Taking preventive measures against infections with a cost in static and dynamic single-group populations

The options for agents susceptible to an infection of taking preventive measures with a cost or to expose themselves to an acceptable level of risk of being infected are incorporated into an epidemic model within a game theoretical framework. A susceptible agent decides on his action by assessing the risk posed by his neighborhood, the cost-to-benefit ratio c∕b and the infection probability. In a well-mixed population, the evolution and long time limit of the densities of infected and susceptible agents are studied for different values of the parameters, and the results can be described well by a set of dynamical equations in all cases. The epidemic model is also studied within a dynamical 1-group situation in a population where only agents who are simultaneously present in the common place can make contact. While the 1-group grows and fragments in time repeatedly, the infection profiles show different behavior: from a resurgent behavior following the 1-group dynamics to one that decouples from the grouping dynamics and everything in between. Formulating the co-evolving dynamics mathematically is a challenging task. We demonstrate the necessity of setting up separate sets of dynamical equations for the growing and fragmented stages. The work sheds light on the debate around whether vaccinations should be imposed and whether unvaccinated students should be allowed to go to school amid the recent measles outbreak, as well as on the proper mathematical formulation of co-evolving problems involving contacts among agents only in a popular place.