes their 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.

For transmissible diseases the average individual risk does not equal the population level risk per individual. You cannot sum up individual risks to get the population risk. The reason for this 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 exposure 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 science of how to do that is just emerging. Developing that science is a principal goal of the Michigan HIV modeling group.

The goal of this lecture is to get you to understand why there are important differences in risk assessment between non-transmissible and transmissible diseases.

There are two major reasons why risks to individuals do not sum up to risks in populations for transmissible diseases.

1 With transmissible diseases, there are indirect effects at the population level of exposures on unexposed individuals. These arise because an individual infected because of one exposure may transmit infection to others.

Indirect effects are important for STDs, enteric infections, respiratory infections, and indeed any type of infection where the agent from an infected individual may eventually be a cause of infection in another individual. 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.

2 Immunity must be considered in assessing an individual's or a population's risk from an exposure. Immunity is usually unmeasured in our studies but the effects of immunity must be considered in our analysis even when immunity is unmeasured. Patterns of immunity in a population will determine how an infection can spread in that population.

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.

Examples of indirect risk factor effects in infectious diseases

A Immunizable diseases:

The person directly exposed to a risk factor or to directly benefit from a protective action like vaccination may not be the only one to suffer or benefit from the exposure. 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, not only the vaccinated individual may benefit, but many people further down the chain who did not get vaccinated may benefit as well. 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 depend upon the extent of contact which 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 left unvaccinated. 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.

B STDs

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%.

C Enteric Infections:

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 with 23 other children. All of those children became infected. 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 species. 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.

D Arthropod borne infections:

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 there are multiple agents circulating each year, there are many more cases of sequences of infection that can cause severe dengue.

The exposure of people to the different Aedes species that can transmit dengue probably did not change that 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 over the interval were not so much direct exposure to mosquitoes, but changes in movement of people that made it more possible for small epidemics among localized villages or neighborhoods to spread beyond their borders and take advantage of the existing mosquito populations that were only rarely invaded by the virus previously.

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.

E Mixed Route Infections:

Most infections can be transmitted by several different routes. Hepatitis B and HIV can both be transmitted sexually or percutaneously. It is quite possible that 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 very 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.

The social dimension in infectious disease epidemiology

The structure of epidemiological data

Standard data structure

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 presented in Chapter 4 and all of the statistics University of Michigan epidemiology students learn from the statistics department if they only take the standard courses assume that this data structure is valid for assessing risk factor effects and controlling such effects for confounding.

Standard Epidemiological Analysis Data Structure

Outcome 1 Outcome 2 Exposure 1 Exposure 2 Exposure 3

Individual 1

Individual 2

Individual 3

Individual 4

Individual 5

Social Network Analysis Data Structure

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.

Social Network Analysis Data Structure

Individual 1 Individual 2 Individual 3 Individual 4 Individual 5

Individual 1

Individual 2

Individual 3

Individual 4

Individual 5

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 we have studied. They lie in this dimension, not in the dimension connecting exposure to disease.

The underlying three dimensional structure of epidemiological processes

The complete structure of epidemiological needed to assess any disease where the connections between individuals matter, that is to say where social components play a role in disease processes, is a three dimensional structure as seen in figure 1.

Figure 1

The effects of exposures cannot be assessed just in the individual effect plane because those exposure effects alter the outcomes of interactions between individuals. Interactions between any two individuals are connected in contact chains and trees with interactions between other individuals. Those interactions are affected by the exposure variables and produce the outcome variables. 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 which specifies the patterns of contact is required. The transmission models we will now examine essentially take this three dimensional structure dynamically through time. The models examined in Chapter 4 assumed that the social dimension was irrelevant. The models in Chapter 5 did not distinguish individuals by their exposure status and assumed away the effects of different arrangements of who contacted whom by assuming that all contacts were made at random and each individual had an equal chance of contacting each other individual. Since individuals were not classified by exposure status in this model, there was no possibility of arranging contacts according to exposure status. The model we now examine is the minimal model that distinguishes both exposure status and who mixes with whom on the basis of exposure status.

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. Despite the fact that those methods make clearly erroneous assumptions, 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 methodology is needed which incorporates the now well established phenomenon whereby interactions between adaptive agents at one system level lead to the emergence of wholly new phenomenon at another system level which cannot be defined solely in terms of causal events in the agents. Many scientists in many different disciplines are now developing appropriate methodologies. The Santa Fe Institute plays an important leadership role in this task. Epidemiologists who want to lead their profession to ever more productive methodologies to define disease control actions should pay attention to these developments.

The first step in developing the methodology for integrating the social dimension into epidemiological analysis is to recognize its potential utility. One step to recognizing its utility is to demonstrate the errors that can arise when the social dimension is not taken into account. Demonstrating errors, however, will not change things. 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 appropriate methods which will put epidemiological inferences on a more solid basis 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.

Can hierarchical analysis models adequately integrate the social dimension

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 Krueger 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 which 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 will flow are inadequate for predicting future risks because the future of infection transmission through any system is very highly dependent upon the past history of that transmission and the patterns of immunity and sources of contagion which 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.

An SIR model with fixed exposure effects

Preferred Mixing

In Chapter 5 we just dealt with populations which mixed randomly and we did not distinguish exposed and unexposed classes of individuals as we did in Chapter 4. Here we now integrate the dimensions of Chapters 4 and 5. When we add exposure classification in a context where interactions between population segments are specified, we must specify the patterns of interaction between the exposed and the unexposed.

There are many ways to formulate contact between different segments of a population. For the purpose of this chapter we do not have to employ those formulations which are causally most meaningful. 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 the preferred mixing formulation which, because it is very tractable mathematically, is widely used by mathematical modelers of infectious diseases. The preferred mixing formulation was first presented in 1988 by the Michigan Transmission Analysis Group in Mathematical Biosciences.

The basic parameter of the preferred mixing formulation is a fraction of contacts that each population segments reserves for contact with its own subgroup. Such reservation is not a meaningful causal process. It should eventually be replaced by the more meaningful causal models such as structured and selective mixing which the Michigan Transmission Analysis Group have also formulated. For the purposes of understanding system phenomenon, however, it is a useful first introduction to non-random mixing.

The preferred mixing formulation assumes that different population segments have defined rates at which individuals are making contact with other individuals. The population segments may be defined by exposure status, age groups, disease status, or any other characteristic. The segments may combine various different compartments. In our model, we will define exposed and unexposed population segments which combine individuals in the various infection status compartments. We will assume that infection status does not affect the rate at which individuals make contact with other individuals.

The total rates at which population segments make contact between individuals is divided into a reserved fraction and a general population fraction which is one minus the reserved fraction. The reserved fraction of the rates are only made with the class of individuals making the reservation. That is to say, there are various homogenous population mixing settings defined within which mixing is random in exactly the same fashion that occurred in the models presented in Chapter 5. The unreserved fraction of contacts are made with the entire population so that different classes of individuals are making contact. The simplest way of formulating this mixing between disparate individuals is the proportionate mixing formulation.

The Proportionate Mixing Formulation

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 another individual are proportionate to the rate at which that individual is making contacts. In the formulation we will present here, it is assumed that the contact has no directionality. That means that the contact is neither classified in terms of who initiated the contact nor in terms of who has the potential to transmit to whom within the contact. Formulations with directionality are quite important for many infectious disease system analyses but they are not necessary for the purposes of this chapter.

In our case in some situations we will model exposed populations that are making contacts at a higher rate than unexposed populations. In that case we formulate proportionate mixing as follows:

Define

N_e = the number of exposed individuals

c_e = the rate at which exposed individuals make contact

N_u = the number of unexposed individuals

c_u = the rate at which unexposed individuals make contact

C_eu = C_ue = the overall rate of contacts in the population which are between individuals where one is exposed and one is unexposed.

C_ee = the overall rate of contacts in the population where both individuals are exposed

C_UU= 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 N_ec_e. These will be distributed proportionately between exposed and unexposed individuals according to the total number of contacts made by exposed and unexposed individuals. of those contacts will be made among exposed individuals and will be made with unexposed individuals. Note that whether we start by calculating the total number of contacts made by exposed individuals and determining what fraction of those will be with unexposed or by calculating the total number of contacts made by unexposed individuals and calculating the fraction of these that are made with exposed individuals, we get the same answer:

C_EU = = C_UE =

The preferred mixing formulation

As stated earlier, in the preferred mixing formulation, besides the general mixing in the population which can be 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. The general formulation of reserved mixing has separate reserved fractions for exposed and unexposed individuals. In this case we have:

For the purposes of this chapter, we need not refine the model with separate reserved fractions for different segments of the population. We will just define a single r that applies to both the exposed and unexposed populations. Thus our formulations will reduce to:

In the model which we will now present we will divide the N_E into S_E, I_E, and R_E in the fashion of the SIR models presented in Chapter 5. We will make a similar division of the unexposed. Infection status will not be a determinant of contact patterns, however, so that the formulas we will present could be collapsed into those just presented.

Model formulation

We now present a standard SIR model without vital dynamics similar to the one presented in Chapter 5 but with division into exposed and unexposed populations. Our initial model will just assume that everyone mixes at random and that 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. As we will discuss later, this may not be a wholly realistic exposure effect. It is a simple exposure effect, however, which allows us to present the basic model before presenting more involved exposure effects. The model is presented in the Model 1 Diagram and equations which follow. It is available in the Public IFS space of Dr. Koopman.

Model 1 Diagram

DIFFERENCE EQUATIONS FOR STOCKS

IE(t) = IE(t - dt) + (NewIE - NewRE) * dt :: INIT IE = .001*(1-ExposEliminat)

NewIE = SE*EForceInf

NewRE = IE/Dur

IU(t) = IU(t - dt) + (NewIU - NewRU) * dt :: INIT IU = .001*(1+ExposEliminat)

NewIU = SU*UForceInf

NewRU = IU/Dur

RE(t) = RE(t - dt) + (NewRE) * dt :: INIT RE = 0

NewRE = IE/Dur

RU(t) = RU(t - dt) + (NewRU) * dt :: INIT RU = 0

NewRU = IU/Dur

SE(t) = SE(t - dt) + (- NewIE) * dt :: INIT SE = .999*(1-ExposEliminat)

NewIE = SE*EForceInf

SU(t) = SU(t - dt) + (- NewIU) * dt :: INIT SU = .999*(1+ExposEliminat)

NewIU = SU*UForceInf

PARAMETERS

ContRt = 1.5

tpGvnCont = .25

Dur = 2

EefctOnSuscept = 3

ExposElimin affects initial division of population into exposed and unexposed

DERIVED VARIABLES

UForceInf = ContRt*tpGvnCont*(IE+IU)/(SE+IE+RE+SU+IU+RU)

EForceInf = ContRt*tpGvnCont*EefctOnSuscept*(IE+IU)/(SE+IE+RE+SU+IU+RU)

FractUInf = (IU+RU)/(SU+IU+RU)

FractEInf = (IE+RE)/(SE+IE+RE)

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

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

---------------------------------------------

Note that we treat both I and R compartments as infected for the purpose of calculating the attack rates FractUInf and FractEInf. These are the attack rates that one would get if the entire epidemic up to a certain point were studied and all cases during the epidemic were counted as infected.

The model I have constructed for your use contains a duplicate model which will not be affected by ExposEliminat. This enables one to compare SIR epidemics with and without the elimination of exposure. In chapter 4, we saw that risk difference and attributable risk measures could be used to predict the effects of eliminating exposures in the model without vital dynamics presented in that chapter. Here we see that the same measures are 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 ExposEliminat = 0) and in the population where all of the individuals are unexposed (a population where ExposEliminat = 1). If the assumptions in the Chapter 4 models held, eliminating the exposure status from the half of the population that was exposed would have caused all of the population to have the disease rates in the unexposed. In Figure 2, however, we see that eliminating exposure from the half of the population that was exposed resulted in a complete elimination of infection.

Figure 2

Model 1 cumulative infection patterns when half of the population is exposed (FractUInf 2 and FractEInf 2) and when now one is exposed (FractUInf)

Homework 6.1

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. Use the model provided to explore model behavior and to find reasonable explanations. Explanations are often helped by constructing further derived variables that isolate some segment of the process. You might want to consider constructing a variable related to R₀ as discussed in Chapter 5.

Homework 6.2

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 which is exposed decreases. Explain why equal percentage decreases in exposure cause different degrees of decrease in the final size of epidemics.

The dimensions of risk factor effects

There are many different types of risk factors for infectious diseases. Some risk factors, such as being part of crowds or forming a high number of sexual partnerships in certain social settings involve increased contact rates. Some increased contact exposures, such as increased number of the same type of sexual partnerships engaged in by others, may affect the total number of contacts without affecting who is contacted. Most contact exposures, however, will involve a special set of individuals. Thus they will affect the nature of the person contacted. Some contact exposures may not affect the total number of contacts made at all. They may just affect who is contacted.

Some risk factors, such as not washing one's hands in the hospital setting or not wearing condoms during sex increase the probability of transmission given that there is a contact.

Some risk factors might affect the biological process of infection. Acquired immunity may affect the infection process so dramatically that infection is so quickly controlled that it becomes unnoticeable. In that case it would appear that the transmission probability given contact would decrease. With very sensitive infection detection, however, it is likely that some infection process would be noticeable no matter how dramatic the immune response. Thus the immune response might affect the duration of infection and the amount of infectious agent produced by the infection. Not having the acquired immunity would put one in the exposed category. Reduced agent production should be associated with decreased transmission probabilities from the infected individual. If infection is cut short with treatment, the duration of infection might decrease without affecting the degree of contagiousness before treatment.

In the file ExpEffSIR found in Koopman's Public IFS directory, parameters for preferred mixing and for all of the above mechanisms of exposure effect are introduced into Model 1. These are seen below in the Model 2 Diagram and equations.

Model 2 Diagram

DIFFERENCE EQUATIONS FOR STOCKS

IE(t) = IE(t - dt) + (NewIE - NewRE) * dt :: INIT IE = .001*(1-ExposEliminat)

NewIE = SE*EForceInf

NewRE = IE/(Dur*EefctOnDurat)

IU(t) = IU(t - dt) + (NewIU - NewRU) * dt :: INIT IU = .001*(1+ExposEliminat)

NewIU = SU*UForceInf

NewRU = IU/Dur

RE(t) = RE(t - dt) + (NewRE) * dt :: INIT RE = 0

NewRE = IE*EefctOnDurat/(Dur)

RU(t) = RU(t - dt) + (NewRU) * dt :: INIT RU = 0

NewRU = IU/Dur

SE(t) = SE(t - dt) + (- NewIE) * dt :: INIT SE = .999*(1-ExposEliminat)

NewIE = SE*EForceInf

SU(t) = SU(t - dt) + (- NewIU) * dt :: INIT SU = .999*(1+ExposEliminat)

NewIU = SU*UForceInf

PARAMETER VALUES

ContRt = 2

tpGvnCont = .25

Dur = 2

EefctOnContact = 2

EefctOnContag = 1

EefctOnDurat = 1

EefctOnSuscept = 1

PrefFract = 0 (defines an initial condition rather than a model parameter)

DERIVED VARIABLES

EForceInf = GenEForceInf+PrefEForceInf

GenEForceInf = (1-PrefFract)*ContRt*tpGvnCont*EefctOnSuscept*EefctOnContact*

(EefctOnContact*EefctOnContag*IE+IU)/((SE+IE+RE)*EefctOnContact+(SU+IU+RU))

PrefEForceInf = PrefFract*ContRt*tpGvnCont*EefctOnContact*EefctOnSuscept*

EefctOnContag*(IE/(IE+RE+SE))

----------------------------------

UForceInf = GenUForceInf+PrefUForceInf

GenUForceInf = (1-PrefFract)*ContRt*tpGvnCont*(EefctOnContact*EefctOnContag*IE+IU)/

((SE+IE+RE)*EefctOnContact+(SU+IU+RU))

PrefUForceInf = PrefFract*ContRt*tpGvnCont*(IU/(IU+RU+SU))

----------------------------------

FractEInf = (IE+RE)/(IE+RE+SE)

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

FractUInf = (IU+RU)/(IU+RU+SU)

Homework 6.3 (To be handed in.)

Describe for any infection that induces immunity any exposure that increases the risk of infection. Describe this exposure as realistically and as completely as possible. Then run Model 2 at the parameter settings you think are most likely for this situation to compare the effect of eliminating exposure that would be calculated using the risk difference to the effect that would be seen on the basis of the transmission model we have constructed. Describe situations where the risk difference would and would not be useful given your model findings.

Effect on contact rates

Increasing contact rates can have a variable effect upon infection risk in the exposed individual and in other individuals depending upon whether contact with a general population or a high risk population is increased. Intuition into what alters the effects of contact exposures can be quite important for making appropriate public health decisions. Intuition is gained not only by experience with the behavior of a system, but by having explanations for that behavior. The following exercise is intended to provide intuition regarding the important role that contact patterns play in mediating the risk of increased contacts. We examine the number of infections prevented by comparing the total number of infections for numerical solutions of the model with 50% of the population is exposed and when exposure has been eliminated from half of those so that only 25% of the population is exposed. Note that we calculate the number of prevented infections only by comparing the immune populations with and without exposure. This is an arbitrary measurement that might be modified if desired.

Figure 3

Number of infections prevented by eliminating exposure from 50% of exposed individuals when the preferred fraction is 0 (1) 0.5 (2) and 1.0 (3)

Homework 6.4

Explain why the curves in figure 3 have the shapes that they do. You will definitely need to explore several different curves of model stocks or derived variables to come up with a good explanation. You may even want to devise some additional derived variables.

Effect on the infection process

Treatment

We simulate treatment of infected individuals by merely reducing the duration of infection to one fourth of its untreated value. We compare populations where no one was treated to populations where 50% or 100% of the infected individuals are treated. We had to go in and modify the starting values so that in the bottom model of ExpEffSIR no one receives treatment and in the top model either 50% or 100% of the population receive treatment. The initial conditions we used were as follows:

INIT IE = 0.0000000000000001

INIT IE_2 = 0.0000000000000001

INIT IU = .001*ExposEliminat

INIT IU_2 = .001

INIT SE = .00000000000001+(2-2*ExposEliminat)

INIT SE_2 = 0.000000000001

INIT SU = 1.999*ExposEliminat

INIT SU_2 = 1.999

We ran from these starting values under the three conditions of preferred mixing, 0% reserved, 50% reserved, and 100% reserved. The model results where 50% of infected individuals are treated is seen in figure 4 and where 100% are treated is seen in figure 5. Note that our intervention does not directly prevent any infections. It only treats those who are infected and thus acts indirectly to prevent infection in those who would have otherwise been infected by these treated individuals.

Figure 4

Number of prevented infections as the result of treating 50% of infected individuals and thus reducing their duration from 2 to 0.5. The three situations simulated are preferred fraction = 0 (1), 0.5 (2), and 1.0 (3). Contact rate = 4, tpGvnCont=0.25

Figure 5

Number of prevented infections as the result of treating 100% of infected individuals and thus reducing their duration from 2 to 0.5. The three situations simulated are preferred fraction = 0 (1), 0.5 (2), and 1.0 (3). Contact rate = 4, tpGvnCont=0.25

Homework 6.5

Explain why the curves if figures 4 and 5 have the shapes and relationships that they do. Of course you will most likely want to play around with the model to come up with the best explanation.

Effect on pre-exposure immunity

We will have an exercise in chapter 7 using this model which will examine the effects of pre-exposure immunity.

	Outcome 1	Outcome 2	Exposure 1	Exposure 2	Exposure 3
Individual 1
Individual 2
Individual 3
Individual 4
Individual 5