# Chewing Khat Transmission Dynamics: A Mathematical Model and Stability Analysis

Authors: Abayneh Kebede Fantaye; Zerihun Kinfe Birhanu
Journal: Journal of Applied Mathematics
 (2022)

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## Abstract

In this study, the authors proposed a nonlinear deterministic model and stability analysis for the transmission dynamics of chewing khat. The model’s solution is proved to be positive and bounded, and the basic reproduction number (R0) is calculated using the next-generation matrix method. Following that, the authors have looked at the local and global stability of the model’s khat-free and endemic equilibrium points. When R0<1, the chewing khat-free equilibrium point is locally and globally asymptotically stable, whereas when R0>1, the endemic equilibrium point is locally and globally asymptotically stable. The simulation results demonstrate the analytical results.


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## Body

## 1. Introduction

Khat (Catha edulis Forsk) is a green plant of the Celesterece family (El-Setouhy et al. [1]). It is a naturally occurring psychoactive stimulant that has been chewed for centuries in Ethiopia, East Africa, and the southern Arabian Peninsula (Gebissa [2]). Different countries have different names for khat such as khat or chat in Ethiopia, miraa in Kenya, jaad in Somalia (Alem et al. [3]), and qat or gat in (Yemen Odenwald et al. [4]). Ethiopia is one of the leading countries that produce khat in East Africa. Khat grows almost in all parts of the country (Rather et al. [5]). It is a green leaf and greenish plant that is chewed to release fluids containing active chemicals that influence the user’s mood (Wubneh et al. [6]).In private or small social events, stimulation is normally produced by chewing the young flowers and vulnerable leaves. Abdeta et al. [7] reported that khat chewing has recently grown popular among high school, college, and university students. Chewing khat was more common among the uneducated, elderly, alcoholics and smokers, Muslims, and professionals, while the lowest income index and divorce were related to a lower risk (Akalu et al. [8]). Furthermore, the leaves of khat are commonly chewed culturally and socially for a variety of reasons, including improving social interaction, increasing alertness, improving mood, and decreasing the need for sleep (Douglas et al. [9]), decreasing appetite, inducing euphoria, staying active while learning, and increase self-esteem (Teni et al. [10]).Khat was also chewed for its euphoric and stimulating effects (Adeoya-Osiguwa and Fraser [11]). Chronic khat usage, on the other hand, has been related to detrimental health effects such as cardiac arrhythmias, sleeplessness, liver damage, mouth cancer, spermatorrhea, hemorrhoids (Yahya et al. [12]), high blood pressure, gastrointestinal problems, inflammation of the esophagus and stomach, mouth ulcers, gum disease, coronary artery narrowing, and myocardial infarction (Al-Habori [13]). Various studies have identified altered stress responses, cognitive impairments, increased depressive symptoms and stress levels, and insomnia in khat users (Yitayih and van Os [14]).Additionally, chewing khat has been related to an increased risk of stroke and mortality (Ali et al. [15]), can impair your ability to drive, and therefore increases the rate of road accidents (Sheikh et al. [16]). In addition to the physical and psychological effects, khat sourcing and chewing consumes a significant amount of time and household income (Milanovic [17]), which severely causes gingivitis, tooth loss, stomach upset, heart complications, male impotence, insomnia, and various psychological problems. According to the WHO, persistent khat use can lead to urinary retention, impotence, oral cancer, dental decay, chronic gastritis, hemorrhoids, paralytic ileus, liver cirrhosis, high blood pressure, and blurred vision, among other health issues (Havlicek et al. [18]).In recent decades, both mathematicians and biologists have constructed numerous mathematical models to study the transmission of various infectious diseases, according to the study of Sunthrayuth et al. [19], Fantaye and Birhanu [20], Li et al. [21], Alqarni et al. [22], Fantaye [23], Fantaye and Muchie [24], Tiwari et al. [25], Sardar et al. [26], Khajanchi et al. [27], Tiwari et al. [28], Das et al. [29], Birhanu and Fantaye [30], Das et al. [31], Misra et al. [32], and Wong et al. [33].Wubneh et al. [6] develop and analyze a mathematical model that demonstrates the dynamics of chewing khat by dividing the population into a group of people who do not chew khat Nt, a group of people who are surrounded by khat chewers but do not chew at the moment but may chew khat in the future Σt, a group of people who chew khat Ct, a group of people includes people who only consumed khat temporarily for social, spiritual ,and recreational purposes Tt, and a group of people who chew khat constantly Ht. The findings demonstrate that increasing the conversion rate of non-khat chewers to exposed groups makes disease eradication more challenging.George et al. [34] construct a deterministic model of miraa addiction based on three compartment classes: susceptible (S), light users (L), and addicted (A). The fundamental reproduction number (R0) was determined, the positivity and boundedness of the solution were studied, and it was discovered that the system of equations is within the feasible range. The findings of this study will provide information on the spread of addiction to stakeholders such as the government, NACADA, rehabilitation institutions, and the general public, allowing them to take appropriate action to address the issues.Motivated by the works of George et al. [34] and Wubneh et al. [6], in this study, we proposed a nonlinear deterministic model to study and analyze the dynamics of chewing khat. Moreover, we subdivides the total population into susceptible St, moderate Et, addicted It, and quitter Rt classes in which taking into account that moderate khat chewers and addicted can be joined to quitting class due to addressing the risk groups by building self-efficacy and peer resistance skills among the young khat chewers and involvement of religious institution. This study is organized as follows. In section two, we develop our new mathematical model for the stability analysis of the transmission dynamics of chewing khat. We analyze the positivity and boundedness of solutions and stability of the chewing khat-free and endemic equilibria in section three. Section four deals with numerical simulation, and finally, conclusion are given in section five.


## 2. Model Formulation

This section proposes a nonlinear deterministic model ofSEIRS to describe the interaction between khat chewer classes. The total population is divided into four compartments: susceptible St, moderate Et, addicted It and quitters of khat chewer Rt individuals. We assumed that:
(i)
Susceptible are those who do not chew khat, and who may become moderate chewers through effective contact with moderate chewers. There is a positive recruitment rateΛ into the susceptible class. It increased with the rate γ and decreased by effective contact with the moderate chewers at a rate of α(ii)
Moderate khat chewers are individuals who chew khat for their psychostimulant and mood uplift, to combat fatigue and act against obesity due to appetite suppression, religious purpose, peer pressure, passing the time, and socialization. They do not often think about chewing khat or often feel the need to chew, but they can influence the susceptible individuals to chew khat. It is increased by susceptible individuals who turn to be moderate chewers atα rate and decreased when moderate chewers become addicted and quit at rate βδ and 1−δβ, respectively(iii)
Addicted are individuals who constantly chew khat. Due to chewing khat, the majority of addicted individuals due to chewing khat begin as susceptible chewers and then turn to moderate chewers. These people’s health, work, family, and social environment are at risk. This class becomes larger as the number of addicted increases by the rateβδ and decreases when they join to quitter class at rate of θ(iv)
Quitters of khat chewers are individuals who are quitting of chewing khat. It is increased with the rates ofθ and 1−δβ and while it decreased by the rate of γ.(v)
There is a positive natural death rateμ for all classes and disease-induced death rate σ in addicted class. The parameters of the model are presented in Table 1Table 1
The parameter values of model.
ParametersDescriptionValueΛRecruitment rate800αEffective contact rate0.06βRate at whichEt joins to It and Rt0.01δProportion of individuals that joins toIt and Rt from Et0.0007θThe recovery rate0.02γThe rate ofRt becomes St0.0012σDisease-induced death rate0.005μNatural death rate0.08Based on the assumption and the flow chart shown in Figure1 for the khat chewing dynamics, we have the following system of nonlinear ordinary differential equations:
(1)dSdt=Λ−αSIN+γR−μS,(2)dEdt=αSIN−β+μE,(3)dIdt=βδE−σ+θ+μI,(4)dRdt=θI+1−δβE−γ+μR,with
(5)S0>0,E0≥0,I0≥0,R0≥0.Figure 1
Flow chart of the model.


## 3. Model Analysis

This section studies the solution of Equation (1) in the epidemiologically feasible region.
(6)Ω=S,E,I,R∈R+4:0≤St+Et+It+Rt≤Λμ.


### 3.1. Positivity of Solution of the Model

Since model Equation (1) deals with the human population, it is necessary to prove that all state variables and associated parameters are positive. The following theorem will recognize this:Theorem 1.
LetS0>0,E0>0,I0>0 and R0>0 be the initial solution of model Equation (1); then, St,Et,It and Rt are positive in R4 for all t>0.Proof.
Consider(7)τ=supt>0:St>0,Et>0,It>0,Rt>0.
SinceSt,Et,It, and Rt are continuous, we deduce that τ>0. If τ=+∞, then positivity holds, but, if 0<τ<+∞, Sτ=0 or Eτ=0 or Iτ=0 or Rτ=0. Now, from the first equation of system (1), we have
(8)dSdt+μ+ctSt=Λ+γR,where ct=αI/N. So that, by integrating and simplifying, Equation (8) becomes
(9)Sτ=D1S0+D1∫0τe∫μ+ctdt.Λ+γRdt>0,where D1=e−μτ+∫0τcwdw>0,S0>0, and from the definition of τ, we have Rt>0, and then, the solution Sτ>0, and hence, Sτ≠0. Again, from the second equation of system (1), we can obtain that
(10)dEdt+β+μE=αct.
Then, by integrating and simplifying Equation (10), we get
(11)Eτ=G1E0+G1∫0τe∫β+μdt.ctStdt>0,where G1=e−βτ+μτ>0,E0>0, and from the above St>0; then, the solution Eτ>0, and hence, Eτ≠0. Similarly, Iτ>0; hence, Iτ≠0 and Rτ>0; hence, Rτ≠0. Thus, based on definition, τ is not finite which means τ=+∞, and as a result, all the solutions of model Equation (1) are nonnegative.


### 3.2. Invariant Region

It is essential to prove that all solution of Equation (1) system with positive initial data will remain positive for all time t≥0. So, we will establish the following lemma:Lemma 2.
All feasible solutionsSt,Et,It, and Rt of the system of Equation (1) are bounded for the system.
(12)Ω=S,E,I,R∈R+4;0≤St+Et+It+Rt:Nt≤Λμ.Proof.
Differentiating the total human population,Nt=St+Et+It+Rt with respect to time of our model Equation (1), we have
(13)dNdt=dSdt+dEdt+dIdt+dRdt.
Substituting all state equations from the system of Equation (1) in Equation (13), we get
(14)dNdt=Λ−μNt−σIt,implies that
(15)dNdt≤Λ−μNt.
By rearranging and then multiplying both sides of Equation (15) by integrating factor eμt and after some simplification, we have
(16)Nt≤Λμ+N0e−μt,where N0 is the initial values of the total human population,
From Equation (16), we have limsupt⟶∞Nt≤Λ/μ. Thus, Ω is positively invariant (all solutions in Ω remain in Ω for all time).Remark 1.
In the regionΩ, our Equation (1) model is mathematically and epidemiologically well posed.


### 3.3. Khat Chewing-Free Equilibrium Point of the Model

The khat chewing-free equilibrium points,E0 of model Equation (1), are steady-state solutions, where there is no khat chewer in the community. In the absence of chewing khat, we have E=0 and I=0. Then, we obtain
(17)E0=S0,E0,I0,R0=Λμ,0,0,0.


#### 3.3.1. The Basic Reproduction Number (R0)

The basic reproduction numberR0 measures the expected number of secondary khat chewers that result from one newly khat chewer individuals introduced into a susceptible population. We can obtain R0 of the system of Equation (1) using the next-generation matrix method as described by Van den Driessche and Watmough [35]. Then, we calculate the square matrix F, which represents the rate of new khat-chewing devices (new infections) and V which represents the rate of individual transmission. Both matrices have a order of n×n, where n is the total number of infected compartments. We obtain a new subsystem and a new infected compartment and transfer compartment, designated as Et and It, respectively.
(18)dEdt=αSIN−β+μE,dIdt=βδE−σ+θ+μI.That is(19)dEdtdIdt=αSIN0−β+μE−βδE+σ+θ+μI.Now, letx=E,I, and then, the above system of equation can be rewritten as
(20)dxdt=Fx−Vx,where
(21)Fx=αSIN0,Vx=β+μE−βδE+σ+θ+μI.The Jacobian matrices ofFx and Vx at disease free equilibrium point, E0, is given by
(22)F=0α00,V=β+μ0−βδσ+θ+μ,V−1=1β+μ0−βδβ+μσ+θ+μ1σ+θ+μ.As a result,FV−1 becomes
(23)FV−1=αβδβ+μσ+θ+μασ+θ+μ00.So that the eigenvalues of matrixFV−1 are λ=0 and λ=αβδ/β+μσ+θ+μ . Using the method of next-generation matrix, the basic reproduction number, R0 is the spectral radius of FV−1 or the dominant eigenvalue of FV−1 and thus, the basic reproduction number R0 is given by
(24)R0=αβδβ+μσ+θ+μ.


### 3.4. Chewing Khat Endemic Equilibrium Point of the Model

Khat chewing will persist in the population, and the chewing khat endemic equilibrium point of the model is given byE1=S∗,E∗,I∗,R∗. Now, a system of Equation (1) can be rewritten as:
(25)Λ−αS∗I∗N+γR∗−μS∗=0,(26)αS∗I∗N−β+μE∗=0,(27)βδE∗−σ+θ+μI∗=0,(28)θI∗+1−δβE∗−γ+μR∗=0.From the second and third equation of system Equation (25), we have
(29)αS∗I∗N−β+μE∗=0,(30)βδE∗−σ+θ+μI∗=0.Then, by dividing Equation (29) by Equation (30), we get
(31)S∗=ΛμR0.Again by adding the first and second equation of system Equation (25), we have
(32)R∗=β+μE∗γ+Λγ1R0−1.From Equation (30), we obtain
(33)I∗=βδE∗σ+θ+μ.Substituting Equation (33) in the last equation of Equation (33), we get
(34)R∗=βδθE∗γ+μσ+θ+μ+1−δβE∗γ+μ.Equating Equations (32) and (34), we get
(35)E∗=γ+μσ+θ+μΛ1−1/R0βγδσ+μ+μσ+θ+μβ+γ+μ.Substituting Equation (35) in Equation (33), we obtain
(36)I∗=βδγ+μΛ1−1/R0βγδσ+μ+μσ+θ+μβ+γ+μ.Again substituting Equation (35) in the last equation of system Equation (34), we obtain
(37)R∗=βδθ+1−δβσ+θ+μγ+μΛ1−1/R0γ+μβγδσ+μ+μσ+θ+μβ+γ+μ.


### 3.5. Local Stability of the Chewing Khat-Free Equilibrium Point

The local stability of the chewing khat-free equilibrium point will be obtained by using the linearized form of the system Equation (1) at the steady state.Theorem 4.
The equilibrium solutionE0 of the nonlinear system of Equation (1) is locally asymptotically stable, if R0<1.Proof.
The Jacobian matrix of system Equation (1) at chewing khat-free equilibrium point E0=Λ/μ,0,0,0 is
(38)JE0=−μ0−αγ0−β+μα00βδ−σ+θ+μ001−δβθ−γ+μ.
The characteristic equation of Equation (29) at chewing khat-free equilibrium point, E0(39)JE0−λI4=−μ−λ0−αγ0−β+μ−λα00βδ−σ+θ+μ−λ001−δβθ−γ+μ−λ=0.Equation (39) can be rearranged and simplified as
(40)−μ−λ−γ+μ−λλ2+k1λ+k2=0.
Here, from Equation (40), we obtain
(41)λ1=−μ<0,λ2=−γ+μ<0.
Here,λ1<0 and λ2<0. Furthermore, using Routh-Hurwitz criteria, the last equation of Equation (40) has strictly negative real part since k1=2μ+β+σ+θ>0 and k2=β+μσ+θ+μ−αβδ=αβδβ+μσ+θ+μ/αβδ−1=αβδ1/R0−1>0 if R0<1. Hence, with R0<1, our system of Equation (1) at the chewing khat-free equilibrium point E0 offers all eigenvalues with real negative part, and thus, it is locally asymptotically stable.


### 3.6. Global Stability of the Khat Chewer Free Equilibrium Point

We used the method proposed by Iggidr et al. [36] to investigate the global stability. Based on Iggidr et al. [36], the system of Equation (1) can be rewritten as the following form:
(42)dXdt=HX−XE0,n+H1Y,(43)dYdt=H2Y.where X=S,R which represents the number of uninfected individuals, while Y=E,I which represents the number of infected individuals, and E0,n represent a vector at disease free equilibrium point pf the same length as X. Based on [20], the chewing khat-free equilibrium point, E0, is globally asymptotically stable if the following two conditions are fulfilled:
(i)
(C1): H should be a matrix with real negative eigenvalues.(ii)
(C2): H2 should be Metzler matrix.Theorem 5.
The chewing khat-free equilibrium point,E0, is globally asymptotically stable if R0≤1.Proof.
From system of Equation (1), we have
(44)X=S,R,Y=E,I,E0=Λμ,0,0,0T,XE0,n=Λμ,0T.
The system of Equation (1) together with Equation (42) can be written as
(45)Λ−αSIN+γR−μSθI+1−δβE−γ+μR=HS−ΛμR+H1EI,αSIN−β+μEβδE−σ+θ+μI=H2EI.
Using uninfected entry of Jacobian matrix of model Equation (1) and the representation in Equation (42), the matrices H,H1,H2 are
(46)H=−μγ0−γ+μ,H1=0−αSN1−δβθ,H2=−β+μαSNβδ−σ+θ+μ.
This implies that the sufficient conditions are satisfied, Therefore, chewing khat-free equilibrium point,E0, is globally asymptotically stable if R0≤1.


### 3.7. Local Stability of Chewing Khat Endemic Equilibrium Point

In this section, we apply the Jacobian stability approach to prove the local stability of chewing khat endemic equilibrium state.Theorem 6.
The chewing khat endemic equilibrium,E1, of the model is locally asymptotically stable when R0>1.Proof.
The Jacobian matrix of system Equation (1) at chewing khat endemic equilibrium point E1=S∗,E∗,I∗,R∗ is
(47)JE1=−αI∗N+μ0−αS∗NγαI∗N−β+μαS∗N00βδ−σ+θ+μ001−δβθ−γ+μ.
The characterstic equation of Equation (47) at E1 is JE1−λI4=.(48)−αI∗N+μ−λ0−αS∗NγαI∗N−β+μ−λαS∗N00βδ−σ+θ+μ−λ001−δβθ−γ+μ−λ.
Equation (48) can be simplified as
(49)Pλ=a4λ4+a3λ3+a2λ2+a1λ+a0,where
(50)a4=1,a3=4μ+σ+θ+β+γ+μαβδγ+μR0−1βδγσ+μ+μσ+θ+μβ+γ+μR0,a2=3μ+σ+θ+β+γμ+2μ+σ+θ+βγ+μ+μαβδ3μ+σ+θ+β+γγ+μR0−1βδγσ+μ+μσ+θ+μβ+γ+μR0,a1=2μ+σ+θ+βγ+μμ+μαβδ2μ+σ+θ+βγ+μ2R0−1βδγσ+μ+μσ+θ+μβ+γ+μR0+μαβ2δγ+μδα+γ1−δR0−1βδγσ+μ+μσ+θ+μβ+γ+μR0,a0=μαβ2δαδγ+μ+δγσ+μ+γσ+θ+μγ+μR0−1βδγσ+μ+μσ+θ+μβ+γ+μR0,a3a2−a1=3μ+σ+θ+β+γ4μ+σ+θ+β+γμ+γ+μ2μ+σ+β+θ+μαβδγ+μ3μ+σ+θ+β4μ+σ+θ+β+γ+δα+γ1−δβR0−1βδγσ+μ+μσ+θ+μβ+γ+μR0+3μ+σ+θ+β+γμαβδγ+μR0−12βδγσ+μ+μσ+θ+μβ+γ+μR02>0,a1a2a3−a0a32+a12=4μ+σ+θ+β+γ2μ+σ+θ+βγ+μμ3μ+σ+β+θ+γμ+2μ+σ+θ+βγ+μ+3μ+σ+β+θ+γμ+2μ+σ+θ+βγ+μμαβδγ+μR0−1βδγσ+μ+μσ+θ+μβ+γ+μR0+μαβδ22μ+σ+β+θ3μ+σ+β+θ+γγ+μ3R0−12βδγσ+μ+μσ+θ+μβ+γ+μR02+μαβδR0−12ασσ+μ+γσ+θ+μ+σγγ+μβδγσ+μ+μσ+θ+μβ+γ+μR02>0.
Using the Routh-Hurwitz criterion, all roots of characteristic polynomial have negative real parts if and only ifa4>0,a3>0,a2>0,a1>0,a0>0,a3a2−a1>0,a1a2a3−a0a32+a12>0 for R0>1. Thus, the chewing khat endemic equilibrium point, E1, is locally asymptotically stable.


## 3.1. Positivity of Solution of the Model

Since model Equation (1) deals with the human population, it is necessary to prove that all state variables and associated parameters are positive. The following theorem will recognize this:Theorem 1.
LetS0>0,E0>0,I0>0 and R0>0 be the initial solution of model Equation (1); then, St,Et,It and Rt are positive in R4 for all t>0.Proof.
Consider(7)τ=supt>0:St>0,Et>0,It>0,Rt>0.
SinceSt,Et,It, and Rt are continuous, we deduce that τ>0. If τ=+∞, then positivity holds, but, if 0<τ<+∞, Sτ=0 or Eτ=0 or Iτ=0 or Rτ=0. Now, from the first equation of system (1), we have
(8)dSdt+μ+ctSt=Λ+γR,where ct=αI/N. So that, by integrating and simplifying, Equation (8) becomes
(9)Sτ=D1S0+D1∫0τe∫μ+ctdt.Λ+γRdt>0,where D1=e−μτ+∫0τcwdw>0,S0>0, and from the definition of τ, we have Rt>0, and then, the solution Sτ>0, and hence, Sτ≠0. Again, from the second equation of system (1), we can obtain that
(10)dEdt+β+μE=αct.
Then, by integrating and simplifying Equation (10), we get
(11)Eτ=G1E0+G1∫0τe∫β+μdt.ctStdt>0,where G1=e−βτ+μτ>0,E0>0, and from the above St>0; then, the solution Eτ>0, and hence, Eτ≠0. Similarly, Iτ>0; hence, Iτ≠0 and Rτ>0; hence, Rτ≠0. Thus, based on definition, τ is not finite which means τ=+∞, and as a result, all the solutions of model Equation (1) are nonnegative.


## 3.2. Invariant Region

It is essential to prove that all solution of Equation (1) system with positive initial data will remain positive for all time t≥0. So, we will establish the following lemma:Lemma 2.
All feasible solutionsSt,Et,It, and Rt of the system of Equation (1) are bounded for the system.
(12)Ω=S,E,I,R∈R+4;0≤St+Et+It+Rt:Nt≤Λμ.Proof.
Differentiating the total human population,Nt=St+Et+It+Rt with respect to time of our model Equation (1), we have
(13)dNdt=dSdt+dEdt+dIdt+dRdt.
Substituting all state equations from the system of Equation (1) in Equation (13), we get
(14)dNdt=Λ−μNt−σIt,implies that
(15)dNdt≤Λ−μNt.
By rearranging and then multiplying both sides of Equation (15) by integrating factor eμt and after some simplification, we have
(16)Nt≤Λμ+N0e−μt,where N0 is the initial values of the total human population,
From Equation (16), we have limsupt⟶∞Nt≤Λ/μ. Thus, Ω is positively invariant (all solutions in Ω remain in Ω for all time).Remark 1.
In the regionΩ, our Equation (1) model is mathematically and epidemiologically well posed.


## 3.3. Khat Chewing-Free Equilibrium Point of the Model

The khat chewing-free equilibrium points,E0 of model Equation (1), are steady-state solutions, where there is no khat chewer in the community. In the absence of chewing khat, we have E=0 and I=0. Then, we obtain
(17)E0=S0,E0,I0,R0=Λμ,0,0,0.


### 3.3.1. The Basic Reproduction Number (R0)

The basic reproduction numberR0 measures the expected number of secondary khat chewers that result from one newly khat chewer individuals introduced into a susceptible population. We can obtain R0 of the system of Equation (1) using the next-generation matrix method as described by Van den Driessche and Watmough [35]. Then, we calculate the square matrix F, which represents the rate of new khat-chewing devices (new infections) and V which represents the rate of individual transmission. Both matrices have a order of n×n, where n is the total number of infected compartments. We obtain a new subsystem and a new infected compartment and transfer compartment, designated as Et and It, respectively.
(18)dEdt=αSIN−β+μE,dIdt=βδE−σ+θ+μI.That is(19)dEdtdIdt=αSIN0−β+μE−βδE+σ+θ+μI.Now, letx=E,I, and then, the above system of equation can be rewritten as
(20)dxdt=Fx−Vx,where
(21)Fx=αSIN0,Vx=β+μE−βδE+σ+θ+μI.The Jacobian matrices ofFx and Vx at disease free equilibrium point, E0, is given by
(22)F=0α00,V=β+μ0−βδσ+θ+μ,V−1=1β+μ0−βδβ+μσ+θ+μ1σ+θ+μ.As a result,FV−1 becomes
(23)FV−1=αβδβ+μσ+θ+μασ+θ+μ00.So that the eigenvalues of matrixFV−1 are λ=0 and λ=αβδ/β+μσ+θ+μ . Using the method of next-generation matrix, the basic reproduction number, R0 is the spectral radius of FV−1 or the dominant eigenvalue of FV−1 and thus, the basic reproduction number R0 is given by
(24)R0=αβδβ+μσ+θ+μ.


## 3.3.1. The Basic Reproduction Number (R0)

The basic reproduction numberR0 measures the expected number of secondary khat chewers that result from one newly khat chewer individuals introduced into a susceptible population. We can obtain R0 of the system of Equation (1) using the next-generation matrix method as described by Van den Driessche and Watmough [35]. Then, we calculate the square matrix F, which represents the rate of new khat-chewing devices (new infections) and V which represents the rate of individual transmission. Both matrices have a order of n×n, where n is the total number of infected compartments. We obtain a new subsystem and a new infected compartment and transfer compartment, designated as Et and It, respectively.
(18)dEdt=αSIN−β+μE,dIdt=βδE−σ+θ+μI.That is(19)dEdtdIdt=αSIN0−β+μE−βδE+σ+θ+μI.Now, letx=E,I, and then, the above system of equation can be rewritten as
(20)dxdt=Fx−Vx,where
(21)Fx=αSIN0,Vx=β+μE−βδE+σ+θ+μI.The Jacobian matrices ofFx and Vx at disease free equilibrium point, E0, is given by
(22)F=0α00,V=β+μ0−βδσ+θ+μ,V−1=1β+μ0−βδβ+μσ+θ+μ1σ+θ+μ.As a result,FV−1 becomes
(23)FV−1=αβδβ+μσ+θ+μασ+θ+μ00.So that the eigenvalues of matrixFV−1 are λ=0 and λ=αβδ/β+μσ+θ+μ . Using the method of next-generation matrix, the basic reproduction number, R0 is the spectral radius of FV−1 or the dominant eigenvalue of FV−1 and thus, the basic reproduction number R0 is given by
(24)R0=αβδβ+μσ+θ+μ.


## 3.4. Chewing Khat Endemic Equilibrium Point of the Model

Khat chewing will persist in the population, and the chewing khat endemic equilibrium point of the model is given byE1=S∗,E∗,I∗,R∗. Now, a system of Equation (1) can be rewritten as:
(25)Λ−αS∗I∗N+γR∗−μS∗=0,(26)αS∗I∗N−β+μE∗=0,(27)βδE∗−σ+θ+μI∗=0,(28)θI∗+1−δβE∗−γ+μR∗=0.From the second and third equation of system Equation (25), we have
(29)αS∗I∗N−β+μE∗=0,(30)βδE∗−σ+θ+μI∗=0.Then, by dividing Equation (29) by Equation (30), we get
(31)S∗=ΛμR0.Again by adding the first and second equation of system Equation (25), we have
(32)R∗=β+μE∗γ+Λγ1R0−1.From Equation (30), we obtain
(33)I∗=βδE∗σ+θ+μ.Substituting Equation (33) in the last equation of Equation (33), we get
(34)R∗=βδθE∗γ+μσ+θ+μ+1−δβE∗γ+μ.Equating Equations (32) and (34), we get
(35)E∗=γ+μσ+θ+μΛ1−1/R0βγδσ+μ+μσ+θ+μβ+γ+μ.Substituting Equation (35) in Equation (33), we obtain
(36)I∗=βδγ+μΛ1−1/R0βγδσ+μ+μσ+θ+μβ+γ+μ.Again substituting Equation (35) in the last equation of system Equation (34), we obtain
(37)R∗=βδθ+1−δβσ+θ+μγ+μΛ1−1/R0γ+μβγδσ+μ+μσ+θ+μβ+γ+μ.


## 3.5. Local Stability of the Chewing Khat-Free Equilibrium Point

The local stability of the chewing khat-free equilibrium point will be obtained by using the linearized form of the system Equation (1) at the steady state.Theorem 4.
The equilibrium solutionE0 of the nonlinear system of Equation (1) is locally asymptotically stable, if R0<1.Proof.
The Jacobian matrix of system Equation (1) at chewing khat-free equilibrium point E0=Λ/μ,0,0,0 is
(38)JE0=−μ0−αγ0−β+μα00βδ−σ+θ+μ001−δβθ−γ+μ.
The characteristic equation of Equation (29) at chewing khat-free equilibrium point, E0(39)JE0−λI4=−μ−λ0−αγ0−β+μ−λα00βδ−σ+θ+μ−λ001−δβθ−γ+μ−λ=0.Equation (39) can be rearranged and simplified as
(40)−μ−λ−γ+μ−λλ2+k1λ+k2=0.
Here, from Equation (40), we obtain
(41)λ1=−μ<0,λ2=−γ+μ<0.
Here,λ1<0 and λ2<0. Furthermore, using Routh-Hurwitz criteria, the last equation of Equation (40) has strictly negative real part since k1=2μ+β+σ+θ>0 and k2=β+μσ+θ+μ−αβδ=αβδβ+μσ+θ+μ/αβδ−1=αβδ1/R0−1>0 if R0<1. Hence, with R0<1, our system of Equation (1) at the chewing khat-free equilibrium point E0 offers all eigenvalues with real negative part, and thus, it is locally asymptotically stable.


## 3.6. Global Stability of the Khat Chewer Free Equilibrium Point

We used the method proposed by Iggidr et al. [36] to investigate the global stability. Based on Iggidr et al. [36], the system of Equation (1) can be rewritten as the following form:
(42)dXdt=HX−XE0,n+H1Y,(43)dYdt=H2Y.where X=S,R which represents the number of uninfected individuals, while Y=E,I which represents the number of infected individuals, and E0,n represent a vector at disease free equilibrium point pf the same length as X. Based on [20], the chewing khat-free equilibrium point, E0, is globally asymptotically stable if the following two conditions are fulfilled:
(i)
(C1): H should be a matrix with real negative eigenvalues.(ii)
(C2): H2 should be Metzler matrix.Theorem 5.
The chewing khat-free equilibrium point,E0, is globally asymptotically stable if R0≤1.Proof.
From system of Equation (1), we have
(44)X=S,R,Y=E,I,E0=Λμ,0,0,0T,XE0,n=Λμ,0T.
The system of Equation (1) together with Equation (42) can be written as
(45)Λ−αSIN+γR−μSθI+1−δβE−γ+μR=HS−ΛμR+H1EI,αSIN−β+μEβδE−σ+θ+μI=H2EI.
Using uninfected entry of Jacobian matrix of model Equation (1) and the representation in Equation (42), the matrices H,H1,H2 are
(46)H=−μγ0−γ+μ,H1=0−αSN1−δβθ,H2=−β+μαSNβδ−σ+θ+μ.
This implies that the sufficient conditions are satisfied, Therefore, chewing khat-free equilibrium point,E0, is globally asymptotically stable if R0≤1.


## 3.7. Local Stability of Chewing Khat Endemic Equilibrium Point

In this section, we apply the Jacobian stability approach to prove the local stability of chewing khat endemic equilibrium state.Theorem 6.
The chewing khat endemic equilibrium,E1, of the model is locally asymptotically stable when R0>1.Proof.
The Jacobian matrix of system Equation (1) at chewing khat endemic equilibrium point E1=S∗,E∗,I∗,R∗ is
(47)JE1=−αI∗N+μ0−αS∗NγαI∗N−β+μαS∗N00βδ−σ+θ+μ001−δβθ−γ+μ.
The characterstic equation of Equation (47) at E1 is JE1−λI4=.(48)−αI∗N+μ−λ0−αS∗NγαI∗N−β+μ−λαS∗N00βδ−σ+θ+μ−λ001−δβθ−γ+μ−λ.
Equation (48) can be simplified as
(49)Pλ=a4λ4+a3λ3+a2λ2+a1λ+a0,where
(50)a4=1,a3=4μ+σ+θ+β+γ+μαβδγ+μR0−1βδγσ+μ+μσ+θ+μβ+γ+μR0,a2=3μ+σ+θ+β+γμ+2μ+σ+θ+βγ+μ+μαβδ3μ+σ+θ+β+γγ+μR0−1βδγσ+μ+μσ+θ+μβ+γ+μR0,a1=2μ+σ+θ+βγ+μμ+μαβδ2μ+σ+θ+βγ+μ2R0−1βδγσ+μ+μσ+θ+μβ+γ+μR0+μαβ2δγ+μδα+γ1−δR0−1βδγσ+μ+μσ+θ+μβ+γ+μR0,a0=μαβ2δαδγ+μ+δγσ+μ+γσ+θ+μγ+μR0−1βδγσ+μ+μσ+θ+μβ+γ+μR0,a3a2−a1=3μ+σ+θ+β+γ4μ+σ+θ+β+γμ+γ+μ2μ+σ+β+θ+μαβδγ+μ3μ+σ+θ+β4μ+σ+θ+β+γ+δα+γ1−δβR0−1βδγσ+μ+μσ+θ+μβ+γ+μR0+3μ+σ+θ+β+γμαβδγ+μR0−12βδγσ+μ+μσ+θ+μβ+γ+μR02>0,a1a2a3−a0a32+a12=4μ+σ+θ+β+γ2μ+σ+θ+βγ+μμ3μ+σ+β+θ+γμ+2μ+σ+θ+βγ+μ+3μ+σ+β+θ+γμ+2μ+σ+θ+βγ+μμαβδγ+μR0−1βδγσ+μ+μσ+θ+μβ+γ+μR0+μαβδ22μ+σ+β+θ3μ+σ+β+θ+γγ+μ3R0−12βδγσ+μ+μσ+θ+μβ+γ+μR02+μαβδR0−12ασσ+μ+γσ+θ+μ+σγγ+μβδγσ+μ+μσ+θ+μβ+γ+μR02>0.
Using the Routh-Hurwitz criterion, all roots of characteristic polynomial have negative real parts if and only ifa4>0,a3>0,a2>0,a1>0,a0>0,a3a2−a1>0,a1a2a3−a0a32+a12>0 for R0>1. Thus, the chewing khat endemic equilibrium point, E1, is locally asymptotically stable.


## 4. Numerical Simulation

In this section, we verify our work numerically using MATLAB ODE solvers. Our simulations examine the effect of different combinations of parameters of the model on the transmission dynamics of chewing khat. The description and values for parameters, which are chosen arbitrarily, are shown on Table1. The simulations and analysis made are based on these initial conditions: S0=10000,E0=1000,I0=400,R0=600.The time series plot of state variables forR0<1 and R0>1 is shown in Figures 2 and 3. From Figure 2, we observe that susceptible and quitter individuals are increasing asymptotically to the disease free equilibrium point, while the moderate and addicted chewers are decreasing asymptotically to the disease free equilibrium point. The existence of such condition is due to the fact that R0=0.0021 which is less than one. This supports theorem that the stability of khat chewer free equilibrium point exists when R0<1, that is, if R0<1, and then on average, one addicted individuals produce less than one newly khat chewer over the course of its chewers period. From Figure 3, we observe that susceptible and quitter individuals are decreased due to influence of moderate and addicted individuals, and then, they become chewers; as a result, the moderate and addicted individuals are increased. Therefore, moderate and addicted khat chewers are increased, and the khat chewer endemic equilibrium point exists and stable. The existence of this condition is due to the fact that R0=6.0217 which is greater than one. This supports theorem that the stability of khat chewers endemic equilibrium point exists when R0>1, that is, if R0>1, each moderate and addicted chewer produces on average more than one new khat chewer; then, khat chewers will be able to spread in the given society.Figure 2
Time series plot of state variables forR0=0.0021<1.Figure 3
Time series plot of state variables forR0=6.0217>1.From Figure4, we observe that as the effective contact rate, α, increases, the moderate chewer individuals are increased, while, the susceptible individuals are decreased due to the influence of moderate chewer individuals. It suggests that as the effective contact rate decreases, moderate individuals who chew khat to combat fatigue and act against obesity due to appetite suppression, peer pressure, passing the time, and socialization decrease, and they cannot influence the susceptible individuals to chew khat, and as a result, the khat chewer individuals are eliminated or removed from the community.Figure 4
Susceptible and moderate khat chewer population with varying effective contact rate (α).Moreover, from Figure5, we observe that as the rate at which Et joins to It and Rt, β, increases, moderate chewer individuals decreases, while addicted individuals are increased. From Figure 6, we observe that as the rate at which recovery rate, θ, increases, the addicted individuals are decreased, while the quitter individuals are increased. This is due to economic, social (isolation) and health concerns, media awareness, and stage-based treatments. Finally, from Figure 7, we observe that as the rate at which Rt becomes to St, γ, increases, the susceptible individuals are increased, while the quitter individuals are decreased.Figure 5
Moderate khat chewer and addicted population with varying rate at whichEt joins to It and Rt, β.Figure 6
Addicted and quitter population with varying recovery rate (θ)Figure 7
Susceptible and quitter population with varying the rate at whichRt becomes St, γ


## 5. Conclusion

In this paper, a deterministic mathematical model on the transmission dynamics of khat chewing was developed. We first showed that there exists a domain where the model is epidemiological and mathematically well posed. The basic reproduction numberR0, chewing khat-free E0, and endemic equilibrium point E1 are computed. Furthermore, we proved that for R0<1, then the chewing khat-free equilibrium point E0 is locally and globally asymptotically stable. We also showed that, chewing khat of an endemic equilibrium point E1 exists and locally and globally asymptotically stable for R0>1. Finally, the simulation result shows that, as the contact rate, α, as well as the rate at which Et joins to It, β, decreases, and as the recovery rate, θ, increases, the addicted individuals decreased from the society. Designing optimal control would be considered in a future study.



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*Source: 3844885-2022-12-19.xml*