# Mathematical Modeling Investigation of Violence and Racism Coexistence as a Contagious Disease Dynamics in a Community

Authors: Shewafera Wondimagegnhu Teklu; Birhanu Baye Terefe
Journal: Computational and Mathematical Methods in Medicine
 (2022)

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## Abstract

Recently, violence, racism, and their coexistence have been very common issues in most nations in the world. In this newly social science discipline mathematical modelling approach study, we developed and examined a new violence and racism coexistence mathematical model with eight distinct classes of human population (susceptible, violence infected, negotiated, racist, violence-racism coinfected, recuperated against violence, recuperated against racism, and recuperated against the coinfection). The model takes into account the possible controlling strategies of violence-racism coinfection. All the submodels and the violence-racism coexistence model equilibrium points are calculated, and their stabilities are analyzed. The model threshold values are derived. As a result of the model qualitative analysis, the violence-racism coinfection spreads under control if the corresponding basic reproduction number is less than unity, and it propagates through the community if this number exceeds unity. Moreover, the sensitivity analysis of the parameter values of the full model is illustrated. We have applied MATLAB ode45 solver to illustrate the numerical results of the model. Finally, from qualitative analysis and numerical solutions, we obtain relevant and consistent results.


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## Body

## 1. Introduction

The World Health Organization defines violence as “the intentional use of physical force or power, threatened or actual, against oneself, against another person or against a group or community, which either results in or has a high likelihood of resulting in injury, death, psychological harm mal-development, or deprivation.” It is considered as a common universal public health issue due to its frequency and consequences against community [1]. Ethnic violence is a comprehensive term for violence that is prompted from hatred or racism or ethnic stresses or ethnic conflict [2]. Violence against females occurs in all types of society almost in the entire world and affects girls and women of all ages and in all stages of life. In western countries, it has not been until quite recently (1979) that the intimate violence partner was institutionally identified and condemned, and its origin is found in feminists in the 1950s [3].Globalization and migration flows induce a rapidly enhancing of ethnic and racism diverseness within many nations in the world [4]. The spreading of racism in a mixed culturally diverse society affects in a significant manner in all aspects of their life. The widespread proliferation of racism can lead to a series of serious hazards, such as social instability, impacts on election results, or large financial losses. It can be considered as mind infection, and its expansion and impact on individuals indicted similar to infectious diseases, like tuberculosis (TB), COVID-19, and pneumonia pathogenic agents [5]. Various social science studies related to individual behaviors such as violence, racism, social media addiction, and corruption have been carried out by many scholars throughout the world [6–15]. Violence and violation are crucially at the heart of racism, and hence, in principle, the coexistence of violence and racism on individuals in a community is assured [14].Any situation, such as individuals’ behavior that can be spreading from human being to human being, can lead to similar unstable epidemiological infectious disease conditions. Indeed, there are a lot of literatures associated with the happening of behavioral contagion related to individuals’ mental health situation. Violence is one condition in which behavioral contagion may happen, and some contagious behaviors have been observed to occur in situations of higher density and in larger groups, consistent with the behavior of infectious epidemics [16].Mathematical modeling has a continuous fundamental role in understanding of the various aspects of dynamical system of real-world situations like [17–21]. It has been formulated and analyzed in different disciplines such as natural sciences as well as social science like [1–5, 9, 22–33]. Many researchers have applied infectious disease dynamics model to violence, racism, social media addition, corruption, and other social situations. From those researchers, some were applied modeling for social media addiction [34], some were used modeling for violence [1–3, 33], some were applied modeling for racism [4, 5, 25], and others used modeling for corruption dynamics [22–24, 26–28, 35–38]. However, to the best of our knowledge, no one has developed and analyzed a mathematical model on violence-racism coexistence on individuals in a given society. Therefore, in this newly proposed violence-racism coexistence model, we are motivated and interested in filling the specified gap, and we attempt to examine this connection by constructing a mathematical model of violence-racism coexistence contagion with controlling strategies.The remaining part of this study is organized as follows. In Section2, we describe and formulate the compartmental mathematical model of violence-racism coexistence. In Section 3, we analyzed the submodels and the main model. We determined the equilibrium points and basic reproduction numbers and analyzed stabilities of the submodels and the main model equilibrium points. In Section 4, we have carried out the sensitivity analysis and numerical simulations. Finally, we have performed discussions and conclusions in Section 5.


## 2. Violence and Racism Coexistence Model Formulation

In this study, we considered both violence and racism as chronic contagious diseases, and we divide the total number of human populationNt in a given time t into eight mutually exclusive social states. Those are susceptible for either violence or racism St, violence infected Vt, negotiated Ut, recuperated from violence R1t, racism infected Rt, recuperated from racism R2t, violence and racism coinfected Ivrt, and recuperated from coinfected R3t such that Nt=St+Vt+Ut+R1t+Rt+R2t+Ivrt+R3t.


### 2.1. Description of Social State Variables of the Model

(i)
Susceptible individuals for both violence and racism are those group of people who are at risk of violence and racism. These individuals have not received, heard, or acted violence and racism spreading activities described bySt(ii)
Violence-infected individuals are those group of people who use physical force to harm, injure, damage, or destroy someone to spreading violence described byVt(iii)
Negotiated individuals are the group of individuals who are ongoing to reach an agreement with a formal discussion between people described byUt(iv)
Recuperated from violence is a group of individuals who made compatible, consistent, or group to become friendly again after an argument described byR1t(v)
Racism-infected is the group of individuals who have received or heard racist information and support the racist activity. These people are actively spreading the racist ideology and described byRt(vi)
Recuperated from racism is a group of individuals who reject the racist ideology byR2t(vii)
Violence and racism coinfected is a group of individuals who are infected by both violence and racism described byIvrt(viii)
Recuperated from coinfected is a group of individuals who made compatible, consistent, or group to become friend again and reject the racist ideology described byR3t


### 2.2. Basic Assumptions of the Model

(i)
Coinfected individuals can transmit violence and racism infections one after the other and do not transmit simultaneously(ii)
Individuals acquire violence infection following effective contacts with people infected with violence (V and Ivr classes) at the force of infection rate given by(1)λv=β1V+θ2Ivr(iii)
Individuals acquire racism infection from (R and Ivr classes) at the force of infection rate given by(2)λr=β2R+θ1Ivr(iv)
Human population is variable and homogeneous(v)
We did not consider racism only controlling mechanisms for the coinfectious individuals in the community, and we used coinfection instead of behavior coexistence


### 2.3. Description of Model Parameters Is Given in Table1

Table 1
Description of biological parameters.
ParameterBiological descriptionΛRecruitment rate of susceptible individualsβThe modification parameterβ1Violence transmission rateβ2Racism transmission rateδThe negotiated rate of violated individualsεThe recuperated rate of negotiated individualsαThe conversion rate of recuperated individual to susceptibleωThe controlling rate of violence from violence and racism coexistence classκThe modification parameterσThe recuperated rate of racist individualsρThe rate of individuals those who stopped racism purelyϕRecuperated rate of violence-racism coexistenceθThe rate of individuals those who stopped both violence and racism purelyμIndividual’s natural death rateUsing the model assumptions and the flow diagram of violence and racism coexistence transmission dynamics given in Figure1, the corresponding dynamical system is given by
(3)dSdt=Λ+αR1+ρR2+θR3–λv+λr+μS,dVdt=λvS–δ+κλr+μV,dUdt=δV–ε+μU,dRdt=λrS+ωIvr−σ+βλv+μR,dIvrdt=βλvR+κλrV−ϕ+ω+μIvr,dR1dt=εU–α+μR1,dR2dt=σR−ρ+μR2,dR3dt=ϕIvr−θ+μR3.Figure 1
The flow diagram of violence and racism coexistence transmission dynamics.


### 2.4. Basic Properties of the Coexistence Model

Since the violence-racism coexistence model (3) deals with human population which cannot be negative, we need to show that all the solutions of system (3) remain positive with positive initial conditions in the bounded region
(4)Ω=S,V,U,R,Ivr,R1,R2,R3∈ℝ+8,N≤Λμ.Theorem 1.
LetS0>0,V0>0,U0>0,R10>0,R0>0, R20>0,Ivr0>0, and R30>0 be the initial solutions of the model (3); then, St,Vt,Ut,R1t,Rt,R2t,Ivrt, and R3t are positive in ℝ+8 for all time t>0.Proof.
Let us defineτ=supt>0:St>0,Vt>0,Ut>0,Rt>0,Ivrt>0,R1t>0,R2t>0,R3t>0.
SinceSt,Vt,Ut,Rt,Ivrt,R1t,R2t, and R3t are continuous, we deduce thatτ>0. If τ=+∞, then positivity holds, but, if 0<τ<+∞,Sτ=0 or Vτ=0 or Uτ=0 or Rτ=0 or Ivrτ=0 or Ivrτ=0, R1τ=0 or R2τ=0, or R3τ=0.
Here, from the first equation of the model differential equation in (3), we do have
(5)dSdt+λv+λr+μS=Λ+αR1+ρR2+θR3.
Then, by integrating using the method of integrating factor, we got(6)Sτ=M1S0+M1∫0τexp∫λvt+λrt+μdtΛ+αR1t+ρR2t+θR3tdt>0,where M1=exp−μτ+∫0τλvw+λrw>0,S0>0,and from the definition of τ, we have R1t>0,R2t>0,R3t>, then the solutionSτ>0 and hence Sτ≠0.
Again, from the second equation of the model differential equation in (3), we do have
(7)dVdt+δ+κλr+μV=λvS,
and we have obtainedSτ=M1S0+M1∫0τexp∫δ+κλr+μdtλvSdt>0,where M1=exp−μτ+δτ+∫0τκλrw>0,S0>0, and from the definition of τ, the solutionSτ>0; hence, Sτ≠0.
Similarly,Vτ>0; hence,Vτ≠0; Uτ>0; hence, Uτ≠0; Rτ>0; hence, Rτ≠0; R1τ>0; hence, R1τ≠0; R2τ>0; hence, R2τ≠0; and R3τ>0; hence, R3τ≠0.
Thus, based on the definition,τ is not finite which means τ=+∞, and hence, all the solutions of the system (3) are nonnegative.Theorem 2.
The regionΩ in system (4) is bounded in the spaceℝ+8.Proof.
The total number of human populationsNt is
(8)Nt=St+Vt+Ut+R1t+Rt+R2t+Ivrt+R3t.
By differentiating both side with respect to time, we get(9)dNdt=dSdt+dVdt+dVdt+dR1dt+dRdt+dR2dt+dIvrdt+dR3dt,dNdt=Λ+αR1+ρR2+θR3–λv+λr+μS+λvS–δ+κλr+μV+δV–ε+μU+εU–α+μR1+λrS+ωIvr−σ+βλv+μ+σR−ρ+μR2+βλvR+κλrV−ϕ+ω+μIvr+ϕIvr−θ+μR3,dNdt=Λ−μS−μV−μU−μR1−μR−μR2−μIvr−μR3,dNdt=Λ−μS+V+U+R1+R+R2+Ivr+R3,dNdt≤Λ−μN.
Since all the state variables are nonnegative by Theorem1, in the absence of infections, we have obtained dN/dt≤Λ−μN. By applying the standard comparison theorem, we have obtained ∫dN/Λ−μN≤∫dt, and integrating both sides gives −1/μlnΛ−μN≤t+c, where c is some constant. After some steps of calculations, we have obtained 0≤Nt≤Λ/μ which means all possible solutions of the system (3) with positive initial conditions enter in the bounded region (4).


## 2.1. Description of Social State Variables of the Model

(i)
Susceptible individuals for both violence and racism are those group of people who are at risk of violence and racism. These individuals have not received, heard, or acted violence and racism spreading activities described bySt(ii)
Violence-infected individuals are those group of people who use physical force to harm, injure, damage, or destroy someone to spreading violence described byVt(iii)
Negotiated individuals are the group of individuals who are ongoing to reach an agreement with a formal discussion between people described byUt(iv)
Recuperated from violence is a group of individuals who made compatible, consistent, or group to become friendly again after an argument described byR1t(v)
Racism-infected is the group of individuals who have received or heard racist information and support the racist activity. These people are actively spreading the racist ideology and described byRt(vi)
Recuperated from racism is a group of individuals who reject the racist ideology byR2t(vii)
Violence and racism coinfected is a group of individuals who are infected by both violence and racism described byIvrt(viii)
Recuperated from coinfected is a group of individuals who made compatible, consistent, or group to become friend again and reject the racist ideology described byR3t


## 2.2. Basic Assumptions of the Model

(i)
Coinfected individuals can transmit violence and racism infections one after the other and do not transmit simultaneously(ii)
Individuals acquire violence infection following effective contacts with people infected with violence (V and Ivr classes) at the force of infection rate given by(1)λv=β1V+θ2Ivr(iii)
Individuals acquire racism infection from (R and Ivr classes) at the force of infection rate given by(2)λr=β2R+θ1Ivr(iv)
Human population is variable and homogeneous(v)
We did not consider racism only controlling mechanisms for the coinfectious individuals in the community, and we used coinfection instead of behavior coexistence


## 2.3. Description of Model Parameters Is Given in Table1

Table 1
Description of biological parameters.
ParameterBiological descriptionΛRecruitment rate of susceptible individualsβThe modification parameterβ1Violence transmission rateβ2Racism transmission rateδThe negotiated rate of violated individualsεThe recuperated rate of negotiated individualsαThe conversion rate of recuperated individual to susceptibleωThe controlling rate of violence from violence and racism coexistence classκThe modification parameterσThe recuperated rate of racist individualsρThe rate of individuals those who stopped racism purelyϕRecuperated rate of violence-racism coexistenceθThe rate of individuals those who stopped both violence and racism purelyμIndividual’s natural death rateUsing the model assumptions and the flow diagram of violence and racism coexistence transmission dynamics given in Figure1, the corresponding dynamical system is given by
(3)dSdt=Λ+αR1+ρR2+θR3–λv+λr+μS,dVdt=λvS–δ+κλr+μV,dUdt=δV–ε+μU,dRdt=λrS+ωIvr−σ+βλv+μR,dIvrdt=βλvR+κλrV−ϕ+ω+μIvr,dR1dt=εU–α+μR1,dR2dt=σR−ρ+μR2,dR3dt=ϕIvr−θ+μR3.Figure 1
The flow diagram of violence and racism coexistence transmission dynamics.


## 2.4. Basic Properties of the Coexistence Model

Since the violence-racism coexistence model (3) deals with human population which cannot be negative, we need to show that all the solutions of system (3) remain positive with positive initial conditions in the bounded region
(4)Ω=S,V,U,R,Ivr,R1,R2,R3∈ℝ+8,N≤Λμ.Theorem 1.
LetS0>0,V0>0,U0>0,R10>0,R0>0, R20>0,Ivr0>0, and R30>0 be the initial solutions of the model (3); then, St,Vt,Ut,R1t,Rt,R2t,Ivrt, and R3t are positive in ℝ+8 for all time t>0.Proof.
Let us defineτ=supt>0:St>0,Vt>0,Ut>0,Rt>0,Ivrt>0,R1t>0,R2t>0,R3t>0.
SinceSt,Vt,Ut,Rt,Ivrt,R1t,R2t, and R3t are continuous, we deduce thatτ>0. If τ=+∞, then positivity holds, but, if 0<τ<+∞,Sτ=0 or Vτ=0 or Uτ=0 or Rτ=0 or Ivrτ=0 or Ivrτ=0, R1τ=0 or R2τ=0, or R3τ=0.
Here, from the first equation of the model differential equation in (3), we do have
(5)dSdt+λv+λr+μS=Λ+αR1+ρR2+θR3.
Then, by integrating using the method of integrating factor, we got(6)Sτ=M1S0+M1∫0τexp∫λvt+λrt+μdtΛ+αR1t+ρR2t+θR3tdt>0,where M1=exp−μτ+∫0τλvw+λrw>0,S0>0,and from the definition of τ, we have R1t>0,R2t>0,R3t>, then the solutionSτ>0 and hence Sτ≠0.
Again, from the second equation of the model differential equation in (3), we do have
(7)dVdt+δ+κλr+μV=λvS,
and we have obtainedSτ=M1S0+M1∫0τexp∫δ+κλr+μdtλvSdt>0,where M1=exp−μτ+δτ+∫0τκλrw>0,S0>0, and from the definition of τ, the solutionSτ>0; hence, Sτ≠0.
Similarly,Vτ>0; hence,Vτ≠0; Uτ>0; hence, Uτ≠0; Rτ>0; hence, Rτ≠0; R1τ>0; hence, R1τ≠0; R2τ>0; hence, R2τ≠0; and R3τ>0; hence, R3τ≠0.
Thus, based on the definition,τ is not finite which means τ=+∞, and hence, all the solutions of the system (3) are nonnegative.Theorem 2.
The regionΩ in system (4) is bounded in the spaceℝ+8.Proof.
The total number of human populationsNt is
(8)Nt=St+Vt+Ut+R1t+Rt+R2t+Ivrt+R3t.
By differentiating both side with respect to time, we get(9)dNdt=dSdt+dVdt+dVdt+dR1dt+dRdt+dR2dt+dIvrdt+dR3dt,dNdt=Λ+αR1+ρR2+θR3–λv+λr+μS+λvS–δ+κλr+μV+δV–ε+μU+εU–α+μR1+λrS+ωIvr−σ+βλv+μ+σR−ρ+μR2+βλvR+κλrV−ϕ+ω+μIvr+ϕIvr−θ+μR3,dNdt=Λ−μS−μV−μU−μR1−μR−μR2−μIvr−μR3,dNdt=Λ−μS+V+U+R1+R+R2+Ivr+R3,dNdt≤Λ−μN.
Since all the state variables are nonnegative by Theorem1, in the absence of infections, we have obtained dN/dt≤Λ−μN. By applying the standard comparison theorem, we have obtained ∫dN/Λ−μN≤∫dt, and integrating both sides gives −1/μlnΛ−μN≤t+c, where c is some constant. After some steps of calculations, we have obtained 0≤Nt≤Λ/μ which means all possible solutions of the system (3) with positive initial conditions enter in the bounded region (4).


## 3. Qualitative Analysis of the Model

### 3.1. Violence Submodel Analysis

In the absence of racism from the community of system (3), the model is said to be violence submodel which is obtained by making R=Ivr=R2=R3=0 and λv=β1V; the violence submodel is
(10)dSdt=Λ+αR1–λv+μS,dVdt=λvS–δ+μV,dUdt=δV–ε+μU,dR1dt=εU–α+μR1,with total population given by N1t=St+Vt+Ut+R1t.


#### 3.1.1. Violence-Free Equilibrium Point

In the absence of violence from the community, the timely independent solution of system (10) is said to be violence-free equilibrium point which is denoted by Ev0 and obtained by making system (10) equal to zero with V=0 is Ev0=Sv0,0,0,0=Λ/μ,0,0,0.That is,(11)dSdt=Λ+αR1–λv+μS=0,dVdt=λvS–δ+μV=0,dUdt=δV–ε+μU=0,dR1dt=εU–α+μR1=0.Then, fromδV–ε+μU=0, U=0, from εU–α+μR1=0, R1=0, Λ+αR1–λv+μS=0, S=Λ/μ.Hence,Ev0=Λ/μ,0,0,0 is the violence-free equilibrium point of system (10).


#### 3.1.2. Basic Reproduction Number of Violence Submodel

In this submodel, we do have one infectious classV and use the method of next generation matrix to determine the basic reproduction number of violence submodel.TakeX=S,V,U,R1T and system (10) rewritten as
(12)dXdt=fi−vi,where
(13)fi=λvS=β1SV,f=β1S0=β1Λμ,vi=δ+μv,v=δ+μ,v−1=1δ+μ,fv−1=β1Λμδ+μ.Thus, the spectral radius (the basic reproduction of violence infection submodel) offv−1 is R0v=β1Λ/μδ+μ.


#### 3.1.3. Violence-Persistence Equilibrium Point

In the presence of violence in the population, the time dependent solution of the system (10) is said to be violence-persistence equilibrium point denoted by Ev∗ and defined as Ev∗=Sv∗,Uv∗,Vv∗,R1v∗, and after some steps of calculations, we have obtained
(14)Sv∗=Λα+με+μδ+μα+μλv∗+με+μδ+μ−αεδλv∗,Uv∗=Λδλv∗α+μα+μλv∗+με+μδ+μ−αεδλv∗,Vv∗=Λλv∗α+με+μα+μλv∗+με+μδ+μ−αεδλv∗,R1v∗=Λδελv∗α+μλv∗+με+μδ+μ−αεδλv∗.Theorem 3.
Violence-persistence equilibrium point of system (10) is unique if and only if R0v>1.Proof.
Using violence force of infection, we have(15)λv∗=β1V∗=β1Λλv∗α+με+μα+μλv∗+με+μδ+μ−αεδλv∗.
The nonzero value ofλv∗ from equation (15) is
(16)λv∗=α+με+μμδ+μβ1Λ/μδ+μ−1α+με+μδ+μ−αεδ=α+με+μμδ+μR0v−1α+με+μδ+μ−αεδ>0ifandonlyifR0v>1.
Hence, violence submodel has unique violence-persistence equilibrium point iffR0v>1.Theorem 4.
The violence-free equilibrium point of system (10) is locally asymptotically stable if R0v<1, otherwise unstable.Proof.
The Jacobian matrix of system (10) at the violence-free equilibrium point is
(17)JEv0=–μ–β1Λμ0α0β1Λ−μδ+μμ000δ–ε+μ000ε–α+μ.
From the Jacobian matrix, the characteristic equation is(18)–μ−λ–β1Λμ0α0β1Λ−μδ+μμ−λ000δ–ε+μ−λ000ε–α+μ−λ=0,–μ−λβ1Λ−μδ+μμ−λ–ε+μ−λ–α+μ−λ=0,
which gives the corresponding eigenvalues(19)λ1=–μ<0,λ2=–ε+μ<0,λ3=–α+μ<0,λ4=δ+μR0v−1<0.
Those all eigenvalues are negative which implies that the violence-free equilibrium point of violence submodel is locally asymptotically stable ifR0v<1.Theorem 5.
The violence-free equilibrium point of system (10) is globally asymptotically stable if R0v<1, otherwise unstable.Proof.
Consider the Lyapunov functionlV=aV, where a=1/δ+μ, therefore,
(20)lV=1δ+μV,dldt=1δ+μdVdt,dldt=1δ+μλvS–δ+μV,dldt≤1δ+μλvSv0–δ+μV,dldt≤μδ+μδ+μβ1Λ/μδ+μ−1μV,dldt≤R0v−1V,dldt<0ifandonlyifR0v<1.
Thus, violence-free equilibrium of system (10) is globally asymptotically stable when R0v<1.


### 3.2. Racism Submodel Analysis

In the absence of violence from the community of system (3), the model is said to be racism submodel which is obtained by making V=U=Ivr=R1=R3=0 and λr=β2R.The racism submodel is(21)dSdt=Λ+ρR2–λr+μS,dRdt=λrS−σ+μR,dR2dt=σR−ρ+μR2,with total population given by N2t=St+Rt+R2t.


#### 3.2.1. Racism-Free Equilibrium Point

In the absence of racism from the community, the timely independent solution of system (21) is said to be racism-free equilibrium point which is denoted by Er0 and obtained by making system (21) equals to zero with R=0 is Er0=Sr0,0,0=Λ/μ,0,0.That is,(22)dSdt=Λ+ρR2–λr+μS=0,dRdt=λrS−σ+μR=0,dR2dt=σR−ρ+μR2=0.Then, fromσR−ρ+μR2=0, R2=0, and from Λ+ρR2–λr+μS=0, S=Λ/μ.Hence,Er0=Λ/μ,0,0 is the racism-free equilibrium point of system (21).


#### 3.2.2. Racism Submodel Basic Reproduction Number

In this submodel, we do have one infectious classR and use the method of next generation matrix approach to determine the basic reproduction number of racism submodel.TakeX=S,R,R2T, and system (21) can be rewritten as
(23)dXdt=fi−vi,where
(24)fi=λrS=β2SR,f=β1Sr0=β2Λμ,vi=σ+μR,v=σ+μ,v−1=1σ+μ,fv−1=β2Λμσ+μ.Thus, the spectral radius (the basic reproduction of violence infection submodel) offv−1 is R0r=β2Λ/μσ+μ.


#### 3.2.3. Racism-Persistence Equilibrium Point of Racism Submodel

In the presence of racism under the population, the timely dependent solution of the system (21) is said to be racism-persistence equilibrium point denoted by Er∗ and given by Er∗=Sr∗,Rr∗,R2r∗, and after some steps of calculations, we have obtained
(25)Sr∗=Λρ+μσ+μρ+μσ+μλr∗+μ−ρσλr∗,Rr∗=Λρ+μσ+μλr∗ρ+μσ+μ2λr∗+μ−σ+μρσλr∗,R2r∗=σΛρ+μσ+μλr∗ρ+μ2σ+μ2λr∗+μ−ρ+μσ+μρσλr∗.Theorem 6.
Racism-persistence equilibrium point of system (21) is unique if and only if R0r>1.Proof.
The racism force of infection isλr=β2R.
Then, substituteRr∗ in λr=β2R as
(26)λr∗=β2Rr∗=β2Λρ+μσ+μλr∗ρ+μσ+μ2λr∗+μ−σ+μρσλr∗.
The nonzero solution of equation (26) is λr∗=ρ+μμσ+μR0r−1/ρ+μσ+μ−ρσ>0 if and only if R0r>1.
Hence, the system (21) has unique racism-persistence equilibrium point if and only if R0r>1.Theorem 7.
The racism-free equilibrium point of system (21) is locally asymptotically stable if R0r<1, otherwise unstableProof.
The Jacobian matrix of system (21) at racism-free equilibrium point is
(27)JEr∗=–μ–β2Λμ0β2Λμβ2Λ−μσ+μμ00σ–ρ+μ.
The characteristic equation ofJEr0 is
(28)–μ−λ–β2Λμ0β2Λμβ2Λ−μσ+μμ−λ00σ–ρ+μ−λ=0,–μ−λβ2Λ−μσ+μμ−λ–ρ+μ−λ+β2Λμ∗β2Λμ–ρ+μ−λ=0,λ1=–ρ+μ,or–μ−λβ2Λ−μσ+μμ−λ+β2Λμ∗β2Λμ=0,λ2+μ−σ+μR0r−1λ+μσ+μ1−R0r+β2Λμβ2Λμ=0.
It can be written asa0λ2+a1λ+a2=0, where
(29)a0=1,a1=μ−σ+μR0r−1>0,iffR0r<1,a2=μσ+μ1−R0r+β2Λμ∗β2Λμ>0,iffR0r<1.
Hence, all eigenvalues are negative if and only ifR0r<1. Thus, racism-free equilibrium point is locally asymptotically stable if R0r<1.Theorem 8.
The racism-free equilibrium point of system (21) is globally asymptotically stable if R0r<1, otherwise unstable.Proof.
Consider the Lyapunov functionlR=aR, where a=1/σ+μ(30)lR=Rσ+μ,dldt=1σ+μλrS−σ+μR,dldt=1σ+μβ2S−σ+μR,dldt≤1σ+μβ2S0−σ+μR,dldt≤1σ+μβ2Λμ−σ+μR,dldt≤1σ+μβ2Λ−μσ+μμR,dldt≤μσ+μσ+μβ2Λ/μσ+μ−1μR,dldt≤μσ+μμσ+μR0r−1R,dldt<0,ifR0r<1.
Thus, the racism-free equilibrium point is globally asymptotically stable ifR0r<1.


### 3.3. Violence and Racism Coexistence Full Model Analysis

#### 3.3.1. Coexistence Free Equilibrium Point

The coexistence free equilibrium point of the full model is obtained by making system (3) equal to zero with V=U=R=R1=R2=R3=0, and it is given by
(31)Evr0=S0,V0,U0,R0,Ivr0,R10,R20,R30=Λμ,0,0,0,0,0,0,0.


#### 3.3.2. Basic Reproduction Number of the Full Model

In this study, we compute the violence-racism coexistence model basic reproduction number denoted byR0vr using next-generation matrix criteria by Van den Driessche and Watmough [39]. In this model, we do have three infectious classes; those are V, R, and Ivr; then, we have
(32)fi=λvSλrSβλvR+κλrV=β1V+θ2IvrSβ2R+θ1IvrSβ1βV+θ2IvrR+β2κR+θ1IvrV,f=β1S00β1θ2S00β2S0β2θ1S0000=β1Λμ0β1θ2Λμ0β2Λμβ2θ1Λμ000,vi=vi−x−vi+x=δ+κλr+μVσ+βλv+μR−ϕ+ω+μIvr,v=δ+μ000σ+μ000ϕ+ω+μ,v−1=1δ+μ0001σ+μ0001ϕ+ω+μ,fv−1=β1Λμδ+μβ1Λμσ+μβ1θ2Λμϕ+ω+μ0β2Λμσ+μβ2θ1Λμϕ+ω+μ000.The nonzero eigenvalues offv−1 are λ1=β1Λ/μδ+μ, or λ2=β2Λ/μσ+μ.Hence, the basic reproduction number of full model (3) isR0vr=maxλ1,λ2=maxR0v,R0r, where R0v=β1Λ/μδ+μ and R0r=β2Λ/μσ+μ.


#### 3.3.3. Coexistence Free Equilibrium Point Local Stability

Theorem 9.
The coexistence free equilibrium point of a full model is locally asymptotically stable ifR0vr<1,otherwise unstable.Proof.
The Jacobian matrix of the full coexistence model is(33)JEo=−μ−β1S00α−β2S0ρD3θ0D10000β1θ2S000δ–ε+μ0000000ε–α+μ00000000D20D400000σ−ρ+μ00000000−ϕ+ω+μ0000000ϕ−θ+μ,where D1=β1S0−δ+μ, D2=β2S0−δ+μ, D3=−β1θ2+β2θ1S0, D4=β2θ1S0+ω, and D5=−ϕ+ω+μ.
The characteristic equation of the Jacobian matrixJEo is
(34)−μ−λ−β1S00α−β2S0ρD3θ0D10000β1θ2S000δ–ε+μ−λ0000000ε–α+μ−λ00000000D20D400000σ−ρ+μ−λ00000000−λ−D50000000ϕ−θ+μ−λ=0,−μ−λβ1S0−δ+μ−λ–ε+μ−λ–α+μ−λβ2S0−δ+μ−λ−ρ+μ−λ−ϕ+ω+μ−λ−θ+μ−λ=0.
Then, after some simplification, we have obtained(35)λ1=−μ<0,λ2=δ+μR0v−1<0ifR0v<1,λ3=–ε+μ<0,λ4=–α+μ<0,λ5=δ+μR0r−1ifR0r<1,λ6=−ρ+μ<0,λ7=−ϕ+ω+μ<0,λ8=−θ+μ<0.
Hence, the coexistence free equilibrium point of system (3) is locally asymptotically stable wheneverR0vr<1.


## 3.1. Violence Submodel Analysis

In the absence of racism from the community of system (3), the model is said to be violence submodel which is obtained by making R=Ivr=R2=R3=0 and λv=β1V; the violence submodel is
(10)dSdt=Λ+αR1–λv+μS,dVdt=λvS–δ+μV,dUdt=δV–ε+μU,dR1dt=εU–α+μR1,with total population given by N1t=St+Vt+Ut+R1t.


### 3.1.1. Violence-Free Equilibrium Point

In the absence of violence from the community, the timely independent solution of system (10) is said to be violence-free equilibrium point which is denoted by Ev0 and obtained by making system (10) equal to zero with V=0 is Ev0=Sv0,0,0,0=Λ/μ,0,0,0.That is,(11)dSdt=Λ+αR1–λv+μS=0,dVdt=λvS–δ+μV=0,dUdt=δV–ε+μU=0,dR1dt=εU–α+μR1=0.Then, fromδV–ε+μU=0, U=0, from εU–α+μR1=0, R1=0, Λ+αR1–λv+μS=0, S=Λ/μ.Hence,Ev0=Λ/μ,0,0,0 is the violence-free equilibrium point of system (10).


### 3.1.2. Basic Reproduction Number of Violence Submodel

In this submodel, we do have one infectious classV and use the method of next generation matrix to determine the basic reproduction number of violence submodel.TakeX=S,V,U,R1T and system (10) rewritten as
(12)dXdt=fi−vi,where
(13)fi=λvS=β1SV,f=β1S0=β1Λμ,vi=δ+μv,v=δ+μ,v−1=1δ+μ,fv−1=β1Λμδ+μ.Thus, the spectral radius (the basic reproduction of violence infection submodel) offv−1 is R0v=β1Λ/μδ+μ.


### 3.1.3. Violence-Persistence Equilibrium Point

In the presence of violence in the population, the time dependent solution of the system (10) is said to be violence-persistence equilibrium point denoted by Ev∗ and defined as Ev∗=Sv∗,Uv∗,Vv∗,R1v∗, and after some steps of calculations, we have obtained
(14)Sv∗=Λα+με+μδ+μα+μλv∗+με+μδ+μ−αεδλv∗,Uv∗=Λδλv∗α+μα+μλv∗+με+μδ+μ−αεδλv∗,Vv∗=Λλv∗α+με+μα+μλv∗+με+μδ+μ−αεδλv∗,R1v∗=Λδελv∗α+μλv∗+με+μδ+μ−αεδλv∗.Theorem 3.
Violence-persistence equilibrium point of system (10) is unique if and only if R0v>1.Proof.
Using violence force of infection, we have(15)λv∗=β1V∗=β1Λλv∗α+με+μα+μλv∗+με+μδ+μ−αεδλv∗.
The nonzero value ofλv∗ from equation (15) is
(16)λv∗=α+με+μμδ+μβ1Λ/μδ+μ−1α+με+μδ+μ−αεδ=α+με+μμδ+μR0v−1α+με+μδ+μ−αεδ>0ifandonlyifR0v>1.
Hence, violence submodel has unique violence-persistence equilibrium point iffR0v>1.Theorem 4.
The violence-free equilibrium point of system (10) is locally asymptotically stable if R0v<1, otherwise unstable.Proof.
The Jacobian matrix of system (10) at the violence-free equilibrium point is
(17)JEv0=–μ–β1Λμ0α0β1Λ−μδ+μμ000δ–ε+μ000ε–α+μ.
From the Jacobian matrix, the characteristic equation is(18)–μ−λ–β1Λμ0α0β1Λ−μδ+μμ−λ000δ–ε+μ−λ000ε–α+μ−λ=0,–μ−λβ1Λ−μδ+μμ−λ–ε+μ−λ–α+μ−λ=0,
which gives the corresponding eigenvalues(19)λ1=–μ<0,λ2=–ε+μ<0,λ3=–α+μ<0,λ4=δ+μR0v−1<0.
Those all eigenvalues are negative which implies that the violence-free equilibrium point of violence submodel is locally asymptotically stable ifR0v<1.Theorem 5.
The violence-free equilibrium point of system (10) is globally asymptotically stable if R0v<1, otherwise unstable.Proof.
Consider the Lyapunov functionlV=aV, where a=1/δ+μ, therefore,
(20)lV=1δ+μV,dldt=1δ+μdVdt,dldt=1δ+μλvS–δ+μV,dldt≤1δ+μλvSv0–δ+μV,dldt≤μδ+μδ+μβ1Λ/μδ+μ−1μV,dldt≤R0v−1V,dldt<0ifandonlyifR0v<1.
Thus, violence-free equilibrium of system (10) is globally asymptotically stable when R0v<1.


## 3.1.1. Violence-Free Equilibrium Point

In the absence of violence from the community, the timely independent solution of system (10) is said to be violence-free equilibrium point which is denoted by Ev0 and obtained by making system (10) equal to zero with V=0 is Ev0=Sv0,0,0,0=Λ/μ,0,0,0.That is,(11)dSdt=Λ+αR1–λv+μS=0,dVdt=λvS–δ+μV=0,dUdt=δV–ε+μU=0,dR1dt=εU–α+μR1=0.Then, fromδV–ε+μU=0, U=0, from εU–α+μR1=0, R1=0, Λ+αR1–λv+μS=0, S=Λ/μ.Hence,Ev0=Λ/μ,0,0,0 is the violence-free equilibrium point of system (10).


## 3.1.2. Basic Reproduction Number of Violence Submodel

In this submodel, we do have one infectious classV and use the method of next generation matrix to determine the basic reproduction number of violence submodel.TakeX=S,V,U,R1T and system (10) rewritten as
(12)dXdt=fi−vi,where
(13)fi=λvS=β1SV,f=β1S0=β1Λμ,vi=δ+μv,v=δ+μ,v−1=1δ+μ,fv−1=β1Λμδ+μ.Thus, the spectral radius (the basic reproduction of violence infection submodel) offv−1 is R0v=β1Λ/μδ+μ.


## 3.1.3. Violence-Persistence Equilibrium Point

In the presence of violence in the population, the time dependent solution of the system (10) is said to be violence-persistence equilibrium point denoted by Ev∗ and defined as Ev∗=Sv∗,Uv∗,Vv∗,R1v∗, and after some steps of calculations, we have obtained
(14)Sv∗=Λα+με+μδ+μα+μλv∗+με+μδ+μ−αεδλv∗,Uv∗=Λδλv∗α+μα+μλv∗+με+μδ+μ−αεδλv∗,Vv∗=Λλv∗α+με+μα+μλv∗+με+μδ+μ−αεδλv∗,R1v∗=Λδελv∗α+μλv∗+με+μδ+μ−αεδλv∗.Theorem 3.
Violence-persistence equilibrium point of system (10) is unique if and only if R0v>1.Proof.
Using violence force of infection, we have(15)λv∗=β1V∗=β1Λλv∗α+με+μα+μλv∗+με+μδ+μ−αεδλv∗.
The nonzero value ofλv∗ from equation (15) is
(16)λv∗=α+με+μμδ+μβ1Λ/μδ+μ−1α+με+μδ+μ−αεδ=α+με+μμδ+μR0v−1α+με+μδ+μ−αεδ>0ifandonlyifR0v>1.
Hence, violence submodel has unique violence-persistence equilibrium point iffR0v>1.Theorem 4.
The violence-free equilibrium point of system (10) is locally asymptotically stable if R0v<1, otherwise unstable.Proof.
The Jacobian matrix of system (10) at the violence-free equilibrium point is
(17)JEv0=–μ–β1Λμ0α0β1Λ−μδ+μμ000δ–ε+μ000ε–α+μ.
From the Jacobian matrix, the characteristic equation is(18)–μ−λ–β1Λμ0α0β1Λ−μδ+μμ−λ000δ–ε+μ−λ000ε–α+μ−λ=0,–μ−λβ1Λ−μδ+μμ−λ–ε+μ−λ–α+μ−λ=0,
which gives the corresponding eigenvalues(19)λ1=–μ<0,λ2=–ε+μ<0,λ3=–α+μ<0,λ4=δ+μR0v−1<0.
Those all eigenvalues are negative which implies that the violence-free equilibrium point of violence submodel is locally asymptotically stable ifR0v<1.Theorem 5.
The violence-free equilibrium point of system (10) is globally asymptotically stable if R0v<1, otherwise unstable.Proof.
Consider the Lyapunov functionlV=aV, where a=1/δ+μ, therefore,
(20)lV=1δ+μV,dldt=1δ+μdVdt,dldt=1δ+μλvS–δ+μV,dldt≤1δ+μλvSv0–δ+μV,dldt≤μδ+μδ+μβ1Λ/μδ+μ−1μV,dldt≤R0v−1V,dldt<0ifandonlyifR0v<1.
Thus, violence-free equilibrium of system (10) is globally asymptotically stable when R0v<1.


## 3.2. Racism Submodel Analysis

In the absence of violence from the community of system (3), the model is said to be racism submodel which is obtained by making V=U=Ivr=R1=R3=0 and λr=β2R.The racism submodel is(21)dSdt=Λ+ρR2–λr+μS,dRdt=λrS−σ+μR,dR2dt=σR−ρ+μR2,with total population given by N2t=St+Rt+R2t.


### 3.2.1. Racism-Free Equilibrium Point

In the absence of racism from the community, the timely independent solution of system (21) is said to be racism-free equilibrium point which is denoted by Er0 and obtained by making system (21) equals to zero with R=0 is Er0=Sr0,0,0=Λ/μ,0,0.That is,(22)dSdt=Λ+ρR2–λr+μS=0,dRdt=λrS−σ+μR=0,dR2dt=σR−ρ+μR2=0.Then, fromσR−ρ+μR2=0, R2=0, and from Λ+ρR2–λr+μS=0, S=Λ/μ.Hence,Er0=Λ/μ,0,0 is the racism-free equilibrium point of system (21).


### 3.2.2. Racism Submodel Basic Reproduction Number

In this submodel, we do have one infectious classR and use the method of next generation matrix approach to determine the basic reproduction number of racism submodel.TakeX=S,R,R2T, and system (21) can be rewritten as
(23)dXdt=fi−vi,where
(24)fi=λrS=β2SR,f=β1Sr0=β2Λμ,vi=σ+μR,v=σ+μ,v−1=1σ+μ,fv−1=β2Λμσ+μ.Thus, the spectral radius (the basic reproduction of violence infection submodel) offv−1 is R0r=β2Λ/μσ+μ.


### 3.2.3. Racism-Persistence Equilibrium Point of Racism Submodel

In the presence of racism under the population, the timely dependent solution of the system (21) is said to be racism-persistence equilibrium point denoted by Er∗ and given by Er∗=Sr∗,Rr∗,R2r∗, and after some steps of calculations, we have obtained
(25)Sr∗=Λρ+μσ+μρ+μσ+μλr∗+μ−ρσλr∗,Rr∗=Λρ+μσ+μλr∗ρ+μσ+μ2λr∗+μ−σ+μρσλr∗,R2r∗=σΛρ+μσ+μλr∗ρ+μ2σ+μ2λr∗+μ−ρ+μσ+μρσλr∗.Theorem 6.
Racism-persistence equilibrium point of system (21) is unique if and only if R0r>1.Proof.
The racism force of infection isλr=β2R.
Then, substituteRr∗ in λr=β2R as
(26)λr∗=β2Rr∗=β2Λρ+μσ+μλr∗ρ+μσ+μ2λr∗+μ−σ+μρσλr∗.
The nonzero solution of equation (26) is λr∗=ρ+μμσ+μR0r−1/ρ+μσ+μ−ρσ>0 if and only if R0r>1.
Hence, the system (21) has unique racism-persistence equilibrium point if and only if R0r>1.Theorem 7.
The racism-free equilibrium point of system (21) is locally asymptotically stable if R0r<1, otherwise unstableProof.
The Jacobian matrix of system (21) at racism-free equilibrium point is
(27)JEr∗=–μ–β2Λμ0β2Λμβ2Λ−μσ+μμ00σ–ρ+μ.
The characteristic equation ofJEr0 is
(28)–μ−λ–β2Λμ0β2Λμβ2Λ−μσ+μμ−λ00σ–ρ+μ−λ=0,–μ−λβ2Λ−μσ+μμ−λ–ρ+μ−λ+β2Λμ∗β2Λμ–ρ+μ−λ=0,λ1=–ρ+μ,or–μ−λβ2Λ−μσ+μμ−λ+β2Λμ∗β2Λμ=0,λ2+μ−σ+μR0r−1λ+μσ+μ1−R0r+β2Λμβ2Λμ=0.
It can be written asa0λ2+a1λ+a2=0, where
(29)a0=1,a1=μ−σ+μR0r−1>0,iffR0r<1,a2=μσ+μ1−R0r+β2Λμ∗β2Λμ>0,iffR0r<1.
Hence, all eigenvalues are negative if and only ifR0r<1. Thus, racism-free equilibrium point is locally asymptotically stable if R0r<1.Theorem 8.
The racism-free equilibrium point of system (21) is globally asymptotically stable if R0r<1, otherwise unstable.Proof.
Consider the Lyapunov functionlR=aR, where a=1/σ+μ(30)lR=Rσ+μ,dldt=1σ+μλrS−σ+μR,dldt=1σ+μβ2S−σ+μR,dldt≤1σ+μβ2S0−σ+μR,dldt≤1σ+μβ2Λμ−σ+μR,dldt≤1σ+μβ2Λ−μσ+μμR,dldt≤μσ+μσ+μβ2Λ/μσ+μ−1μR,dldt≤μσ+μμσ+μR0r−1R,dldt<0,ifR0r<1.
Thus, the racism-free equilibrium point is globally asymptotically stable ifR0r<1.


## 3.2.1. Racism-Free Equilibrium Point

In the absence of racism from the community, the timely independent solution of system (21) is said to be racism-free equilibrium point which is denoted by Er0 and obtained by making system (21) equals to zero with R=0 is Er0=Sr0,0,0=Λ/μ,0,0.That is,(22)dSdt=Λ+ρR2–λr+μS=0,dRdt=λrS−σ+μR=0,dR2dt=σR−ρ+μR2=0.Then, fromσR−ρ+μR2=0, R2=0, and from Λ+ρR2–λr+μS=0, S=Λ/μ.Hence,Er0=Λ/μ,0,0 is the racism-free equilibrium point of system (21).


## 3.2.2. Racism Submodel Basic Reproduction Number

In this submodel, we do have one infectious classR and use the method of next generation matrix approach to determine the basic reproduction number of racism submodel.TakeX=S,R,R2T, and system (21) can be rewritten as
(23)dXdt=fi−vi,where
(24)fi=λrS=β2SR,f=β1Sr0=β2Λμ,vi=σ+μR,v=σ+μ,v−1=1σ+μ,fv−1=β2Λμσ+μ.Thus, the spectral radius (the basic reproduction of violence infection submodel) offv−1 is R0r=β2Λ/μσ+μ.


## 3.2.3. Racism-Persistence Equilibrium Point of Racism Submodel

In the presence of racism under the population, the timely dependent solution of the system (21) is said to be racism-persistence equilibrium point denoted by Er∗ and given by Er∗=Sr∗,Rr∗,R2r∗, and after some steps of calculations, we have obtained
(25)Sr∗=Λρ+μσ+μρ+μσ+μλr∗+μ−ρσλr∗,Rr∗=Λρ+μσ+μλr∗ρ+μσ+μ2λr∗+μ−σ+μρσλr∗,R2r∗=σΛρ+μσ+μλr∗ρ+μ2σ+μ2λr∗+μ−ρ+μσ+μρσλr∗.Theorem 6.
Racism-persistence equilibrium point of system (21) is unique if and only if R0r>1.Proof.
The racism force of infection isλr=β2R.
Then, substituteRr∗ in λr=β2R as
(26)λr∗=β2Rr∗=β2Λρ+μσ+μλr∗ρ+μσ+μ2λr∗+μ−σ+μρσλr∗.
The nonzero solution of equation (26) is λr∗=ρ+μμσ+μR0r−1/ρ+μσ+μ−ρσ>0 if and only if R0r>1.
Hence, the system (21) has unique racism-persistence equilibrium point if and only if R0r>1.Theorem 7.
The racism-free equilibrium point of system (21) is locally asymptotically stable if R0r<1, otherwise unstableProof.
The Jacobian matrix of system (21) at racism-free equilibrium point is
(27)JEr∗=–μ–β2Λμ0β2Λμβ2Λ−μσ+μμ00σ–ρ+μ.
The characteristic equation ofJEr0 is
(28)–μ−λ–β2Λμ0β2Λμβ2Λ−μσ+μμ−λ00σ–ρ+μ−λ=0,–μ−λβ2Λ−μσ+μμ−λ–ρ+μ−λ+β2Λμ∗β2Λμ–ρ+μ−λ=0,λ1=–ρ+μ,or–μ−λβ2Λ−μσ+μμ−λ+β2Λμ∗β2Λμ=0,λ2+μ−σ+μR0r−1λ+μσ+μ1−R0r+β2Λμβ2Λμ=0.
It can be written asa0λ2+a1λ+a2=0, where
(29)a0=1,a1=μ−σ+μR0r−1>0,iffR0r<1,a2=μσ+μ1−R0r+β2Λμ∗β2Λμ>0,iffR0r<1.
Hence, all eigenvalues are negative if and only ifR0r<1. Thus, racism-free equilibrium point is locally asymptotically stable if R0r<1.Theorem 8.
The racism-free equilibrium point of system (21) is globally asymptotically stable if R0r<1, otherwise unstable.Proof.
Consider the Lyapunov functionlR=aR, where a=1/σ+μ(30)lR=Rσ+μ,dldt=1σ+μλrS−σ+μR,dldt=1σ+μβ2S−σ+μR,dldt≤1σ+μβ2S0−σ+μR,dldt≤1σ+μβ2Λμ−σ+μR,dldt≤1σ+μβ2Λ−μσ+μμR,dldt≤μσ+μσ+μβ2Λ/μσ+μ−1μR,dldt≤μσ+μμσ+μR0r−1R,dldt<0,ifR0r<1.
Thus, the racism-free equilibrium point is globally asymptotically stable ifR0r<1.


## 3.3. Violence and Racism Coexistence Full Model Analysis

### 3.3.1. Coexistence Free Equilibrium Point

The coexistence free equilibrium point of the full model is obtained by making system (3) equal to zero with V=U=R=R1=R2=R3=0, and it is given by
(31)Evr0=S0,V0,U0,R0,Ivr0,R10,R20,R30=Λμ,0,0,0,0,0,0,0.


### 3.3.2. Basic Reproduction Number of the Full Model

In this study, we compute the violence-racism coexistence model basic reproduction number denoted byR0vr using next-generation matrix criteria by Van den Driessche and Watmough [39]. In this model, we do have three infectious classes; those are V, R, and Ivr; then, we have
(32)fi=λvSλrSβλvR+κλrV=β1V+θ2IvrSβ2R+θ1IvrSβ1βV+θ2IvrR+β2κR+θ1IvrV,f=β1S00β1θ2S00β2S0β2θ1S0000=β1Λμ0β1θ2Λμ0β2Λμβ2θ1Λμ000,vi=vi−x−vi+x=δ+κλr+μVσ+βλv+μR−ϕ+ω+μIvr,v=δ+μ000σ+μ000ϕ+ω+μ,v−1=1δ+μ0001σ+μ0001ϕ+ω+μ,fv−1=β1Λμδ+μβ1Λμσ+μβ1θ2Λμϕ+ω+μ0β2Λμσ+μβ2θ1Λμϕ+ω+μ000.The nonzero eigenvalues offv−1 are λ1=β1Λ/μδ+μ, or λ2=β2Λ/μσ+μ.Hence, the basic reproduction number of full model (3) isR0vr=maxλ1,λ2=maxR0v,R0r, where R0v=β1Λ/μδ+μ and R0r=β2Λ/μσ+μ.


### 3.3.3. Coexistence Free Equilibrium Point Local Stability

Theorem 9.
The coexistence free equilibrium point of a full model is locally asymptotically stable ifR0vr<1,otherwise unstable.Proof.
The Jacobian matrix of the full coexistence model is(33)JEo=−μ−β1S00α−β2S0ρD3θ0D10000β1θ2S000δ–ε+μ0000000ε–α+μ00000000D20D400000σ−ρ+μ00000000−ϕ+ω+μ0000000ϕ−θ+μ,where D1=β1S0−δ+μ, D2=β2S0−δ+μ, D3=−β1θ2+β2θ1S0, D4=β2θ1S0+ω, and D5=−ϕ+ω+μ.
The characteristic equation of the Jacobian matrixJEo is
(34)−μ−λ−β1S00α−β2S0ρD3θ0D10000β1θ2S000δ–ε+μ−λ0000000ε–α+μ−λ00000000D20D400000σ−ρ+μ−λ00000000−λ−D50000000ϕ−θ+μ−λ=0,−μ−λβ1S0−δ+μ−λ–ε+μ−λ–α+μ−λβ2S0−δ+μ−λ−ρ+μ−λ−ϕ+ω+μ−λ−θ+μ−λ=0.
Then, after some simplification, we have obtained(35)λ1=−μ<0,λ2=δ+μR0v−1<0ifR0v<1,λ3=–ε+μ<0,λ4=–α+μ<0,λ5=δ+μR0r−1ifR0r<1,λ6=−ρ+μ<0,λ7=−ϕ+ω+μ<0,λ8=−θ+μ<0.
Hence, the coexistence free equilibrium point of system (3) is locally asymptotically stable wheneverR0vr<1.


## 3.3.1. Coexistence Free Equilibrium Point

The coexistence free equilibrium point of the full model is obtained by making system (3) equal to zero with V=U=R=R1=R2=R3=0, and it is given by
(31)Evr0=S0,V0,U0,R0,Ivr0,R10,R20,R30=Λμ,0,0,0,0,0,0,0.


## 3.3.2. Basic Reproduction Number of the Full Model

In this study, we compute the violence-racism coexistence model basic reproduction number denoted byR0vr using next-generation matrix criteria by Van den Driessche and Watmough [39]. In this model, we do have three infectious classes; those are V, R, and Ivr; then, we have
(32)fi=λvSλrSβλvR+κλrV=β1V+θ2IvrSβ2R+θ1IvrSβ1βV+θ2IvrR+β2κR+θ1IvrV,f=β1S00β1θ2S00β2S0β2θ1S0000=β1Λμ0β1θ2Λμ0β2Λμβ2θ1Λμ000,vi=vi−x−vi+x=δ+κλr+μVσ+βλv+μR−ϕ+ω+μIvr,v=δ+μ000σ+μ000ϕ+ω+μ,v−1=1δ+μ0001σ+μ0001ϕ+ω+μ,fv−1=β1Λμδ+μβ1Λμσ+μβ1θ2Λμϕ+ω+μ0β2Λμσ+μβ2θ1Λμϕ+ω+μ000.The nonzero eigenvalues offv−1 are λ1=β1Λ/μδ+μ, or λ2=β2Λ/μσ+μ.Hence, the basic reproduction number of full model (3) isR0vr=maxλ1,λ2=maxR0v,R0r, where R0v=β1Λ/μδ+μ and R0r=β2Λ/μσ+μ.


## 3.3.3. Coexistence Free Equilibrium Point Local Stability

Theorem 9.
The coexistence free equilibrium point of a full model is locally asymptotically stable ifR0vr<1,otherwise unstable.Proof.
The Jacobian matrix of the full coexistence model is(33)JEo=−μ−β1S00α−β2S0ρD3θ0D10000β1θ2S000δ–ε+μ0000000ε–α+μ00000000D20D400000σ−ρ+μ00000000−ϕ+ω+μ0000000ϕ−θ+μ,where D1=β1S0−δ+μ, D2=β2S0−δ+μ, D3=−β1θ2+β2θ1S0, D4=β2θ1S0+ω, and D5=−ϕ+ω+μ.
The characteristic equation of the Jacobian matrixJEo is
(34)−μ−λ−β1S00α−β2S0ρD3θ0D10000β1θ2S000δ–ε+μ−λ0000000ε–α+μ−λ00000000D20D400000σ−ρ+μ−λ00000000−λ−D50000000ϕ−θ+μ−λ=0,−μ−λβ1S0−δ+μ−λ–ε+μ−λ–α+μ−λβ2S0−δ+μ−λ−ρ+μ−λ−ϕ+ω+μ−λ−θ+μ−λ=0.
Then, after some simplification, we have obtained(35)λ1=−μ<0,λ2=δ+μR0v−1<0ifR0v<1,λ3=–ε+μ<0,λ4=–α+μ<0,λ5=δ+μR0r−1ifR0r<1,λ6=−ρ+μ<0,λ7=−ϕ+ω+μ<0,λ8=−θ+μ<0.
Hence, the coexistence free equilibrium point of system (3) is locally asymptotically stable wheneverR0vr<1.


## 4. Sensitivity Analysis and Numerical Simulations

In this section, we convey both the sensitivity analysis and numerical simulations to verify the qualitative results of our mathematical model (3). Particularly, some numerical verification is considered to illustrate the qualitative analysis and results of the preceding sections. Here, we have taken some parameter values from literatures and assume some of the parameter values that are not from real data, since there is the lack of mathematical model analysis literatures which have been done to study the dynamics of violence-racism coexistence in the community.


### 4.1. Sensitivity Analysis

Definition 10.
The normalized forward sensitivity index of a variable violence-racism coexistence reproduction numberR0vr for the coexistence model (3) that depends differentially on a parameter ζ is defined as SIp=∂R0vr/∂ζ∗ζ/R0vr [27, 28].
The violence-racism coexistence sensitivity indices allow us to justify the relative importance of various parameters in the violence-racism coexistence incidence and prevalence. The most sensitive parameter has the magnitude of the sensitivity index greater than all other parameters. In this study, we computed the sensitivity index in terms ofR0vr.Taking the values of parameters given in Table2, the sensitivity indices are calculated in Table 3 and Table 4 asTable 2
Parameter values for numerical simulation.
ParameterValueSourceμ0.01[8, 9]Λ50[1, 9]α0.51Assumeε0.60Assumeσ0.50Assumeδ0.40Assumeκ1.3Assumeρ0.61Assumeθ0.72Assumeβ1.2Assumeβ1VariableAssumeβ2VariableAssumeω0.60Assumeϕ0.50Assumeω0.42AssumeTable 3
Sensitivity indices ofR0v.
Sensitivity indexSensitivity indicesSIβ1+1SIΛ+1SIδ-0.81SIμ-0.31Table 4
Sensitivity indices ofR0r.
Sensitivity indexSensitivity indicesSIβ2+1SIΛ+1SIσ-0.86SIμ-0.34In the study, with the given parameter values in Table2, we have computed R0v=3.7 at the violence spreading rate β1=0.003 which imply that violence spreads throughout the community. Also, we have obtained the sensitivity indices given in Table 3. Moreover, sensitivity analysis given in Table 3 explains that the human population recruitment rate Λ and violence transmission rate β1 are highly affecting the violence reproduction numberR0v.Moreover, with the given parameter values in Table2, we have computed R0r=6.9 at the racism spreading rate β2=0.007 which imply that racism spreads throughout the community; also, we have obtained the sensitivity indices given in Table 4. Moreover, sensitivity analysis given in Table 4 explains that the human population recruitment rate Λ and racism transmission rateβ2 are highly affecting the racism reproduction numberR0r.


### 4.2. Numerical Simulations

#### 4.2.1. Simulations of the Model Thresholds

Using Table2 data, we have obtained R0vr=maxR0v,R0r=max3.7,6.9=6.9>1, and from the sensitivity indices calculation results of Table 4, we can identify some parameters that strongly influence the violence-racism coexistence model dynamics. Parameter β2 has a positive impact on the basic reproduction numberR0vr; that is, an increase in β2 implies an increase in R0vr=R0r. Similarly, parameter σ has a negative impact on the basic reproduction numberR0vr; that is, an increase in the value of σ implies a decrement in R0vr=R0r.Figure2 illustrates that whenever the value of the racism transmission rate increases, the coexistence reproduction number R0vr=R0r highly increases.Figure 2
R0vr=R0r versus the transmission rate β2.Figure3 illustrates that whenever the value of the racism recovery rate increases, then the coexistence reproduction number R0vr=R0r decreases.Figure 3
R0vr=R0r versus the recovery rate σ.


#### 4.2.2. Simulations for the Full Dynamical System

Figure4 shows us the trajectory simulation of the violence-racism coexistence model with parameter values given in Table 2 andβ2=0.0002, where the violence-racism coexistence model basic reproduction number is R0vr=R0r=0.32. Meaning, in the long run (after 10 years), we can see that the violence-racism coexistence state eradicates in the community. That means that the solutions of the model converge to the violence-racism coexistence free equilibrium point.Figure 4
Behaviors of the model solutions wheneverR0vr=R0r<1.Figure5 shows us the trajectory simulation of the violence-racism coexistence model with parameter values given in Table 2 andβ2=0.004, where the violence-racism coexistence model basic reproduction number is R0vr=R0r=6.9. Meaning, in the long run (after 12 years), we can see that the violence-racism coexistence state persists in the community and stabilizes in time. That means the solutions of the model converge to the violence-racism coexistence endemic equilibrium point. Furthermore, the simulation shows that the less recovery rate of the violence-racism coinfectious individuals leads to the prevalence of violence-racism coexistence spread.Figure 5
Behaviors of the model solutions wheneverR0vr=R0r>1.


#### 4.2.3. Simulations of Coinfectious Variable

Figure6 illustrates the impact of violence only controlling rate ω on violence-racism coinfectious individualsIvr, which means we set the impact of the rate ω as we increase values 0.6, 0.7, and 0.8. In Figure 6, we can see that the number of violence-racism coinfectious individuals decreases as ω increases. Similarly, Figure 7 illustrates us that the result of violence-racism coinfectious individuals decreases whenever the controlling rate ϕ increases from 0.6 to 0.8.Figure 6
Effect ofω on coinfectious individuals Ivr.Figure 7
Effect ofϕ on coinfectious individuals Ivr.


## 4.1. Sensitivity Analysis

Definition 10.
The normalized forward sensitivity index of a variable violence-racism coexistence reproduction numberR0vr for the coexistence model (3) that depends differentially on a parameter ζ is defined as SIp=∂R0vr/∂ζ∗ζ/R0vr [27, 28].
The violence-racism coexistence sensitivity indices allow us to justify the relative importance of various parameters in the violence-racism coexistence incidence and prevalence. The most sensitive parameter has the magnitude of the sensitivity index greater than all other parameters. In this study, we computed the sensitivity index in terms ofR0vr.Taking the values of parameters given in Table2, the sensitivity indices are calculated in Table 3 and Table 4 asTable 2
Parameter values for numerical simulation.
ParameterValueSourceμ0.01[8, 9]Λ50[1, 9]α0.51Assumeε0.60Assumeσ0.50Assumeδ0.40Assumeκ1.3Assumeρ0.61Assumeθ0.72Assumeβ1.2Assumeβ1VariableAssumeβ2VariableAssumeω0.60Assumeϕ0.50Assumeω0.42AssumeTable 3
Sensitivity indices ofR0v.
Sensitivity indexSensitivity indicesSIβ1+1SIΛ+1SIδ-0.81SIμ-0.31Table 4
Sensitivity indices ofR0r.
Sensitivity indexSensitivity indicesSIβ2+1SIΛ+1SIσ-0.86SIμ-0.34In the study, with the given parameter values in Table2, we have computed R0v=3.7 at the violence spreading rate β1=0.003 which imply that violence spreads throughout the community. Also, we have obtained the sensitivity indices given in Table 3. Moreover, sensitivity analysis given in Table 3 explains that the human population recruitment rate Λ and violence transmission rate β1 are highly affecting the violence reproduction numberR0v.Moreover, with the given parameter values in Table2, we have computed R0r=6.9 at the racism spreading rate β2=0.007 which imply that racism spreads throughout the community; also, we have obtained the sensitivity indices given in Table 4. Moreover, sensitivity analysis given in Table 4 explains that the human population recruitment rate Λ and racism transmission rateβ2 are highly affecting the racism reproduction numberR0r.


## 4.2. Numerical Simulations

### 4.2.1. Simulations of the Model Thresholds

Using Table2 data, we have obtained R0vr=maxR0v,R0r=max3.7,6.9=6.9>1, and from the sensitivity indices calculation results of Table 4, we can identify some parameters that strongly influence the violence-racism coexistence model dynamics. Parameter β2 has a positive impact on the basic reproduction numberR0vr; that is, an increase in β2 implies an increase in R0vr=R0r. Similarly, parameter σ has a negative impact on the basic reproduction numberR0vr; that is, an increase in the value of σ implies a decrement in R0vr=R0r.Figure2 illustrates that whenever the value of the racism transmission rate increases, the coexistence reproduction number R0vr=R0r highly increases.Figure 2
R0vr=R0r versus the transmission rate β2.Figure3 illustrates that whenever the value of the racism recovery rate increases, then the coexistence reproduction number R0vr=R0r decreases.Figure 3
R0vr=R0r versus the recovery rate σ.


### 4.2.2. Simulations for the Full Dynamical System

Figure4 shows us the trajectory simulation of the violence-racism coexistence model with parameter values given in Table 2 andβ2=0.0002, where the violence-racism coexistence model basic reproduction number is R0vr=R0r=0.32. Meaning, in the long run (after 10 years), we can see that the violence-racism coexistence state eradicates in the community. That means that the solutions of the model converge to the violence-racism coexistence free equilibrium point.Figure 4
Behaviors of the model solutions wheneverR0vr=R0r<1.Figure5 shows us the trajectory simulation of the violence-racism coexistence model with parameter values given in Table 2 andβ2=0.004, where the violence-racism coexistence model basic reproduction number is R0vr=R0r=6.9. Meaning, in the long run (after 12 years), we can see that the violence-racism coexistence state persists in the community and stabilizes in time. That means the solutions of the model converge to the violence-racism coexistence endemic equilibrium point. Furthermore, the simulation shows that the less recovery rate of the violence-racism coinfectious individuals leads to the prevalence of violence-racism coexistence spread.Figure 5
Behaviors of the model solutions wheneverR0vr=R0r>1.


### 4.2.3. Simulations of Coinfectious Variable

Figure6 illustrates the impact of violence only controlling rate ω on violence-racism coinfectious individualsIvr, which means we set the impact of the rate ω as we increase values 0.6, 0.7, and 0.8. In Figure 6, we can see that the number of violence-racism coinfectious individuals decreases as ω increases. Similarly, Figure 7 illustrates us that the result of violence-racism coinfectious individuals decreases whenever the controlling rate ϕ increases from 0.6 to 0.8.Figure 6
Effect ofω on coinfectious individuals Ivr.Figure 7
Effect ofϕ on coinfectious individuals Ivr.


## 4.2.1. Simulations of the Model Thresholds

Using Table2 data, we have obtained R0vr=maxR0v,R0r=max3.7,6.9=6.9>1, and from the sensitivity indices calculation results of Table 4, we can identify some parameters that strongly influence the violence-racism coexistence model dynamics. Parameter β2 has a positive impact on the basic reproduction numberR0vr; that is, an increase in β2 implies an increase in R0vr=R0r. Similarly, parameter σ has a negative impact on the basic reproduction numberR0vr; that is, an increase in the value of σ implies a decrement in R0vr=R0r.Figure2 illustrates that whenever the value of the racism transmission rate increases, the coexistence reproduction number R0vr=R0r highly increases.Figure 2
R0vr=R0r versus the transmission rate β2.Figure3 illustrates that whenever the value of the racism recovery rate increases, then the coexistence reproduction number R0vr=R0r decreases.Figure 3
R0vr=R0r versus the recovery rate σ.


## 4.2.2. Simulations for the Full Dynamical System

Figure4 shows us the trajectory simulation of the violence-racism coexistence model with parameter values given in Table 2 andβ2=0.0002, where the violence-racism coexistence model basic reproduction number is R0vr=R0r=0.32. Meaning, in the long run (after 10 years), we can see that the violence-racism coexistence state eradicates in the community. That means that the solutions of the model converge to the violence-racism coexistence free equilibrium point.Figure 4
Behaviors of the model solutions wheneverR0vr=R0r<1.Figure5 shows us the trajectory simulation of the violence-racism coexistence model with parameter values given in Table 2 andβ2=0.004, where the violence-racism coexistence model basic reproduction number is R0vr=R0r=6.9. Meaning, in the long run (after 12 years), we can see that the violence-racism coexistence state persists in the community and stabilizes in time. That means the solutions of the model converge to the violence-racism coexistence endemic equilibrium point. Furthermore, the simulation shows that the less recovery rate of the violence-racism coinfectious individuals leads to the prevalence of violence-racism coexistence spread.Figure 5
Behaviors of the model solutions wheneverR0vr=R0r>1.


## 4.2.3. Simulations of Coinfectious Variable

Figure6 illustrates the impact of violence only controlling rate ω on violence-racism coinfectious individualsIvr, which means we set the impact of the rate ω as we increase values 0.6, 0.7, and 0.8. In Figure 6, we can see that the number of violence-racism coinfectious individuals decreases as ω increases. Similarly, Figure 7 illustrates us that the result of violence-racism coinfectious individuals decreases whenever the controlling rate ϕ increases from 0.6 to 0.8.Figure 6
Effect ofω on coinfectious individuals Ivr.Figure 7
Effect ofϕ on coinfectious individuals Ivr.


## 5. Discussions and Conclusions

In this newly proposed social science discipline mathematical model study, we have constructed and analyzed the first and new compartmental mathematical model on violence and racism coexistence behaviors in a community with the attempting of applying possible control measures. Like infectious diseases, in this study, we have interchangeably applied persistence and coexistence as infection and coinfection, respectively. We have determined and shown the positivity and boundedness of the model solutions in a mathematically and physically meaningfully feasible region. The equilibrium points and the basic reproduction numbers are determined by using next generator operator method. In addition, we analyzed the local and global stability of all equilibria.Using sensitivity analysis and numerical simulations, we have verified the detailed theoretical results such that applying data given in Table2, the basic reproduction number of the coinfection is the maximum of the two submodel reproduction numbers given by R0vr=maxR0v,R0r=max3.7,6.9=R0r=6.9>1 at β1=0.003 and β2=0.007 which implies that the coinfection is spreading through the considered community. Sensitivity analysis shows that transmission rates are the most positively influencing parameters. On the other hands, the recuperated rate σ and the negotiated rate δ are the most negatively influencing parameters.Figure2 reflects that increasing the value of the racism transmission rate increases the basic reproduction number R0vr=R0r of the coinfectious population. Figure 3 reflects that whenever we increase the value of the racism recovery rate by applying the general controlling strategies, the coexistence reproduction number R0vr=R0r decreases. Figure 4 shows the trajectory of the violence-racism coexistence model with given constant parameter values atβ1=0.0001 and β2=0.0002, and the basic reproduction number is R0vr=0.32. It implies that in the long run (after 10 years), the violence-racism coexistence state eradicates from the community. That means that the solutions of the model converge to the violence-racism coexistence free equilibrium point. Figure 5 shows the trajectory simulation of the violence-racism coexistence model with given constant parameter values at β1=0.001 and β2=0.004; the violence-racism coexistence model basic reproduction number is R0vr=R0r=6.9; it means, in the long run (after 20 years), the violence-racism coexistence state persists in the community and stabilizes in time. That means that the solutions of the model converge to the violence-racism coexistence prevalence equilibrium point.Figures6 and 7 illustrate the impact of violence only controlling rate ω and violence-racism coinfection general controlling rateϕ on violence-racism coinfectious individualsIvr, which means if we increase the values of ω and ϕ from 0.6 to 0.8, the violence-racism coinfectious individuals decreases. Finally, we recommend for both the social sciences experts and the public health stakeholders to decrease the spreading rates and to maximize the general controlling mechanisms (negotiated and recovery rates) of the violence-racism coinfectious individuals.Finally, we recommend for the governments of nations to introduce, apply, and ensure antiracism and antiviolence laws and take the bold measures to break the interconnection of violence and racism. We want to remark the whole community to stay united to identify common problems and committed to research and advocacy from societies. The international institutions shall be collaborated for better understanding of these two interlinked problems and set up monitoring and investigation bodies. The limitations of this study are as follows: the next potential researchers can incorporate and extend them; this study will be optimal control approach, stochastic approach, fractional order derivative approach, environmental impacts, and age and spatial structures, whenever possible validate the model by applying appropriate real data.



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