Modeling and simulation of tax evasion dynamics: Optimal control and its cost-effectivenesss

Tax evasion remains a significant challenge that undermines government revenue and economic stability. In this paper, we propose proposes a new fractional-order three compartment tax evasion model that highlights the transition from susceptible individuals to honest taxpayers, capturing voluntary compliance driven by awareness and behavioral reinforcement mechanisms. The fundamental properties such as positivity, boundedness, and the existence and uniqueness of solutions, were established for the model. In this study, basic reproduction number $(R_0)$ was derived to characterize the persistence of tax evasion. Stability analysis showed that the evader-free equilibrium was locally and globally asymptotically stable when the threshold parameter was less than one, and forward bifurcation occurred when the threshold parameter exceeded one, indicating the persistence of tax evasion. Sensitivity analysis using the Partial Rank Correlation Coefficient (PRCC) method identified that the influence rate $(\beta )$ and the transition rate $(\alpha )$ are the dominant factors driving tax evasion, whereas audit effectiveness and reformation rate significantly reduce evasion. The model was numerically investigated using the Modified Fractional Euler Method (MFEM) for various combinations of parameters. Two control measures, namely media awareness campaigns ( $u_1$) and penalties ( $u_2$), are incorporated to mitigate tax evasion. Pareto and efficiency analyses revealed that penalties are effective when implemented individually, whereas the combined intervention strategy is the most cost-effective and economically viable approach for enhancing tax compliance. The results are presented in the form of figures and tables.