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We provide a theoretical foundation for analyzing how social stigma and adopted behavioral traits affect the transmission of HIV across a population. We combine an evolutionary gametheoretic model—based on a relationship signaling stage game—with the SIR (susceptibleinfectedrecovered) model of disease transmission. Our evolutionary model specifies how two types of social stigma—that which accompanies an HIV^{+} condition and that which follows associating with an HIV^{+} partner—influence behavioral propensities to honestly report one’s condition (or not) and to unconditionally accept relationships (or not). With respect to reporting an HIV^{+} condition, we find that condition stigma impedes the fitness of honest reporting, whereas association stigma impedes the relative fitness of concealing an HIV^{+} condition; and both propensities can coexist in a polymorphic equilibrium. By linking our model to the SIR model, we find that condition stigma unambiguously enhances disease transmission by discouraging both honest reporting and a society’s acceptance of AIDS education, whereas association stigma has an ambiguous impact: on one hand it can impede HIV transmission by discouraging concealing behavior and unconditional relationship acceptance, but it also compromises a society’s acceptance of AIDS education. Our relatively simple evolutionary/SIR model offers a foundation for numerous theoretical extensions—such as applications to social network theory—as well as foundation for many testable empirical hypotheses.
The HIV epidemic poses a collectiveaction problem to society because social interactions among individuals—interactions that operate within and respond to prevailing social contexts—affect the transmission of the virus. In particular, the social stigma that often accompanies an individual’s HIV^{+} condition tends to enhance disease transmission by discouraging honest reporting and by reducing a society’s acceptance to AIDS education.
This paper incorporates evolutionary game theory and the SIR (susceptibleinfectedrecovered) model of disease transmission to provide a theoretical foundation for examining how specific elements of social context—notably the social stigma that can accompany either an individual’s HIV^{+} condition or one’s association with an HIV^{+} partner, along with a society’s provision and acceptance of AIDS education—affect the dynamics of HIV transmission within a population. More specifically, this paper develops an evolutionary gametheoretic model that illustrates how multiple relationship interactions within populations influence the prevalence of behavioral traits that affect the transmission of HIV. Using a twoperson relationship signaling game as a stage game, our evolutionary model illustrates interactions between the disease status of members of a population (HIV^{+} or HIV^{–}) and two types of behavioral traits—a propensity to honestly report one’s status (or not) and a propensity to unconditionally accept relationships (or not). We proceed to link specific variables and outcomes from our evolutionary model to the SIR model of disease transmission in order to gain insight on how social context influences the transmission of the HIV virus.
By itself, our evolutionary model allows for a polymorphic equilibrium in which the behavioral trait of honest reporting as well as that of concealment of an HIV^{+} condition (two alternative strategies or phenotypes) can both survive with stable (or perhaps oscillating) population proportions. Moreover, the population proportion of the honestreporting trait within such a mix decreases in the society’s level of
Recent advances in game theory, information economics, and social preference theory facilitate rigorous modeling of fundamentally social phenomenon—in this case factors that affect the rate of HIV transmission within a society. For decades, game theory has enabled systematic modeling of strategic behavior among interdependent agents.
Information economics enhances such analysis by systematically addressing how asymmetrically available information creates opportunities for strategic manipulation—via concealing, selectively revealing, or distorting information related to environments, actions, or attributes (e.g., one’s own disease status) in various social interactions.
A more precise literature, both theoretical and empirical, focuses on rates of disease and/or HIV transmission. Funk
Most fundamentally, the SIR model [
Sources of information—either global or local—affect rates of disease transmission.
The type of available information—either prevalence or belief based—influences rates of transmission, and beliefbased information need not reflect actual disease prevalence.
Human behavior affects disease transmission by influencing the S, I, R,
Analysts can fruitfully use game theory to model relevant decisions (e.g., whether or not to vaccinate).
Behavioral traits related to disease transmission can, themselves, be transferred among individuals; hence such traits and their prevalence evolve over time.
Local social network structure affects the transmission of information, behavioral traits, and, indeed, levels of infection across populations.
Focusing directly on HIV in an analysis that relates to assertions 2, 3, and 5, Schroeder and Rojas [
A related empirical literature examines social factors that affect the disclosure of disease status, and by extension, rates of disease transmission. In terms of Funk
Our paper extends this literature. Our chief contribution involves combining an evolutionary gametheoretic approach with the SIR model. In so doing, we address Funk
Evolutionary game theory (EGT) facilitates modeling socially transmitted behaviors and associated processes of adaptive learning within or across populations; certain practices gain acceptance over time, while others decline or perish. Whereas in classical game theory, players choose strategies via estimated best responses to potential strategies of others, in EGTs, players “inherit” strategies. For the social sciences, such inheritance emerges from education and other forms of cultural (rather than genetic) transmission.
More precisely, in EGT models, randomly matched pairs of agents, as bearers of specific inherited strategies, interact. As in classical game theory, specific strategy combinations generate
In the present context, agents—and by extension populations—may gradually learn that a strategy of either honest signaling or of concealing an HIV^{+} condition offers greater or lesser rewards than the alternative. Likewise, they may learn that that either accepting or rejecting a relationship after either receiving or not receiving a signal offers relative benefits. An evolutionary model can then generate predictions concerning the signaling and acceptance practices that develop among certain populations—with attendant implications on disease transmission.
We now present an evolutionary model of HIV transmission among randomly matched pairs of potential relationship partners. The associated stage game unfolds in four steps:
Nature decides whether each player is an HIV carrier (HIV^{+}), an event that occurs with probability
Each player privately observes nature’s move and then either signals to their potential partner that he or she is HIV^{+} (
Neither player observes the other’s type, but each observes either
If and only if both choose A, a relationship ensues.
Payoffs follow.
To proceed, we use a simplified version of this stage game to illustrate the sequence of interaction and the basic payoffs.
Stage Game from the perspective of an accepting (A)type agent (Pat) interacting with an HIV^{−} agent (Chris).
The payoffs are as follows. Assume that, independent of the presence of HIV, Pat and Chris would benefit equally from having a relationship: each would receive positive payoff
For each of the three steps in the stage game, our evolutionary model establishes two possible phenotypes that agents may exhibit. While there are eight possible combinations of phenotypes, only four such combinations actually affect behavior. Nature’s move in step 1 divides the population into two basic diseasecondition phenotypes: HIV^{+} with probability
Thus, there are four phenotype combinations (two sets of two) that actually affect agents’ behavior:
HIV^{+} agents either signal or not, depending on whether they are RE or IR, but they always accept relationships.
HIV^{−} agents never signal, but depending on whether they are A or Ctypes, they either always accept a relationship or do so only if they have not received an HIV^{+} signal.
In this evolutionary game, agents match randomly in pairs. Using the described payoffs, from
In this table, the cells that represent interactions between two HIV^{+} agents do not follow from
Fitness payoffs from interactions among different phenotypes.
Phenotype (proportion)  HIV^{+ }& RE 
HIV^{+ }& IR 
HIV^{−} & A(1 − 
HIV^{−} & C(1 − 

HIV^{+} & RE  − 



HIV^{+} & IR  
HIV^{−} & A  
(1 − 

HIV^{−} & C  0, − 

(1 − 
To proceed and more fully integrate social context into our analysis, we assume that
We now specify the expected fitness payoff (
Since the most important question with respect to HIV transmission concerns the propensity of HIV^{+} agents to report, we first compare the relative fitness of strategies RE and IR within the HIV^{+} population. Equation (1) states that the fitness of REs equals their
Equation (5) states that REs exhibit greater fitness than IRs when the guilt that IRs experience from their fear of infecting other IRs and all HIV^{−} agents plus the association stigma they receive from relationships with REs exceeds the their relative gains with respect to attaining relationships with HIV^{−} Ctypes plus their avoidance (in all cases) of signaling cost
Turning now to HIV^{−} agents, we compare the fitness of Atypes with that of Ctypes. Equation (3) states that Atype fitness equals payoff
Ctypes exhibit greater relative fitness if the disadvantage to Atypes from risk
We now consider possible evolutionary equilibria that can emerge from this model.
Evolutionary existence and stability conditions.
ESS Outcome  Existence  Stability 

RE is an ESS  (i) (1 − 
a. (i) holds for all 
IR is an ESS  (ii) 
c. (ii) holds for all 
Polymorphic 
(iii) (1 
d. ∂Φ/∂ 
A is an ESS  (iv) 
e. (iii) holds for all 
C is an ESS  (v) 
g. (iii) holds for all 
Polymorphic 
(vi) 
h. ∂Γ/∂ 
In principle, there are nine possible equilibrium phenotype combinations. There are four possible pairings of evolutionary stable phenotypes (corner solutions): (RE, A); (RE, C); (IR, A); and (IR, C).
Corner solutions are of interest only if they are stable. Accordingly, we first examine the stability conditions phenotypes RE and IR. Given the previously discussed derivatives of the
The
Relative fitness or RE and IR phenotypes.
Turning to stability conditions for partial equilibria that include the A and C phenotypes, we see that condition (f) is consistent with our specification of the
Relative fitness of A and C phenotypes.
We thus end up with two possible stable combined evolutionary equilibria among our four phenotypes, (
We now address comparative statics from this model, first examining influences relative fitness of RE
Comparative statics.
A. RE 




(i) ∂Φ/∂ 
Condition stigma (d) 
(ii) ∂Φ/∂ 
Association stigma ( 
(iii) ∂Φ/∂ 
Relationship utility (d) 
(iv) ∂Φ/∂ 
Guilt ( 
(v) ∂Φ/∂ 
The population proportion of Atypes ( 
(vi) ∂Φ/∂ 
The population proportion of HIV^{+} agents ( 




(vii) ∂Γ/∂ 
Informed transmission ( 
(viii) ∂Γ/∂ 
Association stigma ( 
(ix) ∂Γ/∂ 
Relationship value (d) 
First note that results (i) and (ii) imply that condition and association stigma have opposite effects on Φ, since the former applies to REs and the latter to IRs. In (iii), Φ decreases in
The two population results (v and vi.) are somewhat more complicated. In (v), Φ increases in
The A
On this foundation, we now apply our evolutionary model to the SIR model of infection transmission.
One may use the SIR model to depict the time trend in HIV infection over time. Our evolutionary model informs this analysis in two fashions: first, we may incorporate specific variables and functions from our model into the SIR model to derive a number of hypotheses concerning the spread of HIV; in particular, our variables influence the level of the SIR mixing coefficient (
Here is a summary of the original SIR model. Consider a specific strain of virus, such as the Hong Kong flu, that spreads via direct contact, and for which some agents develop immunity after being infected. There are three types of agents: susceptible (S), infected (I), and recovered or immune (R). At time period
The following equations describe the time path of the three types of agents:
Equation (7) states that the pool of susceptible agents declines steadily as more become infected (no longer susceptible). Mathematically, Equation (7) follows from the definitions of
Following the assertion of Funk
We now use variables from our evolutionary model to specify factors that influence the time trends in Equations (8) and (9).
More fundamentally, our evolutionary model can illustrate how social context influences HIV transmission via mixing coefficient
Equation (12) states that the aggregate rate of HIV transmission for a population of size
We immediately see that ∂
Social influences on the SIR (susceptibleinfectedrecovered) model mixing coefficient
Partial derivatives for Φ and Γ from 
Impacts on 
Partial derivatives with respect to 

∂Φ/∂ 
∂ 
(x) ∂ 
∂Φ/∂ 
  (xi) ∂ 
∂Φ/∂ 
  (xii) ∂ 
∂Φ/∂ 
∂ 
(xiii) ∂ 
∂Φ/∂ 
  (xiv) ∂ 
Overall, the HIV mixing coefficient
The first two products in Equation (14) are negative, but the third is positive. Hence the overall influence of
Concerning evolutionary equilibria, our model makes two basic statements. First, at either equilibrium, the polymorphic value
Finally, we consider the overall impact of total social stigma
In circumstances when ∂
This paper has combined an evolutionary gametheoretic model of HIV transmission within a society with the SIR model of disease transmission, noting the related influences of social stigma, and AIDS education on the propensities of individuals to honestly report their HIV status, the propensities to accept or reject relationships on the basis of such reports, the associated withinrelationship risks of transmission, and the ensuing population rate of transmission of the virus.
We now list our key findings, each of which can generate one or more testable hypotheses, in three basic categories:
Our discussion of evolutionary equilibria implies the following:
Among the HIV^{+} population, the behavioral traits of honest and dishonest reporting of an HIV^{+} status (the RE and IR phenotypes) can both survive. More precisely, both phenotypes coexist in a polymorphic evolutionary equilibrium, as signified by a stable (or possibly oscillating) population ratio of REs (
Among the HIV^{−} population, the behavioral traits of unconditional and conditional acceptance of relationships (the A and C phenotypes) can each be evolutionary stable strategies. Moreover, the accompanying evolutionary dynamics likely exhibit a positivefeedback dynamic whereby increased prevalence of phenotype A (C) above (below) critical mass tipping point (unstable internal equilibrium
The evolutionary model thus generates two stable evolutionary equilibrium combinations of phenotypes: a polymorphic mix
Concerning key influences on the nature of the evolutionary equilibria, we find:
The two types of social stigma—condition and association stigma—exert opposing direct influences on the fitness Φ of the RE phenotype. Whereas condition stigma compromises RE fitness, reducing equilibrium proportion
Fitness Φ and proportion
and
Ctype fitness Γ increases in
Combining these findings with the SIR model, we find:
Condition stigma
Association stigma
Either evolutionary equilibrium (
The distinction between the (
Finding 9 notwithstanding, the greater the share of condition stigma within a society’s overall social stigma, the greater the likelihood that overall social stigma enhances the SIR rate of HIV transmission.
This paper offers a preliminary examination of the evolutionary dynamics of HIV transmission. Our approach can be extended in four basic fashions that relate to each of the following: the specification of phenotypes; fuller specification of the entire model; alternative evolutionary approaches; and extensions to network analysis.
Concerning phenotypes, the Ctype could include a more precise criterion for acceptance: it could stipulate that cautious players reject relationships in the presence of a signal (as in the present model) but only accept relationships with a probability equal to HIV^{−} prevalence (1 −
Concerning the full model, a thorough description of the present evolutionary mechanism would involve stipulating and analyzing its dynamic replicator equations, a set of differential equations that would specify time trends for each phenotype. Although quite complicated, such specification would permit a thorough description of any polymorphic IRRE equilibrium (e.g., the precise nature of oscillation) and would likely condition our result that allows both C and Atypes to be evolutionary stable strategies. An alternative, though related, modeling approach for the same set of specifications would be to incorporate the basic dynamics of this model into an agentbased model. A systematic series of simulations could then illustrate multiple evolutionary developments.
As another variant on evolutionary approaches, one could replace our (direct) evolutionary model with an indirect evolutionary approach [
Finally, this model lends itself to a social network analysis of its SIR component (as in the seventh assertion of Funk
In all cases, this paper’s combination of an evolutionary gametheoretic model of HIV transmission with the SIR disease transmission model offers a foundation for subsequent empirical and theoretical examination of society’s collectiveaction problem with respect to HIV transmission. Moreover, our models can also serve as a basis for more general analysis of disease transmission in society.
The authors would like to thank two anonymous referees for extremely useful comments on this manuscript.
Both authors contributed to this paper. The initial idea for the paper including the initial version of the signaling game and basic evolutionary model appeared in a draft written by TKN several years ago. WDF revised and added substantially to this draft several times before and after its initial submission to
The authors declare no conflict of interest.
For an intuitive undergraduate introduction to applied game theory, see Dixit, Skeath and Reiley [
Such manipulation fits Oliver Williamson’s [
On the political economy of collectiveaction problems, see Ferguson [
As in Funk
Evolutionary biologists argue that evolutionary selection can apply to cultural processes and that cultural evolution unfolds much more rapidly than its biological counterpart. See for example, Wilson and Wilson [
For an introduction to evolutionary game theory, see Chapter 13 of Dixit, Skeath and Reiley [
For Bowles, strategies (not individuals) are the “
The dynamic emergence properties of evolutionary models present a new terrain for economic and social scientific analysis; see Epstein [
The full stage game includes all such interactions between all possible pairs, where each agent in a pair may belong to one of four distinct combinations of phenotype, with associated probabilities. In
Here Smeans signal HIV^{+} and NS can imply HIV^{−}. By contrast, in many games, a betterinformed party signals (honestly or not) the presence of some advantageous characteristic (e.g., high qualifications for a job). While the S and NS names could be reversed without affecting outcomes, we choose this terminology because it more accurately reflects stylized relationship dynamics.
For simplicity, we combine AIDs education and access to protection into the variable
In a more complicated model, a Ctype agent could base his probability of acceptance on the population infection rate
In terms of infection, they have nothing to lose. Implicitly we assume that for IR agents,
Norm internalization follows from ethical content Ferguson [
Each member of these pairs is an evolutionary stable strategy (an ESS). A strategy is an ESS if, once it has attained significant predominance in a population (as in 99%), it cannot be successfully invaded by a new “mutant” strategybecause (in that region) the mutant receives lower fitness payoffs. Note that the ESS concept does not specify what would happen in cases with a substantial proportion of mutants (e.g., 30%).
Our brief review of relevant empirical literature clearly suggests a mix of honest reporting and concealment.
A more complete (and far more complicated) model would include full specification of differential replicator equations for all four phenotypes (specified over time) and would consider longterm implications of changes in population ratios
The basin of attraction around an all Ctype equilibrium increases.
We assume that this secondorder effect does not outweigh the firstorder effect shown in
We assume homogeneous mixing among agents (effectively no social network), hence parameter
This relation follows from the requirement that proportions
A more complicated model could add immunization.
It is easy to deduce influences on Equation (7) from our discussion of Equation (9).
The probability of a HIV^{+} HIV^{−} match between two partners is 2
∂
Whether society settles on an equilibrium
Preliminary analysis suggests that the more complex Ctype would not alter key results of a polymorphic IRRE equilibrium and positive feedback dynamics for A and Ctypes. Adding an Ftype as a fifth behavioral phenotype would complicate the model considerably, but a prior analysis suggested this addition did not significantly affect outcomes; but its inclusion could generate a negativefeedback dynamic for relationship acceptance, operating above a tippingpoint level of
For a discussion of agentbased models, see Epstein [
For a dramatic example of how connectedness affected HIV transmission in its early history, see Shilts [