Exploring Voluntary Vaccinating Behaviors Using Evolutionary N-person Threshold Games

Human experiences in combating with infectious diseases show that the large-scale spread of a number of epidemics(such as measles and bird flu)can be suppressed by vaccination.Various vaccination strategies have been proposed,ranging from preemptive mass vaccination to post-outbreak ring vaccination,where voluntary vaccination is still a popular vaccination strategy.However,epidemiological analysis has shown that relying solely on voluntary vaccination,there will be a herd immunity threshold(HIT)in well-mixed populations.When the vaccine coverage reaches the herd immunity threshold,some individuals will not be infected even through they are unvaccinated.This forms a public dilemma: individuals incline to be a free-riders due to the fear of vaccination risk.Moreover,the real-world situation is more complicated: the interaction between individuals forms a complex network structure.At the same time,each individual can participate in multiple groups of activities,which makes it difficult to understand individuals' voluntary vaccination behaviors.To model such social activities,in this work,we propose a locally-mixed complex network model.In doing so,we further explore human voluntary vaccinating behaviors by proposing an evolutionary threshold game.From the perspective of game theory,individuals who decide to vaccinate are treated as cooperators,while individuals who decide not to vaccinate are treated as defectors.Taking into consideration the existence of herd immunity threshold,we propose an evolutionary threshold game to model human vaccinating decision-making processes.The payoffs for both cooperators and defectors are derived based on the relative cost of vaccination and infection.Specifically,in a locally well-mixed social environment,we further derive the risk of infection for uninfected individuals given a specific vaccine coverage in a well-mixed population.Based on the proposed evolutionary threshold game,we then perform an equilibrium analysis in well-mixed populations.Theoretically,we reveal the relationships between the vaccine coverage,the population size,the relative cost of vaccination and infection,and the severity of the disease.In structured populations,we carry out simulations on four types of complex social networks to explore the evolutionary dynamics of the multi-person threshold game.The results show when vaccine costs are free,full coverage of the vaccine can be achieved among the population;the diversity of the network structureand the average population size of each game have a significant impact on the final vaccine coverage.This findings can provide new insight into,as well as new methods for public health authorities in developing vaccination policies.Finally,the proposed game in this work also provides a new type of threshold game model for studying the evolution of cooperation in human society.