# Mathematical Model Analysis on the Diffusion of Violence

Authors: Birhanu Baye Terefe
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

Recently, violence has been a very common and serious public health problem in the world. In this new mathematical modeling tactic study, we formulated and examined the firsthand violence mathematical model with five distinct classes of the human population (susceptible, violence-exposed, violence, negotiated, and reconciled). The model takes into account the diffusion of violence and infection. The violence-free and violence-dominance model equilibrium points are calculated, and their local and global stabilities are analyzed. The model threshold values are obtained. As a result of the model analysis, the violence diffusion is under control if the basic reproduction number is less than unity, and it diffuses through the community if this number exceeds unity. Besides, the sensitivity analysis of the parameter values of the basic reproduction number is demonstrated. We have applied the MATLAB ode45 solver to illustrate the numerical results of the model. Finally, from analytical and numerical solutions, we obtain jointly equivalent and consistent results.


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## Body

## 1. Introduction

Violence is a major and serious behavioral problem that affects all societies, involving physical force intended to hurt, damage, or kill someone in the world [1–3]. Violence is a form of mental pain that causes distress to people physically, psychologically, sexually, emotionally, and economically. It is one of the most common human right violations and a serious global public health concern in developing countries [4–9]. It is a contagious disease among the greatest killers of human beings, which spread from one person to another due to direct or indirect contact quickly or slowly, depending on a host of factors in the world [10–13]. Violence is a political thing between two states that functions as a kind of disciplinary, regulatory, and hierarchical form of militarized law enforcement agents and maldistributions of economic resources to achieve political profit [14–16]. The current political administrative systems facilitate state violence encoded in laws, policies, and schemes that arrange and define people by categories of indigeneity, race, gender, ability, and national origin, which leads to community mobilization from a different perspective [17–19].The mathematical modeling tactics using the deterministic techniques have been systematic efforts to discover real-world situations using mathematical models for the analysis of the dynamics of communicable diseases and can be used for analyzing a number of real-world physical dynamical situations [20–23]. Many researchers have applied the infectious disease dynamics model to analyze violence, corruption, and other social situations. Of those researchers, some applied mathematical modeling analysis on the dynamics of university student animosity towards mathematics with optimal control theory [24], some applied modeling for the dynamics of racism in cyberspace [25], and some applied modeling for violence [26–28], some applied modeling for the stability analysis of corruption [29], and others study universality of political corruption networks [30]. But, to the best of our knowledge, no one has developed and analyzed a mathematical model of the diffusion of violence. Therefore, in this first-hand proposed violence diffusion model, we are motivated and interested in filling the specified break, and we try to examine this connection by constructing a mathematical model of violence diffusion. The remaining part of this study is organized as follows. In Section 2, we formulate the compartmental mathematical model of violence diffusion, equilibrium points, basic reproduction numbers, and the stability analysis of the model. Section 3 presents numerical simulations. Section 4 presents discussions. Finally, we drew conclusions in Section 5.


## 2. Compartmental Model Formulation

In this study, we considered the total number of populationN (t) in a given time t, and we divide it into five mutually exclusive classes. Those are susceptible, violence-exposed, violence, negotiated, and reconciled individuals denoted by S, E, P, H, and R, respectively. Those state variables are described as follows:(i)
Susceptible individual is the number of populations who are free from violence, while they can receive or observe the idea of violent, and we call it susceptible. It is denoted bySt.(ii)
Violence-exposed individuals are the group of population who have close contact with violence and observe different violent activities. These individuals may have been violent while they are in a pause state which does not spread violence, and we call it violently exposed. It is denoted byEt.(iii)
Violently infectious individuals are a group of population who use physical force to harm, injure, damage, or destroy someone to spread violence, and we call it violence. It is denoted byPt.(iv)
Negotiated individuals are a group of population who are going to reach an agreement with a formal discussion between people. It is denoted byHt.(v)
reconciled individual is the group of people who made compatible, consistent, or group decisions to become friendly again after an argument.It is denoted byRt.Basic assumptions and parameter definitions of the model.(1)
Susceptible individuals increase with the recruitmentΛ, decreased by the contact rate β and enter into the violence exposed classes, and decrease with the natural death rate μ.(2)
Violence-exposed classes increase by the contact rateβ and decreased by the reconciled rate δ, the incubation period α, and the natural death rate μ.(3)
Violent individuals increased by the incubation periodα and decreased by the negotiation rate γ1, the reconciled rate γ2, and the natural death rate μ.(4)
The reconciled individual increased by the reconciled rateε from the negotiate classes, increased by the reconciled rate γ2 from the violence classes, and decreased by the conversion rate δ and the natural death rate μ.Based on the model assumptions and descriptions, the flow diagram of the model is as follows:Using the basic assumptions, parameters definitions, and flow diagram of Figure1, the dynamical system of the study is given by(1)dSdt=Λ−βSP−μS,dEdt=βSP−μ+δ+αE,dPdt=αE−μ+γ1+γ2P,dHdt=γ1P−μ+εH.Figure 1
The flow diagram of violence diffusion.We considered that all model parameter values are nonnegative and the initial conditionS0>0,E0≥0,P0≥0,H0≥0, R0≥0.The total number of populations in system (1) is given as(2)Nt=St+Et+Pt+Ht+Rt.Then, the mathematically and biologically feasible domain of system (1) is(3)Q=S,E,P,H,R0<St+Et+Pt+Ht+Rt≤Λμ.The model is both mathematically and biologically meaningful when the following important lemma holds.Lemma 1.
-IfSt>0, Et≥0, Pt≥0, Ht≥0, Rt≥0, then the solution {St, Et, Pt, Ht, Rt} of the system (1) are positive for all time t≥0.Proof.
Using the initial conditions, we can prove the components of the solutions of the system1 are positive by contradiction, that is, there exists a time ti for i=1,2,3,4,5 such that(1)
t1:St1=0,S′t1<0, St>0, Et>0, Pt>0, t>0, Rt>0 'for 0<t<t1(2)
t2:Et2=0,E′t2<0, St>0, Et>0, Pt>0, t>0, Rt>0 for 0<t<t2(3)
t3:Pt3=0,P′t3<0, St>0, Et>0, t>0, Rt>0 for 0<t<t3(4)
t4:Ht4=0,H′t4<0, St>0, Et>0, Pt>0, Rt>0 for 0<t<t4(5)
t5:Rt5=0,R′t5<0, St>0, Et>0, Pt>0, Ht>0 for 0<t<t5
Then, when we evaluate(i)
From the first equationof the system given in (1) at time t1, we have obtained S′t1=Λ−βSt1Pt1−μSt1=Λ>0, which contradict our first assumption. It implies that St>0 for all t≥0.(ii)
The 2nd equation of system (1) at time t2 we get E′t2=βSt2Pt2–μ+δ+αEt2=βSt2Pt2>0, which contradict our second assumption. It implies that Et>0 for all t≥0.(iii)
The 3rd equation of the system (1) at time t3 we get P′t3=αEt3−μ+γ1+γ2Pt3=γEt3>0, which contradict our third assumption. It implies that Pt>0 for all t≥0.(iv)
The 4th equation of the system (1) at time t4 we get H′t4=γ1Pt4−μ+εHt4=γ1Pt4>0, which contradict our fourth assumption. It implies that Ht>0 for all t≥0.(v)
The 5th equation of the system (1) at time t5 we get R′t5=γ2Pt5+εHt5+δEt5−μRt5=γ2Pt5+εHt5+δEt5>0, which contradict our fifth assumption. It implies that At>0 for all t≥0.
Therefore, fromi−v, the solution {St, Et, Pt, Ht, Rt} of system (1) are positive for all time t≥0.Lemma 2.
The total number of populations in system (1) is bounded on 0<Nt≤Λ/μ.Proof.
To show the boundedness of the solution, take the total population(4)Nt=St+Et+Pt+Ht+Rt,by differentiating both sides with respect to time, we get(5)dNdt=dSdt+dEdt+dPdt+dHdt+dRdt,by substituting (1) in (5) and after simple simplification, we get(6)dNdt=Λ−μN,dN/dt+μN=Λ, which is first order ordinary differential equations, and its solution isNt=Λ/μ+ce−μt, then take limit on both sides(7)limt⟶∞Nt=limt⟶∞Λμ+ce−μt=Λμ.
Hence, the solution is bounded on0<Nt≤Λ/μ .


### 2.1. Equilibrium Points of the Model

#### 2.1.1. Violence-Free Equilibrium Point

In the absence of violence, the violence-free equilibrium point of the model (1) is obtained by making dS/dt=dE/dt=dP/dt=dH/dt=dR/dt=0. At the violence-free equilibrium point they have no violently infectious. That is P  = 0, and by substituting those in system (1), we get the violence-free equilibrium point is E0∗ = (Λ/μ, 0, 0, 0, 0).


#### 2.1.2. Basic Reproduction Number of the Model

TakeX=S,E,P,H,RT, and system (1) can be rewritten as(8)dXdt=fi−vi,where(9)fi=βSP0,f=0Λβμ0000000,vi+x=0αEγ1P,vi−x=μ+δ+αEμ+γ1+γ2Pμ+εH,vi=vi−x−vi+x=μ+δ+αEμ+γ1+γ2P−αEμ+εH−γ1P,V=μ+δ+α00−αμ+γ1+γ200−γ1μ+ε,V−1=1μ+δ+α00αμ+δ+αμ+γ1+γ21μ+γ1+γ20αγ1μ+δ+αμ+γ1+γ2μ+εγ1μ+δ+αμ+ε1μ+ε,fV−1=Λαβμμ+δ+αμ+γ1+γ2Λβμμ+γ1+γ20000000.Then, the spectral radius offV−1 is Λαβ/μμ+δ+αμ+γ1+γ2.(10)R0=Λαβμμ+δ+αμ+γ1+γ2.Sensitivity Analysis. The basic reproduction number is the function of seven parameters β,δ,α,Λ,γ1,γ2, and μ. In order to reduce the diffusion of violence in the population, it is necessary to control the parameter values to make R0>1. We are interested in finding the rate of change of R0 due to each parameter value change. The rate of change of R0 for a change in the value of parameters h can be estimated from a normalized sensitivity index, SI h defined as, SI h=h/R0∂R0/∂h [21, 24].The normalized sensitivity indices of the reproduction numberR0 with respect to β,δ,α,b,γ1,γ2, and μ are given as(11)R0=Λαβμμ+δ+αμ+γ1+γ2,SIβ=βR0∂R0∂β=1,SIα=αR0∂R0∂α=μ+δμ+δ+α>0,SIδ=δR0∂R0∂δ=−δμ+δ+α<0,SIγ1=γ1R0∂R0∂γ1=−γ1μ+γ1+γ2<0,SIγ1=γ2R0∂R0∂γ2=−γ2μ+γ1+γ2<0.Then, the sensitivity indices SIβ and SI α are positive, that is, the basic reproduction number R0 is increased as the infectious rate β and incubation period α increase. The remaining indices are negative, which implies that the basic reproduction number R0 is decreased as δ, α, and μ increase. Since all of the indices, except for SI β, are functions of other parameters, the sensitivity indices will change with changes in values of these other parameters.


#### 2.1.3. Violence Dominance Equilibrium Point

In the presence of violence in the population, the timely dependent solution of system (1) is said to be the violence-dominance equilibrium point, denoted by E∗ and given by e∗=S∗,E∗,P∗,H∗,R∗, where after some simplification, we have got(12)S∗=ΛμR0,E∗=μμ+γ1+γ2R0−1αβ,P∗=μR0−1β,H∗=γ1μR0−1βμ+ε,R∗=μ+γ1+γ2μ+ε+αμ+ε+αγ1R0−1αβμ+ε.


### 2.2. Stability Analysis of the Equilibrium Point

Theorem 1.
-The violence-free equilibrium point is locally asymptotically stable whenR0<1.Proof.
The Jacobean matrix of system (1) at the violence-free equilibrium point is(13)JΛμ,0,0,0,0=−μ0−βΛμ000−μ+δ+αβΛμ000α−μ+γ1+γ20000γ1−μ+ε00δγ2ε−μ.
The characteristics equation is(14)−μ−λ0−βΛμ000−μ+δ+α−λβΛμ000α−μ+γ1+γ2−λ0000γ1−μ+ε−λ00δγ2ε−μ−λ=0.
After simple simplification, we have(15)−μ−λ−μ+δ+α−λ−μ+δ+α−λ−μ+γ1+γ2−λ−μ+ε−λ−μ−λ−βαΛμ−μ+ε−λ−μ−λ=0,−μ−λ−μ+δ+α−λ−μ+ε−λ−μ−λλ2+μ+δ+α+μ+γ1+γ2λ+μμ+γ1+γ2μ+δ+αμ1−αβΛμμ+γ1+γ2μ+δ+α=0.which gives negative eigenvalues λ1=−μ, λ2=−μ+δ+α, λ3=−μ+ε or(16)λ2+μ+δ+α+μ+γ1+γ2λ+μμ+γ1+γ2μ+δ+αμ1−αβΛμμ+γ1+γ2μ+δ+α=0,λ2+μ+δ+α+μ+γ1+γ2λ+μμ+γ1+γ2μ+δ+αμ1−R0=0,which have negative root when R0<1.
Therefore, the violence-free equilibrium point is locally stable whenR0<1.Theorem 2.
The violence-dominance equilibrium point is locally asymptotically stable ifR0>1.Proof.
The Jacobean matrix of system (1) at violence-dominance equilibrium point e∗ is(17)Je∗=−μR0−1−μ0−βμΛR000μR0−1−μ+δ+αβμΛR0000α−μ+σ+γ1+γ20000γ1−μ+ε00δγ2ε−μ.
The characteristics equation of the Jacobean matrix is(18)−μR0−1−λ0−βμΛR000μR0−1−μ+δ+α−λβμΛR0000α−μ+σ+γ1+γ2−λ0000γ1−μ+ε−λ00δγ2ε−μ−λ=0−μR0−1−λ−μ+δ+α−λ−μ+γ1+γ2−λ−μ+ε−λ−μ−λ−βαμΛR0−μ+ε−λ−μ−λ−βμΛR0μαR0−1−μ+ε−λ−μ−λ=0,−μR0−1−λ−μ+δ+α−λ−μ+γ1+γ2−λ−μ+ε−λ−μ−λ−βαμbΛ−μ+ε−λ−μ−λ−αβμΛR0μR0−1−μ+ε−λ−μ−λ=0,−μ+ε−λ−μ−λ−μR0−1−λ−μ+δ+α−λ−μ+γ1+γ2−λ−βαμΛR0−αβμ2R0−1ΛR0=0.−μ+ε−λ−μ−λ=0, which implies that(19)λ1=−μ,λ2=−μ+ε.or(20)λ3+μ+δ+α+μ+γ1+γ2+μR0−1λ2+μR0−1μ+δ+α+μ+γ1+γ2+μ+δ+αμ+γ1+γ2−βαμΛR0λ+μR0−1μ+δ+αμ+γ1+γ2−βαμR0+αβμ2R0−1ΛR0=0.
Therefore, using the Routh–Hurwitz stability criteria, the violence-dominance equilibrium point is locally asymptotically stable whenR0>1.


## 2.1. Equilibrium Points of the Model

### 2.1.1. Violence-Free Equilibrium Point

In the absence of violence, the violence-free equilibrium point of the model (1) is obtained by making dS/dt=dE/dt=dP/dt=dH/dt=dR/dt=0. At the violence-free equilibrium point they have no violently infectious. That is P  = 0, and by substituting those in system (1), we get the violence-free equilibrium point is E0∗ = (Λ/μ, 0, 0, 0, 0).


### 2.1.2. Basic Reproduction Number of the Model

TakeX=S,E,P,H,RT, and system (1) can be rewritten as(8)dXdt=fi−vi,where(9)fi=βSP0,f=0Λβμ0000000,vi+x=0αEγ1P,vi−x=μ+δ+αEμ+γ1+γ2Pμ+εH,vi=vi−x−vi+x=μ+δ+αEμ+γ1+γ2P−αEμ+εH−γ1P,V=μ+δ+α00−αμ+γ1+γ200−γ1μ+ε,V−1=1μ+δ+α00αμ+δ+αμ+γ1+γ21μ+γ1+γ20αγ1μ+δ+αμ+γ1+γ2μ+εγ1μ+δ+αμ+ε1μ+ε,fV−1=Λαβμμ+δ+αμ+γ1+γ2Λβμμ+γ1+γ20000000.Then, the spectral radius offV−1 is Λαβ/μμ+δ+αμ+γ1+γ2.(10)R0=Λαβμμ+δ+αμ+γ1+γ2.Sensitivity Analysis. The basic reproduction number is the function of seven parameters β,δ,α,Λ,γ1,γ2, and μ. In order to reduce the diffusion of violence in the population, it is necessary to control the parameter values to make R0>1. We are interested in finding the rate of change of R0 due to each parameter value change. The rate of change of R0 for a change in the value of parameters h can be estimated from a normalized sensitivity index, SI h defined as, SI h=h/R0∂R0/∂h [21, 24].The normalized sensitivity indices of the reproduction numberR0 with respect to β,δ,α,b,γ1,γ2, and μ are given as(11)R0=Λαβμμ+δ+αμ+γ1+γ2,SIβ=βR0∂R0∂β=1,SIα=αR0∂R0∂α=μ+δμ+δ+α>0,SIδ=δR0∂R0∂δ=−δμ+δ+α<0,SIγ1=γ1R0∂R0∂γ1=−γ1μ+γ1+γ2<0,SIγ1=γ2R0∂R0∂γ2=−γ2μ+γ1+γ2<0.Then, the sensitivity indices SIβ and SI α are positive, that is, the basic reproduction number R0 is increased as the infectious rate β and incubation period α increase. The remaining indices are negative, which implies that the basic reproduction number R0 is decreased as δ, α, and μ increase. Since all of the indices, except for SI β, are functions of other parameters, the sensitivity indices will change with changes in values of these other parameters.


### 2.1.3. Violence Dominance Equilibrium Point

In the presence of violence in the population, the timely dependent solution of system (1) is said to be the violence-dominance equilibrium point, denoted by E∗ and given by e∗=S∗,E∗,P∗,H∗,R∗, where after some simplification, we have got(12)S∗=ΛμR0,E∗=μμ+γ1+γ2R0−1αβ,P∗=μR0−1β,H∗=γ1μR0−1βμ+ε,R∗=μ+γ1+γ2μ+ε+αμ+ε+αγ1R0−1αβμ+ε.


## 2.1.1. Violence-Free Equilibrium Point

In the absence of violence, the violence-free equilibrium point of the model (1) is obtained by making dS/dt=dE/dt=dP/dt=dH/dt=dR/dt=0. At the violence-free equilibrium point they have no violently infectious. That is P  = 0, and by substituting those in system (1), we get the violence-free equilibrium point is E0∗ = (Λ/μ, 0, 0, 0, 0).


## 2.1.2. Basic Reproduction Number of the Model

TakeX=S,E,P,H,RT, and system (1) can be rewritten as(8)dXdt=fi−vi,where(9)fi=βSP0,f=0Λβμ0000000,vi+x=0αEγ1P,vi−x=μ+δ+αEμ+γ1+γ2Pμ+εH,vi=vi−x−vi+x=μ+δ+αEμ+γ1+γ2P−αEμ+εH−γ1P,V=μ+δ+α00−αμ+γ1+γ200−γ1μ+ε,V−1=1μ+δ+α00αμ+δ+αμ+γ1+γ21μ+γ1+γ20αγ1μ+δ+αμ+γ1+γ2μ+εγ1μ+δ+αμ+ε1μ+ε,fV−1=Λαβμμ+δ+αμ+γ1+γ2Λβμμ+γ1+γ20000000.Then, the spectral radius offV−1 is Λαβ/μμ+δ+αμ+γ1+γ2.(10)R0=Λαβμμ+δ+αμ+γ1+γ2.Sensitivity Analysis. The basic reproduction number is the function of seven parameters β,δ,α,Λ,γ1,γ2, and μ. In order to reduce the diffusion of violence in the population, it is necessary to control the parameter values to make R0>1. We are interested in finding the rate of change of R0 due to each parameter value change. The rate of change of R0 for a change in the value of parameters h can be estimated from a normalized sensitivity index, SI h defined as, SI h=h/R0∂R0/∂h [21, 24].The normalized sensitivity indices of the reproduction numberR0 with respect to β,δ,α,b,γ1,γ2, and μ are given as(11)R0=Λαβμμ+δ+αμ+γ1+γ2,SIβ=βR0∂R0∂β=1,SIα=αR0∂R0∂α=μ+δμ+δ+α>0,SIδ=δR0∂R0∂δ=−δμ+δ+α<0,SIγ1=γ1R0∂R0∂γ1=−γ1μ+γ1+γ2<0,SIγ1=γ2R0∂R0∂γ2=−γ2μ+γ1+γ2<0.Then, the sensitivity indices SIβ and SI α are positive, that is, the basic reproduction number R0 is increased as the infectious rate β and incubation period α increase. The remaining indices are negative, which implies that the basic reproduction number R0 is decreased as δ, α, and μ increase. Since all of the indices, except for SI β, are functions of other parameters, the sensitivity indices will change with changes in values of these other parameters.


## 2.1.3. Violence Dominance Equilibrium Point

In the presence of violence in the population, the timely dependent solution of system (1) is said to be the violence-dominance equilibrium point, denoted by E∗ and given by e∗=S∗,E∗,P∗,H∗,R∗, where after some simplification, we have got(12)S∗=ΛμR0,E∗=μμ+γ1+γ2R0−1αβ,P∗=μR0−1β,H∗=γ1μR0−1βμ+ε,R∗=μ+γ1+γ2μ+ε+αμ+ε+αγ1R0−1αβμ+ε.


## 2.2. Stability Analysis of the Equilibrium Point

Theorem 1.
-The violence-free equilibrium point is locally asymptotically stable whenR0<1.Proof.
The Jacobean matrix of system (1) at the violence-free equilibrium point is(13)JΛμ,0,0,0,0=−μ0−βΛμ000−μ+δ+αβΛμ000α−μ+γ1+γ20000γ1−μ+ε00δγ2ε−μ.
The characteristics equation is(14)−μ−λ0−βΛμ000−μ+δ+α−λβΛμ000α−μ+γ1+γ2−λ0000γ1−μ+ε−λ00δγ2ε−μ−λ=0.
After simple simplification, we have(15)−μ−λ−μ+δ+α−λ−μ+δ+α−λ−μ+γ1+γ2−λ−μ+ε−λ−μ−λ−βαΛμ−μ+ε−λ−μ−λ=0,−μ−λ−μ+δ+α−λ−μ+ε−λ−μ−λλ2+μ+δ+α+μ+γ1+γ2λ+μμ+γ1+γ2μ+δ+αμ1−αβΛμμ+γ1+γ2μ+δ+α=0.which gives negative eigenvalues λ1=−μ, λ2=−μ+δ+α, λ3=−μ+ε or(16)λ2+μ+δ+α+μ+γ1+γ2λ+μμ+γ1+γ2μ+δ+αμ1−αβΛμμ+γ1+γ2μ+δ+α=0,λ2+μ+δ+α+μ+γ1+γ2λ+μμ+γ1+γ2μ+δ+αμ1−R0=0,which have negative root when R0<1.
Therefore, the violence-free equilibrium point is locally stable whenR0<1.Theorem 2.
The violence-dominance equilibrium point is locally asymptotically stable ifR0>1.Proof.
The Jacobean matrix of system (1) at violence-dominance equilibrium point e∗ is(17)Je∗=−μR0−1−μ0−βμΛR000μR0−1−μ+δ+αβμΛR0000α−μ+σ+γ1+γ20000γ1−μ+ε00δγ2ε−μ.
The characteristics equation of the Jacobean matrix is(18)−μR0−1−λ0−βμΛR000μR0−1−μ+δ+α−λβμΛR0000α−μ+σ+γ1+γ2−λ0000γ1−μ+ε−λ00δγ2ε−μ−λ=0−μR0−1−λ−μ+δ+α−λ−μ+γ1+γ2−λ−μ+ε−λ−μ−λ−βαμΛR0−μ+ε−λ−μ−λ−βμΛR0μαR0−1−μ+ε−λ−μ−λ=0,−μR0−1−λ−μ+δ+α−λ−μ+γ1+γ2−λ−μ+ε−λ−μ−λ−βαμbΛ−μ+ε−λ−μ−λ−αβμΛR0μR0−1−μ+ε−λ−μ−λ=0,−μ+ε−λ−μ−λ−μR0−1−λ−μ+δ+α−λ−μ+γ1+γ2−λ−βαμΛR0−αβμ2R0−1ΛR0=0.−μ+ε−λ−μ−λ=0, which implies that(19)λ1=−μ,λ2=−μ+ε.or(20)λ3+μ+δ+α+μ+γ1+γ2+μR0−1λ2+μR0−1μ+δ+α+μ+γ1+γ2+μ+δ+αμ+γ1+γ2−βαμΛR0λ+μR0−1μ+δ+αμ+γ1+γ2−βαμR0+αβμ2R0−1ΛR0=0.
Therefore, using the Routh–Hurwitz stability criteria, the violence-dominance equilibrium point is locally asymptotically stable whenR0>1.


## 3. Numerical Simulations

In this section, we use the numerical simulations to deliver the analytical solution of our compartmental mathematical model (1). Mostly, some numerical explanations are considered to explain the analytical analysis and the results of numerical outputs. Here, we assume that the parameter values for numerical simulations are not from real data since there is a lack of mathematical modeling analysis literature which has been done to study the diffusion of violence. The initial values of model (1) are positive, i.e., S0>0,E0≥0,P0≥0,H0≥0,H0≥0. To understand the diffusion of violence, we need to assume parameter values and analyze the model (1) and describe how these parameters stimulate the diffusion of violence. In this case, let us consider the mathematical model (1) with the initial condition S0,E0,P0,H0,H0=100,6,4,1,0 and the parameter values as Λ=0.0042, β=0.06, α=0.17, δ=0.03, μ=0.01, γ1=0.12, γ2=0.13, ε=0.11, and the basic reproduction number of model (1) with the estimated data is R0=0.085. Theorem 1 confirms that the violence-free equilibrium point is locally asymptotically stable. Here, from Figure 2, we can explain that the violence-free equilibrium point is both locally and globally asymptotically stable whenever R0=0.085<1, which means that violence will be eradicated. If we consider the parameter values Λ=0.055, β=0.06, α=0.17, δ=0.03, μ=0.01, γ1=0.12, γ2=0.13, ε=0.11, then the basic reproduction number R0=1.1. Theorem 2 confirms that the violence-dominance equilibrium point is locally asymptotically stable, which means that violence will be diffused. The effects of the transmission rate β on the violence-infected individual are given in Figure 3 by taking β=0.1,0.3,0.9. Figure 3 shows that violence will be diffused in the population when the contact rate increases. The effects of the transmission rate β on the negotiated individual are given in Figure 4 by taking β=0.1,0.3,0.9.Figure 4 shows that negotiated individuals will be an increase in the population when the contact rate increases.Figure 2
Solutions of model (1) when R0<1.Figure 3
Effect of the diffusion rateβ on the violently infectious individuals. P.Figure 4
Effect of the diffusion rateβ on the negotiated individuals. H.Figure2 shows us the path simulation of the violence diffusion model with assumed parameter values, where the violence diffusion model basic reproduction number is R0=0.085. It shows that in the long run, we can see that the violently infectious state eliminates from the population. This means that the solutions of the model converge to the violence-free equilibrium point.Figure3 illustrates the impact of the violence diffusion rate β on violently infectious individuals P, which means that we observe the impact of the rate β by increasing their values 0.1, 0.3, and 0.9. Figure 3, shows that the number of violently infectious individuals increases as β increases.Figure4 shows the impact of the violence diffusion rate β on negotiated individuals H, and that means we set the impact of the rate β as we increase values 0.1, 0.3, and 0.9. In Figure 4, we see that the number of negotiated individuals increase as β increases.Figure5 shows the impact of the reconciled rate δ on violently infectious individuals P, which means we set the impact of the rate δ as we increase values 0.1, 0.3, and 0.9. In Figure 5, we see that the number of violently infectious individuals decreases as δ increases.Figure 5
Effect of the reconciled rateδ on the violently infectious individuals. P.Figure6 shows the impact of the reconciled rate δ on negotiated individuals H, and that means we set the impact of the rate δ as we increase values 0.1, 0.5, and 0.8. In Figure 6, we can see that the number of negotiated individuals decreases as δ increases.Figure 6
Effect of the reconciled rateδ on the negotiated individuals. H.Figure7 shows the impact of the reconciled rate δ on reconciled individuals R, and that means we set the impact of the rate δ as we increase values 0.1, 0.5, and 0.9. In Figure 6, we can see that the number of reconciled individuals increase as δ increases.Figure 7
Effect of the reconciled rateδ on the reconciled individuals R.Figure8 shows the impact of the negotiation rate γ1 on violently infectious individuals P, and that means we set the impact of the rate γ1 as we increase values 0.1, 0.5, and 0.9. In Figure 8, we can see that the number of violently infectious individuals decreases as γ1 increases.Figure 8
Effect of the negotiation rateγ1 on the violently infectious individuals P.Figure9 shows the impact of the negotiation rate γ1 on negotiated individuals H, and that means we set the impact of the rate γ1 as we increase values 0.1, 0.5, and 0.9. In Figure 9, we can see that the number of negotiated individuals increase as γ1 increases.Figure 9
Effect of the negotiation rateγ1 on the negotiated individuals H.


## 4. Discussion

We have seen the numerical reflection with the variation of model parameters in each state variable using the estimated data. Figure2 reflects that violence is diffused in the population, the number of populations who are believed by negotiations increases, and the reconciled population also increases over a long period of time. Figure 3 shows that when the diffusion rate of violence increases, the violently infected individual increase within a long period of time. Figure 4 shows when the contact rate of violence increases, the number of negotiated individuals increases. Figure 5 shows that the violently infected individuals decrease when the reconciled rate δ of exposed individuals without negotiation increases. Figure 6 shows that the number of negotiated populations decreased when the reconciled rate δ of exposed individuals without negotiation increases. Figure 7 shows that the number of reconciled individuals increases when the reconciled rate δ of exposed individuals without negotiation increases. Figure 8 reflects that the violently infected population decreases when the negotiation rate γ1 of violently infectious increases. Figure 9 reflects that the number of negotiated populations increases when the negotiation rate γ1 of violently infectious increases.


## 5. Conclusion

In this paper, we formulated a first-hand compartmental model to study the diffusion of violence with the help of a nonlinear system of differential equations. A new SEPHR model is formulated to analyze analytically and numerically. We have shown the positivity and boundedness of the model. The violence-free and violence-dominance equilibrium points are calculated. The basic reproduction number of the model is calculated to determine the diffusion of violence. As the analytically analysis output, the violence-free equilibrium point is locally and globally asymptotically stable whenR0<1, and violence-dominance equilibrium is also locally and globally asymptotically stable when R0<1. The sensitivity analysis of the parameters on the basic reproduction number is performed to determine the more sensitive parameter for the diffusion or elimination of violence. To eliminate violence from the population under the study we have to decrease the most sensitive parameterβ since it is a sensitive parameter. According to the numerical output of figures 3, 4, the violently infectious and negotiated population will decrease by decreasing the contact rate β¯¯. According to Figures 5–9, the population will live together and eliminate violence by increasing the negotiation rate and the reconciled rate with negotiation and without negotiation.


### 5.1. Limitation of the Study

There was a lack of literature study which took numerical simulation of modeling on the diffusion of violence and well-organized standard parameter values for the determination of model parameters. Due to the lack of literature and the current war in the Ethiopian population, we take estimated data because it is difficult to integrate experimental data into the study.


## 5.1. Limitation of the Study

There was a lack of literature study which took numerical simulation of modeling on the diffusion of violence and well-organized standard parameter values for the determination of model parameters. Due to the lack of literature and the current war in the Ethiopian population, we take estimated data because it is difficult to integrate experimental data into the study.



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