# Mathematical Modeling, Analysis, and Optimal Control of Corruption Dynamics

Authors: Haileyesus Tessema Alemneh
Journal: Journal of Applied Mathematics
 (2020)

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## Abstract

In this paper, a nonlinear deterministic model for the dynamics of corruption is proposed and analysed qualitatively using the stability theory of differential equations. The basic reproduction number with respect to the corruption-free equilibrium is obtained using next-generation matrix method. The conditions for local and global asymptotic stability of corruption-free and endemic equilibria are established. From the analysis using center manifold theory, the model exhibits forward bifurcation. Then, the model was extended by reformulating it as an optimal control problem, with the use of two time-dependent controls to assess the impact of corruption on human population, namely, campaigning about corruption through media and advertisement and exposing corrupted individuals to jail and giving punishment. By using Pontryagin’s maximum principle, necessary conditions for the optimal control of the transmission of corruption were derived. From the numerical simulation, it was found that the integrated control strategy must be taken to fight against corruption.


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## Body

## 1. Introduction

Corruption is an illegal activity carried out for private gain and benefit, by misuse of authority or power by public (government) or private (company) officeholders [1]. Different types of corruption are documented in the literature and include public corruption [2], private corruption [3], pervasive corruption, and arbitrary corruption [4]. Corruption can originate from either the demand side or the supply side [5]. In general, corruption faces a major threat to the rule of law, democracy and human rights, fairness, and social justice, hinders economic development, and brings market economies at risk for their proper and fair functioning [2, 6]. It is a serious problem in all countries in the world mainly for developing countries [6]. Even though most countries have anticorruption policies or strategies that are being made to control corruption, it remains an epidemic in the society.Mathematical models with optimal control analysis are an important tool in understanding the corruption transmission dynamics and in decision-making processes regarding intervention programs for corruption control. In the current research, there are only a few quantitative studies on corruption.The study by Abdulrahman [7] proposed and analysed a deterministic model for corruption in a population. They computed the basic reproduction number (BRN), corruption-free equilibrium point, and endemic equilibrium point. Numerical simulations were carried out, and they revealed that corruption can only be reduced to a manageable level but not totally removed. In [2], the authors developed and analysed a mathematical model for corruption dynamics. They determined the basic reproduction number and corruption-free and endemic equilibrium points. The study by Lemecha [8] proposed a mathematical model for corruption by considering awareness created by anticorruption and counseling in jail. The existence of unique corruption-free and endemic equilibrium points was investigated, and the basic reproduction number was computed. The study in [9] developed the differential equation-based models representing either growth or decay laws for corruption. In [10], a difference equation-based model was framed to measure the level of corruption. The author in [11] considered a game theoretic approach to deal with corruption. A corruption prevention model was formulated by Khan [12] and showed that the complete prevention of corruption is possible if the ratio between rate of dismissal and rate of corruption is equal to one. In [13], the authors developed an SIR model for the corruption dynamics. Furthermore, they extended the model to include optimal control with a single optimal control strategy. They concluded that the level of corruption in society can be reduced if efforts to control corruption are increased and put in place through media/punishments. Nathan and Jackob developed an epidemiological compartment model for corruption in Kenya, mainly considering those who take advantage of office holders and political office holders [14].In this paper, an optimal control theory was used to study the effectiveness of all possible combinations of two corruption preventive measures, namely, (i) campaigning about corruption through media and advertisement and (ii) exposing the corrupted individuals to jail and giving punishment. For this, we considered a model for the transmission of corruption and incorporated two time-dependent controls. Firstly, the model has been studied with no control and the stability properties have been investigated. Secondly, we consider the control parameters to be time dependent and examine the impact of different combinations of these measures in controlling corruption. We use Pontryagin’s maximum principle to derive necessary conditions for the optimal control of corruption.The paper is organized as follows. Section2 is devoted for the model description and the underlying assumptions. In Section 3, we carry out mathematical analysis including bifurcation and sensitivity of the model. In Section 4, the optimal control problem is presented. Section 5 is devoted for numerical simulations and discussions of the model. The conclusion is presented in Section 6.


## 2. Model Formulation and Description

The total populationNt is divided into five compartments. Those who are susceptible to corruption are susceptible individuals St, those who are exposed to a corrupted person but do not perform it are exposed individuals Et, those who are performing corruption are corrupted individuals Ct, those who stopped doing corruption are recovered individuals Rt, and those who know the badness of corruption and do not perform it permanently are honest individuals Ht at time t≥0. Assume that there is a positive recruitment Π into the susceptible class by birth or immigration. From this class, κ proportion will join the honest subpopulation which never involve in corruption practices irrespective of the circumstances around them. We consider a positive natural death rate μ for all humans at all time under the study. Susceptible individuals will have a contact rate with the corrupted individuals at rate β with a corruption transmission probability p per contact and moved to the exposed class. From these individuals, α progressed at a rate of δ to the corrupted compartment and the remaining proportion will move to the recovered compartments. The corrupted individuals get knowledge about the effect of corruption in jail and move to the recovered subpopulation at a rate of σ. From these recovered individuals, θ moved at a rate of ε to susceptible compartment and the other proportion joins the honesty compartment. All the descriptions of the parameters are listed in Table 1.Table 1
Description of parameters of the corruption model (1).
ParameterDescriptionpCorruption transmission probability per contactβContact rate of corrupted individual to the susceptible individualδRate at which exposed individuals become corruptedσRate at which corrupted individuals become recoveredεRate at which recovered individuals become honestΛRecruitment rate of susceptible humansκProportion of individuals that joins the honest population from susceptible populationμDeath rate of all humansαProportion of individuals that joins the corrupted subpopulation from the exposed compartmentθProportion of individuals that joins the honest subpopulation from the recovered compartmentWith regard to the above considerations, we have the compartmental flow diagram shown in Figure1. From the flow chart, the model will be governed by the following system of differential equations:
(1)dSdt=Π+1−θεR−pβSC−k+μS,dEdt=pβSC−δ+μE,dCdt=αδE−σ+μC,dRdt=σC+1−αδE−ε+μR,dHdt=κS+θεR−μH,with the initial condition
(2)S0=S0≥0,E0=E0≥0,I0=I0≥0,R0=R0≥0,H0=H0≥0.Figure 1
Compartmental diagram for the transmission dynamics of corruption.


## 3. Model Analysis

### 3.1. Positivity of Solutions

Since the model (1) monitors human population, it is important that all its state variables and associated parameters are positive for future time. This will be established by the following theorem:Theorem 1.
LetΩ=S,E,C,R,H∈R5:S0>0,E0>0,C0>0,R0>0,H0>0. Then, the solution set St,Et,Ct,Rt,Ht of system (1) is positive for all t≥0.Proof.
We letτ=supt>0:S0ν≥0,E0ν≥0,C0ν≥0,R0ν≥0,H0ν≥0forallν∈0,t. Since S0t≥0,E0t≥0,C0t≥0,R0t≥0 and H0t≥0, then τ>0. If τ<∞, then automatically S0t or E0t or C0t or R0t or H0t is equal to zero at τ. Taking the first equation of the model (1)
(3)dSdt=Π+1−θεR−pβSC−k+μS.Then, using the variation of constant formula the solution of equation (3) at τ is given by
(4)Sτ=S0exp−∫0TpβC+k+μsds+∫0tΠ+1−θεR⋅exp−∫stpβC+k+μvdvds>0.Moreover, since all the variables are positive in0,τ, hence Sτ>0. It can be shown in a similar way that Eτ>0, Cτ>0, Rτ>0, and Hτ>0, which is a contradiction. Hence, τ=∞. Therefore, all the solution sets are positive for t≥0.


### 3.2. Invariant Region

We obtained the invariant region, in which the model solution is bounded. To do this, consider the total human populationNt, where Nt=St+Et+Ct+Rt+Ht.
(5)dNdt=dSdt+dEdt+dCdt+dRdT+dHdt.Substituting all state equations from the model (1), we get
(6)dNdt=Π−μN.Integrating equation (6) both sides and taking t⟶∞, we get
(7)Ω=S,E,C,R,H∈R+5:St+Et+Ct+Rt+Ht+≤Πμis positively invariant, inside which the model is considered to be epidemiologically meaningful and mathematically well posed.


### 3.3. Corruption-Free Equilibrium Point (CFEP)

The corruption-free equilibrium point of model (1) is obtained by equating all equations of the model to zero and letting E=0, C=0, and R=0. Then, we get
(8)E0=Πk+μ,0,0,0,kΠμk+μ.


### 3.4. Basic Reproduction Number

The basic reproduction numberR0 measures the expected number of secondary infections that result from one newly infected individual introduced into a susceptible population. We calculate R0 of the model using the next-generation matrix method as described in [15]. The first step to get R0 is rewriting the model equations starting with newly infective classes:
(9)dEdt=pβSC−δ+μE,dCdt=αδE−σ+μC,dRdt=σC+1−αδE−ε+μR,dHdt=κS+θεR−μH.Then, by the principle of next-generation matrix, the Jacobian, at CFEP, we obtained(10)F=0Πβpk+μ00000000000000,V=δ+μ000−αδσ+μ00−1−αδ−σε+μ000−θεμ.Therefore, the basic reproduction number is the spectral radius of the next generation matrixFV−1 and is given by
(11)R0=pβδαΠδ+μσ+μk+μ.


### 3.5. Local Stability of CFEP

Theorem 2.
The CFEP is locally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
To prove this theorem, let us first find the Jacobian matrix of system (1):
(12)J=−βpC−κ+μ0−pβS1−θε0βpC−δ−μpβS000αδ−μ−σ0001−αδσ−ε+μ0κ00θε−μ.Evaluating equation (12) at the corruption-free equilibriumE0=Π/k+μ,0,0,0,Π/μk+μ, we got
(13)J=−κ+μ0−pβΠκ+μ1−θε00−δ−μpβΠκ+μ000αδ−μ+σ0001−αδσ−ε+μ0κ00θε−μ.From the Jacobian matrix, a characteristic polynomial was obtained as(14)−λ−κ+μ−λ−μ−λ−ε+μλ2+ψ1λ+ψ2=0,where
(15)ψ1=σ+2μ+δ,ψ2=−Παβδp+δ+μκ+μσ+μκ+μ.From equation (14), we have
(16)−λ−κ+μ⇒λ1=−κ+μ<0,−λ−μ⇒λ2=−μ<0,−λ−ε+μ⇒λ3=−ε+μ<0.From the last expression of equation (14), we have
(17)λ2+ψ1λ+ψ2=0.The Routh-Hurwitz criteria was applied, and equation (17) has strictly negative real root iff ψ1>0 and ψ2>0. Clearly, we see that ψ1>0 because it is the sum of positive parameters and also
(18)ψ2=−Παβδp+δ+μκ+μσ+μκ+μ=δ+μσ+μ1−R0>0.Hence, the CFEP is locally asymptotically stable ifR0<1.This result implies that corruption can be eliminated if the initial size of the corrupted individuals is in the basin of attraction of the CFEP. On the other hand, to ensure that corruption elimination is independent of the initial population size of the corrupted population, it is necessary to show that the CFEP is globally stable.


### 3.6. Global Stability of CFEP

Theorem 3.
The equilibrium pointE0 of the model (1) is globally asymptotically stable if R0<1.Proof.
Consider the following Lyapunov function(19)V=χ1E+χ2C.Differentiating equation (19) with respect to t gives
(20)dVdt=χ1dEdt+χ2dCdt.SubstitutingdE/dt and dC/dt from the model (1), we get
(21)dVdt=χ1pβδC−δ+μE+χ2αδE−σ+μC=χ1pβδC−χ2σ+μC−χ1δ+μE+χ2αδE.Here, takeχ1=αδ/δ+μχ2, then we have
(22)dVdt=αδδ+μχ2βδC−χ2σ+μC≤pαβδΠδ+μk+μ−σ+μχ2C.Takingχ2=1 and substituting R0, we get
(23)dVdt≤σ+μR0−1C,for C≤C0=π/k+μ and dV/dt≤0 for R0<1 and dV/dt=0 if and only if C=0. This implies that the only trajectory of the system (1) on which dV/dt≤0 is E0. Therefore, by LaSalle’s invariance principle, E0 is globally asymptotically stable in Ω. This supports the forward bifurcation of Section 3.9.


### 3.7. The Endemic Equilibrium Point

The endemic equilibrium point denoted byE∗=S∗,E∗,C∗,R∗,H∗.E∗ is the steady state solution where corruption persists in the population. It can be obtained by equating each equation in (1) to zero:
(24)dSdt=dEdt=dCdt=dRdt=dHdt=0.Then, we obtain(25)S∗=δ+μσ+μαβδp,E∗=σ+με+μ1−R0αδpβϕδ+μσ+μκ+μ,C∗=ε+μ1−R0pβϕδ+μσ+μκ+μ,R∗=μα−1−σR0−1αpβϕδ+μσ+μκ+μ,H∗=κδ+μσ+μαδpβμ+εθμα−1−σR0−1αpβϕμδ+μσ+μκ+μ,where
(26)ϕ=αδμθ−1−ε+μδ+μσ+μ−δ1+θσ+μ.


### 3.8. Local Stability of Endemic Equilibrium

Theorem 4.
The endemic equilibriumE∗ of system (1) is locally asymptotically stable in Ω if R0>1, otherwise unstable.Proof.
Let us first obtain the Jacobian matrix of system (1):
(27)J=−βpC−κ+μ0−pβS1−θε0βpC−δ+μpβS000αδ−μ+σ0001−αδσ−ε+μ0κ00θε−μ.The characteristic polynomial of equation (27) is given by
(28)λ5+ψ1λ4+ψ2λ3+ψ3λ2+ψ2λ+ψ4=0,where
(29)ψ1=σ+με+μR0−1αδϕ+δ+∈+κ+5μ+σ,ψ2=−δ+μσ+μ+δ+ε+σ+4με+μR0−1ϕ+δ+4μ+σ∈+κ+εκ+μ10μ+4σ,ψ3=−δ+μσ+με+κ+3μ+ε+μR0−1ϕδ1−θ+σ+3μ+ε+3μ2μ+σ−αδθ−1+σ+3μδκ+εκ+δε+2μ+σ3εμ−3κμ−6δμ+δεκ+2μ25μ+σ,ψ4=2βpCαδθσ+μ+αδμ+σθ+1+σ+μδε+1+βpCδμ3μ+2σ−2θ+δσθ+εμ3μ+2σ+μ24μ+3σ+2μ+σδεκ+3μ+2σδεμ+δκp+εκp−μ24μ+3σε+κ+δ+μ35μ+4σ−δ+μσ+μεκ+2εμ+2κμ+3μ2,ψ5=−με+μR0−1ϕμ+2σαδθ−1−σ+μβδε+μ−θ+βμε+μ+δ−κ+μδ+μσ+μμεββμ+βκμ+σ+μμεκβ+μ+μδκ+ε+μ2δ+ε+κ+μ,Using the Routh-Hurwitz criterion, all roots of characteristic polynomial have negative real parts if and only ifψ1>0,ψ2>0,ψ3>0,ψ4>0,ψ5>0,ψ1ψ2ψ3>ψ32+ψ12ψ4 and ψ1ψ4−ψ5ψ1ψ2ψ3−ψ32+ψ12ψ4>ψ5ψ1ψ2−ψ32+ψ1ψ52forR0>1. Hence, the endemic equilibrium E∗ is locally asymptotically stable.


### 3.9. Bifurcation Analysis

In order to determine the direction of the bifurcation and to prove the stability of the endemic equilibrium point, we make use of the bifurcation theory, which is based on the center manifold theory [16] (as described by Theorem 6 of Castillo-Chavez and Song [16]).Theorem 5.
The model in the system (1) exhibits forward bifurcation at R0=1 and a unique endemic equilibrium exists.Proof.
To conduct this analysis, we assume thatS=z1,E=z2,C=z3,R=z4,H=z5 and from R0=1, we have
(30)β=β∗=δ+μσ+μk+μpδαΠ.The new the model in equation (1) is given by
(31)dz1dt=Π+1−θεz4−pβz1z3−k+μz1,dz2dt=pβz1z3−δ+μz2,dz3dt=αδz2−1−μz3,dz4dt=σz3+1−αδz2−ε+μz4,dz5dt=κz1+θεz4−μz5.The Jacobian of the model system (31) around the corruption-free equilibrium point E0=z1=Π/κ+μ,0,0,0,z5=κΠ/μκ+μ evaluated at R0=1 is
(32)J=−κ+μ0−pβΠκ+μ1−θε00−δ−μpβΠκ+μ000αδ−μ+σ0001−αδσ−ε+μ0κ00θε−μ.The right eigenvector,w=w1,w2,w3,w4,w5T, associated with this simple zero eigenvalue can be obtained from Jw=0.
(33)w1=1−θε1−αδα+μ+σδασ+με+μκ+μ−δ+μκ+μw2,w2=w2>0,w3=αδμ+σw2,w4=1−αεσ+μ+σδαμ+σε+μw2,w5=κμ1−θε1−αδσ+μ+σδασ+με+μκ+μ−δ+μκ+μw2+θε1−αδσ+μ+σδαμμ+σε+μw2.The left eigenvector,v=v1,v2,v3,v4,v5, associated with this simple zero eigenvalue can be obtained from vJ=0 and given by
(34)v1=v4=v5=0,v2=v2>0,v3=δ+μαδv2.Since the first, fourth, and fifth components ofv are zero, we do not need the derivatives of f1, f4, and f5. From the derivatives of f2 and f3, the only ones that are nonzero are the following:
(35)∂2f2∂z2∂z3=∂2f2∂z3∂z2=pβ,∂2f2∂z2∂β=pz1∗,and all the other partial derivatives are zero. The direction of the bifurcation at R0=1 is determined by the signs of the bifurcation coefficients a and b, obtained from the above partial derivatives, given, respectively, by
(36)a=2v2w1w3=2v2w22pβαδσ+μ2L−W<0,b=v2w3pz1∗=v2δαΠk+μσ+μ>0,where
(37)L=εδ1−θσ+μ+θα2σ+με+μκ+μ,W=εαδμ+ε+μδ+μ2ε+μκ+μ.Since the coefficientb is positive and the sign of the coefficient a is negative, the model undergoes forward bifurcation, which implies that R0<1 is a necessary and sufficient condition for corruption elimination.


### 3.10. Sensitivity Analysis

In order to see the relative effect of each parameter to the corruption transmission, we performed sensitivity analysis. To go through, we followed the approach defined by [17] as done in [18, 19] which used the definition of normalized forward sensitivity index defined as a variable, R0, that depends differentiably on a parameter,ℓ, defined as
(38)ΛℓR0=∂R0∂ℓ×ℓR0,for ℓ represents all the basic parameters and R0=αpβδΠ/δ+μσ+μκ+μ.
(39)ΛκR0=∂R0∂β×βR0=1≥0,ΛκR0=∂R0∂κ×βR0=−κκ+μ≤0,ΛσR0=∂R0∂σ×βR0=−σσ+μ≤0.And we do this in a similar fashion for the remaining parameters.


### 3.11. Interpretation of Sensitivity Indices

The sensitivity indices of parameters are found in Table2. Parameters that have positive indices β,α,andδ have great impact on expanding the corruption in the community when their values increase. Parameters with negative indices κ,σ,andμ minimize the burden of corruption in the community as their values increase.Table 2
Sensitivity index table.
Parameter symbolSensitivity indicesβ+veδ+veα+veκ-veμ-veσ-veTherefore, the model sensitivity analysis demonstrated that policy makers are supposed to decrease positive index parameters and increase negative index parameters to combat corruption in a population. From this result, to identify the optimal strategy to fight against corruption, an optimal control model analysis is considered in the next section.


## 3.1. Positivity of Solutions

Since the model (1) monitors human population, it is important that all its state variables and associated parameters are positive for future time. This will be established by the following theorem:Theorem 1.
LetΩ=S,E,C,R,H∈R5:S0>0,E0>0,C0>0,R0>0,H0>0. Then, the solution set St,Et,Ct,Rt,Ht of system (1) is positive for all t≥0.Proof.
We letτ=supt>0:S0ν≥0,E0ν≥0,C0ν≥0,R0ν≥0,H0ν≥0forallν∈0,t. Since S0t≥0,E0t≥0,C0t≥0,R0t≥0 and H0t≥0, then τ>0. If τ<∞, then automatically S0t or E0t or C0t or R0t or H0t is equal to zero at τ. Taking the first equation of the model (1)
(3)dSdt=Π+1−θεR−pβSC−k+μS.Then, using the variation of constant formula the solution of equation (3) at τ is given by
(4)Sτ=S0exp−∫0TpβC+k+μsds+∫0tΠ+1−θεR⋅exp−∫stpβC+k+μvdvds>0.Moreover, since all the variables are positive in0,τ, hence Sτ>0. It can be shown in a similar way that Eτ>0, Cτ>0, Rτ>0, and Hτ>0, which is a contradiction. Hence, τ=∞. Therefore, all the solution sets are positive for t≥0.


## 3.2. Invariant Region

We obtained the invariant region, in which the model solution is bounded. To do this, consider the total human populationNt, where Nt=St+Et+Ct+Rt+Ht.
(5)dNdt=dSdt+dEdt+dCdt+dRdT+dHdt.Substituting all state equations from the model (1), we get
(6)dNdt=Π−μN.Integrating equation (6) both sides and taking t⟶∞, we get
(7)Ω=S,E,C,R,H∈R+5:St+Et+Ct+Rt+Ht+≤Πμis positively invariant, inside which the model is considered to be epidemiologically meaningful and mathematically well posed.


## 3.3. Corruption-Free Equilibrium Point (CFEP)

The corruption-free equilibrium point of model (1) is obtained by equating all equations of the model to zero and letting E=0, C=0, and R=0. Then, we get
(8)E0=Πk+μ,0,0,0,kΠμk+μ.


## 3.4. Basic Reproduction Number

The basic reproduction numberR0 measures the expected number of secondary infections that result from one newly infected individual introduced into a susceptible population. We calculate R0 of the model using the next-generation matrix method as described in [15]. The first step to get R0 is rewriting the model equations starting with newly infective classes:
(9)dEdt=pβSC−δ+μE,dCdt=αδE−σ+μC,dRdt=σC+1−αδE−ε+μR,dHdt=κS+θεR−μH.Then, by the principle of next-generation matrix, the Jacobian, at CFEP, we obtained(10)F=0Πβpk+μ00000000000000,V=δ+μ000−αδσ+μ00−1−αδ−σε+μ000−θεμ.Therefore, the basic reproduction number is the spectral radius of the next generation matrixFV−1 and is given by
(11)R0=pβδαΠδ+μσ+μk+μ.


## 3.5. Local Stability of CFEP

Theorem 2.
The CFEP is locally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
To prove this theorem, let us first find the Jacobian matrix of system (1):
(12)J=−βpC−κ+μ0−pβS1−θε0βpC−δ−μpβS000αδ−μ−σ0001−αδσ−ε+μ0κ00θε−μ.Evaluating equation (12) at the corruption-free equilibriumE0=Π/k+μ,0,0,0,Π/μk+μ, we got
(13)J=−κ+μ0−pβΠκ+μ1−θε00−δ−μpβΠκ+μ000αδ−μ+σ0001−αδσ−ε+μ0κ00θε−μ.From the Jacobian matrix, a characteristic polynomial was obtained as(14)−λ−κ+μ−λ−μ−λ−ε+μλ2+ψ1λ+ψ2=0,where
(15)ψ1=σ+2μ+δ,ψ2=−Παβδp+δ+μκ+μσ+μκ+μ.From equation (14), we have
(16)−λ−κ+μ⇒λ1=−κ+μ<0,−λ−μ⇒λ2=−μ<0,−λ−ε+μ⇒λ3=−ε+μ<0.From the last expression of equation (14), we have
(17)λ2+ψ1λ+ψ2=0.The Routh-Hurwitz criteria was applied, and equation (17) has strictly negative real root iff ψ1>0 and ψ2>0. Clearly, we see that ψ1>0 because it is the sum of positive parameters and also
(18)ψ2=−Παβδp+δ+μκ+μσ+μκ+μ=δ+μσ+μ1−R0>0.Hence, the CFEP is locally asymptotically stable ifR0<1.This result implies that corruption can be eliminated if the initial size of the corrupted individuals is in the basin of attraction of the CFEP. On the other hand, to ensure that corruption elimination is independent of the initial population size of the corrupted population, it is necessary to show that the CFEP is globally stable.


## 3.6. Global Stability of CFEP

Theorem 3.
The equilibrium pointE0 of the model (1) is globally asymptotically stable if R0<1.Proof.
Consider the following Lyapunov function(19)V=χ1E+χ2C.Differentiating equation (19) with respect to t gives
(20)dVdt=χ1dEdt+χ2dCdt.SubstitutingdE/dt and dC/dt from the model (1), we get
(21)dVdt=χ1pβδC−δ+μE+χ2αδE−σ+μC=χ1pβδC−χ2σ+μC−χ1δ+μE+χ2αδE.Here, takeχ1=αδ/δ+μχ2, then we have
(22)dVdt=αδδ+μχ2βδC−χ2σ+μC≤pαβδΠδ+μk+μ−σ+μχ2C.Takingχ2=1 and substituting R0, we get
(23)dVdt≤σ+μR0−1C,for C≤C0=π/k+μ and dV/dt≤0 for R0<1 and dV/dt=0 if and only if C=0. This implies that the only trajectory of the system (1) on which dV/dt≤0 is E0. Therefore, by LaSalle’s invariance principle, E0 is globally asymptotically stable in Ω. This supports the forward bifurcation of Section 3.9.


## 3.7. The Endemic Equilibrium Point

The endemic equilibrium point denoted byE∗=S∗,E∗,C∗,R∗,H∗.E∗ is the steady state solution where corruption persists in the population. It can be obtained by equating each equation in (1) to zero:
(24)dSdt=dEdt=dCdt=dRdt=dHdt=0.Then, we obtain(25)S∗=δ+μσ+μαβδp,E∗=σ+με+μ1−R0αδpβϕδ+μσ+μκ+μ,C∗=ε+μ1−R0pβϕδ+μσ+μκ+μ,R∗=μα−1−σR0−1αpβϕδ+μσ+μκ+μ,H∗=κδ+μσ+μαδpβμ+εθμα−1−σR0−1αpβϕμδ+μσ+μκ+μ,where
(26)ϕ=αδμθ−1−ε+μδ+μσ+μ−δ1+θσ+μ.


## 3.8. Local Stability of Endemic Equilibrium

Theorem 4.
The endemic equilibriumE∗ of system (1) is locally asymptotically stable in Ω if R0>1, otherwise unstable.Proof.
Let us first obtain the Jacobian matrix of system (1):
(27)J=−βpC−κ+μ0−pβS1−θε0βpC−δ+μpβS000αδ−μ+σ0001−αδσ−ε+μ0κ00θε−μ.The characteristic polynomial of equation (27) is given by
(28)λ5+ψ1λ4+ψ2λ3+ψ3λ2+ψ2λ+ψ4=0,where
(29)ψ1=σ+με+μR0−1αδϕ+δ+∈+κ+5μ+σ,ψ2=−δ+μσ+μ+δ+ε+σ+4με+μR0−1ϕ+δ+4μ+σ∈+κ+εκ+μ10μ+4σ,ψ3=−δ+μσ+με+κ+3μ+ε+μR0−1ϕδ1−θ+σ+3μ+ε+3μ2μ+σ−αδθ−1+σ+3μδκ+εκ+δε+2μ+σ3εμ−3κμ−6δμ+δεκ+2μ25μ+σ,ψ4=2βpCαδθσ+μ+αδμ+σθ+1+σ+μδε+1+βpCδμ3μ+2σ−2θ+δσθ+εμ3μ+2σ+μ24μ+3σ+2μ+σδεκ+3μ+2σδεμ+δκp+εκp−μ24μ+3σε+κ+δ+μ35μ+4σ−δ+μσ+μεκ+2εμ+2κμ+3μ2,ψ5=−με+μR0−1ϕμ+2σαδθ−1−σ+μβδε+μ−θ+βμε+μ+δ−κ+μδ+μσ+μμεββμ+βκμ+σ+μμεκβ+μ+μδκ+ε+μ2δ+ε+κ+μ,Using the Routh-Hurwitz criterion, all roots of characteristic polynomial have negative real parts if and only ifψ1>0,ψ2>0,ψ3>0,ψ4>0,ψ5>0,ψ1ψ2ψ3>ψ32+ψ12ψ4 and ψ1ψ4−ψ5ψ1ψ2ψ3−ψ32+ψ12ψ4>ψ5ψ1ψ2−ψ32+ψ1ψ52forR0>1. Hence, the endemic equilibrium E∗ is locally asymptotically stable.


## 3.9. Bifurcation Analysis

In order to determine the direction of the bifurcation and to prove the stability of the endemic equilibrium point, we make use of the bifurcation theory, which is based on the center manifold theory [16] (as described by Theorem 6 of Castillo-Chavez and Song [16]).Theorem 5.
The model in the system (1) exhibits forward bifurcation at R0=1 and a unique endemic equilibrium exists.Proof.
To conduct this analysis, we assume thatS=z1,E=z2,C=z3,R=z4,H=z5 and from R0=1, we have
(30)β=β∗=δ+μσ+μk+μpδαΠ.The new the model in equation (1) is given by
(31)dz1dt=Π+1−θεz4−pβz1z3−k+μz1,dz2dt=pβz1z3−δ+μz2,dz3dt=αδz2−1−μz3,dz4dt=σz3+1−αδz2−ε+μz4,dz5dt=κz1+θεz4−μz5.The Jacobian of the model system (31) around the corruption-free equilibrium point E0=z1=Π/κ+μ,0,0,0,z5=κΠ/μκ+μ evaluated at R0=1 is
(32)J=−κ+μ0−pβΠκ+μ1−θε00−δ−μpβΠκ+μ000αδ−μ+σ0001−αδσ−ε+μ0κ00θε−μ.The right eigenvector,w=w1,w2,w3,w4,w5T, associated with this simple zero eigenvalue can be obtained from Jw=0.
(33)w1=1−θε1−αδα+μ+σδασ+με+μκ+μ−δ+μκ+μw2,w2=w2>0,w3=αδμ+σw2,w4=1−αεσ+μ+σδαμ+σε+μw2,w5=κμ1−θε1−αδσ+μ+σδασ+με+μκ+μ−δ+μκ+μw2+θε1−αδσ+μ+σδαμμ+σε+μw2.The left eigenvector,v=v1,v2,v3,v4,v5, associated with this simple zero eigenvalue can be obtained from vJ=0 and given by
(34)v1=v4=v5=0,v2=v2>0,v3=δ+μαδv2.Since the first, fourth, and fifth components ofv are zero, we do not need the derivatives of f1, f4, and f5. From the derivatives of f2 and f3, the only ones that are nonzero are the following:
(35)∂2f2∂z2∂z3=∂2f2∂z3∂z2=pβ,∂2f2∂z2∂β=pz1∗,and all the other partial derivatives are zero. The direction of the bifurcation at R0=1 is determined by the signs of the bifurcation coefficients a and b, obtained from the above partial derivatives, given, respectively, by
(36)a=2v2w1w3=2v2w22pβαδσ+μ2L−W<0,b=v2w3pz1∗=v2δαΠk+μσ+μ>0,where
(37)L=εδ1−θσ+μ+θα2σ+με+μκ+μ,W=εαδμ+ε+μδ+μ2ε+μκ+μ.Since the coefficientb is positive and the sign of the coefficient a is negative, the model undergoes forward bifurcation, which implies that R0<1 is a necessary and sufficient condition for corruption elimination.


## 3.10. Sensitivity Analysis

In order to see the relative effect of each parameter to the corruption transmission, we performed sensitivity analysis. To go through, we followed the approach defined by [17] as done in [18, 19] which used the definition of normalized forward sensitivity index defined as a variable, R0, that depends differentiably on a parameter,ℓ, defined as
(38)ΛℓR0=∂R0∂ℓ×ℓR0,for ℓ represents all the basic parameters and R0=αpβδΠ/δ+μσ+μκ+μ.
(39)ΛκR0=∂R0∂β×βR0=1≥0,ΛκR0=∂R0∂κ×βR0=−κκ+μ≤0,ΛσR0=∂R0∂σ×βR0=−σσ+μ≤0.And we do this in a similar fashion for the remaining parameters.


## 3.11. Interpretation of Sensitivity Indices

The sensitivity indices of parameters are found in Table2. Parameters that have positive indices β,α,andδ have great impact on expanding the corruption in the community when their values increase. Parameters with negative indices κ,σ,andμ minimize the burden of corruption in the community as their values increase.Table 2
Sensitivity index table.
Parameter symbolSensitivity indicesβ+veδ+veα+veκ-veμ-veσ-veTherefore, the model sensitivity analysis demonstrated that policy makers are supposed to decrease positive index parameters and increase negative index parameters to combat corruption in a population. From this result, to identify the optimal strategy to fight against corruption, an optimal control model analysis is considered in the next section.


## 4. Extension of the Model into Optimal Control

The optimal level that would be needed to control corruption is minimizing the number of exposed and corrupted population. An optimal control model was developed by including the following two controls:(i)
u1: campaigning about corruption through media and advertisement(ii)
u2: exposing the corrupted individuals to jail and giving punishmentAdding the two controls on the model (1), the optimal control model is given by
(40)dSdt=Π+1−θεR−1−u2pβSC−k+μ+u1S,dEdt=1−u2pβSC−δ+μ+u1E,dCdt=1−u1αδE−σ+μ+u2C,dRdt=σ+u2C+1−αδ+u2E−ε+μR,dHdt=k−u1S+θεR−μH.The control variablesu1 and u2 minimize the optimal control model (40) subject to the objective functional defined as
(41)J=∫0tfa1E+a2I+12w1u12+w2u22dt,where tf is the final time, a1 and a2 are weight constants of the exposed and corrupted population, respectively, while w1 and w2 are weight coefficients for each individual control measure. We choose a nonlinear cost on the controls based on the assumption that the cost takes nonlinear form [20]. Optimal control function u1∗,u2∗ needs to be found such that
(42)Ju1∗,u2∗=minJu1,u2u1,u2∈U,where U=u1,u2uitislebesguemeasurableon0,tf,0≤uit≤1,i=1,2 is the closed set.


### 4.1. The Hamiltonian and Optimality System

The optimal control must satisfy the necessary conditions that are formulated by Pontryagin’s maximum principle ([21]). This principle converts the system of equations (40) and (41) into a problem of minimizing point-wise a Hamiltonian (M), with respect to u1t and u2t as
(43)M=a1E+a2C+12w1u12+w2u22+w3u32+λ1Π+1−θεR−1−u2pβSC−k+μ+u1S+λ21−u2pβSC−δ+μ+u1E+λ31−u1αδE−σ+μ+u2C+λ4σ+u2C+1−αδ+u2E−ε+μR+λ5k+u1S+θεR−μH,where λi,i=1,⋯,5 are the adjoint variable functions to be determined. Because of the convexity of the integrand of J with respect to u1and u2, a priori boundaries of the state solutions and the Lipschitz property of the state system relating to the state variables, the existence of the optimal control was proved using Corollary 4.1 in the work of Fleming and Rishel [22].Theorem 6.
Suppose we have optimal controlsu1∗,u2∗ and S,E,C,R,H solutions of the respective state system that minimizes J over U, there exist adjoint variables, λ1,⋯,λ5 such that
(44)dλ1dt=λ11−u2pβC+κ+μ+u1−λ21−u2pβC−λ5κ+u1,dλ2dt=−a1+λ2δ+μ+u1−λ3αδ1−u1−λ41−αδ+u1,dλ3dt=−a2+λ11−u2pβS−λ21−u2pβS+λ3σ+μ+u2−λ4σ+u2,dλ4dt=λ1∈1−θ+λ4∈+μ−λ5θ∈,dλ5dt=μλ5.
With transversality conditions,λitf=0,i=1,⋯,5. Furthermore, we obtained the control set u1∗,u2∗ characterized by
(45)u1∗=max0,min1,λ1−λ5S+λ2+αδλ3−1−αλ4Ew1,u2∗=max0,min1,pβCSλ2−λ1+λ4−λ3Cw2.Proof.
The adjoint equation and transversality conditions are standard results obtained from Pontryagin’s maximum principle [21]. Differentiate the Hamiltonian with respect to states S,E,C,R, and H, respectively, then we got the adjoint equations as
(46)dλ1dt=−∂M∂S=λ11−u2pβC+κ+μ+u1−λ21−u2pβC−λ5κ+u1,dλ2dt−∂M∂E=−a1+λ2δ+μ+u1−λ3αδ1−u1−λ41−αδ+u1,dλ3dt−∂M∂C=−a2+λ11−u2pβS−λ21−u2pβS+λ3σ+μ+u2−λ4σ+u2,dλ4dt−∂M∂R=−λ1∈1−θ+λ4∈+μ−λ5θ∈,dλ5dt−∂M∂H=μλ5.With transversality conditions,λitf=0,i=1,⋯,5. Differentiate the Hamiltonian equation with respect to u1 and u2 and obtain the characterization of optimal controls u1∗,u2∗ from ∂M/∂u1=0, ∂M/∂u2=0. Thus, we got the characteristic equation in standard control arguments form involving the bounds on the controls as follows:
(47)u1∗=ψ1,if0<ψ1<1,0,ifψ1≤0,1,ifψ1≥1,u2∗=ψ2,if0<ψ2<1,0,ifψ2≤0,1,ifψ2≥1.In compacted notation:(48)u1∗=max0,min1,ψ1,u2∗=max0,min1,ψ2,where
(49)ψ1=λ1−λ5S+λ2+αδλ3−1−αλ4Ew1,ψ2=pβCSλ2−λ1+λ4−λ3Cw2.The optimality system is formed from the optimal control system (the state system) and the adjoint variable system by incorporating the characterized control set under initial and transversal condition:(50)dSdt=∏+1−θεR−1−u2∗pβSC−k+μ+u1∗S,dEdt=1−u2∗pβSC−δ+μ+u1∗E,dCdt=1−u1∗αδE−α+μ+u2∗C,dRdt=σ+u2∗C+1−αδ+u2∗E−ε+μR,dHdt=k+u1∗S+θεR−μH,dλ1dt=λ11−u2∗pβC+k+μ+u1∗−λ21−u2∗pβC−λ5k+u1∗,dλ2dt=−a1+λ2δ+μ+u1∗−λ3αδ1−u1∗−λ41−αδ+u1∗,dλ3dt=−a2+λ11−u2∗pβS−λ21−u2∗pβS+λ3σ+μ+u2∗−λ4σ+u2∗,dλ4dt=−λ1∈1−θ+λ4∈+μ−λ5θε,dλ5dt=μλ5,u1∗=max0,min1,λ1−λ5S+λ2+αδλ3−1−αλ4Ew1,u2∗=max0,min1,pβCSλ2−λ1+λ4−λ3Cw2,λitf=0,i=1,⋯,5,S0=S0,E0=E0,C0=C0,R0=R0,H0=H0.


## 4.1. The Hamiltonian and Optimality System

The optimal control must satisfy the necessary conditions that are formulated by Pontryagin’s maximum principle ([21]). This principle converts the system of equations (40) and (41) into a problem of minimizing point-wise a Hamiltonian (M), with respect to u1t and u2t as
(43)M=a1E+a2C+12w1u12+w2u22+w3u32+λ1Π+1−θεR−1−u2pβSC−k+μ+u1S+λ21−u2pβSC−δ+μ+u1E+λ31−u1αδE−σ+μ+u2C+λ4σ+u2C+1−αδ+u2E−ε+μR+λ5k+u1S+θεR−μH,where λi,i=1,⋯,5 are the adjoint variable functions to be determined. Because of the convexity of the integrand of J with respect to u1and u2, a priori boundaries of the state solutions and the Lipschitz property of the state system relating to the state variables, the existence of the optimal control was proved using Corollary 4.1 in the work of Fleming and Rishel [22].Theorem 6.
Suppose we have optimal controlsu1∗,u2∗ and S,E,C,R,H solutions of the respective state system that minimizes J over U, there exist adjoint variables, λ1,⋯,λ5 such that
(44)dλ1dt=λ11−u2pβC+κ+μ+u1−λ21−u2pβC−λ5κ+u1,dλ2dt=−a1+λ2δ+μ+u1−λ3αδ1−u1−λ41−αδ+u1,dλ3dt=−a2+λ11−u2pβS−λ21−u2pβS+λ3σ+μ+u2−λ4σ+u2,dλ4dt=λ1∈1−θ+λ4∈+μ−λ5θ∈,dλ5dt=μλ5.
With transversality conditions,λitf=0,i=1,⋯,5. Furthermore, we obtained the control set u1∗,u2∗ characterized by
(45)u1∗=max0,min1,λ1−λ5S+λ2+αδλ3−1−αλ4Ew1,u2∗=max0,min1,pβCSλ2−λ1+λ4−λ3Cw2.Proof.
The adjoint equation and transversality conditions are standard results obtained from Pontryagin’s maximum principle [21]. Differentiate the Hamiltonian with respect to states S,E,C,R, and H, respectively, then we got the adjoint equations as
(46)dλ1dt=−∂M∂S=λ11−u2pβC+κ+μ+u1−λ21−u2pβC−λ5κ+u1,dλ2dt−∂M∂E=−a1+λ2δ+μ+u1−λ3αδ1−u1−λ41−αδ+u1,dλ3dt−∂M∂C=−a2+λ11−u2pβS−λ21−u2pβS+λ3σ+μ+u2−λ4σ+u2,dλ4dt−∂M∂R=−λ1∈1−θ+λ4∈+μ−λ5θ∈,dλ5dt−∂M∂H=μλ5.With transversality conditions,λitf=0,i=1,⋯,5. Differentiate the Hamiltonian equation with respect to u1 and u2 and obtain the characterization of optimal controls u1∗,u2∗ from ∂M/∂u1=0, ∂M/∂u2=0. Thus, we got the characteristic equation in standard control arguments form involving the bounds on the controls as follows:
(47)u1∗=ψ1,if0<ψ1<1,0,ifψ1≤0,1,ifψ1≥1,u2∗=ψ2,if0<ψ2<1,0,ifψ2≤0,1,ifψ2≥1.In compacted notation:(48)u1∗=max0,min1,ψ1,u2∗=max0,min1,ψ2,where
(49)ψ1=λ1−λ5S+λ2+αδλ3−1−αλ4Ew1,ψ2=pβCSλ2−λ1+λ4−λ3Cw2.The optimality system is formed from the optimal control system (the state system) and the adjoint variable system by incorporating the characterized control set under initial and transversal condition:(50)dSdt=∏+1−θεR−1−u2∗pβSC−k+μ+u1∗S,dEdt=1−u2∗pβSC−δ+μ+u1∗E,dCdt=1−u1∗αδE−α+μ+u2∗C,dRdt=σ+u2∗C+1−αδ+u2∗E−ε+μR,dHdt=k+u1∗S+θεR−μH,dλ1dt=λ11−u2∗pβC+k+μ+u1∗−λ21−u2∗pβC−λ5k+u1∗,dλ2dt=−a1+λ2δ+μ+u1∗−λ3αδ1−u1∗−λ41−αδ+u1∗,dλ3dt=−a2+λ11−u2∗pβS−λ21−u2∗pβS+λ3σ+μ+u2∗−λ4σ+u2∗,dλ4dt=−λ1∈1−θ+λ4∈+μ−λ5θε,dλ5dt=μλ5,u1∗=max0,min1,λ1−λ5S+λ2+αδλ3−1−αλ4Ew1,u2∗=max0,min1,pβCSλ2−λ1+λ4−λ3Cw2,λitf=0,i=1,⋯,5,S0=S0,E0=E0,C0=C0,R0=R0,H0=H0.


## 5. Numerical Simulations and Discussions

In this section, numerical simulations are performed on the model (1). Maple 18 software is used to check the effect of some parameters in the spread and control of corruption. The parameter values in Table 3 and the initial conditions S0=10000,E0=0,C0=100,R0=0,H0=100 are used.Table 3
Values of parameter used in the numerical simulation.
ParameterValueSourceΠ85Assumedp0.036[7, 8]β0.0234[7, 8]δ0.2Assumedσ0.007Assumedε0.35Assumedκ0.03[8]μ0.0160[8]α0.3[8]θ0.1[8]


### 5.1. Effect ofα and δ on the Corrupted Population

As we see in Figure2(a), we experimented the effect of changing α on the number of corrupted population by keeping the other parameters constant. It can be seen from Figure 2(a) that, as the α value increases, the number of corrupted population increases. Similarly, in Figure 2(b), we investigated the effect of changing δ on the number of corrupted population by keeping the other parameters constant. It is evident from Figure 2(b) that, when the value of δ increases, the number of corrupted population also increases. Therefore, public policy makers must concentrate on minimizing the contact rate of corrupted individuals to susceptible population and reduce the transition of exposed individuals to the corrupted population.Figure 2
Effect ofα and δ on the corrupted population.
(a)(b)


### 5.2. Effect ofκ and σ on the Corrupted Population

In Figure3(a), we can see that κ plays an important role in reducing the number of corrupted population. When the rate at which individuals join the honest subpopulation from susceptible class increases from 0.03 to 0.4, the number of corrupted population decreases. From Figure 3(b), the rate at which corrupted individuals join the recovered population through education and/or jail sentence increases from 0.007 to 0.1; the number of corrupted population decreases. Therefore, public policy makers must concentrate on maximizing the number of populations that join the honest subpopulation and corrupted population to recover from doing it.Figure 3
Effect ofκ and σ on the corrupted population.
(a)(b)To simulate the optimality system in (50), iterative technique is applied which gives the optimal solution. Forward fourth-order Runge-Kutta method is used to solve the state system, and backward fourth-order Runge-Kutta method is used to solve the adjoint system by considering the initial condition of the state system and the final condition of the adjoint systems. The controls continue to be updated by combining from the previous result of the controls with the characterization. The solution of the state and adjoint system is repeated by the updated controls. This condition continues repeatedly until consecutive iteration is close enough each other [23]. We proposed the following three strategies for numerical simulation of the corruption model:
(i)
Strategy A: campaigning through media and advertisement about corruptionu1≠0andu2≠0(ii)
Strategy B: exposing the corrupted individuals to jail and giving punishmentu1≠0andu2≠0(iii)
Strategy C: combination of both strategies A and Bu1≠0andu2≠0


### 5.3. Strategy A: Campaigning through Media and Advertisement about Corruption

In this strategy, controlsu1 were used to optimize the objective function J while we set control u2 to zero. In Figure 4, we see that exposed and corrupted population significantly reduced when there is control compared to situation with no control. From this, we conclude that applying these strategy can reduce corruption along with time from the community but not possible to avoid it in a specified period of time. Hence, it is not an effective strategy to combat corruption from the population.Figure 4
Simulations of the corruption model showing the effect of the optimal strategiesu1≠0.


### 5.4. Strategy B: Exposing the Corrupted Individuals to Jail and Giving Punishment

Under this strategy, we optimize the objective functionJ using the control u2 while u1 is set to zero. It is evident from Figure 5 that the number of exposed individuals cannot reduce using the strategy and the corrupted population seems eliminated in the first year but grows in number due to the increase in number of exposed individuals. Therefore, the strategy is not effective in fighting and reducing corruption in the population even though it reduces the number of corrupted individuals when compared with control from without control.Figure 5
Simulations of the corruption model showing the effect of the optimal strategiesu2≠0.


### 5.5. Strategy C: Combination of Both Strategies A and B

Under this strategy, both the control measuresu1andu2 are used to optimize the objective functional J. The results of the applications of the strategy are shown in Figure 6. It can be seen from Figure 6 that the exposed and corrupted populations decrease as compared to no control. Therefore, the intervention strategy is effective in bringing down the exposed and corrupted population in the specified period of time.Figure 6
Simulations of the corruption model showing the effect of the optimal strategiesu1≠0 and u2≠0.


## 5.1. Effect ofα and δ on the Corrupted Population

As we see in Figure2(a), we experimented the effect of changing α on the number of corrupted population by keeping the other parameters constant. It can be seen from Figure 2(a) that, as the α value increases, the number of corrupted population increases. Similarly, in Figure 2(b), we investigated the effect of changing δ on the number of corrupted population by keeping the other parameters constant. It is evident from Figure 2(b) that, when the value of δ increases, the number of corrupted population also increases. Therefore, public policy makers must concentrate on minimizing the contact rate of corrupted individuals to susceptible population and reduce the transition of exposed individuals to the corrupted population.Figure 2
Effect ofα and δ on the corrupted population.
(a)(b)


## 5.2. Effect ofκ and σ on the Corrupted Population

In Figure3(a), we can see that κ plays an important role in reducing the number of corrupted population. When the rate at which individuals join the honest subpopulation from susceptible class increases from 0.03 to 0.4, the number of corrupted population decreases. From Figure 3(b), the rate at which corrupted individuals join the recovered population through education and/or jail sentence increases from 0.007 to 0.1; the number of corrupted population decreases. Therefore, public policy makers must concentrate on maximizing the number of populations that join the honest subpopulation and corrupted population to recover from doing it.Figure 3
Effect ofκ and σ on the corrupted population.
(a)(b)To simulate the optimality system in (50), iterative technique is applied which gives the optimal solution. Forward fourth-order Runge-Kutta method is used to solve the state system, and backward fourth-order Runge-Kutta method is used to solve the adjoint system by considering the initial condition of the state system and the final condition of the adjoint systems. The controls continue to be updated by combining from the previous result of the controls with the characterization. The solution of the state and adjoint system is repeated by the updated controls. This condition continues repeatedly until consecutive iteration is close enough each other [23]. We proposed the following three strategies for numerical simulation of the corruption model:
(i)
Strategy A: campaigning through media and advertisement about corruptionu1≠0andu2≠0(ii)
Strategy B: exposing the corrupted individuals to jail and giving punishmentu1≠0andu2≠0(iii)
Strategy C: combination of both strategies A and Bu1≠0andu2≠0


## 5.3. Strategy A: Campaigning through Media and Advertisement about Corruption

In this strategy, controlsu1 were used to optimize the objective function J while we set control u2 to zero. In Figure 4, we see that exposed and corrupted population significantly reduced when there is control compared to situation with no control. From this, we conclude that applying these strategy can reduce corruption along with time from the community but not possible to avoid it in a specified period of time. Hence, it is not an effective strategy to combat corruption from the population.Figure 4
Simulations of the corruption model showing the effect of the optimal strategiesu1≠0.


## 5.4. Strategy B: Exposing the Corrupted Individuals to Jail and Giving Punishment

Under this strategy, we optimize the objective functionJ using the control u2 while u1 is set to zero. It is evident from Figure 5 that the number of exposed individuals cannot reduce using the strategy and the corrupted population seems eliminated in the first year but grows in number due to the increase in number of exposed individuals. Therefore, the strategy is not effective in fighting and reducing corruption in the population even though it reduces the number of corrupted individuals when compared with control from without control.Figure 5
Simulations of the corruption model showing the effect of the optimal strategiesu2≠0.


## 5.5. Strategy C: Combination of Both Strategies A and B

Under this strategy, both the control measuresu1andu2 are used to optimize the objective functional J. The results of the applications of the strategy are shown in Figure 6. It can be seen from Figure 6 that the exposed and corrupted populations decrease as compared to no control. Therefore, the intervention strategy is effective in bringing down the exposed and corrupted population in the specified period of time.Figure 6
Simulations of the corruption model showing the effect of the optimal strategiesu1≠0 and u2≠0.


## 6. Conclusions

In this paper, a mathematical model for the transmission dynamics of corruption in a population was formulated. The basic reproduction numberR0 was computed, and the stability of equilibrium points was investigated. Through Lyapunov’s theory, the corruption-free equilibrium point is globally asymptotically stable whenever R0<1 was proven. When R0=1, bifurcation analysis of the model was proven and it exhibits forward bifurcation. The local stability of the unique endemic equilibrium whenever R0>1 was demonstrated. Using the definition of normalized forward sensitivity, the sensitivity parameters were determined. It can be inferred that positive indices should be decreased while increasing negative indices to minimize/control corruption in a population. Then, the optimal control model was formulated by adding two time-dependent controls (campaigning about corruption through media and advertisement and exposing the corrupted individuals to jail and giving punishment). Optimal control theory was used to establish conditions under which the spread of corruption can be stopped and to examine the impact of a possible combination of these two controls on the transmission of corruption. The characterization of the optimal control was obtained by the application of the Pontryagin’s maximum principle. It is evident from the results of the numerical simulations that the integrated control strategy should be taken to fight against corruption. The cost-effectiveness analysis of the integrated strategy would be considered in a future study.



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