GLOBAL SUSTAINABILITY Mathematical Modeling and Optimal Control of Intervention Strategies for A Banditry Model

This paper presents an optimal control of intervention strategies for the menace of Banditry taking into account media campaign against Banditry 2 ( ) , u t rehabilitation of Bandits detainees 5 ( ) , u t and use of military force against Banditry 6 ( ) , u t as control strategies. The Banditry free equilibrium, Banditry present equilibrium and the basic reproduction number of Banditry 0 B R were obtained. The stability analysis results suggest that the Banditry free equilibrium is locally asymptotically stable when 0 1 B R  and otherwise when 0 1 B R  . The Boko Haram presence equilibrium is globally asymptotically stable when 0 1 B R  and unstable if 0 1 B R  . We used the three control strategies and updated the Banditry menace model. The optimal control issue was resolved using Pontryagin's Maximum Principle (PMP).It was discovered that there is a significant decrease in the population of Bandits and increase in the number of rehabilitated Bandits and detained Bandits when the control measures are implemented compared to the case without control. We solved the optimality control using a forward-backward sweep strategy implemented in MATLAB for numerical simulation. Additionally, we saw that the number of people detained fluctuates as the number of people receiving rehabilitation rises. We argue that in order to lessen or completely erase the menace caused by Bandits in society, the government should fund media campaigns and rehabilitation initiatives.

R were obtained. The stability analysis results suggest that the Banditry free equilibrium is locally asymptotically stable when 0 1 B R  and otherwise when 0 1 B R  . The Boko Haram presence equilibrium is globally asymptotically stable when 0 1 B R  and unstable if 0 1 B R  . We used the three control strategies and updated INTRODUCTION 1.1 Introduction of the Problem According to Walker, (2012), a bandit is a robber or armed thief that engages in banditry by attacking others while traveling in quest of money and occasionally other valuable objects. Banditry is an organized crime committed by outlaws utilizing threats or violence. In the northwest of Nigeria, armed banditry has become a significant security issue, terrifying local communities, killing and displacing thousands, impeding socioeconomic development, and endangering democratic governance.
We found it worthy to carry out this research in order to search for the means to curtail the activities of banditry which happens to be rampant in our societies. Some control strategies were applied in controlling banditry with the intention to trace the most effective ones to so that they can be adopted. It was done through seeking the opinions of some other scholars on banditry.
Bandits target rural residents, often ambushing drivers and killing those resisting kidnapping or refusing ransom. Banditry is a violent crime involving armed robberies and threats. In northern Nigerian states, over 21 million people live in areas with banditry crimes. The discovery of gold mines and illegal miners' activities have worsened armed ISSN: 2488-9229 FEDERAL UNIVERSITY GUSAU-NIGERIA https://doi.org/ 10.57233/ijsgs.v9i2.467 https://fugus-ijsgs.com.ng bandits' presence in the northwest (Bandits, 2021& Onuoha, 2014. There are six main causes cited for why people (youths) join the deadly gang (Banditry): unemployment and poverty, challenging child rearing, high levels of illiteracy, excessive military responses, and widespread corruption. The reasons why young people join terrorism (Banditry) include peer pressure, family and ethnic pressure, marriage, dissatisfaction with the government, a desire for status and prestige, a lack of education, a desire to belong to a group, a desire to make money, a desire to be respected, and other factors (Nadama, 2019) and (Onuoha, 2014).
This essay presents a mathematical model of a socioeconomically stratified society prone to crime. It examines the impact of a non-zero crime rate on a society free of criminals. The model defines a criminal-prone society as unstable under minor changes in socioeconomic conditions. The essay compares two strategies to control crime: enhancing police efficiency or reducing criminal appeal or promoting social classes at risk. (Nu˜no et al, 2011) and (Gambo & Ibrahim, 2020).
A mathematical model examines the dynamics of drug misuse and banditry in the population, determining stable states and assessing reproduction rates. The characteristic equation shows a locally asymptotically stable equilibrium at basic reproduction numbers less than unity, while the center manifold theory of bifurcation reveals a locally asymptotically stable equilibrium at higher reproduction numbers. Bifurcation analysis at the threshold, 0 1 R = is investigated.
Furthermore, it is demonstrated that the banditry and drug abuse model's local asymptotic dynamics display backward bifurcation. Additionally, a sensitivity analysis is carried out to look at how the model settings affect the basic reproduction number of the drug misuse and banditry threat and how that affects the propagation of the threat (Odinkala 2018) and (Olaniyi & Obabiyi, 2013).
Armed banditry and kidnapping have increased in Nigeria's North-West states, causing significant negative consequences on people's lives and property. A study in Katsina state found that armed banditry negatively impacts poverty, unemployment, food security, education, health, income, and overall standard of living. The study suggests tight border security patrols, pro-active measures, universal basic education, skills acquisition centers, and establishing Fulani settlements in under-govern forests. It also calls for adequate farming inputs, food security, and a focus on reviving agricultural production (Muhammad & Bashir, 2022).

Importance to the Problem
It was anticipated that the expansion of the model by Gambo and Ibrahim (2020), would help to reduce the number of people recruited into banditry, hence decreasing the damage they inflicted on people and property. It will help the government to choose the best control strategy to end the threat of banditry in the societies.
In accordance with their area of concern, nongovernmental organizations (NGOs) may also utilise the research's findings. They can apply the approach from this research to focus on Bandits.
Multinational enterprises are required in order to cooperatively combat an insurgency across multiple nations. The findings of this study will also help academic institutions decide what control strategies to implement for the curtailing the banditry, as well as their lethal operations, to a minimum.

Literature Review
A bandit organization treats the residents of the North Western region similarly. On bicycles equipped with guns, they also attack innocent people and law enforcement officers. Using whatever kind of transportation at their disposal, they attack and cause mayhem. Nigerians and even the international community have grave worries regarding the cycle of violence committed by the Islamist group Boko Haram and Bandits against Nigerians (Ademola, 2021) and (Okpaga et al, 2012).
Bandits are also made up of angry young adults, unemployed high school and university graduates, https://doi.org/ 10.57233/ijsgs.v9i2.467 https://fugus-ijsgs.com.ng and orphans, mostly but not solely from northern Nigeria. They also include some wealthy, intelligent, and well-known individuals. Bank robberies, ransom from kidnappings, stop and searches on roadways, cattle rustling, etc. are the main ways that bandits get money. They also demand money from locals in the regions they control and from those they have intimidated into staying away from them (Nadama, 2019) and (Onuoha, 2014).
The study analyzes global health and social issues of illegal narcotics and banditry, focusing on their stability and dynamics. It shows a supercritical bifurcation property and global asymptotically stable equilibria with and without banditry due to the creation of appropriate Lyapunov function. Sensitivity analysis and numerical simulations confirm the findings. Control measures are needed to decrease illegal narcotic use and reduce banditry burden. Akanni & Abidemi, 2023  Using the above assumptions in mind, the model diagram in Figure 1 was constructed and the system of differential equations in 1 were obtained: (1 ) ( ) The population is increased by the number of Bandits who willingly surrendered and moved to the Bandits rehabilitation centre at the rate (1 )  − . The population here reduced due to progression of rehabilitated individuals to the susceptible population at the rate 2 r . The population also reduced due to natural death at the rate .
 Thus
The population of military forces is increased by government reinforcement of military personnel and other defensive weapons at the rate 2  . The population here is reduced due to natural death  and killing of military men by bandits at the rate . ) .
The variables, parameters and descriptions were given in table 1 and 3.

Basic Properties of the Model
In this section, we prove the basic properties of the Banditry model 1.

Theorem 1. Let the initial solution set be
(10) Solving equation (9) by the method of separating the variables, integrating the result and taking the exponential, we have For all 0 t  .
Similar technique can be applied to formulate that   ( ), ( ), ( ), ( )  is positively invariant and attracting with respect to the model (1) then, Substituting model (1) into (17) and simplifying, we get Using the comparison theorem on equation (18) Using separation of variables technique on (20), integrating both sides of the result and multiply both sides of the result by () Taking the exponent of both sides of (22), we have where C ke  − = is taken to be constant in (23). By applying the initial condition Solving for N in equation (25), we have ( ) Thus, the feasible solution set of the system of the model (1) is positively invariant in the region ,  hence it is mathematically well-posed. Therefore, it is adequate to study the dynamic values of the model in the region .
 .  (1) and verifies the type of stability that model (1) displays

Equilibrium points
Setting system (1) to zero yields the equilibrium points derived by model (1) as

Basic reproduction number of banditry model
We derived the basic reproduction number, From equation (38) Furthermore, we also observed from equation (39) Applying Routh-Hurwitz criterion implies the roots of polynomial (39) Determining the derivative of along the solution of the system (41) directly, Replacing the derivatives of the variables with respect to , we have ( ) Now factorizing equation (45) Since the geometric mean is less than the arithmetic mean (Maia, 2015), the following inequality from equation (46) (Korobeinikov & Wake, 2002); (Pykh, 2001) and (Shuai &Van Den , 2013). This stated that if a bandit were to happen to be in a potential population, he could recruit additional individuals if 0 1 B R  , to join him in committing crimes. As a result, the threat persists as practically everyone eventually engages in banditry (Onuoha, 2014).

Sensitivity Analysis
We used the parameter values on table 4.2 to calculate this sensitivity index of Banditry submodel. The sensitivity indices of the basic reproduction number are calculated in order to determine how important each parameter is in the initiation of Banditry: that is, in the control of Banditry, the parameters that have the greatest influence on the basic reproduction number, as well as intervention strategies that target such parameter, should be used.
We adopted the local sensitivity analysis based on the normalized forward sensitivity index 0 .
The remaining sensitivity indices were calculated in a similar way. The results are given in table 3.

Optimal Control Analysis
The optimal control interventions for effective control of the Banditry menace is presented in this section. From the result of the sensitivity analysis carried out, we observed that some parameters in (1 ) .
The steps in Pontryagin's (1962)   The necessary conditions that an optimal control must satisfy were derived from PMP (Pontryagin et al, 1962). and the existence result for optimal control from the adjoint variable of the state variables satisfy the following set of differential equations. This principle converts system (51) where are adjoint variables and Theorem 5. Let and be optimal controls and , the solutions of the optimal control problems of (49) and (51)  where for are the adjoint variables and the controls and obey the optimal conditions, such that 2,5,6 1 2 2 5 3 6 0 Proof. To obtain the differential equations that governs the adjoint variable, we assessed the differential Hamiltonian functions at the optimal control. Hence (1 ) ( ) with transversality conditions: Also, the optimal functions * * * 2 5 6 ,, u u u satisfy 0, 2,5, 6.

DISCUSSION
The impact of interventions on the proliferation of banditry within a population is discussed in this section. To quantitatively solve the optimal control problem (59) and the adjoint variable (51), we employed the approach used by control (Momoh etal (2021) and (Suzanne & John, 2007).
These methods employ the Runge-Kutta forward and backward technique. In order to lower or eradicate the number of bandits, we use the variables and parameter values in Tables 2 and 4. One hundred (100)    Bandits are at top level, but that of rehabilitated Bandits, just below average number. When the method is put into practice (indicated by red line) compared to the scenario without control, we see a drop in the population of Bandits and detained Bandits but the population of rehabilitated Bandits, remained unchanged as shown in Figure 5.5(a,b,c) respectively. The control profile for strategy A shown in Figure 5.5 (d), The control curve for the media campaign against banditry remains at the lower bound for the entire duration of the study period while the curve for the rehabilitation of Bandits remains at the upper bound for the targeted days (100 days) before descending to the lower bound. that of rehabilitated Bandits is very low, respectively. It was observed from Figure 5.6 (c) that there is decline in the population of detained Bandits and sharp increase in the population of rehabilitated bandits, but after 10 days, the rehabilitated bandits declined as the days increase when there is control as indicated in the red line as compared to if there is no control. The increase is probably due to movement of rehabilitated Bandits to susceptible compartment. Figure 5.6 (d) shows the control profile for strategy B where we observed that the control curve for the use of military force raised sharply to 60% implementation and after one day, drop back to 48% . After a day again, it raised sharply to 100% implementation and stay at the upper bound for the targeted days before declining to the lower bound.

Strategy C: Rehabilitation of Bandits and use of Military Forces
The optimal control system (49) for the implementation of the rehabilitation of Bandits and the deployment of Military force is presented in intervention plan C, with the media campaign against banditry is set at zero. In contrast to the situation where there is no control, as indicated by blue line, it can be seen from Figure  5.7(a) that the population of bandits is high, In Figure 5.7(b), the population of detained bandits is minimal while in Figure 5.7(c), there is very scanty rehabilitated Bandits . When we employed the strategies using red lines, the population of bandits drops sharply as shown in Fig. 5.3(a), but the population of detained bandits and the rehabilitated Bandits increases as depicted in Figure.5.7 (b,c), respectively. Figure  5.7 (d) displays the control profile for strategy C. In our research, we discovered that the control curve for the rehabilitation of Bandits falls from 45% to 0% implementation and remains at the lower bound throughout the entire study time. After five days, the implementation curve for using of military force increased from 0% to 100% The execution of a media campaign against Bandits, the rehabilitation of bandits, and the employment of military force are presented in intervention strategy D's simulation of the optimal control system (49). When the strategy is put into practice, we see a sharp increase in the population of rehabilitated bandits and a sharp decrease in the population of detained bandits in Figure 5.8. However, after 10 days, the rehabilitated bandits start to decline as the days go on as opposed to the case where there is no control. Figure 5.8 (d) displays the control profile for strategy D. We discovered that the media campaign against bandits' control curve (red curve) began at 50% and increased to 100% implementation. After staying there for 60 days, it returns to the lower bound. The curve for the rehabilitation of bandits (blue) drops from a 45% implementation to a 0% implementation and remains at the lower bound for the entire study period. The curve for the application of military force (green colour) increased from 1% to 100%, and after 5 days, it remained at the upper bound for 99 days before descending to the lower bound.

CONCLUSION
The research work proposed and examined a mathematical model and the optimal control for preventing the spread of Banditry menace activities in our societies, including media campaigns, rehabilitation programs, detention, and the use of military force against Banditry. By combining the previously indicated control methods, the study enhanced the work of [9] and (Shuai, & Van Den, 2013). The positivity and uniqueness of the model equations' solutions were established, and it was found that they do indeed exist and are distinct. =   +   By rigorously examining these, we get to the conclusion that the susceptible population and military forces alter accordingly with their recruitment rate to their mortality rate when there is no Banditry. The basic reproduction number of the Banditry model was also analyzed and obtained to be In order to examine and analyze the ideal level for regulating the widespread media campaign against banditry, the rehabilitation of captured bandits, and the employment of military forces as control variables, we utilized the optimal control approach employing Pontryagins' Maximum Principle. The findings indicate that combining all three tactics is the most effective way to stop the banditry threat.
6.0 ACKNOWLEDGEMENTS I am immensely grateful to Almighty God for giving me the ability, strength, and wisdom I needed to finish this book. I want to start by expressing my gratitude to my mentor, Dr. A. A. Momoh, for all of the assistance, guidance, and sage advice he provided me with when I was writing my paper. Throughout this time, he has been quite supportive, spending a lot of time teaching me and introducing me to new ideas. I also want to express my gratitude to my co-authors, Dr. A. M. Alkali, and Mr. Ali Michael Inalegwu for their prompt and valuable assistance in making this research a success. Additionally, I want to express my sincere gratitude to my beloved wife, Mrs. Usaku Jacob Solomon, as well as to my friends Mr. Maxwell Mika'ilu, Sylvester Shehu, and my cousin, Engr. Biyama Joseph Offah, for their financial and emotional support in helping me make this publication a success