Mathematical Model and Analysis of Corruption Dynamics with Optimal Control
Abayneh Kebede Fantaye, Zerihun Kinfe Birhanu
Abstract
In this study, a deterministic mathematical model that explains the transmission dynamics of corruption is proposed and analyzed by considering social influence on honest individuals. Positivity and boundedness of solution of the model are proved and basic reproduction number <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mfenced open="(" close=")"> <a:mrow> <a:msub> <a:mrow> <a:mi>R</a:mi> </a:mrow> <a:mrow> <a:mn>0</a:mn> </a:mrow> </a:msub> </a:mrow> </a:mfenced> </a:math> is computed using the next-generation matrix method. The analysis shows that corruption-free equilibrium is locally and globally asymptotically stable whenever <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M2"> <e:msub> <e:mrow> <e:mi>R</e:mi> </e:mrow> <e:mrow> <e:mn>0</e:mn> </e:mrow> </e:msub> <e:mo><</e:mo> <e:mn>1</e:mn> </e:math> . Also, the endemic equilibrium point is locally and globally asymptotically stable whenever <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M3"> <g:msub> <g:mrow> <g:mi>R</g:mi> </g:mrow> <g:mrow> <g:mn>0</g:mn> </g:mrow> </g:msub> <g:mo>></g:mo> <g:mn>1</g:mn> </g:math> . Then, the model was extended to optimal control, and some numerical simulations with and without optimal control are also performed to verify the theoretical analysis using MATLAB. Numerical simulation of optimal control model shows that the prevention and punishment strategy is the most effective strategy to reduce the dynamic transmission of corruption.