Trust game is commonly used to study the evolution of trust among unrelated individuals. It offers valuable insights into human interactions in a range of disciplines, including economics, sociology and psychology. Previous research has revealed that reward and punishment systems can effectively promote the evolution of trust. However, these investigations overlook the gaming environment, leaving unresolved the optimal conditions for employing distinct incentives to effectively facilitate trust level. To bridge this gap, we introduce a transformation incentive mechanism in an N -player trust game, where trustees are given different forms of incentives depending on the number of trustees in the group. Using the Markov decision process approach, our research shows that as incentives increase, the level of trust rises continuously, eventually reaching a high level of coexistence between investors and trustworthy trustees. Specifically, in the case of smaller incentives, rewarding trustworthy trustees is more effective. Conversely, in the case of larger incentives, punishing untrustworthy trustees is more effective. Additionally, we find that moderate incentives have a positive impact on increasing the average payoff within the group.

A reaction-diffusion (R-D) heroin epidemic model with relapse and permanent immunization is formulated. We use the basic reproduction numberR0to determine the global dynamics of the models. For both the ordinary differential equation (ODE) model and the R-D model, it is shown that the drug-free equilibrium is globally asymptotically stable ifR0≤1, and ifR0>1, the drug-addiction equilibrium is globally asymptotically stable. Some numerical simulations are also carried out to illustrate our analytical results.