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Common methods of risk assessment such as attributable risk assume away infection transmission by assuming independence of outcomes. The nonlinear dynamics of infection transmission, however, make the effects of controlling risk factors dramatically different from what would be deduced using attributable risk logic. Yet the inappropriate use of attributable risk logic is still very frequent such as the vaccine effect evaluation examples in a current textbook.1
We address here risk assessment for decisions about focusing control efforts on a high-risk group. Rose2 has argued on the basis of attributable risk calculations that a small risk reduction in the general population is commonly more beneficial than a large risk reduction on a small high-risk group. To provide insights and general principles that can inform decisions about whether to focus infection control interventions on high-risk groups, we analyze simple deterministic compartmental models of infection transmission involving high- and low-risk groups. Our objectives encompass only the preliminary stage of gaining insights within the broader scope of infection modeling that creates a structure for a science of infection transmission.3
The Transmission System Model
Our transmission model is a deterministic compartmental model formulated by a system of differential equations. We describe it simply here and in more detail in the supporting online material available with the electronic version of this article.
Because our intention is to provide general heuristics rather than contribute to any one particular decision, we have made our model abstractly simple. We modeled an endemic infection in which once individuals recover, they become completely susceptible again. In our baseline settings, infection lasts one time unit, on average; thus, at equilibrium, incidence equals prevalence. There are no births or deaths. We believe that most of the behavior we illustrate will be found in models in which endemic infection is sustained in the face of immunity—either through eventual loss of immunity or through death and replacement of immune individuals by births of susceptible individuals.
The model divides the population into a low-risk group and a high-risk group. Risk status may be determined by increased susceptibility or increased contagiousness. Increased duration of infection affects the pattern of endemic infections in a manner similar to increased contagiousness, and so we confine this presentation to contagiousness. Increased contact rates are not presented because they have several complexities and we seek to simplify our message. The broad general conclusions about susceptibility and contagiousness risk factors, however, also hold for risk factors affecting contact rates.
Figure 1 provides a schematic of the model. The large rectangles contain the high- and low-risk groups, and within those groups, the population is divided into susceptible and infectious individuals. The fraction α of the population that is high risk is 0.05 in our baseline settings and 0.1 in sensitivity analyses. Contact between individuals can be conceptualized as taking place at 3 sites—a high-risk site h where only high-risk individuals go, a low-risk site l where only low-risk individuals go, and an opposite mixing site o where both high- and low-risk individuals go. All individuals have the same total number of contacts per unit time so that c = cHo + cHh = cLo + cLl where the first subscript refers to the risk group and the second to the mixing site. Contact patterns can be described by a single parameter, the fraction μ of contacts made by the high-risk group with low-risk individuals. μ=cHo/cHo+cHh because contact is symmetric, which requires that αcHo = (1−α)cLo, setting μ fixes both cHo and cLo. At the assortative mixing extreme of μ = 0, high-risk individuals only contact other high-risk individuals and low-risk individuals likewise only contact others like themselves. At the disassortative extreme of μ = 1, the high-risk individuals have contact only with the low-risk individuals but because there are more total low-risk individuals, they make contacts with other low-risk individuals aswell.
The risk factors for infectious diseases that we examine affect transmission probabilities. We express the direction of transmission between risk groups using subscripts; the first subscript corresponds to the susceptible individual and the second to the infectious individual. Thus, βhl is the transmission probability in a contact between an infectious low-risk individual and a susceptible high-risk individual.
Joint Effects of Intervention Effects With Other Factors
We formulate the joint effects of background and high-risk-generating factors on infection incidence assuming a multiplicative relationship. This corresponds to risk factors and background factors having different actions along the pathway to disease.4 For example, the background risk factors might affect the survival of an organism on skin, whereas the high-risk-generating factors affect either susceptibility or contagiousness of individuals. In the supporting online material, we also examine additive joint effects that arise when the background and high-risk factors affect different pathways to disease.4 The joint-effects formulation determines how the total reduction of risk or transmission probability focused on the high-risk group gets distributed when that same total is applied in a general program.
Making Focused and Unfocused Interventions Comparable
We now present the relationships between the size of individual intervention effects in the focused program and the unfocused program that make the total intervention efforts the same as in the 2 programs. Define Y as the fraction decrease in individual risk-factor effects. For noninfectious diseases, that reduction acts directly on risk. For infectious diseases, it acts on susceptibility or contagiousness in the high-risk group. Define X as the corresponding fraction decrease in both high- and low-risk individuals in unfocused interventions. Given multiplicative joint effects between risk-factor reduction and background and high-risk-generating factors, the reduction in risk-factor effects is set equal for focused and unfocused interventions using the following formulation.
Equation (Uncited)Image Tools
ϵN is the multiplicative effect of factors that generate high risk.
α is the fraction of the population that is high risk.
For noninfectious diseases, the ϵN act directly on disease risk of individuals. See Table 1 for the risk formulations. For infectious diseases, the risks act on transmissions between individuals rather than directly on individuals and ϵN is replaced by either ϵC (for effects on contagiousness) or ϵS (for effects on susceptibility). The resulting effects on transmission probabilities are detailed in Table 2.
Model Analyses Methods
We formulate our models so that, in the absence of intervention, the equilibrium prevalence and incidence in the high- and low-risk groups are the same for all values of μ. Epidemiologists commonly assess infection incidence or prevalence in high- and low-risk groups but they do not assess the effects of mixing on these infection levels. By examining identical infection levels at different mixing settings, we make clear the importance of such assessment. The formulation that allows us to do this is presented in the supporting online material.
We make 2 comparisons of the effects of interventions focused on the high-risk group to interventions affecting everyone evenly. First, we compare the minimal total effects needed to stop transmission in the population. This level is determined by finding effects that get the basic reproduction number just below one. Second, we compare the effects on endemic infection levels of small interventions using equation 1 to equalize the focused and unfocused interventions.
The outcome we examine for the second comparison is the relative reduction in total population incidence (RRInc).
πH and πL are the endemic equilibrium incidence in the high- and low-risk group before intervention and ρH and ρL are the endemic equilibrium incidence after intervention. The equilibrium values are found by numeric solution of the differential equations underlying the model.
Equation (Uncited)Image Tools
Intervention Effects for Noninfectious Diseases
For noninfectious diseases, the population RRInc is unaffected by who mixes with whom. At α = 0.05, πL = 0.03, ϵN = 5 (thus πH = ϵNπL = 0.15), the preintervention population incidence at equilibrium is
When Y = 0.05 the equilibrium incidence after the focused intervention, ρH, is ϵN ρL (1−Y). The population incidence at equilibrium is thus
Equation (Uncited)Image Tools
For the unfocused intervention, the population incidence at equilibrium remains the same at
Equation (Uncited)Image Tools
Thus, RRInc = 0.0104 for both the focused and unfocused interventions.
Equation (Uncited)Image Tools
Patterns of Transmission Model Parameters Generating the Same Incidence in Both Groups
Risk factors affecting infection transmission act during interactions between individuals rather than directly on individuals like in the calculations shown here. Thus, we need a transmission model to analyze risk factor effects. To help the reader understand how the transmission model generates fixed endemic incidence as the mixing parameter is varied, we present in Figure 2 the parameter values for c and β that generate incidences of 0.15 and 0.03 in the absence of any intervention at each mixing level μ.
First consider high-risk generated by increased susceptibility. As μ increases, more high-risk individuals contact low-risk individuals. This situation would raise the incidence in the low-risk group. Consequently, we reduce the contact rate to maintain the low-risk group incidence at 0.03. The incidence in the high-risk group would fall both because fewer of their contacts will be infected and because the c is falling. Thus, to keep the high-risk group incidence at 0.15, the susceptibility of this must increase as μ increases. This is accomplished by increasing the ϵS parameter.
When contagiousness is generating the high risk, a similar phenomenon takes place. The low-risk group experiences even greater increased risk with each increase in mixing because the high-risk group is more contagious. Thus, the contact parameter must decrease even more than when susceptibility generates high risk. In a corresponding fashion, the contagiousness parameter also has to compensate more because it is acting on a smaller fraction of the contacts of the high-risk group. For contagiousness to generate high risk in a group with increased contagiousness, assortative mixing is required. That is why contagiousness risk factors are not plotted at the higher end of the scale in the figures. More details on the formulations of c, ϵS, and ϵC are presented in the supporting online material.
Effort Required to Eradicate Infection
For each of the 8 interventions in Table 2, we determined the average percentage reduction in either susceptibility or contagiousness per person in the overall population that is required to eradicate infection given initial equilibrium incidence values of 0.15 and 0.03 and 5% of the population in the 0.15 group. The results are seen in Figure 3. Note that interventions focused on the high-risk group require much less effort to achieve eradication than the unfocused interventions over most of the mixing range. The unfocused program has a slight advantage only at the very high end of the mixing scale.
Effects of 5% Risk Reductions in the High-Risk Group or an Equivalent Amount in Everyone
To understand better how these various factors change the dynamics of an intervention, we examine the effects of small interventions equal to those for noninfectious diseases. The interventions are small enough so that they never eradicate infection. In Figure 4, we examine the population relative incidence reduction for the focused and unfocused interventions, given the same starting values of 0.15 and 0.03 incidence and 5% in the high-risk group. Like in the eradication analysis, focused interventions have advantages over unfocused interventions. This analysis, however, reveals effects that the eradication analysis does not.
First, note that at every point on the mixing scale, both focused and unfocused intervention effects are considerably greater than the approximately 1% effect seen for noninfectious diseases. This larger effect for infectious diseases occurs because each direct effect for infectious disease control results in indirect effects both on those who do and who do not directly receive benefits from the intervention.
Second, note that at the completely assortative point on the mixing scale, the unfocused interventions have a very small advantage. That advantage occurs because the low-incidence population has a basic reproduction number (R0) closer to one than the high-incidence population and, as one gets closer to an R0 of one, eradication is easier to achieve.
Third, note that the contagiousness interventions in the high-risk group are more effective than the susceptibility interventions regardless of whether susceptibility or contagiousness has generated the high risk. This is due to “differential consumption of susceptibles.” When the high-risk group receives a susceptibility intervention, there is now a reduced differential between the rates at which they and the low-risk group become infected. Thus, at equilibrium, the high-risk-group susceptibles are not consumed as fast, and the consumption of susceptibles has less of an effect on equilibrium levels of infection. With the contagiousness intervention, the consumption of susceptibles is unchanged. Thus, the contagiousness intervention has the larger effect. The reason there is no difference between contagiousness and susceptibility interventions with regard to eradication levels is because no susceptibles are being consumed at the point of eradication.
Fourth, note that focused interventions have greater effects than unfocused programs over most of the mixing range. Unfocused programs compete reasonably with the focused programs only at proportionate mixing levels (μ = 0.95) or higher. In the additive joint effects formulation presented in the supporting online material, the differences between focused and unfocused interventions extend further up the mixing scale.
System Parameter Sensitivities
We explored a range of parameter settings and found some interesting implications. These did not change our general conclusions. When the fraction of the population in the high-risk group is raised to 10%, the unfocused interventions look much like those in Figure 4, but they rise faster and surpass the focused programs at slightly lower levels of mixing. When incidence rates are lowered but have the same ratio between high-risk and low-risk groups, the differential between focused and unfocused interventions increases markedly (regardless of whether susceptibility or contagiousness causes the high risk). However, even in this case, at the upper end of the mixing scale, the unfocused programs surpass the focused programs.
When susceptibility causes the increased risk, at 5-fold higher incidence levels or at 5-fold higher prevalence levels (determined by a 5-times longer duration of infection), there is less difference at the lower end of the mixing scale between focused and unfocused programs. Furthermore, the unfocused programs surpass the focused programs at a somewhat lower level of mixing. In contrast, when contagiousness generates the high risk, there are greater differences than those in Figure 4.
When the difference between high-risk-group and low-risk-group incidence is diminished, the effects of focused interventions are increased when contagiousness is generating the high risk, and they are never surpassed by unfocused infections. The effects of all other programs, however, have the same relationships as in Figure 4 but with lower effects for the focused programs. In this case, the curves of the unfocused programs again edge above those of the focused programs at the high end of the mixing scale.
We have demonstrated that focusing infectious disease control efforts on high-risk groups has a considerable advantage that is not seen for noninfectious diseases. The degree of mixing between high-risk and low-risk groups strongly modifies this advantage. We only examined infections that do not lead to acquired immunity, in which duration of infection is always constant and where contacts are symmetric. In work not reported here, we have found that the general conclusion about the increased effects of focused interventions for infectious diseases also holds for risk factors that affect contact rates or infection duration. The effects of risk factors affecting the duration of infection are similar to those affecting contagiousness.
We have also demonstrated that controlling contagiousness in high-risk groups reduces population levels of infection more than reducing susceptibility or shortening the duration of infection. The greater effects of reducing contagiousness are because differential consumption of susceptibles reduces the effects of susceptibility or infection-duration reductions. Our preliminary explorations indicate that this effect will be even stronger for infections that induce acquired immunity. It seems likely to us that there is more controllable variation in risk factors affecting contagiousness than there is in risk factors affecting susceptibility. This variation, together with the greater effects of reducing contagiousness, should lead infection control to focus on contagiousness. Standard epidemiologic analyses, however, focus on comparing the risk experience of different individuals. They are thus completely incapable of detecting contagiousness effects and consequently do not lead public health efforts in the most productive direction.
To overcome this deficiency, studies should be designed in social units where transmission occurs. The analysis of such studies should use transmission models rather than individual risk-based models that assume away transmission.
Specific Infection Control Implications
Consider the control of HIV transmission in developing countries with the limited resources available. To best protect all the people, our results suggest that it may prove wise to focus on controlling the contagiousness of groups with high risk. Prioritization of contact-tracing efforts, treatment resources, and behavior change efforts should take this into account.
The best way to stop transmission, however, may be vaccines that reduce the contagiousness of early HIV infection. High contagiousness during early HIV infection in high-risk groups has been shown theoretically to have an especially amplifying effect on population circulation of HIV.5,6 Consequently, vaccines that lower contagiousness only moderately during the crucial early stage of infection might have dramatic effects on transmission at the population level. Thus, vaccine trials must be designed in a way to detect such effects. There are many ways to do this.7–17 In each, a transmission unit rather than an individual must be the unit of study. Retrospective partnership trials7 could be an effective trial design in populations in which partnership change is frequent. Where concurrent partnerships are more frequent, prospective designs could be used.8,13,14 Separate populations represent another transmission unit that can be used.17
Another vaccine issue relates to the control of nontypeable Haemophilus influenzae (NTHi), a cause of acute otitis media, bronchitis, sinusitis, and pneumonia. The endemic prevalence of this agent is high. Consequently, most pediatricians and vaccinologists have assumed that an NTHi vaccine must protect against disease given colonization. We have shown that, in fact, vaccine protection that reduces transmission will reduce otitis more than vaccine protection against disease given colonization.18 Immunity effects that reduce contagiousness of colonized individuals can have very dramatic effects on population levels of infection. NTHi vaccine trials must thus have transmission units as their basic study entities.
Another example relates to Cryptosporidium control in developed countries. Because of the devastating effect of this infection on people with AIDS, control of Cryptosporidium transmission through water is becoming more urgent. However, the costs of different approaches to control of water contamination can run into many billions of dollars. We have modeled this issue.19,20 The role that water plays in a transmission system depends heavily on how much transmission is occurring through other modes. A water intervention is a costly way to achieve a small decrease in transmission risk evenly across an entire population. An implication of our analysis here is that cheaper interventions affecting hygiene in high-transmission groups could be more effective in reducing population levels of infection than improved water treatment. On the other hand, we have shown that the disseminating role of water transmission might greatly amplify its importance.21 General insights like those we seek to present in this article cannot resolve this sort of issue. However, such insights do make it evident that a formal evaluation of transmission effects for various focused or general interventions should be a key element contributing to decisions on Cryptosporidium control efforts.
The analysis presented here extends analyses pursued by others. In their classic text, Anderson and May22 lay the foundations of intervention strategies for heterogeneous populations. Their chapter 9 introduces matrices of “Who Acquires Infection From Whom,” which present the effective contact rates between groups. The analysis we present elaborates on this foundation by explicitly formulating causes for high risk and by specifying interventions affecting either susceptibility or contagiousness.
Woolhouse et al23 similarly build on Anderson and May. Arguing that “heterogeneous contact is likely to be an important determinant of the epidemiology of vector-borne diseases and STDs,” they analyze 10 datasets of vectorborne diseases and HIV. They conclude that “control programs targeted at the ‘core’ 20% group are potentially highly effective.” They close with a list of “higher-order heterogeneities” that may further increase R0. These include “patterns of contact between highly sexually active and less sexually active groups” in this list (the only one without references). We have shown here how and why that is the case.
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