# Dynamics of Terrorism in Contemporary Society for Effective Management

Authors: Kazeem Babatunde Akande; Julius Adebowale Makinde; Mohammed Olanrewaju Ibrahim
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

In the contemporary world, the effect of faith in religion cannot be underestimated or overemphasized. In the olden days, traditional religion/faith of a particular locality was the only practice obtainable; however, new faiths emerged and are being absorbed in recent times. Extremism in the newly absorbed faith began to cause the indigenous religion to collapse and increase violence against innocent ones. This paper investigated the interaction between the extremism of faith leading to the act of terror and a susceptible individual (members of the society) to guide the policymakers and decision implementers to embrace the proposed model for counterterrorism for effective management of the insurgency. Mathematical modelling of epidemiology was conceptualized for the model formulation, and the resulting autonomous differential equations were critically analyzed with the Lipschitz condition, next generation matrix, and Bellman and Cooke’s criteria for the management of insurgency in the society. Thresholds were obtained to curtail recruitment into the fanatical groups, and the results of the simulated proposed model identified critical factors (parameters) to be considered for the complete eradication of violence in human society.


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## Body

## 1. Introduction

The society’s aims to wage a war against terrorism and counterterror measures may range from invading certain territories for assassinations of terrorists to freezing assets of organizations with links to potential terrorists [1–3]. The dynamics of bicycles and space shuttles evolve according to Newton’s law, but people’s movements back and forth between different states of activity are driven by far more complex influences [4]. One might think it is impossible to model systems in which humans are the key moving parts, but that misinterprets the definition of modelling [5, 6]. Model is a simplified representation of reality [7]. Models of human systems simplified by omitting many factors may not produce the required result, and there may be more judgment involved in determining what constitutes a reliable model when analyzing human systems (complexity in modelling real-life events), but that makes systems like terrorism better not a worse topic for a research. The inherent complexity of human systems ironically implies that there is particular wisdom in keeping the model simple enough that one can understand intuitively “what makes the model work” [5, 7, 8]. Therefore, it is of interest to model the cause of the act of terrorism in human population using a system of ordinary differential equation (ODE) for effective management. The study builds on the work of Professor Emeritus Castillio-Chavez and Banks in [9], where fanatics are treated as infection, and here, fanatics progress to cause terrorism. This act of terrorism could be international or domestic in nature [10]. Extremism in faith causing terrorism for purposes such as self-protection, politically motivated, personal gain, or ideology towards mobilizing for political violence [11–14] is here considered. Many researchers have used the classical fourth-order Runge–Kutta (RK45) method due to the method’s efficiency and accuracy in solving ODEs and optimal control problems [15–18]. Hence, this study presents the mathematical dynamical model of terrorism and simulates using RK45 in Maple 18 to numerically picture the analytical solutions of the model.


## 2. Model Formulation and Assumptions

### 2.1. Model Formulation

A mathematical dynamics model of terrorism is formulated using the approach of mathematical epidemiology. The entire population is referred to as noncore (typically large), denoted asG(t), while the core population is a subset of G(t), which is further subdivided into three, as shown in Figure 1, and in the following hierarchical order: the naive (susceptible/vulnerable) subpopulation, which includes individuals who are yet to embrace ideology, denoted as S(t), semi-fanatic (exposed to fanatics) subgroup, E(t), and the extreme fanatic subpopulation, Ft, with assumption that only fanatics commit terror; example of such fanaticism is the case of religious parents that refused blood transfusion and almost caused violence against their child (the details of the story are given in [19, 20] and the court injunction is given in [13]). The mathematical model is presented in Figure 2. The formulation is based on the work of Castillio-Chavez and Banks [9].Figure 1
Stratified model population.Figure 2
Block of mathematical notation for the interaction between extremism and susceptible individuals.


### 2.2. Model Assumptions

The following are the assumptions made in formulating the dynamical model:(1)
It is universally agreed that the government refers to and declares any group of faith (extremist) that causes violence against innocent members of the community as a terrorist group [21].(2)
A general compartment,G(t), is assumed to be a very large population that does not discriminate her members in any form, be it religion, ethnicity, or tribalism.(3)
It is assumed that the introduction of an extremist in the general population generates discrimination.(4)
Repentance from the vulnerable class (S(t)) may return to their normal way of life, possibly, due to government policy on extremism or community norms, culture, and value, e.g., US Federal Bureau of Investigation partners with other operatives to help dismantle extremist network worldwide [10].(5)
The model considered that the susceptible and the semifanatics classes return (repent) to noncore class, perhaps, due to self-relection or sanction or strictness of government policy about terrorism.(6)
The model further assumes that individuals with extreme ideology will only be removed, either by natural death or extreme ideology (induced death due to their beliefs or terror). However, law gives freedom to practice religion [13] of choice, provided the person is in his/her rightful senses.(7)
The model indeed considers no permanent resident in the terror compartment. Although extremist moves out from the fanatical compartment to commit terror (e.g., Boko Haram) and some return afterward, the government tags this set of people extremists or terrorists. The model considered Boko Haram as a fanatical group and agreed that they move out of their hideout to commit terror.(8)
The general population can never be zero and has no age barrier.The progression fromGt to St is q, i.e., q is the successful contact for recruitment of members to the susceptible class. β is the contact probability with the core group member, and this lies between zero and one 0≤β≤1. ρ is treated as the conversion rate of susceptible class to Et. The natural death rate is µ, and it is assumed equal for all compartments and parameter value in a case study of Nigeria, in which its inverse is an average lifespan and there exists death due to extreme fanatics (d or δ) in this study, and both parameters will be used interchangeably.It is also assumed that after a while, due to personal reasoning or phobia of sanction, S(t) andEt return to Gt at the rates α and γ, respectively. The proportion of Et to Ft is σ, and this is treated with standard incidence function because of complex interaction in humans. The term η is used to denote the proportion of Ft that commits terror, and y is the proportion of return fanatics after committing terror.In view of the formulation and assumptions above, we present Figure2, the block diagram of dynamics of the model, followed by model equations.


### 2.3. Model Diagram and Equations

Model equations:(1)dGdt=Λ−qβGCN+αS+γE−μG,dSdt=qβGCN−ρSE+FC−μ+αS,dEdt=ρSE+FC−σEFC−μ+γE,dFdt=σEFC−μ+d+ηF+yT,dTdt=ηF−μ+y+dT,with initial condition(2)G0=G0>0,S0=S0≥0,E0=E0≥0,F0=F0≥0,T0≥0.,Nt=Gt+Ct,Ct=St+Et+Ft+Tt.


## 2.1. Model Formulation

A mathematical dynamics model of terrorism is formulated using the approach of mathematical epidemiology. The entire population is referred to as noncore (typically large), denoted asG(t), while the core population is a subset of G(t), which is further subdivided into three, as shown in Figure 1, and in the following hierarchical order: the naive (susceptible/vulnerable) subpopulation, which includes individuals who are yet to embrace ideology, denoted as S(t), semi-fanatic (exposed to fanatics) subgroup, E(t), and the extreme fanatic subpopulation, Ft, with assumption that only fanatics commit terror; example of such fanaticism is the case of religious parents that refused blood transfusion and almost caused violence against their child (the details of the story are given in [19, 20] and the court injunction is given in [13]). The mathematical model is presented in Figure 2. The formulation is based on the work of Castillio-Chavez and Banks [9].Figure 1
Stratified model population.Figure 2
Block of mathematical notation for the interaction between extremism and susceptible individuals.


## 2.2. Model Assumptions

The following are the assumptions made in formulating the dynamical model:(1)
It is universally agreed that the government refers to and declares any group of faith (extremist) that causes violence against innocent members of the community as a terrorist group [21].(2)
A general compartment,G(t), is assumed to be a very large population that does not discriminate her members in any form, be it religion, ethnicity, or tribalism.(3)
It is assumed that the introduction of an extremist in the general population generates discrimination.(4)
Repentance from the vulnerable class (S(t)) may return to their normal way of life, possibly, due to government policy on extremism or community norms, culture, and value, e.g., US Federal Bureau of Investigation partners with other operatives to help dismantle extremist network worldwide [10].(5)
The model considered that the susceptible and the semifanatics classes return (repent) to noncore class, perhaps, due to self-relection or sanction or strictness of government policy about terrorism.(6)
The model further assumes that individuals with extreme ideology will only be removed, either by natural death or extreme ideology (induced death due to their beliefs or terror). However, law gives freedom to practice religion [13] of choice, provided the person is in his/her rightful senses.(7)
The model indeed considers no permanent resident in the terror compartment. Although extremist moves out from the fanatical compartment to commit terror (e.g., Boko Haram) and some return afterward, the government tags this set of people extremists or terrorists. The model considered Boko Haram as a fanatical group and agreed that they move out of their hideout to commit terror.(8)
The general population can never be zero and has no age barrier.The progression fromGt to St is q, i.e., q is the successful contact for recruitment of members to the susceptible class. β is the contact probability with the core group member, and this lies between zero and one 0≤β≤1. ρ is treated as the conversion rate of susceptible class to Et. The natural death rate is µ, and it is assumed equal for all compartments and parameter value in a case study of Nigeria, in which its inverse is an average lifespan and there exists death due to extreme fanatics (d or δ) in this study, and both parameters will be used interchangeably.It is also assumed that after a while, due to personal reasoning or phobia of sanction, S(t) andEt return to Gt at the rates α and γ, respectively. The proportion of Et to Ft is σ, and this is treated with standard incidence function because of complex interaction in humans. The term η is used to denote the proportion of Ft that commits terror, and y is the proportion of return fanatics after committing terror.In view of the formulation and assumptions above, we present Figure2, the block diagram of dynamics of the model, followed by model equations.


## 2.3. Model Diagram and Equations

Model equations:(1)dGdt=Λ−qβGCN+αS+γE−μG,dSdt=qβGCN−ρSE+FC−μ+αS,dEdt=ρSE+FC−σEFC−μ+γE,dFdt=σEFC−μ+d+ηF+yT,dTdt=ηF−μ+y+dT,with initial condition(2)G0=G0>0,S0=S0≥0,E0=E0≥0,F0=F0≥0,T0≥0.,Nt=Gt+Ct,Ct=St+Et+Ft+Tt.


## 3. Properties and Analysis of the Model

It is essential to investigate the behaviour of model formulated before recommending it for use. The properties of the proposed model to be examined are positive invariant, regions of feasibility, reproductive number (recruitment threshold into extremism), equilibria states, and stability analysis (dynamics of human behaviour and criteria for management). All these properties and analyses were applied in many studies [18, 22, 23].Theorem 1.
Region of feasibility and positivity of solution: ifSt,Et,Ft,Tt,Gt have nonnegative initial condition, then there exists a region(3)Ω=SFt,EFt,Ft,Tt,Gt∈R+5:0<SFt+EFt+Ft+Tt+Gt≤Λμ,for all t,SFt≥0,EFt≥0,Ft≥0,Tt≥0,Gt≥0.Proof.
The model concerns humans; therefore, parameters of the models are assumed to be nonnegative for all timet. Furthermore, we consider(4)Nt=SFt+EFt+Ft+Tt+GtdNtdt=SFtdt+EFtdt+Ftdt+Ttdt+Gtdt,=Λ−μSFt+EFt+Ft+Tt+Gt−dT+F=Λ−μNt−dT+Flimt⟶∞dNtdt=Λ−μNt,and this implies(5)Nt≤Λμ,and dynamics of the model are(6)Nt=Λμ1−e−μt+N0e−μt.
Hence, this proved the first hypothesis of the theorem, and for the second hypothesis, consider the first expression in equation (1):(7)dGdt=Λ−qβGCN+γEF+αSF−μG≥−qβCN+μG.
Integrate both sides to have(8)Gt≥G0e−∫0∞qβC/N−μdt>0,for allG0>0.
Similar argument holds for(9)dSFdt=qβGCN−ρSFEF+FC−μ+αSF≥−μ+αSF.
Integrating this gives(10)SFt≥SF0e−μ+αt≥0∀t.
In the same way forEFt, Ft, and Tt, we showed that all are greater than or equal to zero, and this completes the proof.


### 3.1. Interpretation

The result obtained in (6) showed that the population cannot be zero. Hence, validate the integral claim of assumption nine (8).


### 3.2. Existence and Uniqueness of the Solution

In mathematics, posing a problem is not enough if a solution does not exist and the uniqueness of such a solution is essential in some instances. Here, the existence and uniqueness of the model shall be tested using the Lipschitz condition.Theorem 2.
Suppose (1) is written in compact form as Fit,x,xt0=x0,i=1,2,…,5, and there exists a unique solution; then, (1) satisfies the Lipschitz condition.Proof.
Rewrite model equation (1) as(11)F1t,G=Λ−qβGCN+γEF+αSF−μG,F2t,SF=qβGCN−ρSFEF+FC−μ+αSF,F3t,EF=ρSFEF+FC−σEFFC−μ+γEF,F4t,F=σEFFC−η+μ+dF+yT,F5t,T=ηF−μ+d+yT,and define equation (1) in the form Xi′=Fit,x,xt0=x0,i=1,2,…,5; then,(12)Fit,x2−Fit,x1≤Mx2−x1,and by illustration,(13)F1t,G2−F1t,G1=−qβC2N+μG2−−qβC2N+μG1=−qβC2N+μG2−G1≤−qβC2N+μG2−G1,F2t,SF1−F2t,SF1=−−ρEF+FC+μ+αSF2−−ρEF+FC+μ+αSF1≤−−ρEF+FC+μ+αSF2−SF1,F3t,EF2−F3t,EF1=−ρSF−σFC−μ+γEF2−ρSF−σFC−μ+γEF1≤ρSF−σFC−μ+γEF2−EF1.
For argument sake, letF′ be denoted as the usual F for a fanatic to differentiate between itself and Lipschitz function.(14)F4t,F2′−F4t,F1′=σEFC−μ+η+d+yF2′−σEFC−μ+η+d+yF1′≤−σEFC−μ+η+d+yF2′−F1′,F5t,T2−F5t,T1=−ηF′+μ+y+dT2−ηF′+μ+y+dT1≤ηF′+μ+y+dT2−T1.Remark 1.
(i)
A system of ODEs is said to exist and is said to be unique in the solution if it satisfies the Lipschitz condition.(ii)
Lipschitz conditions cater for continuity and boundedness. According to [24], first-order ODE is said to exist and is said to be unique if the partial derivatives of the function exist. This is stated in the next theorem relevant to the proposed model. Hence, the proposed model is said to be mathematically well-posed.Theorem 3.
LetΩ be denoted the region stated in Theorem 1 and bounded in 0≤R<∞. It suffices to show that ∂fit,x/∂x,i=115,x=G,SF,EF,F,T are continuous in Ω.Proof.
Rewrite equation (1) as(15)F1t,G,SF,EF,F,T=Λ−qβGCN+γEF+αSF−μG,F2t,G,SF,EF,F,T=qβGCN−ρSFEF+FC−μ+αSF,F3t,G,SF,EF,F,T=ρSFEF+FC−σEFFC−μ+γEF,F4t,G,SF,EF,F,T=σEFFC−η+μ+dF+yT,F5t,G,SF,EF,F,T=ηF−μ+d+yT,(16)∂F1∂G=−qβCN−μ<∞,∂F1∂SF=α<∞,∂F1∂EF=γ<∞,∂F1∂F=α<∞,∂F1∂T=α<∞,∂F2∂G=qβCN<∞,∂F2∂SF=ρEF+FC+μ+α<∞,∂F2∂EF=−ρSFC<∞,∂F2∂F=−ρSFC<∞,∂F2∂T=|0|<∞,∂F3∂G=|0|<∞,∂F3∂SF=ρEF+FC<∞,∂F3∂EF=ρSF−σFC−μ+γ<∞,∂F3∂F=ρSF−σEFC<∞,∂F3∂T=|0|<∞,∂F4∂G=0<∞,∂F4∂SF=0<∞,∂F4∂EF=|σFC|<∞,∂F4∂F=σEFC−η+μ+d<∞,∂F4∂T=y<∞,∂F5∂G=0<∞,∂F5∂SF=0<∞,∂F5∂EF=0<∞,∂F5∂F=0<∞,∂F5∂T=−μ+d+y<∞.
Hence,∂Fi/∂x∈R<∞,fori=1,2,…,5andx=G,SF,EF,F,T exist and have solution in feasible region.
This completes the proof.


#### 3.2.1. Interpretation of the Result

The result of existence and uniqueness is that the problem of extremism is in the human population and the solution also lies therein.


### 3.3. Threshold for Recruitment into the Ideology

The long-term sustainability of the core subpopulation is analyzed by using the concept of basic reproduction number (BRN) in epidemiology [25]. With this BRN, one can deduce that for certain parameter values, the model predicts the extinction of the core population ct, and effective recruitment shows that the core population will persist (endemic in epidemiology) [16]. Next theorem gives the thresholds for recruitment into the ideology.Theorem 4.
(i)
Assumen=minα,γ and R0,min≡qβ/μ+n. Then, limt⟶∞Ct⟶0.(ii)
Effective reproductive ratioRe=ρ/μ+γ.Proof.
SinceC=S+E+F+T, then(17)dCdt=qβ1−CC−μ+αS−μ+γE−μ+δF−μ+δT,=qβ1−CC−μ+nC−δF+T,wheren=α,γ,≤qβ1−C−μ+nC,=−μ+nC1−qβ2μ+n<0,and by the hypothesis of the theorem,(18)R0≡qβμ+n.
Hence,dC/dt=−μ+nC1−R0<0.
It could be seen from the above inequality thatCt decays exponentially fast to zero. If R0<1, it implies that the core population cannot be established, but from the assumptions of the model, the general population can never be extinct, so there will always be people vulnerable to fanatics. Reducing resistance to vulnerability increases the core population, or the longer the residence time 1/n, the higher the possibility of increased recruitment (i.e., R0≥1).
The approach of next generation matrix (NGM) is employed for determining effective threshold (Re) for recruitment of fanatics. From (1), we extract F and V, which, respectively, denote new infection and other transmissions in the model, i.e.,(19)F=ρSFCρSFC0000000,V=γ+μ000β+μ+d−y0−βμ+d+y,V−1=1γ+μ000ac0bd,where “a” =μ+β+d/μ+d+y+βμ+d, “b” =−β/μ+d+y+βμ+d, “c” =−y/μ+d+y+βμ+d, and “d” =μ+d+y/μ+d+y+βμ+d; then, with little algebraic simplification, FV−1 is obtained to be(20)FV−1=ρμ+γaρcρ000000.
Hence,Re is the spectral radius of NGM given by(21)Re=ρFV−1=ρμ+γ.


### 3.4. Equilibria and Stability Analysis

#### 3.4.1. Equilibrium State

Investigating long-time dynamics of the model, the process starts by equating (1) to zero (i.e., at equilibrium for the model):(22)Λ−qβGCN+γEF+αSF−G=0,ρSFEF+FC−σEFFC−μ+γEF=0,ρSFEF+FC−σEFFC−μ+γEF=0,σEFFC−η+μ+dF+yT=0,ηF−μ+d+yT=0.Also, the biological interpretation of this is that for a society free of extremists, the following results were obtained:(23)E0∗=G,SF,EF,F,T=Λμ,0,0,0,0,E1∗=1R0,Λμ−R0,0,0,0,E2∗=Λμ−CR0,CR0,0,0,0,E3∗=Λμ+Cμ+αρμ+αα−γμ−R0Re,CRe,Cμ+αR0Re−1ρ,0,0,E4∗=ΛNμCR0+N,ΛCR0μCR0+N,0,0,0,E5∗=NCμα−γ+ΛρμReCqβ+Nμ+γ,1ReC,CΛqβρ−μμ+γCqβ+Nμ+αμρCqβ+Nμ+γ,0,0.Six equilibrium states were obtained. The first expression in (23) is interpreted as a state whereby the society will be completely free of terrorism, and the general population alone exists. In summary of result (23), for all dynamical states, the model proved that the act of terrorism can be completely eradicated through the threshold parameters R0 and Re and R0<Re.


#### 3.4.2. Stability Analysis (Factor(s) for Abolishment of Ideologies)

To investigate the abolishment of the ideology (stability analysis), we linearize the model (Jacobian matrix) and obtain the eigenvalues similar to the work of Okoye et al. [26]. For the stability analysis for the fanatic free equilibrium (FFE), the following Jacobian is derived from (1):(24)J=−μαγ00qβCN−μ+α−ρSFC−ρSFC000−μ+γ00000−μ+η+dy000η−μ+d+y.λ1=−μ,λ2=−μ+d,λ3=−η+μ+d+y,λ4=−qβSF+α+μ, and λ5=ρ−μ+γ Clearly, the first four eigenvalues of (24) are found to be negative and the last eigenvalue is found to be negative, λ5=ρ−μ+γ<0, which implies that the removal/recovery from ideology must be higher than recruitment; otherwise, the society will not enjoy yearning peace. This is the Routh–Hurwitz criterion for the stability [27] of FFE of the Jacobian matrix J.For fanatic persistent equilibrium (FPE),(25)J=−qβCN+μαγ00qβCN−ρEF+FC+μ+α−ρSFC−ρSFC200ρEF+FCρSFC−σFC−μ+γρSFC−σFC000σFC−μ+η+dy000η−μ+d+y,and for simplicity, several assumptions were made on N and C in line with the equilibria stated above. To analyze the stability of persistence of the extremist in the society, the concept of Bellman and Cooke is employed. Next is the statement of the result by Bellman and Cooke cited in [28, 29].Theorem 5 (Bellman and Cooke, 1963).
LetHz=Pz,ex where Pz,w is a polynomial with principal term.
SupposeHiy,y∈R, is separated into real and imaginary parts:(26)Hiy=Fy+iGy.
If the zeros ofHy have negative real parts, then the zeros of Fy and Gy are real, simple, and alternate, and(27)F0G′0−G′0G0>0,∀y∈R.
Conversely, all zeros ofHz will be in the left half plane provided that either of the following conditions is satisfied:(i)
The zeros ofFy and Gy are real, simple, and alternate, and inequality (24) is satisfied at least for one y.(ii)
All zeros ofFy are real, and for each zero, relation (24) is satisfied.(iii)
All zeros ofGy are real, and for each zero, relation (24) is satisfied.Proof.
Since (1) has Jacobian matrix in (22), which has the characteristic polynomial that can be separated into real and imaginary (23), by the hypothesis of the theorem,(28)Hiλ=Fλ+iGλ,where(29)Fλ=λ5+Fσw−SFρx−aqβ+αqβ2+bqβ+cqβ−dqβ−μqβ−qyβ+ab+ac−b±b2−4ac2a−ad−2aμ−ay−bc+bd+2bμ+by+ηy+cd+2cμ+cy−dμ−μ2−μyλ3+−FSFqρσxβ−FSFdρσx−2FSFμρσx−FSFρσxy+Faqσwβ−Fαqσwβ+Fdqσwβ+Fμqσwβ+Fqσwyβ+SFcqρxβ−SFdqρxβ−SFμqρxβ−SFqρxyβ+Fadσw+2Faμσw+Faσwy+Fdμσw+Fμ2σw+Fμσwy+SFηρxy+SFcdρx+2SFcμρx+SFcρxy−SFdμρx−SFμ2ρx−SFμρxy−abcqβ+abdqβ+abμqβ+abqyβ+aβqyβ+acdqβ+acμqβ+acqyβ+αbcqβ−αbdqβ−bcμqβ−bcqyβ−cγqxβ+dγqxβ+γμqxβ+γqxyβ−abηy−abcd−2abcμ−abcy+abdμ+abμ2+abμy+aημy+acdμ+acμ2+acμy−bημy−bcdμ−bcμ2−bcμyλ,Gλ=−qβ−a+b+c−d−2μ−yλ4+−FSFρσx+Fqσwβ−SFqρxβ+Faσw+Fdσw+2Fμσw+Fσwy+SFcρx−SFdρx−2SFμρx−SFρxy+abqβ+acqβ−adqβ−aμqβ−aqyβ−αbqβ−αcqβ+αdqβ+αμqβ+αqyβ−bcqβ+bdqβ+bμqβ+bqyβ+βqyβ+cdqβ+cμqβ+cqyβ+γqxβ−abc+abd+2abμ+aby+aηy+acd+2acμ+acy−adμ−aμ2−aμy−bηy−bcd−2bcμ−bcy+bdμ+bμ2+bμy+ημy+cdμ+cμ2+cμyλ2+FSFdqρσxβ+FSFμqρσxβ+FSFqρσxyβ+FSFdμρσx+FSFμ2ρσx+FSFμρσxy−Fadqσwβ−Faμqσwβ−Faqσwyβ+Fαdqσwβ+Fαμqσwβ+Fαqσwyβ−SFβqρxyβ−SFcdqρxβ−SFcμqρxβ−SFcqρxyβ−Fadμσw−Faμ2σw−Faμσwy−SFημρxy−SFcdμρx−SFcμ2ρx−SFcμρxy+abβqyβ+abcdqβ+abcμqβ+abcqyβ−αbβqyβ−αbcdqβ−αbcμqβ−αbcqyβ+βγqxyβ+cdγqxβ+cγμqxβ+cγqxyβ+abημy+abcdμ+abcμ2+abcμy.
The characteristic polynomial could simply be written as(30)Hiλ=a0λ5+a1λ4+a2λ3+a3λ2+a4λ+a5.
It can be separated into real and imaginary parts as(31)Fλ+iGλ,where a1,a3, and a5 are the coefficients of F(t) and a0,a2, and a4 are the coefficients of G(t).
Then,(32)F0G′0−F′0G0=a4a5>0,where(33)a4=−FSFqρσxβ−FSFdρσx−2FSFμρσx−FSFρσxy+Faqσwβ−Fαqσwβ+Fdqσwβ+Fμqσwβ+Fqσwyβ+SFcqρxβ−SFdqρxβ−SFμqρxβ−SFqρxyβ+Fadσw+2Faμσw+Faσwy+Fdμσw+Fμ2σw+Fμσwy+SFηρxy+SFcdρx+2SFcμρx+SFcρxy−SFdμρx−SFμ2ρx−SFμρxy−abcqβ+abdqβ+abμqβ+abqyβ+aβqyβ+acdqβ+acμqβ+acqyβ+αbcqβ−αbdqβ−αbμqβ−αbqyβ−αβqyβ−αcdqβ−αcμqβ−αcqyβ−bβqyβ−bcdqβ−bcμqβ−bcqyβ−cγqxβ+dγqxβ+γμqxβ+acμy−bημy−bcdμ−bcμ2−bcμy,a5=FSFdqρσxβ+FSFμqρσxβ+FSFqρσxyβ+FSFdμρσx+FSFμ2ρσx+FSFμρσxy−Fadqσwβ−Faμqσwβ−Faqσwyβ+Fαdqσwβ+Fαμqσwβ+Fαqσwyβ−SFβqρxyβ−SFcdqρxβ−SFcμqρxβ−SFcqρxyβ−Fadμσw−Faμ2σw−Faμσwy−SFημρxy−SFcdμρx−SFcμ2ρx−SFcμρxy+abβqyβ+abcdqβ+abcμqβ+abcqyβ−αbβqyβ−αbcdqβ−αbcμqβ−αbcqyβ+βγqxyβ+cdγqxβ+cγμqxβ+cγqxyβ+abημy+abcdμ+abcμ2+abcμy.
Multiplyinga4 by a5 satisfies the inequality of (27).
Hence, the terrorism persistence equilibrium stability is established as stable, and this completes the proof.
Next is a numerical simulation of the model, and Table1 contains the parameters and values for the simulation.Table 1
Description of parameters and variables of the model and value used for numerical simulation.
Parameter/variableDescriptionValueSourceΛRecruitment into the general population (birth/immigration)500AssumedβContact rate of core member withG(t)0–1.0RandomnessαPer capita recovery rate ofS(t) to G(t)0.1AssumedρThe force of embrace of ideology toEt, the semi-fanatic class (conversion rate to E(t))0.2AssumedμNatural death rate (assumed to be constant)0.0181[30]σRate of progression ofEt to Ft0.3AssumedγMeasure of recovery of semi-fanatics toG(t)0.1AssumedqSuccessful contact rate to recruitment of members to the class of core group (susceptible)0.5EstimatedyThe proportion of survived fanatics returning to their settlement (fanatics) after committing terror0.35EstimatedηProportion of extremistsFt that move to commit terror0.4EstimateddorδInduced death rate due to extremism of ideology0.05AssumedGNoncore group500EstimatedSFThe naive (susceptible/vulnerable) population to fanatic group85AssumedFIndividuals who have completely internalized themselves in the ideology (fanatic)40AssumedESemi-fanatic group50AssumedTGroup of extremists who commit terror acts60AssumedNGeneral population735Estimated from [31]


### 3.5. Numerical Simulation and Results

#### 3.5.1. Parameterization and General Dynamics

While some parameter values were picked from existing literature, the World Bank data for Nigeria population were estimated for the state variables for numerical simulation. Between year 2002 and 2021, the difference in Nigeria population is 72,535,692, and we set the model general population to 735 million (approximate). Then, 1% of 735,000,00 of Nigeria population (735 in 000) is used for simulation.According to [32], an exact number of Boko Haram troops is unknown but estimated to be at least 15,000 with the fact that in January 2015, the group took complete control of 15 local governments in the northeastern Nigeria, and BBC News [33] reported 90 armed assaults with 59 suicide attacks in 2017. Except Boko–Haram, some other religious extremist and domestic and commercial violence perpetrated by individuals or group of people are yet to be officially recognized and declared as terrorist despite height of violent to either self or other members of the society. Thus, we include these people in our assumed values for simulation and the assumed value for ft,Et,Tt to be 40, 50, and 60, respectively.For the stability analysis at persistent equilibrium and when simulated with the parament value in Table1, the Jacobian in (25) is(34)J−λI=−0.1319727891−λ0.10.1000.03197278912−0.2765957447−λ−0.07234042553−0.07234042553000.07659574468−0.1787234043−λ0.021276595740000.05106382979−0.35−λ0.350000.2−0.50−λ,and the characteristic equation is(35)J−λI=λ5+1.437291938λ4+0.7149792438λ3+0.1624877869λ2+0.01713983038λ+0.000676661147=0.By the hypothesis of the theorem and in comparison,a4=0.000676661147 and a5=0.01713983038.We established that(36)a4∗a5=1.159786E−5>0.Also, the persistence equilibrium is stable.Figure 3
Model simulation. (a) The effect of increase in efforts of fanatical (β2=0,0.2,0.4,0.6,0.8,1.0) for recruitment into ideology on G(t). At start, many people seem to be embracing the ideology (i.e., the first 10-20 months of preaching the ideology, recruitment is growing), but beyond 20 months of preaching, recruitment becomes asymptotic for around 3000 people limit as seen in (a) (this proves assumption 4 of the propose model). (b) Success rate of fanatical recruiting into extreme fanatics. The simulation assumed ineffective implementation of government policy to curtail the menace will consequentially cause the fall in G(t) below threshold; 3000 population in (a). (c) The behaviour of the core groups in the absence of extremism, β[2] and ρ[2] are zero (i.e., effective implementation of government policy will eradicate extremism in the society, and the environment will be peaceful), while (d) dynamics of core populations at successful contact rate on the noncore group and successful embracement of ideology by fanatics (i.e., β[2] and ρ[2], respectively, are not zero, but at 20% minimum efforts).
(a)(b)(c)(d)


## 3.1. Interpretation

The result obtained in (6) showed that the population cannot be zero. Hence, validate the integral claim of assumption nine (8).


## 3.2. Existence and Uniqueness of the Solution

In mathematics, posing a problem is not enough if a solution does not exist and the uniqueness of such a solution is essential in some instances. Here, the existence and uniqueness of the model shall be tested using the Lipschitz condition.Theorem 2.
Suppose (1) is written in compact form as Fit,x,xt0=x0,i=1,2,…,5, and there exists a unique solution; then, (1) satisfies the Lipschitz condition.Proof.
Rewrite model equation (1) as(11)F1t,G=Λ−qβGCN+γEF+αSF−μG,F2t,SF=qβGCN−ρSFEF+FC−μ+αSF,F3t,EF=ρSFEF+FC−σEFFC−μ+γEF,F4t,F=σEFFC−η+μ+dF+yT,F5t,T=ηF−μ+d+yT,and define equation (1) in the form Xi′=Fit,x,xt0=x0,i=1,2,…,5; then,(12)Fit,x2−Fit,x1≤Mx2−x1,and by illustration,(13)F1t,G2−F1t,G1=−qβC2N+μG2−−qβC2N+μG1=−qβC2N+μG2−G1≤−qβC2N+μG2−G1,F2t,SF1−F2t,SF1=−−ρEF+FC+μ+αSF2−−ρEF+FC+μ+αSF1≤−−ρEF+FC+μ+αSF2−SF1,F3t,EF2−F3t,EF1=−ρSF−σFC−μ+γEF2−ρSF−σFC−μ+γEF1≤ρSF−σFC−μ+γEF2−EF1.
For argument sake, letF′ be denoted as the usual F for a fanatic to differentiate between itself and Lipschitz function.(14)F4t,F2′−F4t,F1′=σEFC−μ+η+d+yF2′−σEFC−μ+η+d+yF1′≤−σEFC−μ+η+d+yF2′−F1′,F5t,T2−F5t,T1=−ηF′+μ+y+dT2−ηF′+μ+y+dT1≤ηF′+μ+y+dT2−T1.Remark 1.
(i)
A system of ODEs is said to exist and is said to be unique in the solution if it satisfies the Lipschitz condition.(ii)
Lipschitz conditions cater for continuity and boundedness. According to [24], first-order ODE is said to exist and is said to be unique if the partial derivatives of the function exist. This is stated in the next theorem relevant to the proposed model. Hence, the proposed model is said to be mathematically well-posed.Theorem 3.
LetΩ be denoted the region stated in Theorem 1 and bounded in 0≤R<∞. It suffices to show that ∂fit,x/∂x,i=115,x=G,SF,EF,F,T are continuous in Ω.Proof.
Rewrite equation (1) as(15)F1t,G,SF,EF,F,T=Λ−qβGCN+γEF+αSF−μG,F2t,G,SF,EF,F,T=qβGCN−ρSFEF+FC−μ+αSF,F3t,G,SF,EF,F,T=ρSFEF+FC−σEFFC−μ+γEF,F4t,G,SF,EF,F,T=σEFFC−η+μ+dF+yT,F5t,G,SF,EF,F,T=ηF−μ+d+yT,(16)∂F1∂G=−qβCN−μ<∞,∂F1∂SF=α<∞,∂F1∂EF=γ<∞,∂F1∂F=α<∞,∂F1∂T=α<∞,∂F2∂G=qβCN<∞,∂F2∂SF=ρEF+FC+μ+α<∞,∂F2∂EF=−ρSFC<∞,∂F2∂F=−ρSFC<∞,∂F2∂T=|0|<∞,∂F3∂G=|0|<∞,∂F3∂SF=ρEF+FC<∞,∂F3∂EF=ρSF−σFC−μ+γ<∞,∂F3∂F=ρSF−σEFC<∞,∂F3∂T=|0|<∞,∂F4∂G=0<∞,∂F4∂SF=0<∞,∂F4∂EF=|σFC|<∞,∂F4∂F=σEFC−η+μ+d<∞,∂F4∂T=y<∞,∂F5∂G=0<∞,∂F5∂SF=0<∞,∂F5∂EF=0<∞,∂F5∂F=0<∞,∂F5∂T=−μ+d+y<∞.
Hence,∂Fi/∂x∈R<∞,fori=1,2,…,5andx=G,SF,EF,F,T exist and have solution in feasible region.
This completes the proof.


### 3.2.1. Interpretation of the Result

The result of existence and uniqueness is that the problem of extremism is in the human population and the solution also lies therein.


## 3.2.1. Interpretation of the Result

The result of existence and uniqueness is that the problem of extremism is in the human population and the solution also lies therein.


## 3.3. Threshold for Recruitment into the Ideology

The long-term sustainability of the core subpopulation is analyzed by using the concept of basic reproduction number (BRN) in epidemiology [25]. With this BRN, one can deduce that for certain parameter values, the model predicts the extinction of the core population ct, and effective recruitment shows that the core population will persist (endemic in epidemiology) [16]. Next theorem gives the thresholds for recruitment into the ideology.Theorem 4.
(i)
Assumen=minα,γ and R0,min≡qβ/μ+n. Then, limt⟶∞Ct⟶0.(ii)
Effective reproductive ratioRe=ρ/μ+γ.Proof.
SinceC=S+E+F+T, then(17)dCdt=qβ1−CC−μ+αS−μ+γE−μ+δF−μ+δT,=qβ1−CC−μ+nC−δF+T,wheren=α,γ,≤qβ1−C−μ+nC,=−μ+nC1−qβ2μ+n<0,and by the hypothesis of the theorem,(18)R0≡qβμ+n.
Hence,dC/dt=−μ+nC1−R0<0.
It could be seen from the above inequality thatCt decays exponentially fast to zero. If R0<1, it implies that the core population cannot be established, but from the assumptions of the model, the general population can never be extinct, so there will always be people vulnerable to fanatics. Reducing resistance to vulnerability increases the core population, or the longer the residence time 1/n, the higher the possibility of increased recruitment (i.e., R0≥1).
The approach of next generation matrix (NGM) is employed for determining effective threshold (Re) for recruitment of fanatics. From (1), we extract F and V, which, respectively, denote new infection and other transmissions in the model, i.e.,(19)F=ρSFCρSFC0000000,V=γ+μ000β+μ+d−y0−βμ+d+y,V−1=1γ+μ000ac0bd,where “a” =μ+β+d/μ+d+y+βμ+d, “b” =−β/μ+d+y+βμ+d, “c” =−y/μ+d+y+βμ+d, and “d” =μ+d+y/μ+d+y+βμ+d; then, with little algebraic simplification, FV−1 is obtained to be(20)FV−1=ρμ+γaρcρ000000.
Hence,Re is the spectral radius of NGM given by(21)Re=ρFV−1=ρμ+γ.


## 3.4. Equilibria and Stability Analysis

### 3.4.1. Equilibrium State

Investigating long-time dynamics of the model, the process starts by equating (1) to zero (i.e., at equilibrium for the model):(22)Λ−qβGCN+γEF+αSF−G=0,ρSFEF+FC−σEFFC−μ+γEF=0,ρSFEF+FC−σEFFC−μ+γEF=0,σEFFC−η+μ+dF+yT=0,ηF−μ+d+yT=0.Also, the biological interpretation of this is that for a society free of extremists, the following results were obtained:(23)E0∗=G,SF,EF,F,T=Λμ,0,0,0,0,E1∗=1R0,Λμ−R0,0,0,0,E2∗=Λμ−CR0,CR0,0,0,0,E3∗=Λμ+Cμ+αρμ+αα−γμ−R0Re,CRe,Cμ+αR0Re−1ρ,0,0,E4∗=ΛNμCR0+N,ΛCR0μCR0+N,0,0,0,E5∗=NCμα−γ+ΛρμReCqβ+Nμ+γ,1ReC,CΛqβρ−μμ+γCqβ+Nμ+αμρCqβ+Nμ+γ,0,0.Six equilibrium states were obtained. The first expression in (23) is interpreted as a state whereby the society will be completely free of terrorism, and the general population alone exists. In summary of result (23), for all dynamical states, the model proved that the act of terrorism can be completely eradicated through the threshold parameters R0 and Re and R0<Re.


### 3.4.2. Stability Analysis (Factor(s) for Abolishment of Ideologies)

To investigate the abolishment of the ideology (stability analysis), we linearize the model (Jacobian matrix) and obtain the eigenvalues similar to the work of Okoye et al. [26]. For the stability analysis for the fanatic free equilibrium (FFE), the following Jacobian is derived from (1):(24)J=−μαγ00qβCN−μ+α−ρSFC−ρSFC000−μ+γ00000−μ+η+dy000η−μ+d+y.λ1=−μ,λ2=−μ+d,λ3=−η+μ+d+y,λ4=−qβSF+α+μ, and λ5=ρ−μ+γ Clearly, the first four eigenvalues of (24) are found to be negative and the last eigenvalue is found to be negative, λ5=ρ−μ+γ<0, which implies that the removal/recovery from ideology must be higher than recruitment; otherwise, the society will not enjoy yearning peace. This is the Routh–Hurwitz criterion for the stability [27] of FFE of the Jacobian matrix J.For fanatic persistent equilibrium (FPE),(25)J=−qβCN+μαγ00qβCN−ρEF+FC+μ+α−ρSFC−ρSFC200ρEF+FCρSFC−σFC−μ+γρSFC−σFC000σFC−μ+η+dy000η−μ+d+y,and for simplicity, several assumptions were made on N and C in line with the equilibria stated above. To analyze the stability of persistence of the extremist in the society, the concept of Bellman and Cooke is employed. Next is the statement of the result by Bellman and Cooke cited in [28, 29].Theorem 5 (Bellman and Cooke, 1963).
LetHz=Pz,ex where Pz,w is a polynomial with principal term.
SupposeHiy,y∈R, is separated into real and imaginary parts:(26)Hiy=Fy+iGy.
If the zeros ofHy have negative real parts, then the zeros of Fy and Gy are real, simple, and alternate, and(27)F0G′0−G′0G0>0,∀y∈R.
Conversely, all zeros ofHz will be in the left half plane provided that either of the following conditions is satisfied:(i)
The zeros ofFy and Gy are real, simple, and alternate, and inequality (24) is satisfied at least for one y.(ii)
All zeros ofFy are real, and for each zero, relation (24) is satisfied.(iii)
All zeros ofGy are real, and for each zero, relation (24) is satisfied.Proof.
Since (1) has Jacobian matrix in (22), which has the characteristic polynomial that can be separated into real and imaginary (23), by the hypothesis of the theorem,(28)Hiλ=Fλ+iGλ,where(29)Fλ=λ5+Fσw−SFρx−aqβ+αqβ2+bqβ+cqβ−dqβ−μqβ−qyβ+ab+ac−b±b2−4ac2a−ad−2aμ−ay−bc+bd+2bμ+by+ηy+cd+2cμ+cy−dμ−μ2−μyλ3+−FSFqρσxβ−FSFdρσx−2FSFμρσx−FSFρσxy+Faqσwβ−Fαqσwβ+Fdqσwβ+Fμqσwβ+Fqσwyβ+SFcqρxβ−SFdqρxβ−SFμqρxβ−SFqρxyβ+Fadσw+2Faμσw+Faσwy+Fdμσw+Fμ2σw+Fμσwy+SFηρxy+SFcdρx+2SFcμρx+SFcρxy−SFdμρx−SFμ2ρx−SFμρxy−abcqβ+abdqβ+abμqβ+abqyβ+aβqyβ+acdqβ+acμqβ+acqyβ+αbcqβ−αbdqβ−bcμqβ−bcqyβ−cγqxβ+dγqxβ+γμqxβ+γqxyβ−abηy−abcd−2abcμ−abcy+abdμ+abμ2+abμy+aημy+acdμ+acμ2+acμy−bημy−bcdμ−bcμ2−bcμyλ,Gλ=−qβ−a+b+c−d−2μ−yλ4+−FSFρσx+Fqσwβ−SFqρxβ+Faσw+Fdσw+2Fμσw+Fσwy+SFcρx−SFdρx−2SFμρx−SFρxy+abqβ+acqβ−adqβ−aμqβ−aqyβ−αbqβ−αcqβ+αdqβ+αμqβ+αqyβ−bcqβ+bdqβ+bμqβ+bqyβ+βqyβ+cdqβ+cμqβ+cqyβ+γqxβ−abc+abd+2abμ+aby+aηy+acd+2acμ+acy−adμ−aμ2−aμy−bηy−bcd−2bcμ−bcy+bdμ+bμ2+bμy+ημy+cdμ+cμ2+cμyλ2+FSFdqρσxβ+FSFμqρσxβ+FSFqρσxyβ+FSFdμρσx+FSFμ2ρσx+FSFμρσxy−Fadqσwβ−Faμqσwβ−Faqσwyβ+Fαdqσwβ+Fαμqσwβ+Fαqσwyβ−SFβqρxyβ−SFcdqρxβ−SFcμqρxβ−SFcqρxyβ−Fadμσw−Faμ2σw−Faμσwy−SFημρxy−SFcdμρx−SFcμ2ρx−SFcμρxy+abβqyβ+abcdqβ+abcμqβ+abcqyβ−αbβqyβ−αbcdqβ−αbcμqβ−αbcqyβ+βγqxyβ+cdγqxβ+cγμqxβ+cγqxyβ+abημy+abcdμ+abcμ2+abcμy.
The characteristic polynomial could simply be written as(30)Hiλ=a0λ5+a1λ4+a2λ3+a3λ2+a4λ+a5.
It can be separated into real and imaginary parts as(31)Fλ+iGλ,where a1,a3, and a5 are the coefficients of F(t) and a0,a2, and a4 are the coefficients of G(t).
Then,(32)F0G′0−F′0G0=a4a5>0,where(33)a4=−FSFqρσxβ−FSFdρσx−2FSFμρσx−FSFρσxy+Faqσwβ−Fαqσwβ+Fdqσwβ+Fμqσwβ+Fqσwyβ+SFcqρxβ−SFdqρxβ−SFμqρxβ−SFqρxyβ+Fadσw+2Faμσw+Faσwy+Fdμσw+Fμ2σw+Fμσwy+SFηρxy+SFcdρx+2SFcμρx+SFcρxy−SFdμρx−SFμ2ρx−SFμρxy−abcqβ+abdqβ+abμqβ+abqyβ+aβqyβ+acdqβ+acμqβ+acqyβ+αbcqβ−αbdqβ−αbμqβ−αbqyβ−αβqyβ−αcdqβ−αcμqβ−αcqyβ−bβqyβ−bcdqβ−bcμqβ−bcqyβ−cγqxβ+dγqxβ+γμqxβ+acμy−bημy−bcdμ−bcμ2−bcμy,a5=FSFdqρσxβ+FSFμqρσxβ+FSFqρσxyβ+FSFdμρσx+FSFμ2ρσx+FSFμρσxy−Fadqσwβ−Faμqσwβ−Faqσwyβ+Fαdqσwβ+Fαμqσwβ+Fαqσwyβ−SFβqρxyβ−SFcdqρxβ−SFcμqρxβ−SFcqρxyβ−Fadμσw−Faμ2σw−Faμσwy−SFημρxy−SFcdμρx−SFcμ2ρx−SFcμρxy+abβqyβ+abcdqβ+abcμqβ+abcqyβ−αbβqyβ−αbcdqβ−αbcμqβ−αbcqyβ+βγqxyβ+cdγqxβ+cγμqxβ+cγqxyβ+abημy+abcdμ+abcμ2+abcμy.
Multiplyinga4 by a5 satisfies the inequality of (27).
Hence, the terrorism persistence equilibrium stability is established as stable, and this completes the proof.
Next is a numerical simulation of the model, and Table1 contains the parameters and values for the simulation.Table 1
Description of parameters and variables of the model and value used for numerical simulation.
Parameter/variableDescriptionValueSourceΛRecruitment into the general population (birth/immigration)500AssumedβContact rate of core member withG(t)0–1.0RandomnessαPer capita recovery rate ofS(t) to G(t)0.1AssumedρThe force of embrace of ideology toEt, the semi-fanatic class (conversion rate to E(t))0.2AssumedμNatural death rate (assumed to be constant)0.0181[30]σRate of progression ofEt to Ft0.3AssumedγMeasure of recovery of semi-fanatics toG(t)0.1AssumedqSuccessful contact rate to recruitment of members to the class of core group (susceptible)0.5EstimatedyThe proportion of survived fanatics returning to their settlement (fanatics) after committing terror0.35EstimatedηProportion of extremistsFt that move to commit terror0.4EstimateddorδInduced death rate due to extremism of ideology0.05AssumedGNoncore group500EstimatedSFThe naive (susceptible/vulnerable) population to fanatic group85AssumedFIndividuals who have completely internalized themselves in the ideology (fanatic)40AssumedESemi-fanatic group50AssumedTGroup of extremists who commit terror acts60AssumedNGeneral population735Estimated from [31]


## 3.4.1. Equilibrium State

Investigating long-time dynamics of the model, the process starts by equating (1) to zero (i.e., at equilibrium for the model):(22)Λ−qβGCN+γEF+αSF−G=0,ρSFEF+FC−σEFFC−μ+γEF=0,ρSFEF+FC−σEFFC−μ+γEF=0,σEFFC−η+μ+dF+yT=0,ηF−μ+d+yT=0.Also, the biological interpretation of this is that for a society free of extremists, the following results were obtained:(23)E0∗=G,SF,EF,F,T=Λμ,0,0,0,0,E1∗=1R0,Λμ−R0,0,0,0,E2∗=Λμ−CR0,CR0,0,0,0,E3∗=Λμ+Cμ+αρμ+αα−γμ−R0Re,CRe,Cμ+αR0Re−1ρ,0,0,E4∗=ΛNμCR0+N,ΛCR0μCR0+N,0,0,0,E5∗=NCμα−γ+ΛρμReCqβ+Nμ+γ,1ReC,CΛqβρ−μμ+γCqβ+Nμ+αμρCqβ+Nμ+γ,0,0.Six equilibrium states were obtained. The first expression in (23) is interpreted as a state whereby the society will be completely free of terrorism, and the general population alone exists. In summary of result (23), for all dynamical states, the model proved that the act of terrorism can be completely eradicated through the threshold parameters R0 and Re and R0<Re.


## 3.4.2. Stability Analysis (Factor(s) for Abolishment of Ideologies)

To investigate the abolishment of the ideology (stability analysis), we linearize the model (Jacobian matrix) and obtain the eigenvalues similar to the work of Okoye et al. [26]. For the stability analysis for the fanatic free equilibrium (FFE), the following Jacobian is derived from (1):(24)J=−μαγ00qβCN−μ+α−ρSFC−ρSFC000−μ+γ00000−μ+η+dy000η−μ+d+y.λ1=−μ,λ2=−μ+d,λ3=−η+μ+d+y,λ4=−qβSF+α+μ, and λ5=ρ−μ+γ Clearly, the first four eigenvalues of (24) are found to be negative and the last eigenvalue is found to be negative, λ5=ρ−μ+γ<0, which implies that the removal/recovery from ideology must be higher than recruitment; otherwise, the society will not enjoy yearning peace. This is the Routh–Hurwitz criterion for the stability [27] of FFE of the Jacobian matrix J.For fanatic persistent equilibrium (FPE),(25)J=−qβCN+μαγ00qβCN−ρEF+FC+μ+α−ρSFC−ρSFC200ρEF+FCρSFC−σFC−μ+γρSFC−σFC000σFC−μ+η+dy000η−μ+d+y,and for simplicity, several assumptions were made on N and C in line with the equilibria stated above. To analyze the stability of persistence of the extremist in the society, the concept of Bellman and Cooke is employed. Next is the statement of the result by Bellman and Cooke cited in [28, 29].Theorem 5 (Bellman and Cooke, 1963).
LetHz=Pz,ex where Pz,w is a polynomial with principal term.
SupposeHiy,y∈R, is separated into real and imaginary parts:(26)Hiy=Fy+iGy.
If the zeros ofHy have negative real parts, then the zeros of Fy and Gy are real, simple, and alternate, and(27)F0G′0−G′0G0>0,∀y∈R.
Conversely, all zeros ofHz will be in the left half plane provided that either of the following conditions is satisfied:(i)
The zeros ofFy and Gy are real, simple, and alternate, and inequality (24) is satisfied at least for one y.(ii)
All zeros ofFy are real, and for each zero, relation (24) is satisfied.(iii)
All zeros ofGy are real, and for each zero, relation (24) is satisfied.Proof.
Since (1) has Jacobian matrix in (22), which has the characteristic polynomial that can be separated into real and imaginary (23), by the hypothesis of the theorem,(28)Hiλ=Fλ+iGλ,where(29)Fλ=λ5+Fσw−SFρx−aqβ+αqβ2+bqβ+cqβ−dqβ−μqβ−qyβ+ab+ac−b±b2−4ac2a−ad−2aμ−ay−bc+bd+2bμ+by+ηy+cd+2cμ+cy−dμ−μ2−μyλ3+−FSFqρσxβ−FSFdρσx−2FSFμρσx−FSFρσxy+Faqσwβ−Fαqσwβ+Fdqσwβ+Fμqσwβ+Fqσwyβ+SFcqρxβ−SFdqρxβ−SFμqρxβ−SFqρxyβ+Fadσw+2Faμσw+Faσwy+Fdμσw+Fμ2σw+Fμσwy+SFηρxy+SFcdρx+2SFcμρx+SFcρxy−SFdμρx−SFμ2ρx−SFμρxy−abcqβ+abdqβ+abμqβ+abqyβ+aβqyβ+acdqβ+acμqβ+acqyβ+αbcqβ−αbdqβ−bcμqβ−bcqyβ−cγqxβ+dγqxβ+γμqxβ+γqxyβ−abηy−abcd−2abcμ−abcy+abdμ+abμ2+abμy+aημy+acdμ+acμ2+acμy−bημy−bcdμ−bcμ2−bcμyλ,Gλ=−qβ−a+b+c−d−2μ−yλ4+−FSFρσx+Fqσwβ−SFqρxβ+Faσw+Fdσw+2Fμσw+Fσwy+SFcρx−SFdρx−2SFμρx−SFρxy+abqβ+acqβ−adqβ−aμqβ−aqyβ−αbqβ−αcqβ+αdqβ+αμqβ+αqyβ−bcqβ+bdqβ+bμqβ+bqyβ+βqyβ+cdqβ+cμqβ+cqyβ+γqxβ−abc+abd+2abμ+aby+aηy+acd+2acμ+acy−adμ−aμ2−aμy−bηy−bcd−2bcμ−bcy+bdμ+bμ2+bμy+ημy+cdμ+cμ2+cμyλ2+FSFdqρσxβ+FSFμqρσxβ+FSFqρσxyβ+FSFdμρσx+FSFμ2ρσx+FSFμρσxy−Fadqσwβ−Faμqσwβ−Faqσwyβ+Fαdqσwβ+Fαμqσwβ+Fαqσwyβ−SFβqρxyβ−SFcdqρxβ−SFcμqρxβ−SFcqρxyβ−Fadμσw−Faμ2σw−Faμσwy−SFημρxy−SFcdμρx−SFcμ2ρx−SFcμρxy+abβqyβ+abcdqβ+abcμqβ+abcqyβ−αbβqyβ−αbcdqβ−αbcμqβ−αbcqyβ+βγqxyβ+cdγqxβ+cγμqxβ+cγqxyβ+abημy+abcdμ+abcμ2+abcμy.
The characteristic polynomial could simply be written as(30)Hiλ=a0λ5+a1λ4+a2λ3+a3λ2+a4λ+a5.
It can be separated into real and imaginary parts as(31)Fλ+iGλ,where a1,a3, and a5 are the coefficients of F(t) and a0,a2, and a4 are the coefficients of G(t).
Then,(32)F0G′0−F′0G0=a4a5>0,where(33)a4=−FSFqρσxβ−FSFdρσx−2FSFμρσx−FSFρσxy+Faqσwβ−Fαqσwβ+Fdqσwβ+Fμqσwβ+Fqσwyβ+SFcqρxβ−SFdqρxβ−SFμqρxβ−SFqρxyβ+Fadσw+2Faμσw+Faσwy+Fdμσw+Fμ2σw+Fμσwy+SFηρxy+SFcdρx+2SFcμρx+SFcρxy−SFdμρx−SFμ2ρx−SFμρxy−abcqβ+abdqβ+abμqβ+abqyβ+aβqyβ+acdqβ+acμqβ+acqyβ+αbcqβ−αbdqβ−αbμqβ−αbqyβ−αβqyβ−αcdqβ−αcμqβ−αcqyβ−bβqyβ−bcdqβ−bcμqβ−bcqyβ−cγqxβ+dγqxβ+γμqxβ+acμy−bημy−bcdμ−bcμ2−bcμy,a5=FSFdqρσxβ+FSFμqρσxβ+FSFqρσxyβ+FSFdμρσx+FSFμ2ρσx+FSFμρσxy−Fadqσwβ−Faμqσwβ−Faqσwyβ+Fαdqσwβ+Fαμqσwβ+Fαqσwyβ−SFβqρxyβ−SFcdqρxβ−SFcμqρxβ−SFcqρxyβ−Fadμσw−Faμ2σw−Faμσwy−SFημρxy−SFcdμρx−SFcμ2ρx−SFcμρxy+abβqyβ+abcdqβ+abcμqβ+abcqyβ−αbβqyβ−αbcdqβ−αbcμqβ−αbcqyβ+βγqxyβ+cdγqxβ+cγμqxβ+cγqxyβ+abημy+abcdμ+abcμ2+abcμy.
Multiplyinga4 by a5 satisfies the inequality of (27).
Hence, the terrorism persistence equilibrium stability is established as stable, and this completes the proof.
Next is a numerical simulation of the model, and Table1 contains the parameters and values for the simulation.Table 1
Description of parameters and variables of the model and value used for numerical simulation.
Parameter/variableDescriptionValueSourceΛRecruitment into the general population (birth/immigration)500AssumedβContact rate of core member withG(t)0–1.0RandomnessαPer capita recovery rate ofS(t) to G(t)0.1AssumedρThe force of embrace of ideology toEt, the semi-fanatic class (conversion rate to E(t))0.2AssumedμNatural death rate (assumed to be constant)0.0181[30]σRate of progression ofEt to Ft0.3AssumedγMeasure of recovery of semi-fanatics toG(t)0.1AssumedqSuccessful contact rate to recruitment of members to the class of core group (susceptible)0.5EstimatedyThe proportion of survived fanatics returning to their settlement (fanatics) after committing terror0.35EstimatedηProportion of extremistsFt that move to commit terror0.4EstimateddorδInduced death rate due to extremism of ideology0.05AssumedGNoncore group500EstimatedSFThe naive (susceptible/vulnerable) population to fanatic group85AssumedFIndividuals who have completely internalized themselves in the ideology (fanatic)40AssumedESemi-fanatic group50AssumedTGroup of extremists who commit terror acts60AssumedNGeneral population735Estimated from [31]


## 3.5. Numerical Simulation and Results

### 3.5.1. Parameterization and General Dynamics

While some parameter values were picked from existing literature, the World Bank data for Nigeria population were estimated for the state variables for numerical simulation. Between year 2002 and 2021, the difference in Nigeria population is 72,535,692, and we set the model general population to 735 million (approximate). Then, 1% of 735,000,00 of Nigeria population (735 in 000) is used for simulation.According to [32], an exact number of Boko Haram troops is unknown but estimated to be at least 15,000 with the fact that in January 2015, the group took complete control of 15 local governments in the northeastern Nigeria, and BBC News [33] reported 90 armed assaults with 59 suicide attacks in 2017. Except Boko–Haram, some other religious extremist and domestic and commercial violence perpetrated by individuals or group of people are yet to be officially recognized and declared as terrorist despite height of violent to either self or other members of the society. Thus, we include these people in our assumed values for simulation and the assumed value for ft,Et,Tt to be 40, 50, and 60, respectively.For the stability analysis at persistent equilibrium and when simulated with the parament value in Table1, the Jacobian in (25) is(34)J−λI=−0.1319727891−λ0.10.1000.03197278912−0.2765957447−λ−0.07234042553−0.07234042553000.07659574468−0.1787234043−λ0.021276595740000.05106382979−0.35−λ0.350000.2−0.50−λ,and the characteristic equation is(35)J−λI=λ5+1.437291938λ4+0.7149792438λ3+0.1624877869λ2+0.01713983038λ+0.000676661147=0.By the hypothesis of the theorem and in comparison,a4=0.000676661147 and a5=0.01713983038.We established that(36)a4∗a5=1.159786E−5>0.Also, the persistence equilibrium is stable.Figure 3
Model simulation. (a) The effect of increase in efforts of fanatical (β2=0,0.2,0.4,0.6,0.8,1.0) for recruitment into ideology on G(t). At start, many people seem to be embracing the ideology (i.e., the first 10-20 months of preaching the ideology, recruitment is growing), but beyond 20 months of preaching, recruitment becomes asymptotic for around 3000 people limit as seen in (a) (this proves assumption 4 of the propose model). (b) Success rate of fanatical recruiting into extreme fanatics. The simulation assumed ineffective implementation of government policy to curtail the menace will consequentially cause the fall in G(t) below threshold; 3000 population in (a). (c) The behaviour of the core groups in the absence of extremism, β[2] and ρ[2] are zero (i.e., effective implementation of government policy will eradicate extremism in the society, and the environment will be peaceful), while (d) dynamics of core populations at successful contact rate on the noncore group and successful embracement of ideology by fanatics (i.e., β[2] and ρ[2], respectively, are not zero, but at 20% minimum efforts).
(a)(b)(c)(d)


## 3.5.1. Parameterization and General Dynamics

While some parameter values were picked from existing literature, the World Bank data for Nigeria population were estimated for the state variables for numerical simulation. Between year 2002 and 2021, the difference in Nigeria population is 72,535,692, and we set the model general population to 735 million (approximate). Then, 1% of 735,000,00 of Nigeria population (735 in 000) is used for simulation.According to [32], an exact number of Boko Haram troops is unknown but estimated to be at least 15,000 with the fact that in January 2015, the group took complete control of 15 local governments in the northeastern Nigeria, and BBC News [33] reported 90 armed assaults with 59 suicide attacks in 2017. Except Boko–Haram, some other religious extremist and domestic and commercial violence perpetrated by individuals or group of people are yet to be officially recognized and declared as terrorist despite height of violent to either self or other members of the society. Thus, we include these people in our assumed values for simulation and the assumed value for ft,Et,Tt to be 40, 50, and 60, respectively.For the stability analysis at persistent equilibrium and when simulated with the parament value in Table1, the Jacobian in (25) is(34)J−λI=−0.1319727891−λ0.10.1000.03197278912−0.2765957447−λ−0.07234042553−0.07234042553000.07659574468−0.1787234043−λ0.021276595740000.05106382979−0.35−λ0.350000.2−0.50−λ,and the characteristic equation is(35)J−λI=λ5+1.437291938λ4+0.7149792438λ3+0.1624877869λ2+0.01713983038λ+0.000676661147=0.By the hypothesis of the theorem and in comparison,a4=0.000676661147 and a5=0.01713983038.We established that(36)a4∗a5=1.159786E−5>0.Also, the persistence equilibrium is stable.Figure 3
Model simulation. (a) The effect of increase in efforts of fanatical (β2=0,0.2,0.4,0.6,0.8,1.0) for recruitment into ideology on G(t). At start, many people seem to be embracing the ideology (i.e., the first 10-20 months of preaching the ideology, recruitment is growing), but beyond 20 months of preaching, recruitment becomes asymptotic for around 3000 people limit as seen in (a) (this proves assumption 4 of the propose model). (b) Success rate of fanatical recruiting into extreme fanatics. The simulation assumed ineffective implementation of government policy to curtail the menace will consequentially cause the fall in G(t) below threshold; 3000 population in (a). (c) The behaviour of the core groups in the absence of extremism, β[2] and ρ[2] are zero (i.e., effective implementation of government policy will eradicate extremism in the society, and the environment will be peaceful), while (d) dynamics of core populations at successful contact rate on the noncore group and successful embracement of ideology by fanatics (i.e., β[2] and ρ[2], respectively, are not zero, but at 20% minimum efforts).
(a)(b)(c)(d)


## 4. Discussion of Results and Conclusion

This paper presents the novel model dynamics of fanatical group progress on terrorism, and it improved the work Chavez and Song [25]. The model conceptualized the mathematical modelling of infectious disease by fanatics, in which extremism is considered as infection of the susceptible group. The model was checked by some mathematical properties, e.g., the existence and uniqueness of solution, positivity, and region of feasibility of solution. This is necessary for merging real-life events with the abstraction of mathematics. The proposed model is analyzed and synthesized using simple algebra and numercial concepts for understanding the model dynamics. The important results obtained include equilibrium points (i.e., equation 23), threshold parameter (equation 18), and stability analyses in equations 24 and 32. All these results are essential for the society to coexist peacefully and help the decision maker gain good insight on the dynamics of terrorism for effective management. (result in Section 3.5.1 and Table 2). Figures 3 and 4 discuss the findings from simulation. Figure 3(a) presents the noncore human population of the model. The variation in the plots is because of introducing at least a fanatic into the population, who interacts and tries to recruit members into the ideology. The percentage increase in the efforts of fanatics (β2=00.21), which does not yield positive recruitment into fanatics (ρ=0), is shown in Figure 3(a) while Figure 3(b) illustrates the dynamics in the noncore population with the successful effort of fanatics for recruitment (β2=ρ=00.21) and its effect on the general population. Consequentially, there is recruitment into fanatics in proportion to the successful contact made. Thus, this is alarming and if proactive measures are not taken, the entire world may be in chaos. Figures 4(a)–4(d) show the dynamics of the consequence of successful recruitment into the core population from the general population at different percentages.Table 2
Simulation for stability analysis (this is investigated by varying adequate contact rate of fanatics for recruitment).
Λβαγφμσqyμda4a5J=a4×a5Remark5000.000.100.100.200.010.300.500.340.400.05−9.926×10−5−1.560×10−61.548×10−10Stable5000.100.100.100.200.010.300.500.340.400.05−103.353×10−5−11.646×10−50.0120×10−6Stable5000.200.100.100.200.020.300.500.330.400.05−247.338×10−5−5.202×10−50.129×10−6Stable5000.300.100.100.200.030.300.500.320.400.05−450.498×10−5−13.136×10−50.592×10−6Stable5000.400.100.100.200.040.400.500.310.400.05−714.396×10−5−27.207×10−51.944×10−6Stable5000.500.100.100.200.050.400.500.300.400.05−1054.503×10−5−47.345×10−54.993×10−6Stable5000.600.100.100.200.060.400.500.290.400.05−1462.088×10−5−74.934×10−510.956×10−6Stable5000.700.100.100.200.070.400.500.280.400.05−1936.447×10−5−110.945×10−521.484×10−6Stable5000.800.100.100.200.080.500.500.270.400.05−2477.360×10−5−162.819×10−540.336×10−6Stable5000.900.100.100.200.090.500.500.260.400.05−3081.179×10−5−220.547×10−567.954×10−6Stable5001.000.100.100.200.100.500.500.250.400.05−3739.408×10−5−289.164×10−5108.130×10−6StableFigure 4
Revealing the output of simulation when successful efforts of fanatical is gaining sympathy and recruitment into the ideology. (a) simulation at double efforts at a scenario of Figure3(d) (i.e. 40% effort and success rate) resulted in drastically increase in the dynamics of fanatics and terrorism, (b) Presents growth rate in fanatics and terrorism group as efforts and success rate grows to 60%, (c) dynamics of core group with progression in contact and success rate of recruitment, (d) displays adequate contact rate (β2) and successful recruitment into fanaticism (ρ). This illustrates that for adequate contact rate will cause a progressive growth in fanaticism and consequentially, the society will note enjoy a stable peace.
(a)(b)(c)(d)Findings revealed the following:(i)
The model of abstraction of the real-life event (terrorism) agrees with the mathematical sense by the mathematical properties investigated.(ii)
Six (6) equilibrium points were obtained, which tells the complexity of handling acts of terrorism.(iii)
Threshold parameter (R0) of the core ideological group and effective recruitment (Re) of fanatics were obtained and sensitive for the contemporary society to enjoy peace (equations (16) and (17)).(iv)
The model is stable and close to equilibrium points (Table2).(v)
Numerical simulation supports the analytical solution of the threshold parameters, meaning that for the society to be terrorism free, there is a need to cut the chain or channel of recruitment into extremism.In conclusion, the study proved that fanatics are the source of terrorism, specifically when the efforts of the fanatics gained successful sympathy of masses and recruitment. Thus, for the contemporary society to enjoy peace, serious effort is required on the class of fanatics to prevent excessiveness. Hence, an appropriate measure and the class of population for which the implementation is required for curbing excessiveness are subjects for further study.


### 4.1. Recommendation

The model formulation and assumptions in the work are related to Nigeria, and further study should investigate appropriateness of the model for other countries.It is worthy of recommendation for decision makers and policy implementers to consider and carefully be attentive and proactively respond to any form(s) of innovation in their society without violation of human rights. This is in line with the threshold obtained in the study.


## 4.1. Recommendation

The model formulation and assumptions in the work are related to Nigeria, and further study should investigate appropriateness of the model for other countries.It is worthy of recommendation for decision makers and policy implementers to consider and carefully be attentive and proactively respond to any form(s) of innovation in their society without violation of human rights. This is in line with the threshold obtained in the study.



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*Source: 9056841-2022-05-10.xml*