Dynamics of Terrorism in Contemporary Society for Effective Management

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.


Introduction
e 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][2][3]. e dynamics of bicycles and space shuttles evolve according to Newton's law, but people's movements back and forth between di erent states of activity are driven by far more complex in uences [4]. One might think it is impossible to model systems in which humans are the key moving parts, but that misinterprets the de nition of modelling [5,6]. Model is a simpli ed representation of reality [7]. Models of human systems simpli ed 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. e 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]. erefore, it is of interest to model the cause of the act of terrorism in human population using a system of ordinary di erential equation (ODE) for e ective management.
e 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. is 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][12][13][14] is here considered. Many researchers have used the classical fourth-order Runge-Kutta (RK45) method due to the method's e ciency and accuracy in solving ODEs and optimal control problems [15][16][17][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.

Model Formulation.
A mathematical dynamics model of terrorism is formulated using the approach of mathematical epidemiology. e entire population is referred to as noncore (typically large), denoted as G(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, F(t), 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]). e mathematical model is presented in Figure 2. e formulation is based on the work of Castillio-Chavez and Banks [9].

Model Assumptions.
e 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) e 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) e 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) e 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. e model considered Boko Haram as a fanatical group and agreed that they move out of their hideout to commit terror. (8) e general population can never be zero and has no age barrier. e progression from G(t) to S(t) 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 E(t). e 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) and E(t) return to G(t) at the rates α and c , respectively. e proportion of E(t) to F(t) is σ, and this is treated with standard incidence function because of complex interaction in humans. e term η is used to denote the proportion of F(t) that commits terror, and y is the proportion of return fanatics after committing terror.
In view of the formulation and assumptions above, we present Figure 2, the block diagram of dynamics of the model, followed by model equations.

Model Diagram and Equations. Model equations:
with initial condition Non Core (G (t)) Core (C) Naive (S(t))

Properties and Analysis of the Model
It is essential to investigate the behaviour of model formulated before recommending it for use. e 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: if S(t), E(t), F(t), T(t), G(t) have nonnegative initial condition, then there exists a region
Proof. e model concerns humans; therefore, parameters of the models are assumed to be nonnegative for all time t. Furthermore, we consider and this implies and dynamics of the model are Hence, this proved the first hypothesis of the theorem, and for the second hypothesis, consider the first expression in equation (1): Integrate both sides to have Similar argument holds for Integrating this gives In the same way for E F (t), F(t), and T(t), we showed that all are greater than or equal to zero, and this completes the proof. e result obtained in (6) showed that the population cannot be zero. Hence, validate the integral claim of assumption nine (8).

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.

, and there exists a unique solution; then, (1) satisfies the Lipschitz condition.
Proof. Rewrite model equation (1) as and by illustration, For argument sake, let F ′ be denoted as the usual F for a fanatic to differentiate between itself and Lipschitz function.
□ 4 International Journal of Mathematics and Mathematical Sciences 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. is is stated in the next theorem relevant to the proposed model. Hence, the proposed model is said to be mathematically well-posed.

Theorem 3. Let Ω be denoted the region stated in eorem 1 and bounded in
Proof. Rewrite equation (1) as Hence,  e result of existence and uniqueness is that the problem of extremism is in the human population and the solution also lies therein.

reshold for Recruitment into the Ideology.
e longterm 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 International Journal of Mathematics and Mathematical Sciences of the core population c(t), 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) Assume n � min α, c and R 0,min ≡ qβ/μ + n. en, and by the hypothesis of the theorem, It could be seen from the above inequality that C(t) decays exponentially fast to zero. If R 0 < 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., R 0 ≥ 1). e approach of next generation matrix (NGM) is employed for determining effective threshold (R e ) for recruitment of fanatics. From (1), we extract F and V, which, respectively, denote new infection and other transmissions in the model, i.e., where "a" � (μ Hence, R e is the spectral radius of NGM given by  (1) to zero (i.e., at equilibrium for the model): 6 International Journal of Mathematics and Mathematical Sciences Also, the biological interpretation of this is that for a society free of extremists, the following results were obtained: Six equilibrium states were obtained. e 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 R 0 and R e and R 0 < R e .

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): and λ 5 � ρ − (μ + c) Clearly, the first four eigenvalues of (24) are found to be negative and the last eigenvalue is found to be negative, λ 5 � ρ − (μ + c) < 0, which implies that the removal/recovery from ideology must be higher than recruitment; otherwise, the society will not enjoy yearning peace. is is the Routh-Hurwitz criterion for the stability [27] of FFE of the Jacobian matrix J.

International Journal of Mathematics and Mathematical Sciences
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). Let H(z) � P(z, e x ) where P(z, w) is a polynomial with principal term.
Suppose H(iy), y ∈ R, is separated into real and imaginary parts: If the zeros of H(y) have negative real parts, then the zeros of F(y) and G(y) are real, simple, and alternate, and

Conversely, all zeros of H(z) will be in the left half plane provided that either of the following conditions is satisfied: (i) e zeros of F(y) and G(y) are real, simple, and alternate, and inequality (24) is satisfied at least for one y. (ii) All zeros of F(y) are real, and for each zero, relation (24) is satisfied. (iii) All zeros of G(y) 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, where International Journal of Mathematics and Mathematical Sciences − abcy + ab dμ + abμ 2 + abμy + aημy + ac dμ + acμ 2 + acμy − bημy − bc dμ − bcμ 2 − bcμy λ, e characteristic polynomial could simply be written as It can be separated into real and imaginary parts as where a 1 , a 3 , and a 5 are the coefficients of F(t) and a 0 , a 2 , and a 4 are the coefficients of G(t). en, where Multiplying a 4 by a 5 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 Table 1 contains the parameters and values for the 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. us, we include these people in our assumed values for simulation and the assumed value for f(t), E(t), T(t) to be 40, 50, and 60, respectively.

Parameterization and General
For the stability analysis at persistent equilibrium and when simulated with the parament value in Table 1, the Jacobian in (25) and the characteristic equation is By the hypothesis of the theorem and in comparison, a 4 � 0.000676661147 and a 5 � 0.01713983038.
We established that Also, the persistence equilibrium is stable.

Discussion of Results and Conclusion
is paper presents the novel model dynamics of fanatical group progress on terrorism, and it improved the work Chavez and Song [25]. e model conceptualized the mathematical modelling of infectious disease by fanatics, in which extremism is considered as infection of the susceptible group. e model was checked by some mathematical properties, e.g., the existence and uniqueness of solution, positivity, and region of feasibility of solution.
is is necessary for merging real-life events with the abstraction of mathematics. e proposed model is analyzed and synthesized using simple algebra and numercial    [31] concepts for understanding the model dynamics.
Model dynamic for which β [2] and ρ [2] is zero percent Dynamics for which β [2] and ρ [2] is at 20% 20 30 40 ) 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. e 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) e 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).  Dynamics for which β [2] and ρ [2] is at 40% 30 40 Dynamics for which β [2] and ρ [2] is at 100%  Figure 3(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 (β 2 ) and successful recruitment into fanaticism (ρ). is illustrates that for adequate contact rate will cause a progressive growth in fanaticism and consequentially, the society will note enjoy a stable peace.
(ii) Six (6) equilibrium points were obtained, which tells the complexity of handling acts of terrorism. (iii) reshold parameter (R 0 ) of the core ideological group and effective recruitment (R e ) of fanatics were obtained and sensitive for the contemporary society to enjoy peace (equations (16) and (17)). (iv) e model is stable and close to equilibrium points ( Table 2). (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.
us, 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.

Recommendation.
e 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. is is in line with the threshold obtained in the study.

Data Availability
No data used for this study except parameters values picked from literature and have been adequately cited.

Conflicts of Interest
e authors declare that they have no conflicts of interest.