1 Introduction

1.1 Global crime

In 2011, the United Nations Office on Drugs and Crime (UNODC) estimated the total number of annual deaths by homicides to be 468,000 globally. Over a third (36%) of this occurrence happened on the African continent. Similarly in 2011, gender-based violence affect most women worldwide and poses a severe threat to mankind, particularly the peaceful coexistence human beings in the society [1]. According to the UNODC, the global male homicide rate is put at 9.7% as against the female homicide rate 2.7% in 2013, with the highest in the Americas (29.3 per 100,000 males). The increase in the degrees of homicide are linked to organized crime and gangs as coordinated wrongdoing in the Americas is seen to be more than other regions whose homicide rates are put together at 4.5 per 100,000 males. It is troubling to report that 43% of all homicide victims globally are young males aged 15–29 [2]. According to the UNODC homicide report in 2019, the Americas continue to report high homicide rates among young men with a homicide rate for men aged 18–19 estimated at 46 per 100,000. By contrast, Europe has seen a decrease in the homicide rate by 63% and 38% since 2002 and 1990 respectively. Similarly, there is a huge decline rate (36%) in Asia since 1990 [3]. In 2021, it was reported that Latin America witnessed a relatively high unrest in 2019 while the novel coronavirus (COVID-19) ripped through the region in 2020, upending everything from commercial trade to the operations of local gangs and transnational criminal organizations. It is, however, important to note that the pandemic may have impacted levels of violence in some of the countries in the region [4]. Consequently, it is therefore important to take the issues of crime seriously if we are prepared to make the world a better place.

1.2 Crime in Africa

African nations suffer from poorly-resourced criminal justice systems by having the world’s least favorable police–and judge-to-population ratios. This eventually affects prosecution and sentencing rates; regardless of whether the security agencies perform optimally, criminals are considerably less liable to be penalized than offenders in other parts of the world. Africa has consistently been at the forefront of global statistics on crime: out of 437,000 deaths caused by intentional homicide globally in 2012, 31% happened in Africa, consequently, suggesting that Africa has a high homicide rate among the countries of the world [5]. Conflicts in recent times focus on mentally destroying the people through extreme cruelty and brutality as the young ones are often at the receiving end. Among the Angolan children interviewed by the UNODC in 2015, two-thirds of these children had witnessed people murdered in cold blood. Similarly, 56% of Rwanda’s children during the genocide had witnessed other children murder people, while 80% had lost immediate family members. While it is harsh to affirm that survivors of brutality are mechanistically destined to visit it upon others, exposure to brutality has been discovered by crime analysts to be a typical component in the upbringings of offenders. All over the world, teenagers and young adults (males) involve in criminal activities more than their female counterparts, and Africa’s young population of about 43% falls within this pool of potential offenders. Sadly, most of these young populations are either out of school or unemployed [6]. According to the South African Police report on crime statistics in 2021, contact crimes (sexual offenses, murder) and all other categories of assault registered a 60.6% increase, compared to the corresponding period of the previous year [7]. It is, however, important to note that crime statistics in Africa during the pandemic took different dimensions and irregular variation to the crime trends.

1.3 Crime in Nigeria

According to the National Bureau of Statistics (NBS), a total of 125,790 and 134,663 criminal cases in 2016 [8] and 2017 [9] respectively were reported across Nigeria. These high numbers could be attributed to the causative factors of criminal behaviors. Specifically, peer pressure, parental imitation, and neglect, hereditary or natural factor, poor education, media violence, poverty, child abuse, etc [10]. Nigeria has many children identified as out-of-school, homeless, in poverty, orphaned, or single-parent, especially those born out-of-wedlock. The current trend of the Almajiri system in the North; the Alaye or Area Boys in the South-West, particularly the Lagos-Ibadan axis; the Yandaba in Kano are all indications of the breakdown of community norms and value systems. These groups of delinquent youths have many things in common: they are mostly unemployed or not engaged; homeless, poor, and poorly educated, if at all. They are mostly found in slums, on the streets, and under the bridges of cities, and tend to be a complete nuisance [10]. Research has shown that these poorly managed youths were susceptible to contracting the COVID-19 [11, 12].

Furthermore, popular gangs whose members are usually teens and young adults are beginning to emerge. One of such is the yahoo boys gang which has spread across Nigeria. These groups are usually independent but engage in diverse criminal activities which includes internet scam, hacking bank accounts, money ritual amongst others. Unidentified criminal gangs periodically perpetrate other criminal evils which include but are not limited to robbery, rape, kidnapping, cattle rustling, banditry, militancy, and other criminal activities that may result in the loss of lives of residents’ loved ones and breadwinners [13]. Another contributing factor to Nigeria crime rate is the unemployment among the youth population. It is estimated to have grown from 12.6% to 21.5% between 2012 and 2015 as it is believed to have contributed significantly to the increase in social vices and insecurity. One more important reason is the lack of political will of previous administration to address the problems of poverty, unemployment, and inequitable distribution of wealth among the country [14]. This is observed to have compounded the problem and make it more complicated for new administration to manage.

The aforementioned factors must have triggered the recent report by the Nigeria Police Force–a total number of 1210 stolen vehicles and 1075 rape cases were recorded in 2017. In 2020, the National Bureau of Statistics (NBS) reported a total number 1890 and 1173 of trafficked persons in 2017 and 2018 respectively with a minimal decrease in 2019 which is stood at 1152 [15]. Nigeria needs an holistic approach which encompasses scientific and non-scientific measures in order to end this rising insecurity as this research will explore the former in addressing the criminal issues in one of the most populous countries of the world.

1.4 Paper organization

This paper is arranged in the following ways to address the sociological problem discussed earlier from the mathematical perspective. Section 2 presents the formulation of the optimal control model. The stability analysis and the numerical simulation of the optimal control model are considered in Sects. 3 and 4 receptively. The cost-effective analysis is also presented in Sect. 5 while the discussion and conclusion are in Sects. 6 and 7 respectively.

2 Optimal control model

2.1 Model formulation

Several mathematical techniques have been employed to study the dynamics of epidemic models from optimal control perspective, see [16,17,18,19]. More recently, mathematical modeling have been used to study the dynamics of criminal gangs, see, [13, 20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]. As it can rarely be found in the literature, our interest is to investigate the impact of optimal control strategy on the dynamics of the age-structured criminal gangs, in a limited-resource setting. This idea is informed by the previous work in [37]. We are set to answer this open research question in this paper by properly modifying the deterministic mathematical models for the dynamics of the criminal gang as studied in [13]. Consider the age-structured model in [13], while drawing inspiration from the sociological and criminological works in [37,38,39,40], we introduce two time dependent control functions, \(u_1(t)\) and \(u_2(t)\) to all the age-structured classes.

The time-dependent control function: \(u_{1}(t)\), is bounded Lebesgue integrable. The function \(u_{1}(t)\) is a control that supports enlightenment campaign programs and empowerment programs (door-to-door, community square talks, television and radio jingles, print media, youth empowerment, access to loan, and other prevention strategies of crime as listed in [41] report) by government agencies and non-governmental organization (NGO)’s during the criminally active period. The expression \(1 - u_1 (t)\) denotes the drive that supports the enlightenment campaign programs and empowerment programs during the criminally active period. The control \(u_{1}(t) \rightarrow 1\) indicates the impact intensive enlightenment campaigns and consistent empowerment programs will have on gang formation and their activities in the community. In other words, it implies the enlightenment campaign and consistent empowerment programs that prevent the susceptible individuals (\(S_2\) and \(S_3\)) from joining the gang classes (\(G_1\), \(G_2\), and \(G_3\)). The control \(u_{1}(t) \rightarrow 0\) indicates that the enlightenment campaign and empowerment program are loose and no longer effective. Hence, there will be high progression rate of susceptible individuals (\(S_2\) and \(S_3\)) to the gang classes (\(G_1\), \(G_2\) and \(G_3\)).

The time-dependent control function: \(u_{2}(t)\) is bounded Lebesgue integrable. The function \(u_{2}(t)\) is a case-finding control that represents the proportion of initiated persons who are arrested, law-enforced, and imprisoned in the correctional center for correction and the prevention of close interaction with susceptible individuals, in a criminally active population. The term \(1 + u_2 (t)\) represents the effort that sustains the police hunting at criminals and correctional policy that ’holds down’ the convicts serving a sentencing term for proper correction.

Here, we minimize the objective functional

$$\begin{aligned} G(u_1,u_2)&= \int _0 ^{t_{final}} \left( G_1 (t)+G_2 (t) + G_3 (t) - C_1 (t) \right. \nonumber \\&\quad \left. - C_2 (t) - C_3 (t)+ \frac{B_1}{2} u_1 ^{2} (t) + \frac{B_2}{2} u_2 ^{2} (t) \right) \mathrm{d}t \end{aligned}$$
(2.1)

In (2.1), we minimize the population with delinquent behavior in classes \(G_1\) and \(G_2\) through the NGOs and police hunting strategies. Also, we minimize the population with delinquent behavior in class \(G_3\) through police hunting strategies only. In the same manner, we desire to increase the effort that takes criminals from the street for appropriate corrections in the correctional centers, this is of great interest in this work as well. Thus, we desire to minimize functional (2.1) that implies a trade-off required in minimizing the population exhibiting status offenses (eg., tobacco consumption, curfew violation, having delinquent friends, alcohol consumption, underage voting, assault, breaking, and illegal use of banned drugs, etc) in class \(G_1\), and non-status offenses (murder, robbery, illegal sale of drugs, human trafficking, money laundering, kidnapping, illegal smuggling of goods, assassinations, weapon, and drug trafficking, rape among, etc) in classes \(G_2\) and \(G_3\) respectively while increasing the number of arrested, prosecuted and isolated individuals for corrections, with minimal associated cost in their respective classes \(C_1\), \(C_2\) and \(C_3\).

The constants, \( B_1\) and \(B_2\), are the positive weights for balancing cost factors in the functional (2.1). We work with the notion that the \(B_2 > B_1\) because the cost associated with control \(u_2\) involves searching for criminal gangs, picking them for arrest, and prosecuting at the court of competent jurisdiction, then sentencing them for correction may require a huge cost. Hence the government will have more funds to use in this situation and citizens selfless effort is also involved in this regard. The expenses linked with control \(u_1\) entails government and NGOs embarking on enlightenment and empowerment programs only.

The \(t_{final}\) is the final time of the intervention. We note that the set \(U = \{ (u_1, u_2)\) is measurable and \(0 \le u_1, u_2 \le 1\) for \(t \in [0, t_{final}] \}\) is the control set. Thus,

$$\begin{aligned} G(u_1^*, u_2^*) = \mathrm{min} \{G(u_1^*, u_2^*) : u_1^*, u_2^* \in U \}, \end{aligned}$$
(2.2)

where the set U can further be expressed as

$$\begin{aligned} U = \{ (u_1, u_2) \in L^1(0, t_{\mathrm{final}} ) \times L^1(0, t_{\mathrm{final}} ) | a_i \le u_i \le b_i \}. \end{aligned}$$
(2.3)

where \(a_1, a_2, b_1, b_2\) are fixed constants such that \(a_1>0, a_2>0, b_1>0, b_2>0\).

We note that the convention used in setting up the functional (2.1) is exclusively subject to the specific application and goal to be considered. This implies that different types of the functional (2.1) could be defined. Furthermore, it is well established that optimal control results are tied to the optimal control model, how the controls are used in the model and how the associated objective functional is defined [42] As a result, this will genuinely guide us on how the controls presented herein are being placed in the constraints to be formulated. The constraint equations subjected to the objective function \(G(u_1,u_2)\) is given by

$$\begin{aligned} \left. \begin{array}{lll} S_{1}'= &{} \Lambda - (\mu + \alpha _{1})S_{1} \\ S_{2}' =&{} \alpha _{1}S_{1} - (\mu + \alpha _{2})S_{2} - (1-u_1) \beta _{1} (1 + \sigma _{1})G_{1}S_{2} -(1-u_1) \beta _{2} (1 + \sigma _{2})G_{2}S_{2}+ \tau _{1}C_{1} \\ S_{3}' =&{} \alpha _{2}S_{2} -\mu S_{3} - (1-u_1) \beta _{3} (1 + \sigma _{3})G_{3}S_{3} + \tau _{2}C_{3} \\ G_{1}'=&{} (1-u_1) \beta _{1} (1 + \sigma _{1})G_{1}S_{2}+ (1-u_1) p \beta _{2} (1 + \sigma _{2})G_{2}S_{2} -((1+u_2)\gamma _{1}+\mu +\delta _{1})G_{1} \\ G_{2}'=&{} (1-u_1) (1-p)\beta _{2} (1 + \sigma _{2})G_{2}S_{2} -((1+u_2) \gamma _{2}+\alpha _{3}+\mu +\delta _{2})G_{2} \\ G_{3}'=&{} (1-u_1) \beta _{3} (1 + \sigma _{3})G_{3}S_{3} -((1+u_2)\gamma _{3}+\mu +\delta _{3})G_{3}+\alpha _{3}G_{2}\\ C_{1}'=&{} (1+u_2)\gamma _{1}G_{1} -(\mu +\delta _{4}+ \tau _{1})C_{1}\\ C_{2}'=&{} (1+u_2) \gamma _{2}G_{2} -(\mu +\delta _{5}+\alpha _{4})C_{2}\\ C_{3}'=&{} (1+u_2)\gamma _{3} G_{3} +\alpha _{4} C_{2}- (\mu +\delta _{6}+\tau _{2})C_{3}. \end{array} \right\} \end{aligned}$$
(2.4)

In system (2.4), \(S_{1}'\), represents the rate of change in susceptible individuals within the age bracket 0–7 years (they are resistant to crime) with respect to time t; \(S_{2}'\), means the rate of change in susceptible individuals within the age bracket 8–17 years (they are exposed to delinquent behaviors from their peers) with respect to time t; \(S_{3}'\), represents the rate of change in susceptible individuals within the age bracket 18–above years (they are expose to delinquent behaviors from their peers) with respect to time t; \(G_{1}'\), implies the rate of change in criminal gang population (within the age bracket 8–17 years) with status offenses (eg., tobacco consumption, alcohol consumption, assault, breaking and entering, etc [43, 44]), with respect to time t; \(G_{2}'\), represents the rate of change in criminal gang population (within the age bracket 8–17 years) that exhibit capital offenses (murder), with respect to time t; \(G_{3}'\), represents the rate of change in criminal gang population (within the age bracket 18–above years) with non-status offenses (eg., murder, robbery, fraud, illegal sale of drugs, human trafficking, money laundering, kidnapping, illegal smuggling of goods, weapon and drug trafficking, domestic violence, rape, etc) [45], with respect to time t; \(C_{1}'\), means the rate of change in the individuals under rehabilitation (individual within the age bracket 8–17 years) with respect to time t; \(C_{2}'\), implies the rate of change in the individuals in government remand homes (individual within the age bracket 8–17 years) with respect to time t; \(C_{3}'\), represents the rate of change in the prosecuted and law-enforced gang members (individual within the age bracket 18 - above years) serving jail terms with respect to time t; N is the total population at any given time t. The parameters are as well defined in the Table 1.

Table 1 Description of parameters
Full size table

Furthermore, the terms \((1-u_1) \beta _{1} (1 + \sigma _{1})G_{1}S_{2}, (1-u_1) \beta _{2} (1 + \sigma _{2})G_{2}S_{2}, (1-u_1) \beta _{3} (1 + \sigma _{3})G_{3}S_{3}\) in (2.4) represent the effort \((1-u_1)\) required to sustain the enlightenment campaign programs and empowerment programs in classes \(S_2\) and \(S_3\), respectively, during the criminally active period. Also, the terms \((1+u_2) \gamma _1 G_1, (1+u_2) \gamma _2 G_2, (1+u_2) \gamma _3 G_3\) in (2.4) represent the effort \((1+u_2)\) required to acquire the rates \(\gamma _1, \gamma _2,\gamma _3 \) at which gang members in classes \(G_1, G_2, G_3\) are being law-enforced (arrested, prosecuted and sent to the correctional center \(C_3\)) by police during the criminally active period. The bounds are \(u_1\): \(0 \le u_1 \le 1\) following in [16, 42] and \(u_2\): \(0 \le u_2 \le \frac{1}{\gamma _1+\gamma _2+\gamma _3}\) for numerical simulation as implied from [42]. We have chosen that of \(u_2\) because of the rates of enforcement and prosecution in classes \(G_1, G_2, G_3\) influences the level of control that should be enforced to get criminal gangs corrected in the correctional centers \(C_1, C_2, C_3\).

3 Analysis of the optimal control functions

It follows from the work in [46] that the necessary conditions that an optimal pair must satisfy comes from Pontryagin’s Maximum Principles (PMP). This principle converts (2.1, 2.2, 2.4), into the problem of minimizing a Hamiltonian (H), pointwisely with respect to the controls, \(u_1(t)\) and \(u_2(t)\):

$$\begin{aligned} H&= G_1 (t)+G_2 (t) + G_3 (t) - C_1 (t) - C_2 (t)- C_3 (t) \nonumber \\&\quad + \dfrac{B_1}{2}u_1^2 (t)+ \dfrac{B_2}{2}u_2^2 (t) + \sum _{i = 1} ^{9} \lambda ^*_{i} f_i, \end{aligned}$$
(3.1)

where \(f_i\), \(i = 1,\ldots ,9\) is the right hand side (RHS) of the system of DEs of the \(i-th\) state variable. Since the work of [18] guarantee the existence of optimal control model, then PMP can be used and invoke the existence result for the control pairs from the work in [18]. The function H is formed by allowing each of the adjoint variables to correspond to each of the state variables accordingly and combining the results with the objective functional leading to

$$\begin{aligned} \left. \begin{array}{lll} H =&{} G_1 (t)+G_2 (t) + G_3 (t) - C_1 (t) - C_2 (t)- C_3 (t) + \dfrac{B_1}{2}u_1^2 (t)+ \dfrac{B_2}{2}u_2^2 (t)\\ &{}+ \lambda ^{*}_{1} \left[ \Lambda - (\mu + \alpha _{1})S_{1} \right] \\ &{}+ \lambda ^{*}_2 \left[ \alpha _{1}S_{1} - (\mu + \alpha _{2})S_{2} - (1-u_1) \beta _{1} (1 + \sigma _{1})G_{1}S_{2} -(1-u_1) \beta _{2} (1 + \sigma _{2})G_{2}S_{2}+ \tau _{1}C_{1} \right] \\ &{}+ \lambda ^{*}_3 \left[ \alpha _{2}S_{2} -\mu S_{3} - (1-u_1) \beta _{3} (1 + \sigma _{3})G_{3}S_{3} + \tau _{2}C_{3} \right] \\ &{}+ \lambda ^{*}_4 \left[ (1-u_1) \beta _{1} (1 + \sigma _{1})G_{1}S_{2}+ (1-u_1) p \beta _{2} (1 + \sigma _{2})G_{2}S_{2} -((1+u_2)\gamma _{1}+\mu +\delta _{1})G_{1} \right] \\ &{}+ \lambda ^{*}_5 \left[ (1-u_1) (1-p)\beta _{2} (1 + \sigma _{2})G_{2}S_{2} -((1+u_2) \gamma _{2}+\alpha _{3}+\mu +\delta _{2})G_{2}\right] \\ &{}+ \lambda ^{*}_6 \left[ (1-u_1) \beta _{3} (1 + \sigma _{3})G_{3}S_{3} -((1+u_2)\gamma _{3}+\mu +\delta _{3})G_{3}+\alpha _{3}G_{2} \right] \\ &{}+\lambda ^{*}_7 \left[ (1+u_2)\gamma _{1}G_{1} -(\mu +\delta _{4}+ \tau _{1})C_{1} \right] \\ &{}+ \lambda ^{*}_8 \left[ (1+u_2) \gamma _{2}G_{2} -(\mu +\delta _{5}+\alpha _{4})C_{2} \right] \\ &{}+ \lambda ^{*}_9\left[ (1+u_2)\gamma _{3} G_{3} +\alpha _{4} C_{2}- (\mu +\delta _{6}+\tau _{2})C_{3} \right] . \end{array} \right\} \end{aligned}$$
(3.2)

Hence, we state the following Theorem:

Theorem 3.1

There exists an optimal control pair \(u_1^{*} (t), u_2^{*} (t)\) and the corresponding solutions \(S_1^{*}, S_2^{*}, S_3^{*}, G_1^{*}, G_2^{*}, G_3^{*}, C_1^{*}, C_2^{*}, C_3^{*}\) that minimizes \(G(u_1(t), u_2(t))\) over U. Additionally, there exists adjoint functions (co-state variable): \(\lambda ^{*}_{1}, \lambda ^{*}_2, \lambda ^{*}_3, \lambda ^{*}_4, \lambda ^{*}_5\), \(\lambda ^{*}_6\), \(\lambda ^{*}_7\), \(\lambda ^{*}_8\), and \(\lambda ^{*}_9\) satisfying

$$\begin{aligned} \left. \begin{array}{lll} \dfrac{d\lambda ^{*}_{1}}{\mathrm{d}t} =&{} - \left( -\lambda ^{*}_{1} \left[ \mu + \alpha _{1} \right] + \lambda ^{*}_{2}[ \alpha _1] \right) , \\ \dfrac{\mathrm{d}\lambda ^{*}_2}{\mathrm{d}t}= &{} -\left( \lambda ^{*}_2\left[ - (\mu + \alpha _{2}) - (1-u_1) \beta _{1} (1 + \sigma _{1})G_{1}^{*} -(1-u_1) \beta _{2} (1 + \sigma _{2})G_{2}^{*} \right] + \right. \\ &{} \left. + \lambda ^{*}_3 [\alpha _2] + \lambda ^{*}_4 \left[ (1-u_1) \beta _{1} (1 + \sigma _{1})G_{1}^{*}+ (1-u_1) p \beta _{2} (1 + \sigma _{2})G_{2}^{*}\right] + \right. \\ &{} \left. + \lambda ^{*}_5 \left[ (1-u_1) (1-p)\beta _{2} (1 + \sigma _{2})G_{2}^{*} \right] \right) ,\\ \dfrac{\mathrm{d}\lambda ^{*}_3}{\mathrm{d}t}=&{} - \left( \lambda ^{*}_3 \left[ -\mu - (1-u_1) \beta _{3} (1 + \sigma _{3})G_{3}^{*} \right] \right. \\ &{} \left. + \lambda ^{*}_6 \left[ (1-u_1) \beta _{3} (1 + \sigma _{3})G_{3}^{*} \right] \right) ,\\ \dfrac{d\lambda ^{*}_4}{\mathrm{d}t}=&{} - \left( 1 + \lambda ^{*}_{2} \left[ - (1-u_1) \beta _{1} (1 + \sigma _{1})S_{2}^{*} \right] \right. \\ &{} \left. + \lambda ^{*}_4 \left[ (1-u_1) \beta _{1} (1 + \sigma _{1})S_{2}^{*}-((1+u_2) \gamma _1 + \mu + \delta _{1} \right] +\lambda ^{*}_7 \left[ (1+u_2)\gamma _{1} \right] \right) ,\\ \dfrac{\mathrm{d}\lambda ^{*}_5}{\mathrm{d}t}=&{} -\left( 1 + \lambda ^{*}_2 \left[ -(1-u_1) \beta _{2} (1 + \sigma _{2})S_{2}^{*}+ \right] + \lambda ^{*}_4 \left[ (1-u_1) p \beta _{2} (1 + \sigma _{2})S_{2}^{*} \right] \right. \\ &{} \left. +\lambda ^{*}_5 \left[ (1-u_1) (1-p)\beta _{2} (1 + \sigma _{2})S_{2}^{*} -((1+u_2) \gamma _{2}+\alpha _{3}+\mu +\delta _{2})\right] \right. \\ &{} \left. + \lambda ^{*}_6 \left[ \alpha _{3} \right] + \lambda ^{*}_8 \left[ (1+u_2) \gamma _{2} \right] \right) ,\\ \dfrac{\mathrm{d}\lambda ^{*}_6}{\mathrm{d}t}=&{} -\left( 1 + \lambda ^{*}_3 \left[ - (1-u_1) \beta _{3} (1 + \sigma _{3})S_{3}^{*}\right] \right. \\ &{} \left. +\lambda ^{*}_6 \left[ (1-u_1) \beta _{3} (1 + \sigma _{3})S_{3}^{*} -((1+u_2)\gamma _{3}+\mu +\delta _{3}) \right] + \lambda ^{*}_9\left[ (1+u_2)\gamma _{3} \right] \right) , \\ \dfrac{\mathrm{d}\lambda ^{*}_7}{\mathrm{d}t}=&{} - \left( -1+ \lambda ^{*}_2 \left[ \tau _{1} \right] +\lambda ^{*}_7 \left[ -(\mu +\delta _{4}+ \tau _{1}) \right] \right) , \\ \dfrac{\mathrm{d}\lambda ^{*}_8}{\mathrm{d}t}=&{} - \left( -1 + \lambda ^{*}_8 \left[ -(\mu +\delta _{5}+\alpha _{4}) \right] + \lambda ^{*}_9\left[ \alpha _{4}\right] \right) , \\ \dfrac{\mathrm{d}\lambda ^{*}_9}{\mathrm{d}t}=&{} -\left( -1 + \lambda ^{*}_3 \left[ \tau _{2}\right] + \lambda ^{*}_9\left[ - (\mu +\delta _{6}+\tau _{2}) \right] \right) . \end{array} \right\} \end{aligned}$$
(3.3)

Further simplifying (3.3) yields

$$\begin{aligned} \left. \begin{array}{lll} \dfrac{\mathrm{d}\lambda ^{*}_{1}}{\mathrm{d}t} =&{} \lambda ^{*}_{1} \left[ \mu + \alpha _{1} \right] - \lambda ^{*}_{2}[ \alpha _1], \\ \dfrac{\mathrm{d}\lambda ^{*}_2}{\mathrm{d}t}= &{} \lambda ^{*}_2\left[ (\mu + \alpha _{2}) + (1-u_1) \beta _{1} (1 + \sigma _{1})G_{1}^{*} +(1-u_1) \beta _{2} (1 + \sigma _{2})G_{2}^{*} \right] \\ &{} - \lambda ^{*}_3 [\alpha _2] - \lambda ^{*}_4 \left[ (1-u_1) \beta _{1} (1 + \sigma _{1})G_{1}^{*}+ (1-u_1) p \beta _{2} (1 + \sigma _{2})G_{2}^{*}\right] \\ &{} - \lambda ^{*}_5 \left[ (1-u_1) (1-p)\beta _{2} (1 + \sigma _{2})G_{2}^{*} \right] ,\\ \dfrac{\mathrm{d}\lambda ^{*}_3}{\mathrm{d}t}=&{} \lambda ^{*}_3 \left[ \mu + (1-u_1) \beta _{3} (1 + \sigma _{3})G_{3}^{*} \right] - \lambda ^{*}_6 \left[ (1-u_1) \beta _{3} (1 + \sigma _{3})G_{3}^{*} \right] , \\ \dfrac{\mathrm{d}\lambda ^{*}_4}{\mathrm{d}t}=&{} - 1 + \lambda ^{*}_{2} \left[ (1-u_1) \beta _{1} (1 + \sigma _{1})S_{2}^{*} \right] \\ &{} - \lambda ^{*}_4 \left[ (1-u_1) \beta _{1} (1 + \sigma _{1})S_{2}^{*}-((1+u_2) \gamma _1 + \mu + \delta _{1} )\right] -\lambda ^{*}_7 \left[ (1+u_2)\gamma _{1} \right] ,\\ \dfrac{\mathrm{d}\lambda ^{*}_5}{\mathrm{d}t}=&{} -1+ \lambda ^{*}_2 \left[ (1-u_1) \beta _{2} (1 + \sigma _{2})S_{2}^{*} \right] - \lambda ^{*}_4 \left[ (1-u_1) p \beta _{2} (1 + \sigma _{2})S_{2}^{*} \right] \\ &{} -\lambda ^{*}_5 \left[ (1-u_1) (1-p)\beta _{2} (1 + \sigma _{2})S_{2}^{*} -((1+u_2) \gamma _{2}+\alpha _{3}+\mu +\delta _{2})\right] \\ &{}- \lambda ^{*}_6 \left[ \alpha _{3} \right] - \lambda ^{*}_8 \left[ (1+u_2) \gamma _{2} \right] ,\\ \dfrac{\mathrm{d}\lambda ^{*}_6}{\mathrm{d}t}=&{} -1+ \lambda ^{*}_3 \left[ (1-u_1) \beta _{3} (1 + \sigma _{3})S_{3}^{*}\right] \\ &{} -\lambda ^{*}_6 \left[ (1-u_1) \beta _{3} (1 + \sigma _{3})S_{3}^{*} -((1+u_2)\gamma _{3}+\mu +\delta _{3}) \right] - \lambda ^{*}_9\left[ (1+u_2)\gamma _{3} \right] , \\ \dfrac{\mathrm{d}\lambda ^{*}_7}{\mathrm{d}t}=&{} 1- \lambda ^{*}_2 \left[ \tau _{1} \right] +\lambda ^{*}_7 \left[ (\mu +\delta _{4}+ \tau _{1}) \right] , \\ \dfrac{\mathrm{d}\lambda ^{*}_8}{\mathrm{d}t}=&{} 1 + \lambda ^{*}_8 \left[ (\mu +\delta _{5}+\alpha _{4}) \right] - \lambda ^{*}_9\left[ \alpha _{4}\right] , \\ \dfrac{\mathrm{d}\lambda ^{*}_9}{\mathrm{d}t}=&{} 1 - \lambda ^{*}_3 \left[ \tau _{2}\right] + \lambda ^{*}_9\left[ (\mu +\delta _{6}+\tau _{2}) \right] \end{array} \right\} \end{aligned}$$
(3.4)

with transversality conditions

$$\begin{aligned} \lambda _i^* (t_{final}) = 0, \quad i = 1,\ldots ,9, \end{aligned}$$
(3.5)

and \(N^{*} = S_1^{*}+S_2^{*}+S_3^{*}+G_1^{*}+G_2^{*}+G_3^{*}+C_1^{*}+C_2^{*}+C_3^{*}.\)

Furthermore, the following characterization holds:

$$\begin{aligned} \left. \begin{array}{lll} u_1^* (t) =&{} min \left( max \left( a_1, \dfrac{1}{B_1} T_1 \right) , b_1 \right) ,\\ u_2^* (t) =&{} min \left( max \left( a_2, \dfrac{1}{B_2} T_2 \right) , b_2 \right) . \end{array} \right\} \end{aligned}$$
(3.6)

where

$$\begin{aligned} \left. \begin{array}{lll} T_1 = &{} \beta _1 (1+\sigma _1)G_1^* (\lambda _2^*-\lambda _4^*)+\beta _2 (1+\sigma _2)G_2^* (\lambda _2^*-\lambda _5^*)\\ &{}+p\beta _2 (1+\sigma _2)G_2^* (\lambda _5^*-\lambda _4^*) + \beta _1 (1+\sigma _1)S_1^* (\lambda _2^*-\lambda _4^*)+ \beta _2 (1+\sigma _2)S_2^* (\lambda _2^*-\lambda _5^*) \\ &{} +p\beta _2 (1+\sigma _2)S_2^* (\lambda _5^*-\lambda _4^*)+ \beta _3 (1+\sigma _3)S_3^* (\lambda _3^*-\lambda _6^*)+ \beta _3 (1+\sigma _3)G_3^* (\lambda _3^*-\lambda _6^*),\\ T_2 = &{} \gamma _1 (\lambda _4^*-\lambda _7^*) + \gamma _2 (\lambda _5^*-\lambda _8^*)+ \gamma _3 (\lambda _6^*-\lambda _9^*). \end{array} \right\} \end{aligned}$$
(3.7)

The proof of the theorem (3.1) is given thus:

Proof

It follows from the Corollary 4.1 in [18], that the convexity of the integrand of the objective functional G with respect to (\(u_1, u_2\)) ensures the existence of the control pair (\(u_1, u_2\)), and the Lipschitz property of the state system with regards to \(S_1, S_2, S_3, G_1, G_2, G_3, C_1, C_2, C_3\). Through the PMP, the adjoint equations and transversality conditions are established, then we have:

$$\begin{aligned} \left. \begin{array}{lll} \dfrac{\mathrm{d}\lambda _1^*}{\mathrm{d}t} =&{} -\dfrac{\partial H}{\partial S_1}, \quad \lambda _1^* (t_{final}) = 0,\\ \dfrac{\mathrm{d}\lambda _2^*}{\mathrm{d}t} =&{} -\dfrac{\partial H}{\partial S_2}, \quad \lambda _2^* (t_{final}) = 0,\\ \dfrac{\mathrm{d}\lambda _3^*}{\mathrm{d}t} =&{} -\dfrac{\partial H}{\partial S_3}, \quad \lambda _3^* (t_{final}) = 0,\\ \dfrac{\mathrm{d}\lambda _4^*}{\mathrm{d}t} =&{} -\dfrac{\partial H}{\partial S_4}, \quad \lambda _4^* (t_{final}) = 0,\\ \dfrac{\mathrm{d}\lambda _5^*}{\mathrm{d}t} =&{} -\dfrac{\partial H}{\partial S_5}, \quad \lambda _5 (t_{final}) = 0,\\ \dfrac{\mathrm{d}\lambda _6^*}{\mathrm{d}t} =&{} -\dfrac{\partial H}{\partial S_6}, \quad \lambda _6^* (t_{final}) = 0,\\ \dfrac{\mathrm{d}\lambda _7^*}{\mathrm{d}t} =&{} -\dfrac{\partial H}{\partial S_7}, \quad \lambda _7^* (t_{final}) = 0,\\ \dfrac{\mathrm{d}\lambda _8^*}{\mathrm{d}t} =&{} -\dfrac{\partial H}{\partial S_8}, \quad \lambda _8^* (t_{final}) = 0,\\ \dfrac{\mathrm{d}\lambda _9^*}{\mathrm{d}t} =&{} -\dfrac{\partial H}{\partial S_9}, \quad \lambda _9^* (t_{final}) = 0.\\ \end{array} \right\} \end{aligned}$$
(3.8)

We put into consideration the optimality conditions

$$\begin{aligned} \dfrac{\partial H}{\partial u_1} = 0, \quad \text {and} \quad \dfrac{\partial H}{\partial u_2} = 0, \end{aligned}$$
(3.9)

and the control pair is determined, subject to \(S_1, S_2, S_3, G_1, G_2,\) \(G_3, C_1, C_2, C_3\). Having obtained the characterization in (3.6), then we have for the \(u_1^*\)

$$\begin{aligned} \dfrac{\partial H}{\partial u_1} = B_1 u_1 + T_1=0 \end{aligned}$$
(3.10)

which implies that

$$\begin{aligned} u_1^* (t) = \dfrac{1}{B_1} T_1 \end{aligned}$$
(3.11)

on the set \(\{t: a_1< u_1^*(t) < b_1\}\). Similarly for \(u_2^*\), we have

$$\begin{aligned} \dfrac{\partial H}{\partial u_2} = B_2 u_2 + T_2=0 \end{aligned}$$
(3.12)

which implies that

$$\begin{aligned} u_2^* (t) = \dfrac{1}{B_2} T_2, \end{aligned}$$
(3.13)

over the set \(\{t: a_2< u_2^*(t) < b_2\}\).

It is important to note that a few limitations ought to be forced on the period up to \(t_f\), to guarantee the uniqueness of the optimality system. Such limitations can be attributed to the contrary time orientation of (2.4), (3.4), (3.5) since the variables \(S_1, S_2, S_3, G_1, G_2, G_3, C_1, C_2, C_3\) have starting values while the adjoint equations have output values [17, 47].

Table 2 Description of variables
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Table 3 Parameters and values
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4 Numerical simulation

In this present section, the optimal strategy for effective criminal gang control, consisting of controls that supports enlightenment campaign programs and empowerment programs (which includes door-to-door, community square talks, television and radio jingles, print media, youth empowerment, access to loan, and other prevention strategies of crime) and a case finding control that represent the fraction of initiated persons who are arrested, law-enforced and imprisoned in the correctional center for proper correction and prevention of contacts with susceptible individuals. The system is solved by employing the Runge–Kutta scheme, implemented in a forward-backward sweep fashion. The control functions baseline range is given on Table 2 while the parameter values are given in Table 3. We take \(N = 100{,}000\). The value for N is so chosen because most sociological and criminological data on crime incidence in human population are reported in 100,000 s.

Fig. 1
figure 1

Optimal control strategies when \(\beta _1, \beta _2,\) and \(\beta _3\) are varied

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From Fig. 1, the \(u_1\) and \(u_2\) are plotted where the contact rates \(\beta _1, \beta _2\) and \(\beta _3\) were increased. In Fig. 1, changing contact rates \(\beta _1, \beta _2\) and \(\beta _3\) does not have a significant effect on the two values of the control \(u_2\), however, it does have a significant effect on the two results of the control \(u_1\); for the values of contact rates \(\beta _1, \beta _2\) and \(\beta _3\), we observe in (a) that the control \(u_1\) experienced a fleeting increase (and decrease) for about 1.5 years but remained close to the lower bound for the remaining part of the 10 years of the simulation. This is indicative of the fact that there is a small proportion of susceptible population gaining from the campaign program and that criminal gang’s activities have a severe negative impact on the campaign or the intervention is no longer needed. In (b), the control \(u_2\) remains in the upper bound for almost 6-year of the simulation before experiencing a drop to the lower bound for the remaining 4-year period to minimize the number of criminal gangs in the population.

Figure 2 represents the susceptible class \(S_2\) when we had an increase in the contact rates. From Fig. 2 a, we observe that an increase in the contact rate does not impact the susceptible population. Interestingly, Fig. 2a implies a significant drop in the susceptible population who would have proceeded to join criminal gangs \(G_1\) and \(G_2\). Hence, using the first strategy \(u_1\) in class \(S_2\), we were able to prevent 2945/100,000 \(S_2\) susceptible individuals from being initiated by criminal gangs \(G_1\) and \(G_2\) at the end of 10 years period.

Figure 2b implies the susceptible class \(S_3\) when we had an increase in the contact rates. From Fig. 2b, we observe that an increase in the contact rate does not impact the susceptible population. Hence, Fig. 2b implies a significant drop in the susceptible population who would have proceeded to join criminal gangs \(G_3\). Thus, using the first strategy \(u_1\) on class \(S_3\), we were able to prevent 2071/100,000 \(S_2\) susceptible individuals from being initiated by criminal gangs \(G_3\) at the end of 10 years period.

Fig. 2
figure 2

Population \(S_2\) and \(S_3\) when \(\beta _1, \beta _2,\) and \(\beta _3\) are varied

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Fig. 3
figure 3

Optimal control strategies when \(\gamma _1, \gamma _2,\) and \(\gamma _3\) are varied

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Summarily, the study suggests that if the first strategy \(u_1\) is sustained throughout the 10 years or more, we can prevent more susceptible individuals in both the adolescent and adult populations from being initiated into gangs.

Fig. 4
figure 4

Population \(G_1\), \(G_2\) and \(G_3\) when \(\gamma _1, \gamma _2,\) and \(\gamma _3\) are varied

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From Fig. 3, the controls functions are plotted while the arrest and prosecution rates \(\gamma _1, \gamma _2,\) and \(\gamma _3\) were increased from 0.1 to 1. In Fig. 3, changing the arrest and prosecution rates \(\gamma _1, \gamma _2,\) and \(\gamma _3\) have a significant effect on control \(u_2\) and the control \(u_1\) respectively; for the values of the arrest and prosecution rates \(\gamma _1, \gamma _2,\) and \(\gamma _3\), we observe in (a) that the control \(u_1\) at \(\gamma _1= \gamma _2 = \gamma _3 = 0.1\) stayed close to the lower bound for the 10-year period. Then, \(u_1\) at \(\gamma _1= \gamma _2 = \gamma _3 = 1\) experienced a momentary increase from about 0 to about 1.8 years but dropped down to the lower bound for the remaining part of the 10-year period of simulation. Also, for the values of the arrest and prosecution rates \(\gamma _1, \gamma _2,\) and \(\gamma _3\) at different levels, we observe in (b) that the \(u_2\) stayed close to the upper bound for almost 6-year period before dropping to the lower bound at \(\gamma _1= \gamma _2 = \gamma _3 = 1\) while the control \(u_2\) stayed close to the lower bound for almost 10-year period at \(\gamma _1= \gamma _2 = \gamma _3 = 0.1\).

Summarily, an increase in the arrest and prosecution rates \(\gamma _1, \gamma _2,\) and \(\gamma _3\) can sustain the control strategy for a longer time in the population thereby suggesting a significant clampdown (minimization) on the number of criminal gangs in the population.

Figures 4 and 5 represents the criminal gang classes \(G_1, G_2\) and \(G_3\) and correctional center classes \(C_1, C_2\) and \(C_3\) when we had an increase in the rates of arrest and prosecution. From Fig. 4, we observe that an increase in the arrest and prosecution rates, decreases the gang population. Interestingly, Fig. 4 shows a decrease in the gang population has resulted into increase in the correctional center population as evident in Fig. 5. By this, we mean that alot of criminals are taken a way from the society through arrested and prosecution that leads to sentencing or remanding period.

Fig. 5
figure 5

Population \(C_1\), \(C_2\) and \(C_3\) when \(\gamma _1, \gamma _2,\) and \(\gamma _3\) are varied

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Fig. 6
figure 6

Susceptible population (\(S_2\) and \(S_3\)) with control (\(u_1 \ne 0\)) and without control (\(u_1 = u_2 = 0\))

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Thus, using the strategy \(u_2\) in class \(G_1\), we can reduce 5,460/100,000 \(G_1\) criminals from the population at the end of 10 years; using this strategy \(u_2\) in class \(G_2\), we were able to reduce 209/100,000 \(G_2\) criminals from the population at the end of 10 years period; using this strategy \(u_2\) in class \(G_3\), we were able to reduce 6,748/100,000 \(G_3\) criminals at the end of 10 years in the population.

Fig. 7
figure 7

Criminal gang population (\(G_1\), \(G_2\) and \(G_3\)) with control (\(u_2 \ne 0\)) and without control (\(u_1 = u_2 = 0\))

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Hence, using the strategy \(u_2\) in class \(C_1\), we were able to increase the population of the correctional center \(C_1\) by 4,575/100,000 at the end of 10 years; using the strategy \(u_2\) in class \(C_2\), we were able to increase the population of the correctional center \(C_2\) by 171/100,000 at the end of 10 years; using the strategy \(u_2\) in class \(C_3\), we were able to increase the population of the correctional center \(C_3\) by 5,724/100,000 at the end of 10 years using this strategy \(u_2\) in class \(G_3\).

In summary, we stressed that by adopting the second strategy \(u_2\), we can reduce criminals from the street for appropriate corrections in the correctional centers \(C_1, C_2\) and \(C_3\) at the end of 10 years in the population.

Remark

\(u_1(t)\): The values of cases averted (and implemented) are obtained by just taking the difference in the quantity of susceptible (and gang) population when both contact rates are utilized; \(u_2(t)\): the difference in the number of gang (and inmates) population when arrest and prosecution rates are utilized.

Furthermore, considering the criminal gang model (2.4) with and without controls, we note that the total number of susceptible population (\(S_2\) and \(S_3\)) prevented from being initiated into criminal gang population ((\(G_1\) and \(G_2\)) and (\(G_3\))) respectively in Fig. 6; total number of criminal gang population ((\(G_1\) and \(G_2\)) and (\(G_3\))) taken away (arrested and prosecuted) from the society to the correctional centers (\(C_1\), \(C_2\), \(G_3\)) Fig. 7. To be precised further, applying the control \(u_1\) on the criminal gang model, we observed that we were able to avert almost 3980/100,000 new gang initiations in the susceptible adolescent population (\(S_2\)) and 2831/100,000 new gang initiations in susceptible adult population (\(S_3\)). Interestingly, we have established that the control \(u_1\) is more effective in the susceptible adolescent population (\(S_2\)) compared to the susceptible adult population (\(S_3\)). Also, applying the control \(u_2\), we have arrested and prosecuted more criminals by reducing their population and isolating/sentencing them in/to the correctional centers. That is, we have prevented a total of 3,203/100,000 new criminal cases in the adolescent population (\(G_1\)), a total of 1319/100,000 new criminal cases in the adolescent population (\(G_2\)) and a total of 5864/100,000 new criminal cases in criminal gang (adult) population (\(G_3\)). Evidently, the control \(u_2\) is more effective in adult criminal gang population than the adolescent criminal gang population. This is quite instructive of the direction and concentration of the limited-resources, as the control \(u_1\) that deals with the preventive approach is more effective in the adolescent susceptible population while the control \(u_2\) that deals with curative approach is more effective in the adult criminal gang population.

Table 4 Criminal incidence strategies
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Table 5 Criminal incidence strategies
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5 Cost-effectiveness analysis

This section presents a cost-effectiveness analysis for crime-fighting and control interventions to justify the expenses of the optimal control model strategies \(u_1(t)\) and \(u_2(t)\) respectively, and both simultaneously. This approach has been explored in [20, 50] for some mathematical models. For our age-structured optimal control model, we carry out the quantitative analysis by comparing the differences among the criminal incidence (effect of contact rates and arrest and prosecution rates) outcomes, and enormous costs of these interventions; achieved by calculating the average cost-effectiveness ratio (ACER) and incremental cost-effectiveness ratio (ICER), which is defined as the cost per crime control outcome.

We assume here that, the cost of implementing case finding control that represents the fraction of initiated persons who are arrested, law-enforced, and imprisoned in the correctional center for adequate correction and prevention from interacting with susceptible individuals is a lot higher than the cost of executing the control that supports enlightenment campaign programs and empowerment programs. That is, we assume the weight constants \(B_1 = 50\), \(B_2 = 100\). We apply the cost functions \(\frac{1}{2}B_1 u_1(t), \frac{1}{2}B_2 u_2(t)\), over time, to calculate the total cost for the strategies that were executed.

Strategy \(M_1\): Cost of implementing the control (\(u_1(t)\)) that supports enlightenment campaign programs and empowerment programs

We calculate the total number of gang initiations averted and the total cost of the strategies employed as seen in Table 4. We have shown in the previous section that by implementing the optimal control strategy \(u_1(t)\), we prevented:

  • 2,945/100,000 \(S_2\) from criminal gang (\(G_1\) and \(G_2\)) initiations,

  • 2,071/100,000 \(S_3\) from criminal gang (\(G_3\)) initiations.

Strategy \(M_2\): Case finding control (\(u_2(t)\)) that represent the proportion of initiated persons who are law-enforced and imprisoned in the correctional center

We calculate the total number of criminals convicted and the total cost of the strategies applied as seen in Tables 4. We have shown in the previous section that by implementing the optimal control strategy \(u_2(t)\), we increased the number convicted criminals by sentencing:

  • 4575/100,000 \(G_1\) to the correctional center \(C_1\),

  • 171/100,000 \(G_2\) to correctional center \(C_2\),

  • 5724/100,000 \(G_3\) to correctional center \(C_3\).

Strategy \(M_3\): Cost of implementing the control (\(u_1(t)\)) and cost of implementing the control (\(u_2(t)\)) simultaneously

We calculate the total number of criminal incidents averted, and the total cost of the strategies applied as seen in Tables 4. This has been stated in Strategy A and Strategy B respectively.

Implementation of the strategies

As a matter of necessity, we establish the most cost-effective strategy among the \(M_1\), \(M_2\) and \(M_3\) strategies considered in this section. To accomplish this, the cost-effectiveness analysis is obtained by computing the ACER and ICER, adopting the idea in [20, 50]. The outcomes from the mathematical simulation executed, Table 4 gives the ranking of \(M_1\), \(M_2\) and \(M_3\) strategies.

$$\begin{aligned} ACER(M_1)&= \dfrac{50}{5,016} = 0.00997\\ ACER(M_2)&= \dfrac{100}{10,470} = 0.00955\\ ICER(M_1)&= \dfrac{50}{5,016} = 0.00997\\ ICER(M_2)&= \dfrac{100-50}{10,470-5,016} = 0.00916 \end{aligned}$$

From ICER (\(M_1\)) and ICER(\(M_2\)), we notice a cost saving of 0.00917 for procedure \(M_2\) than procedure \(M_1\). This infers that procedure \(M_1\) unequivocally overshadow procedure \(M_2\), showing that procedure \(M_1\) is more costly and less effective compared to procedure \(M_2\). Consequnetly, procedure \(M_1\) is taken out from subsequent ICER calculations, as evident in Table 5. Now, we compare procedures \(M_2\) and \(M_3\).

$$\begin{aligned} ICER(M_2)&= \dfrac{100}{10,470} = 0.00955\\ ICER(M_3)&= \dfrac{150-100}{15,486-10,470} = 0.00997 \end{aligned}$$

Comparing procedures \(M_2\) and \(M_3\), we see that ICER(\(M_3\)) > ICER(\(M_2\)), showing that procedure \(M_3\) strongly dominated procedure \(M_2\) and is more costly and less effective compared to procedure \(M_2\). Consequently, procedure \(M_2\) (arrest and sentencing rates) has the least ICER and is the most cost-effective of all the control strategies for crime prevention and control in this study. This additionally corroborates with the result of the ACER technique in Table 4 that procedure \(M_2\) is the most cost-effective strategy. It is also deserving of note to see that this strategy averts more criminal incidences and reduces crime rates than any other control strategy implemented. This is so because the severity of arresting and punishing criminals serves as a deterrent to others in the society.

The results from the control model indicate how a cost-effective analysis could help policymakers gain further insight into the study of crime by considering the cost of implementing the model parameters, in particular, contact rates (\(\beta _1,\beta _2,\beta _3\)); arrest and prosecution rates (\(\gamma _1,\gamma _2,\gamma _3\)). Conclusively, this research result suggests that the crime control programs that adopt these strategies explained herein can help reduce new gang initiations and criminal activities.

6 Disscusion

Based on the results obtained from this new approach for studying the dynamics of criminal gangs, the initiation rate between susceptible individuals and gang members is a determining factor in controlling the activities of criminals in a limited-resource setting. By reducing the initiation rates through empowerment and enlightenment programs among these individuals, we can potentially decrease the initiation rate and thus minimize the gang population.

We performed a further study (cost-effectiveness analysis) on the optimal control model by considering three strategies (\(M_1, M_2, M_3\)). Consequently, we established that the singular implementation of strategy \(M_1\) (empowerment and enlightenment campaign program) is more costly and less effective among the three strategies considered. Thus, the implication of this is that the implementation of strategy \(M_2\) (the arrest and sentencing criminals for corrective measure) and strategy \(M_3\) (the simultaneous implementation of strategies \(M_1\) and \(M_2\)), are less costly and more effective than the singular implementation of strategy \(M_1\) (the empowerment and enlightenment campaign strategy) only.

7 Conclusion

A criminal gang optimal control model is proposed and studied herein. The main contribution of this work, as it can rarely be found in the literature, is the formulation of optimal control models that would help during crime fighting and control in a limited-resource setting. In addition, we have introduced cost-effective analysis to help us gain further insight on the best strategy for handling the activities of criminal gangs in a limited-resource setting. We hereby note that the major limitation of the study is our inability to validate this current work with real-life data due to lack of access to primary data that will fit into all the variables. However, we rely on secondary data from reputable literature such as a recent publication in Ibrahim et al. [13], United Nations [48] and World Bank [49].

The result from the current study has shown that if the Nigerian government, non-governmental organizations, religious bodies and other stakeholders as highlighted in Sect. 2 can intensify effort in lifting a reasonable number of Nigerians from poverty through empowerment programs and job creation; reducing out of school children by taking these adolescents from the street for proper education and engagement, then we could have a society devoid of rancor and acrimony. Interestingly, our research result is in agreement with the postulate in [14] which has extensively identified unemployment as a root cause of crime in Nigeria and possible solutions.

We have also shown that the least costly and most effective strategy is the arrest and sentencing of criminals for corrective measures in limited-resource setting. This strategy is observed to be more realistic in combating crime since the criminal activities is on the inside and there is inadequate funding for security personals and institutions. The criminal elements then capitalize on this by take advantage of the loop holes in the system. Becker’s economic theory of criminal behavior is also corroborates this assertion by stating that, potential criminals are economically rational and respond significantly to the deterring incentives by the criminal justice system. Thus, if we are immediately interested in combating crime in Nigeria, the government should invest more in security by providing sophisticated weapons and infrastructure for the police and correctional center staff as evident in the cost-effective analysis. The arrest and sentencing criminals for corrective measures strategy, as observed to be the most effective strategy in this study, is currently being tested and seen to be effective in some African countries such as Rwanda. Could the Nigerian situation be an exception?

In a society like Nigeria, being the most populous nation in Africa, the principal intervention as evident in this research is seen to be adequate funding for the police, judiciary and correctional center staff since the majority of the arrested and prosecuted criminals are detained and managed by these individuals/institutions. Specifically, the welfare of the staff of these institutions must be of necessity so that criminal elements do not explore the loop holes to entice and corrupt the government personnel. When the well-being of the staff become the government’s top priority, the personnel will be motivated to use and work with the available resources (weapon and instruments) for crime-fighting and correction).