Mathematical Modeling and Optimal Control of Corruption Dynamics

Abstract

Several research reports have shown that corruption is an impediment to growth, as itmainly constitutes hindrance to investment. It has adverse impacts on the economy andon well-being of a society. In this study, we propose a mathematical model for corruptiondynamics by considering awareness created by anti-corruption and counciling in jail.The model is proved to be both epidemiologically and mathematically well posed. Wehave shown that all solutions of the model are positive and bounded with initial conditionsin a certain meaningful set. The existence of unique corruption-free and endemicequilibrium points are investigated and the basic reproduction number is computed.Then we study the local asymptotic stability of these equilibrium points. The analysisshows that the system has a locally asymptotically stable corruption-free equilibriumpoint when the reproduction number is less than one and locally asymptotically stableendemic equilibrium point for the reproduction number is greater than one. Thesimulation result shows the agreement with the analytical results. Further, we applyoptimal control techniques to a corruption controlled mathematical model to determinethe optimal control strategy in order to minimize the number of susceptible and corruptpopulations. The control strategies are based on education campaign (awarenesscreation) and law enforcement. We then proved the existence of optimal control problem,determined the necessary conditions for optimality, and then performed numericalsimulations. The numerical results showed that the control strategy that involve twocontrol measures have significant impact in reducing corruption dynamics.