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Responding to Infectious Diseases Using Game Theory Table of Contents: Abstract: Impact: Potential Applications: Question/Hypothesis: Background Research: Game Theory: Epidemic Simulation: Virus Model: Strategies Investigated: Public Health Strategies: Viral Strategies: Payoff Calculations: Viral Payoff: Public Health Payoff: Data and Results: Simulation Results: The Sniffles: An Iterative Dominance Game: Spattergroit: A Prisoner’s Dilemma Game: Conclusion: Future Research: 2 2 2 2 2 3 4 4 5 6 6 7 8 9 11 12 1 Abstract The research analyzed interactions between human public health organizations and infectious diseases (specifically viruses) in order to attempt to investigate game theory’s applications to conflicts and interactions between humans and infectious diseases. The research set up games to mimic the interaction between humans and infectious disease by treating both as rational players with discreet payoffs. The procedure consisted of running simulations utilizing the Framework for Reconstructing Epidemiological Dynamics (FRED) from the University of Pittsburgh and varying parameters to set up strategies for the virus and public health organizations and determine the resulting payoffs. The data retrieved consisted primarily of a pair of games, one solvable through iterative dominance and the other an example of the Prisoner’s Dilemma. The iterative dominance game provided an example of a way in which game theory can provide a route to solutions previously not considered by treating the virus as a rational player that will make the best choice for its reproductive success. In addition, it showed the cycle that is exemplified in an extreme form by symbiotic enterobacteriaceae that adopt an asymptomatic and even beneficial strategy while humans choose not to attempt to prevent their further spread. The Prisoner’s Dilemma game suggested that game theory could possibly lead to cycles of cooperation with a disease that could be either continued indefinitely or used to weaken a virus until a sudden defection from cooperation by health organizations could destroy it. Impact Better analysis of public health strategies by treating viruses as rational players Mitigation of Viral symptoms through a Nash Equilibrium Opportunities for cooperation with and defection against a virus in order to advance public health goals Potential Applications Quarantines and Bird Flu Travel restrictions and Ebola Measles and vaccination policy Predicting the prevalence of Flu strains for vaccination Question/Hypothesis Game Theory can be applied among humans to fields from economics to international relations. Likewise, Evolutionary Game Theory can be applied to explain many biological behaviors that otherwise are enigmatic. However, there have been relatively few studies attempting to combine the two. By studying interactions between human public health organizations and diseases through Game Theory, this research investigates a new field of potential interactions. Background Research: Game Theory Game Theory was first developed by John von Neumann in his studies of two-player zero sum games, or games in which any gain by one player must result in a loss by 2 the other player. Since then, it has been expanded to include cooperative games (in which a good outcome for one player is not a bad outcome for another) and games with three or more players. The central concept of game theory is the Nash Equilibrium, an outcome for which neither player has an incentive to unilaterally deviate (switch strategies by themselves) from their chosen strategies. If both players play perfectly (i.e. rationally), the two strategies will result in the Nash Equilibrium. In order to find Nash Equilibria, Game Theorists use a payoff matrix, which details the rewards for each player given that each player chooses a certain strategy. The payoff matrix for a Prisoner’s Dilemma, an oft-studied game, is shown below. Player A Player B Cooperate Defect Cooperate 2 , 2 0 , 4 Defect 4 , 0 1 , 1 In the Prisoner’s Dilemma, each player chooses between one of two strategies, either to cooperate or to defect. The payoff is represented as a pair of values, the first generally for the player on the left of the matrix and the second for the player above the matrix. The payoff for a certain pair of strategies is given by the pair of values in the cell on the same row as Player A’s strategy and the same column as Player B’s strategy. Note that at when both players choose Defect, neither player has an incentive to unilaterally deviate to Cooperate from their current payoff of one to a payoff of zero, despite the fact that if each chose Cooperate, each player would have a better payoff. Thus, (Defect, Defect) is a Nash Equilibrium, while (Cooperate, Cooperate) is not a Nash Equilibrium, because at (Cooperate, Cooperate), each player has an incentive to deviate to Defect and gain a payoff of four. Also, note that when determining the Nash Equilibria studied in this research, the magnitude of each value does not matter save for with respect to the other values. If (Cooperate, Cooperate) produced payoffs of (3, 3), the game would remain the same, because the change between (2, 2) and (3, 3) did not cause the payoffs of (Cooperate, Cooperate) to be greater or less than any other pair of strategies that (Cooperate, Cooperate) was not greater or less than before the change in payoffs. Epidemic Simulation The Framework for Reconstructing Epidemiological Dynamics (FRED) is a simulation created by the University of Pittsburgh Public Health Dynamics Laboratory in collaboration with the Pittsburgh Supercomputing Center and the 3 School of Computer Science at Carnegie Mellon University. FRED is designed to simulate the spread of infectious diseases and the interplay between public health organizations, viral evolution, and personal health behavior. FRED uses agents representing the population of a county to simulate the spread of infectious diseases. Agent statistics are determined by census data, and each individual agent actively moves through environments such as their neighborhood, workplace, or school. Each agent is individually simulated, as are the infection events resulting from the interaction between an infected and an uninfected agent. The simulation allows for parameters changing the behavior of agents, changing the infectivity (in both asymptomatic and symptomatic stages), dormancy time, and other traits of the virus, and changing the responses by public health organizations to mitigate the virus’s spread. Virus Model The simulated virus followed the sequential SEiIR model of infection in each agent: Susceptible Exposed infectious (asymptomatic) Infectious (symptomatic) Recovered An important variable in the model is the asymptomatic infectivity factor, which is the ratio of the virus infectivity in the asymptomatic phase to that in the symptomatic phase. Strategies Investigated Public Health Strategies Two strategies were investigated for the public health organization: (A) A Laissez-faire policy where each symptomatic agent may choose whether to stay home from work on a given day with a likelihood of 50% (B) A Quarantine policy where agents were required to stay home when symptomatic. 4 Viral Strategies Two strategies were investigated for the viral strain: (A) An aggressive strategy in which the virus remains asymptomatic and symptomatic for an equal period of days on average Days after 1 2 3 4 5 6 7 previous stage Probability of 0.0 0.0 0.0 0.3 0.7 0.9 1.0 asymptomatic period ending Probability of 0.0 0.0 0.0 0.3 0.7 0.9 1.0 symptomatic period ending (B) A hiding strategy in which the virus remains asymptomatic for a period two days longer on average than (A) and is symptomatic for two days fewer. Days after 1 2 3 4 5 6 7 8 9 previous stage Probability of 0.0 0.0 0.0 0.0 0.0 0.3 0.7 0.9 1.0 asymptomatic period ending Probability of 0.0 0.3 0.7 0.9 1.0 ----symptomatic period ending The graph above displays the most likely number of days that the virus will spend in the asymptomatic and symptomatic phases of the SEiIR infection model based on the strategy that the virus adopts. 5 Payoff Calculations Viral Payoff The viral payoff was calculated as the sum of the number of individuals infected, representing the virus’s reproductive success. Public Health Payoff Initially, it might seem as if the public health payoff clearly would be inverse to the number of infections. This produces a zero-sum game, which, while simple, results in obvious dominant strategies for both sides, meaning policy-makers do not need to rely on game theory to determine whether to use a quarantine in such a situation. A quarantine is always better in terms of minimizing infections than no quarantine. Examining the economic effects as a payoff, however, produces more relevant outcomes and interesting games. (I) Minimize Economic Impact – The public health payoff was defined so as to value strategies that are best able to limit the economic effects of the epidemic. Economic effects were determined by the effective work lost. For the Quarantine strategy the effective work lost is evaluated simply by the number of agents infected since this is directly proportional to the number of symptomatic workers who were required to stay home. For the Laissez-faire strategy the effective work lost is evaluated using the equation: æ ì Effective ü ç ï ï í Work ý = - ç ï Lost ï ç î þ è ì# Infected ï í Workers ï at Home î æ ü ì Infected ü ï ï ï ç ý + ç 1- í Worker ý ï ï Productivity ï ç þ î þ è öì# Infected ü ï ÷ï ÷í Workers ý ÷ï at Work ï þ øî ö ÷ ÷ ÷ ø The number of infected workers at home is directly proportional to half the number of agents infected (because of the 0.5 probability that an symptomatic agent will opt to stay home). The number of infected workers at work is also evaluated as half the total number of infected agents. The infected worker productivity quantifies to what degree a worker can complete their work duties while symptomatic. This parameter can vary from 0 (no work accomplished) to 1 (no effect on productivity). Infected worker productivity was varied in this research to examine how the resulting Nash equilibria change depending upon its value. 6 Data and Results Simulation Results Daily Infections by Public Health Strategy (Viral Hiding Strategy, "Sniffles" Iterative Dominance Game) 60000 Daily Infections 50000 40000 30000 Public Health Quarantine Strategy 20000 Public Health Laissez-Faire Strategy 10000 0 0 5 101520253035404550556065707580859095 Days Into Epidemic Daily Infections by Viral Strategy (LaissezFaire Public Health Strategy, "Sniffles" Iterative Dominance Game) 60000 Daily Infections 50000 40000 30000 Viral Aggressive Strategy 20000 Viral Hiding Strategy 10000 0 0 5 101520253035404550556065707580859095 Days Into Epidemic 7 Two graphs of data selected from those produced by the simulations are displayed above. Note that infections are higher for the viral Hiding strategy than for the viral Aggressive Strategy, demonstrating that the virus has no incentive to deviate from the viral Hiding strategy’s Nash Equilibrium. Based off the earlier simulations and payoff calculations, two games were selected. The Sniffles - An Iterative Dominance Game The Iterative Dominance Game occurred in the case where workers had moderately disabling symptoms and were moderately infectious when asymptomatic. The key parameters in the simulations were: Infected worker productivity = 50%. Asymptomatic infectivity factor=50% These conditions resulted in the following raw payoffs: This game can be reduced for each player to a simpler format while maintaining the value of each payoff relative to the other payoffs, in effect ranking each payoff in terms of desirability for each player, with 4 being the most desirable and 1 the least desirable. 8 At first, the public health organization appears to have no clear dominant strategy, with Quarantine being the better strategy if the virus adopts the Aggressive strategy and Laissez-faire being the better strategy if the virus adopts the Hiding strategy. However, by treating the virus as a rational player it is clear that the virus will always gets a better outcome by adopting the Hiding strategy, and therefore can be expected to choose this strategy. The public health organization should therefore choose the Laissez-faire strategy and secure for itself the higher payoff. This results in a Nash Equilibrium at (Laissez-faire, Hiding). If the public health organization attempts to go for the most valuable payoff at (Quarantine, Aggressive) without considering the virus’s role as a rational player, the organization will end up with its least desirable payoff. Spattergroit - A Prisoner’s Dilemma Game A Prisoner’s Dilemma Game occurred in the case where workers had highly disabling symptoms and were weakly infectious when asymptomatic. The key parameters in the simulations were: Infected worker productivity = 30%. Asymptomatic infectivity factor=38% These conditions resulted in the raw payoffs on the following page: 9 The game can be reduced to a simpler format using the same ranking approach as before. For both the virus and the public health organizations, this game has a dominant strategy, to quarantine and to hide, respectively, giving a Nash Equilibrium at (Quarantine, Hiding). Note, however, the presence of the (Laissez-Faire, Aggressive) strategy pair that results in higher payoffs for both players than the outcome at the Nash equilibrium. This produces an interesting result in that the solution to the Prisoner’s Dilemma (the classification of a game in which the Nash equilibrium is not the most favorable outcome for the group) relies on either communication, trust, changing the payoffs, or a repeated game. While the first two are impossible with a virus, the third and fourth pose interesting sets of solutions. Through the innate repetition of the game from outbreak to outbreak and the use of antivirals and other means of slowing the spread of a virus, it could be possible to enter into a cycle of cooperation with a virus in which each party chose strategies favorable to both. 10 Conclusion Game Theory provides a useful tool for evaluating the interaction between public health organizations and infectious disease. By considering infectious diseases as rational players that make best strategy choices for their reproductive success, effective public health strategies can be reached in situations that at first might seem ambiguous. By considering economic costs in the game theoretic analysis of these situations, non-zero sum games arise with interesting implications for public health policy. In the Sniffles scenario, game theory provided public health organizations with a more clear strategy in a situation without a clear dominant strategy, demonstrating the usefulness of this approach. In addition, the number of microorganisms that have developed a symbiotic relationship with humans can show this game’s relevance. The game’s result illustrates a possible cause for such an occurrence in which microorganisms chose to increase their asymptomatic period and decrease the severity of symptoms, while humans, cognizant of the infection’s limited severity, chose not to restrict the microorganisms’ spread. In the Spattergroit scenario, it was shown that the dominant strategies for both the virus and human players resulted in an outcome favorable to neither, in a reflection of the famous Prisoner’s Dilemma game. Games similar to the Prisoner’s Dilemma have often been studied in order to devise a cooperative outcome better for both players. It has been shown that in most cases a Prisoner’s Dilemma can be solved either through communication between players, trust, changing the payoffs, or repetition. While communication and trust are impossible to form with a virus, the last two provide interesting methods to elicit cooperation (in the sense of both players receiving a favorable outcome). Changing the payoffs might be accomplished by using antivirals targeted against virus strains that failed to choose a cooperative strategy, thereby encouraging the virus to choose an Aggressive strategy over Hiding. In a repetitive game in which the number of repetitions is indefinite, there is a decided advantage in choosing a cooperative strategy each round that the game is played, if the other player also appears to be willing to cooperate. By combining these two strategies, it is possible that health organizations could create cycles of cooperation with viruses in which humans had minimal economic damages and viruses had a chance to reproduce symptomatically, - a series of interactions that could either be continued in a state of mutual cooperation or end in a sudden and devastating defection by health organizations in order to eliminate the virus once and for all. 11 Future Research Future research into using game theory in the fight against infectious disease should initially include lab studies to determine the ability of viral organisms to react to a human player. While the data showing that viruses and other organisms often interact following game-theoretic principles among one another, games should still be set up in laboratories in order to practically test the results shown here. In addition, in the field of further simulations, it was posited that a game opposite to the example shown with enterobacteriaceae could exist in which human interference causes increases in the severity of bacterial symptoms, a potentially interesting game. Should the methods be confirmed by the further study detailed, game theory should begin to be applied to judge the proper response to epidemics and infectious disease outbreaks and thereby minimize the threat that infectious disease poses on society and maximize public health’s payoffs. By treating viruses as rational players, the potential benefit of using game theory to develop strategies that prevent epidemics could be enormous and critical to the success of public health organizations worldwide. 12