MATHEMATICAL MODELING OF CORRUPTION DYNAMICS: EXAMINING THE REINTEGRATION OF FORMERLY CORRUPT INDIVIDUALS

Corruption is a global menace that undermines the foundations of societies, including the rule of law, fairness, human rights, democracy, and economic growth. This research aims to comprehensively understand the dynamics of corruption and explore strategies for its prevention and control. Specifically, it focuses on the re-integration of individuals who have recovered from corrupt practices back into the population. By evaluating the effectiveness of rehabilitation programs, the potential for relapse, and the influence of societal support systems, the study seeks to determine whether the re-integration of formerly corrupt individuals contributes to a reduction in corruption or reintroduces corrupt practices. The research employs mathematical models and analyses to investigate the transmission of corruption within the population. It examines the stability properties of uncontrolled corruption models and explores the effectiveness of different combinations of corruption prevention measures. By studying these factors, the research aims to gain insights into the underlying dynamics of corruption and identify strategies that can effectively mitigate its prevalence. The findings of this research will contribute to a deeper understanding of corruption dynamics and provide valuable insights for designing intervention programs. By informing policies and strategies, this research aims to combat corruption and foster a society that upholds integrity and ethical practices. The goal is to create a framework that supports the eradication of corruption and the promotion of transparency and accountability in all aspects of society.


INTRODUCTION
Corruption entails the exploitation of authority or power by individuals occupying public or private offices for personal gain. It takes various forms, including public corruption, private corruption, pervasive corruption, and arbitrary corruption, as described in the literature (Binuyo and Akinsola, 2020). Corruption can originate from either the supply side or the demand side (Alemneh, 2020). This pervasive problem poses serious threats to the rule of law, democracy, human rights, fairness, social justice, economic growth, and the proper functioning of market economies (Legesse and Shiferaw, 2018;Nathan and Jackob, 2019). Although many nations have implemented anti-corruption regulations and strategies, corruption continues to plague society, particularly in emerging countries (Lemecha, 2018;Binuyo and Akinsola, 2020). To understand the dynamics of corruption transmission and develop effective intervention programs, mathematical models are used, but there is a lack of quantitative studies on corruption in current literature. One study proposed a deterministic model that analyzed corruption within a population, identifying the endemic equilibrium point, corruption-free equilibrium point, and basic reproduction number (BRN) (Egudam et al., 2017). Numerical simulations demonstrated that while corruption cannot be eliminated, it can be reduced to a tolerable level. Another mathematical model examined corruption dynamics, calculating the fundamental reproduction number, as well as corruption-free and endemic equilibrium points (Legesse and Shiferaw, 2018). In a study by (Danford et al., 2020), a mathematical model was constructed to examine corruption, considering the influence of anti-corruption campaigns and in-prison counseling in raising awareness about the issue. The existence of distinct corruption-free and endemic equilibrium points was explored, and the fundamental reproduction number was calculated. (Nathan and K. O. Jackob, 2020) created differential equation-based models to represent the growth or decay of corruption. Additionally, (Castillo Chavez and Song, 2004) investigated the use of game theory to combat corruption. (Binuyo and Akinsola, 2020;Li et al., 2016) proposed a corruption prevention model that demonstrated how corruption can be completely prevented if the rate of dismissal is equal to the rate of corruption, focusing on individuals exploiting positions of authority. The aim of this research is to examine the effect of recycling individuals who have recovered from corrupt practices back into the population and to assess its impact on the overall levels of corruption. It seeks to understand whether the reintegration of formerly corrupt individuals contributes to the reduction of corruption or if it poses a risk of reintroducing corrupt practices within society. By investigating factors such as the success rate of rehabilitation programs, the likelihood of relapse into corrupt behaviors, and the influence of societal attitudes and support systems, this research aims to provide insights into the dynamics of reintegrating recovered individuals and to inform strategies for effectively managing corruption in the population. The essay is structured as follows: Section 2 provides a description of the model and its underlying assumptions. Mathematical analysis, including bifurcation and model sensitivity, is performed in Section 3. Section 4 discusses the implications of recycling corrupt individuals. Numerical simulations are presented in Section 5, while Section 6 offers a discussion and conclusion.
populations: susceptible (S), exposed (E), corrupt (C), prison (P), and recovered (R). Throughout the study, the total population (N) is assumed to remain constant, with a natural death rate (µ). The recruitment rate is denoted as (Λ), while the contact rate is represented by (β). The rate at which exposed individuals develop clinical symptoms is ξ, γ is the rate of recovery. It is assumed in this paper that corrupt individual when caught will be imprison at the rate (ζ) and the rate at which the prisoner gets out of prison to join the recovered class is (θ).It is possible that recovered individuals acquire certain level of immunological memory for a certain duration (Nathan et al., 2021;Rwat and Atinah, 2023) and we assume that recovered individuals are losing immunological memory at a rate of (1 − ) where 0 < < 1 .The flowchart of this SECPRE model is shown in Figure 1. The set of ordinary differential equations that describe the system is as follows:

Corruption Free Equilibrium Point
The corruption-free equilibrium (CFE) state, 0 , of the system model (1) is obtained by setting the right-hand side of the equation equal to zero and let E = C =P = R = 0, we solve for S and we get: Basic Reproduction Number ℜ To calculate the basic reproduction number, we applied the next generation matrix method (Rwat and Atinah, 2023) to the system, utilizing the matrix F to represent the newly corrupt terms and the matrix H to represent the transition terms. Therefore, we have the following expression: Where, 4 ℱ 2 = 0 ℱ 2 = 0 and ℋ 1 = ( + ) ℋ 2 = − + ( + + ) ℋ 3 = − + ( + ) Therefore, our next generation matrices are

MATHEMATICAL MODELING OF CORR…
The inverse of ℋ is With solutions 1 = 2 = 0 and 3 = ( + )( + + ) Hence the basic reproduction number is

Sensitivity Analysis of the Parameters of the Basic Reproduction Number
Considering the uncertainty associated with certain parameter values, conducting a sensitivity analysis becomes valuable to assess the impact of these parameters on the basic reproduction number. This analysis allows us to identify which control measure, among the constant controls, results in the most significant decrease in the reproduction number and thus proves to be the most effective in curbing the progression of corruption. To achieve this, we calculate the normalized forward sensitivity index of the reproduction number concerning these variables. This index evaluates the correlation between the relative change of a variable and corresponding parameter variations. Following the methodology proposed in (Emmanuel et al., 2023;Rwat and Atinah 2023) the forward normalized sensitivity index can be calculated using the following formula: With respect to the fundamental reproduction number, we compute the sensitivity indices for each parameter, and the results are displayed in table 2 below:

Interpretation of the Sensitivity Indices of the Parameters
If the values of the parameters in table 2 above with positive sensitivity indices are increased while the values of the other parameters are held constant, the spread of the disease is significantly impacted, especially by parameter β, π and α. This is so because as their levels rise, the average number of secondary infections also rises. Their values must be reduced in order to eradicate the diseases. The sensitivity analysis shows that even though the corruption-induced death rate d and the natural mortality rate μ have negative sensitivity indices, raising them to control the disease is not physiologically logical. Another potentially sensitive parameter with a negative sensitivity index that must be enhanced for the disease to be effectively controlled is the recovery rate of corruption. A bar chart depicting the parameter sensitivity indices is shown in Figure 2 below. In order to eradicate infections, parameters with their bars pointing upward must be decreased and those with their bars pointing downward must be increased. Notwithstanding the fact that the bars on μ and γ are pointing downward, it is not physiologically tenable to claim that we should increase the rates of natural death and corruptionrelated mortality in order to eliminate the disease because our goal in doing so is to save life, not to take it away.

Global Stability Analysis of the Corruption-Free Equilibrium Point
We shall investigate the global stability of the system using the techniques implemented by Castillo-Chavez and Song. Where represent the in-corrupted population, that is = { , } while represent infected population, that is Z = { , , }. The corruption free equilibrium point of the model is denoted by U = (X * ,0). The point U= (X * ,0) is globally asymptotically stable equilibrium for the model provided that ℜ < 1, which is locally asymptotically stable, and the following conditions must be met: (ℍ 1 ): For = F(X,0), X * is globally asymptotically stable.
(ℍ 2 ):G(X, Z) = AZ − G(X, Z), G(X, Z) ≥ 0 for (X, Z) ∈ Ω If model (1) meets the given two criteria, then the following theorem holds Theorem 4: The corruption-free equilibrium point U=(X * ,0) is globally asymptotically stable provided < 1 and conditions (ℍ 1 ) and (ℍ 2 ) are satisfied. Proof: from the system we can get F(X,Z) and G(X, Z): From the above system we see that X * =( Λ , 0) is globally asymptotic point. This can be verified from the solutions, namely Which implies the global convergence of (2) in Ω and this satisfies condition ℍ 1 . Now from ℍ 2 we have that G(X, Z) = AZ − G(X, Z), G(X, Z) ≥ 0 for (X, Z) ∈ Ω Therefore, G(X, Z) = AZ − G(X, Z).
Where A is an nxn matrix, Z is a column vector and G(X, Z) is a column vector formed from the corruption equations. The first partial derivative of G(X, Z) with respect to , computed at the corruption free equilibrium point gives matrix A. From G(X, Z),we can see that ̃1 (X, Z) ≤ 0,̃2(X, Z) = 0,which leads to G(X, Z) ≤ 0. That means the second condition (ℍ 2 ) is not satisfied. So, the system of equation may not be globally asymptotically stable when ℜ cor < 1

Determination of Backward or Forward Bifurcation
To explore the possibility of backward or forward bifurcation of the model system (1) we use the center manifold theory. The backward or forward bifurcation phenomena will help us determine the local stability of the system at the endemic equilibrium point. Now from the basic reproduction number
Substitute the values of ∂ 2 g 2 ∂ 3 ∂β , 2 and 3 into the expression for ℬ, we got

MATHEMATICAL MODELING OF CORR…
That is, the value of becomes negative while the value of ℬ is positive. This satisfies the condition of center manifold theorem for forward bifurcation. ℬ = ξβΛ μ(μ + ξ) 3 3 > 0 Also when we differentiate the bifurcation coefficient with respect to , we get Let (1 − )β = Then = −1 = − β(2ξ + μ)( (μ + θ) + ) (μ + ξ)(μ + θ) 3 3 2 Therefore, This shows that the bifurcation coefficient is an increasing function of the re-cycling coefficient, hence increasing the recycling coefficient will increase the bifurcation coefficient. Hence the recycling coefficient need to be decreased to decrease the bifurcation coefficient to produce a forward bifurcation. This is shown in figure (4b)  Sabastine et al., 2022)

DISCUSSION
In this studies we presented a mathematical models for corruption dynamics. The model considered re-cycling of the recovered individuals after some times when they have repented from the act of corruption. The mathematical analysis of the model suggests two different types of bifurcation phenomenon. The model exhibits a backward bifurcation at various values of re-cycling parameter while its exhibits global forward bifurcation when the recycling parameter is removed. Using parameter values in articles associated with the corruption situation, we showed how reinfection can trigger backward bifurcation in the model. A backward bifurcation will make it a lot difficult to control corruption because a critical value of the reproduction number which is much less than 1 is needed to achieve corruption-free state.
The basic reproduction number R cor for the model often defined as the ratio of contact rate to the recovery rate is one of the main indicators used to measure transmissibility of an epidemic in epidemiology. This is due to the believe that the basic reproduction number is necessary and sufficient to determine the fate of an epidemic. However, re-cycling of corruption individuals creates the possibility of a backward bifurcation as shown in figure 3 in this work. Therefore, controlling the basic reproduction number (i.e., controlling contact rate and recovery rate) alone will not be sufficient to control the transmission of corruption. We should therefore control the possibility of re-cycling people that have recovered from corrupt practices into the population in order to eradicate the corrupt individual from the society.

CONCLUSION
In figure 4a we start with the value of the scaling factor (δ) from one and keep on decreasing the values till we reach 0 (1.0, 0.9, 0.6, 0.3, 0.0). It is observed that the population of the corrupt population keep on increasing. Please note that at δ = 0, the re-cycling rate is equal to the exposure rate to corruption, but as the value of δ increases from zero to one, the re-cycling rate decreases. As the value of δ becomes 1, the re-cycling rate becomes zero and therefore no re-cycling occurs. In figure 4b, as the value of δ increases from δ = 0.0 to 0.6, the model exhibits backward bifurcation. At δ = 1 the bifurcation becomes forward.