The goal of contact tracing is to reduce the likelihood of transmission, particularly to individuals who are at greatest risk for developing complications of infection, as well as identifying individuals who are in need of medical treatment of other interventions. In this paper, we develop a simple mathematical model of contact investigations among a small group of individuals and apply game theory to explore conflicts of interest that may arise in the context of perceived costs of disclosure. Using analytic Kolmogorov equations, we determine whether or not it is possible for individual incentives to drive noncooperation, even though cooperation would yield a better group outcome. We found that if all individuals have a cost of disclosure, then the optimal individual decision is to simply not disclose each other. With further analysis of (1) completely offsetting the costs of disclosure and (2) partially offsetting the costs of disclosure, we found that all individuals disclose all contacts, resulting in a smaller basic reproductive number and an alignment of individual and group optimality. More data are needed to understand decision making during outbreak investigations and what the real and perceived costs are.
Contact investigation (contact tracing) is the identification of individuals who have come into contact with an infectious case and may be infected. The goals of contact tracing arise to reduce the likelihood of transmission (particularly to those individuals who are at greatest risk for developing complications of infection) and to identify individuals who are in need of medical treatment or other interventions [
Mathematical models have been used previously to evaluate the impact of contact investigations on the spread of infectious disease generally [
Individuals face costs—real or perceived—of contact disclosure. Real costs include time spent in interviews and in the effort spent recalling contacts. While contact investigations are and must be conducted in a manner that protects confidentiality, interviewees may perceive disclosure as a privacy risk, which may create a perceived cost. While the perceived and real costs of disclosure and their impacts on early contact investigation have been documented, the effects have not been explored thoroughly. If the disclosure of contacts provides a public benefit for disease control, but individuals perceive a cost for disclosing contacts, then there may be a conflict between real or perceived individual interests and the public good.
Mathematical models of contact tracing and ring vaccination—which requires contact tracing—have explored the effect of success rates of contact tracing that are less than unity and thus incorporate less than complete cooperation with contact elicitation [
Our analysis is based on a stochastic, continuoustime process taking place in a small social group; we formulate the model in general terms and restrict our analysis to a group of size 3. Such a model of a small group may, for instance, describe small groups within a model featuring more transmission between members of the same group, but allowing betweengroup transmission for any two individuals in the population [
We assume a standard SEIR model for the untreated natural history of the disease [
Infectious individuals are always assumed to be diagnosed and isolated or treated, and we assume that such individuals are no longer causing new infections in the population. Such individuals may be undergoing treatment which reduces or eliminates infectivity or may be isolating themselves from others during the time of infectiousness. In the absence of contact investigation, the process may be described by the following states:
The following equations describe a single group in the absence of contact investigation. Let
For the case
We denote the transmission coefficient by
The full set of equations for a single small group can be written in a more compact form. Let
Before extending the simple model (
Here, all individuals are susceptible, but we assume that individuals are either unknown to the investigation (
For disease transmission, we assume that disease may be transmitted between any two people. For contact investigation, we do not assume that every person is willing to disclose any other person; any identified person will be asked to name all contacts but may choose not to do so. Let
We model the rate at which persons unknown to the investigation become newly known as follows. Suppose that person
In this setting of a small group of three people without any infection,
We will add the contact investigation model from the previous subsection to the simple SEIR transmission model. One way for individuals to become known to the investigation is to be disclosed by another known individual who is willing to disclose him or her, as in the previous subsection. We now assume, additionally, that reporting insures that all diagnosed individuals are known to the investigation, and we ignore reporting delays. Newly diagnosed individuals are the only way that a contact investigation can become initiated; the first diagnosed individual inaugurates the first contact investigation, regardless of whether any other individuals have been infected or diagnosed and regardless of whether or not the first diagnosed individual was the first infectious case. (We do not assume that any individuals are known to the investigation at the outset (
When an individual is investigated, several events occur in addition to begin queried about his or her contacts (who will then be investigated at rate
When a susceptible individual is investigated, he or she may take protective measures to reduce the chance of infection. Also, when an exposed individual is contacted, he or she may receive postexposure protective measures. Such measures may include vaccination (as in the case of measles or smallpox) or the provision of immunoglobulin (as in the case of measles, for instance). Thus, susceptible individuals who are known to the investigation are assumed to have a smaller risk of infection, and both susceptible and exposed individuals known to the investigation have a rate of vaccination or other protective actions which may prevent them from becoming cases. For an individual in state
Finally, we assume that any exposed person (state
The state space of the model now may be written (see Figure
State space for a single individual, according to (
Specifically, an individual in state
For the case
We write for all states
Here, for
Individuals in both
For investigation, we assume that person 1 becomes investigated at rate
Equation (
Example of state transitions within a small group, according to (
At time 0,
The nature of the costs or disutilities associated with either disclosure or disease is not specified. Disclosure in some settings is an undesirable outcome, and we wish to compare this to the costs of disease. It is not necessary that a person actually incurs any harm from the investigation, because, for some individuals, even a confidential disclosure of an illicit contact may be uncomfortable and undesirable. In principle, it may be possible to estimate such costs using willingness to pay data or timetradeoff data, but we do not pursue this here.
We assume that the cost of disclosure is
Our assumptions imply that the payoff for each person may be computed from the final state of the system. For any final state represented by
Alternatively, we may assume that the cost for each person is
If we assume that each individual infects
The system of ordinary differential equations given by (
Equation (
The transition rates from each state of the system to each other state of the entire system constitute the generator
In practice, we expressed the elements of the jump matrix
For each strategy choice of all three individuals, we determined the probability that person 3 (Charlie) was infected. Assuming fixed strategies for the other two individuals, how does the infection probability for Charlie change if he chooses to disclose other individuals? The results are summarized in Table
Reduction in infection for Charlie due to disclosures by Charlie, assuming given strategies for the other two individuals. For each row, Alice is assumed to disclose either Bob or Charlie or both, as indicated in the first two columns; Bob is assumed to disclose either Alice or Charlie or both, as given in the next two columns. The next column (Alice versus none) shows how much the infection probability for Charlie is reduced by disclosing Alice instead of disclosing no one. The column labeled “Bob versus none” shows the reduction by disclosing Bob instead of no one and so forth. Analytic expressions for the quantities
Alice 
Bob 
Reduction in infection probability for Charlie comparing disclosure choices of  

Bob  Charlie  Alice  Charlie  Alice versus none  Bob versus none  Both versus none  Both versus Alice  Both versus Bob 
N  N  N  N  0  0  0  0  0 
Y  N  N  N  0  0  0  0  0 
N  Y  N  N  0 



0 
Y  Y  N  N  0 



0 
N  N  Y  N  0  0  0  0  0 
Y  N  Y  N  0  0  0  0  0 
N  Y  Y  N  0 



0 
Y  Y  Y  N  0 



0 
N  N  N  Y 

0 

0 

Y  N  N  Y 

0 

0 

N  Y  N  Y 





Y  Y  N  Y 





N  N  Y  Y 

0 

0 

Y  N  Y  Y 

0 

0 

N  Y  Y  Y 





Y  Y  Y  Y 





Equations (
Based on (
Numerical scenarios showing the percent reduction in disease transmission achievable by contact investigation and postexposure prophylaxis, assuming complete disclosure. The latency column provides the ratio of the expected duration of the latent period relative to the infectious period, the tracing column is the ratio of the duration of the infectious period to the expected waiting time to be found from a single disclosure, and the prophylaxis column is the ratio of the duration of the latent period to the waiting time to postexposure prophylaxis following contact investigation. The percent decline in transmission within the group
Scenario  Latency  Tracing  Prophylaxis  Reduction (%) 






1  0.1  0.1  0.1  0.543%, 1.15%, 0.0832% 
2  10  0.1  0.1  6.89%, 12.7%, 7.01% 
3  0.1  10  0.1  4.36%, 13.7%, 3.43% 
4  10  10  0.1  12%, 21.6%, 11.9% 
5  0.1  0.1  10  0.791%, 1.76%, 0.267% 
6  10  0.1  10  47.8%, 53.1%, 49.8% 
7  0.1  10  10  8.64%, 19%, 9.71% 
8  10  10  10  82.8%, 84.9%, 83.1% 
Table
Each individual—Alice (the index case), Bob, or Charlie—may choose to disclose or not to disclose each of his or her two contacts. Thus, for example, Alice has four possibilities: (1) disclosing neither Bob nor Charlie, (2) disclosing Bob but not Charlie, (3) disclosing Charlie but not Bob, or (4) disclosing both Bob and Charlie. Each individual has four possible choices, and thus three individuals yield a total of
How does the total number of transmitted cases
Expected number of secondary cases for different disclosure choices (Scenario 8, Table
Charlie discloses  Bob discloses  Bob discloses  Bob discloses  Bob discloses  

Neither  Alice only  Charlie only  Both  
Alice discloses neither  Neither  1  0.998  0.835  0.835 
Alice only  0.998  0.997  0.834  0.834  
Bob only  0.835  0.834  0.671  0.671  
Both  0.835  0.834  0.671  0.671  


Alice discloses Bob only  Neither  0.508  0.506  0.158  0.158 
Alice only  0.506  0.505  0.157  0.157  
Bob only  0.506  0.505  0.157  0.157  
Both  0.506  0.505  0.157  0.156  


Alice discloses Charlie only  Neither  0.508  0.506  0.506  0.506 
Alice only  0.506  0.505  0.505  0.505  
Bob only  0.158  0.157  0.157  0.157  
Both  0.158  0.157  0.157  0.156  


Alice discloses both  Neither  0.156  0.155  0.154  0.154 
Alice only  0.155  0.153  0.152  0.152  
Bob only  0.154  0.152  0.152  0.151  
Both  0.154  0.152  0.151  0.151 
Is it, in general, possible for an individual to reduce his or her probability of disease by disclosure of others? By assumption, such a reduction is not possible for the first person infected in the group (Alice). Without loss of generality, we may consider the decrease in disease probability Charlie experiences if he (Charlie) discloses Alice, discloses Bob, or discloses both Alice and Bob. Since Alice and Bob each have four choices (disclosure or not of each of the other two people), a total of 16 possible combinations of these choices are available. For each specific choice of what Alice and Bob choose, we compare the infection probability when Charlie discloses Alice to the disease probability when Charlie discloses no one. The difference is the amount by which Charlie reduces his or her probability of disease by disclosing Alice compared to no one. Several salient facts are obtained from these expressions for contact investigations in a group of size 3.
First, in the threeperson group, if Alice discloses no one, then Charlie can never reduce his likelihood of disease unless Bob discloses him. If neither Alice nor Bob is willing to disclose Charlie, then Charlie will never be known to the investigation before diagnosis. The only person Charlie should disclose to obtain benefit is Alice; since Alice is not disclosing Bob, Bob only discloses Charlie after he (Bob) is diagnosed and removed from transmission. But there is a possibility that Alice, who infected Bob, still has not been diagnosed yet; disclosure of Alice yields a possibility of benefit. The ability of Charlie, therefore, to benefit from disclosure depends on the choices made by the other persons in the network.
Equations (
Moreover, if
Finally, the expressions in the appendix show that (unsurprisingly), if there is no transmission (
We now explore the model to determine the effect of costs of disease, disclosure, and participation. We first assume no overall participation costs or incentives (
Whenever the costs for disclosure are negative (there is a benefit to disclosure), the best strategy for each individual is to disclose all other individuals. Under these circumstances, disease prevention attains the maximum possible value. We assume a disease cost
Where each individual may face costs for disclosing other individuals, the possibility of a conflict of interest arises. We will again assume the same numerical scenario as in the previous analysis (Scenario 8, intermediate transmission), except that we now add a small cost
When we assume a reward for disclosing at least one contact (
Nash equilibrium strategies resulting when
Alice  Bob  Charlie  Total transmission ( 


Discloses  Discloses  Discloses  
1  Bob only  Charlie only  Alice only  0.157 
2  Charlie only  Alice only  Bob only  0.157 
For the first strategy in Table
Bob and Charlie have the same incentives that Alice has to disclose exactly one other person. For Bob and Charlie, however, disclosing others may affect the probability of disease, and so the best depends not only on the costs or incentives for disclosure, but also on disease transmission. If the strategy of Alice is to disclose Bob, then the best strategy for Bob is to disclose Charlie and not Alice. Alice is the index case and willing to disclose Bob, and so it would frequently be wasteful for Bob to disclose Alice—Alice is likely to have already been diagnosed. Similarly, if Alice is choosing to disclose Bob, Charlie benefits more from disclosing Alice than Bob. If Bob (but not Alice) is using the strategy of disclosing Charlie, then Charlie could infer that whenever he has been investigated before infection, it was the result of Bob's disclosure and that disclosing Bob again is counterproductive—Alice is the better choice.
In the preceding section, individuals are assumed to use the same strategy for disclosure all the time, whether or not the person was identified before he or she became a case, or after. In the latter case, the individual has no chance to prevent her or himself from becoming diseased. We next suppose that each individual could make a different choice about disclosure depending on whether or not the person was originally identified as a result of seeking health care (diagnosed from state
If no one discloses after diagnosis
We also examined the case
In this paper, we analyzed a simple model of costbenefit tradeoffs in a stylized model of contact investigation and disclosure, reflecting public health circumstances in which individuals may not wish to disclose other individuals in their contact network. Such circumstances may arise if such contacts reflect illicit activity, undocumented presence in the country, or other reasons related to privacy. We therefore assumed a cost for each such disclosure. The model assumes that individuals may benefit by disclosing other individuals in their network of contacts and that the sole such benefit is a reduction in infection risk resulting from earlier diagnosis of other individuals in the group. We assumed a specific simple form where individuals use a fixed strategy, for which any specific contact may or may not be disclosed, and that this did not depend on the progression of the epidemic. We assumed a simple stochastic epidemic model where individuals could be protected after exposure by vaccination, and that once an individual is diagnosed, he or she is removed and will transmit no more infection. Finally, our analysis was restricted to the case of a simple three person cluster. We developed the analytic Kolmogorov equations for the stochastic process, and solved these equations to determine the expected payoffs.
In this setting, we found that if all individuals have a cost of disclosure with no participation incentives, then the optimal individual decision is to simply not disclose others. Contact investigation is unsuccessful, and more transmission within the group results. Also, the population
However, if there is some benefit to disclosing—some incentive to remove all or part of these costs—a different structure emerges. We examined additional cases:
We also examined a case of partially offset costs, in which a person should disclose one such contact, but not both. In this case, we found two solutions. Using the conventional names Alice (for the index case), Bob, and Charlie, these are as follows. If Alice disclosed Bob, then Bob should disclose Charlie, and Charlie should disclose Alice. Similarly, if Alice disclosed Charlie, then Charlie should disclose Bob, and Bob should disclose Alice; the same pattern is seen, with the roles of Bob and Charlie reversed.
We examined an extended version of the model in which individuals could make a different choice depending on whether they were identified in time to prevent illness. This model found that the direct benefits of prevention could outweigh small disclosure costs, favoring disclosure. While this threshold for favoring disclosure may be larger for alternative or more realistic model structures, we believe that direct immediate prevention benefits should not be relied on to provide sufficient incentives for participation. Reducing costs—including perceived costs—is crucial.
In real outbreaks, individuals lack the information necessary to weigh the risks and benefits of disclosure. Individuals do not, in general, know the extent of their exposure, the benefits of vaccination at different times, nor the benefit they would receive by disclosure. Thus, the solutions to the game model are idealized optimal strategies realizable under perfect information. Importantly, this analysis, focusing as it does only on the small group (of size 3) and not beyond, does not fully reflect the epidemiology of novel pathogen introduction. Here, failure to prevent transmission early may lead to widespread transmission beyond the small group. The analysis presented above only includes transmission within the small contact group and could be straightforwardly extended to take into account the benefits—epidemiological and otherwise—of stopping a large epidemic. We also note that real decision making could take into account a much richer strategic set, so that individuals could have a different strategy depending on whether or not they know how many cases there have been, what other individuals have done, or other factors (e.g. [
Our model does show how decisionmaking based solely on reducing an individual's direct risk of disease can lead to noncompliance and an overall unfavorable outcome for the group. Moreover, it suggests that the ability of an individual to reduce his or her own risk would, under these assumptions, be expected to reduce compliance with contact investigation. The findings highlight the central importance of reducing costs of contact investigation for all participants, perhaps through incentives. Further work will be needed to assess the robustness of these conclusions. Empirical data on perceptions of the risks and benefits of contact investigation and the reasons for compliance and noncompliance are urgently needed.
The extended equations for the analysis in Section
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge support from the US NIH NIGMS MIDAS Program, 1U01GM087728. They acknowledge the use of the free software TeXshop for document preparation, R (