# Mathematical Modeling and Analysis of Khat-Chewing Dynamics

Authors: Kahsay Godifey Wubneh; Fitsum Mulaw Desta; Hafte Amsalu Kahsay
Journal: Journal of Mathematics
 (2021)

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## Abstract

Khat is a green leaf and greenish plant where its branches and leaves are chewed to discharge liquid having active chemicals that change the user’s mood. The purpose of this article is to develop and analyze a mathematical model that can be used to understand the dynamics of chewing Khat. The proposed model monitors the dynamics of five compartments, namely, a group of people who do not chew Khat, designated asNt; a group of people who are surrounded by Khat chewers but do not chew at present and may chew Khat in the future, denoted this as Σt; a group of people who chew Khat, which is represented in Ct; a group of people contains individuals who consumed Khat quite temporarily for social, spiritual, and recreational purposes, and we describe this group in Tt; and a group of people those who constantly chew Khat, and they are denoted by Ht. We determined the Khat chewing generation number Rc0 using the next-generation matrix method, and we have examined the biological meaningfulness, mathematical wellposedness, and stability of both Khat chewing-free and Khat chewing-present equilibrium points of the model analytically. Numerical simulations were presented by solving our dynamical system using Matlabode45 to check the analytical results by considering parameter estimations. The results of this study show that, for Rc0=.00039, the Khat chewing-free equilibrium point is stable, and it is unstable for Rc0=1.194, and the Khat chewing-present equilibrium point is stable if Rc0=1.194, and it is unstable if Rc0=.00039. The stability of both equilibrium points implies that, for a high rate of conversion from non-Khat chewer to exposed groups ρ, the inflow of an insignificant number of Khat chewers to the community produces a significant number of Khat chewers , and if the return back from Khat chewing to the exposed group because of socio-economic, environmental, and religious influencesα2 grows exponentially, the inflow of an insignificant number of Khat chewers to the community produces an insignificant number of Khat chewers. It is found that increasing the rate of conversion from non-Khat chewer to exposed groups ρ makes the disease eradication more challenging. We, therefore, strongly urge religious leaders, social committee leaders, elders, and health experts to teach their followers to reduce their Khat-chewing habits.


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## Body

## 1. Introduction

Khat has different names in different countries. For example, chat in Ethiopia, Qat in Yemen [1], Miraa in Kenya, and Jaad in Somalia [2]. Khat is also known as the “Leaf of Allah,” [3] and others express it as a‟ flower of paradise” [4]. Khat (Catha Edulis) is a wild green leaf and greenish plant that grows commercially in East Africa and the Arabian Peninsula whose branches and leaves are chewed to release liquids having the active chemicals cathinone and cathine, which alter the user’s mood [4, 5]. Chewing Khat has a long history dating back to the early 14th century and is commonly used for social and spiritual purposes [6]. Although initially Muslim prayers used Khat for ceremonies, it is also widely used by Christian followers for recreational purposes [7]. Ethiopia is currently a growing producer of Khat [8–10]. Land for Khat production from 2001/2002 to 2014/2015 increased by 160%, and new areas of the country, which had not been used before, cultivated rapidly [11]. It plays a significant role in the social life of many people in most parts of Ethiopia and is economically beneficial to producers.The practice of chewing Khat in the Horn of Africa and the Arabian Peninsula has been a popular practice for centuries. Particularly in Ethiopia, the number of Khat users has increased dramatically in recent years, with an average distribution of 15.3% [12, 13]. In the new era, for example, in Harar, even children under the age of five harvest Khat [2]. However, the average age of Khat users in Dessie, Ethiopia, is 19.23 years [14]. Regular users of this plant will find comfort, care, and a feeling of joy. Postchewing effects are often accompanied by sleeplessness, irritability, and poor concentration [4]. Excessive consumption of Khat can lead to serious family, health, and financial problems [15, 16]. Unfortunately, this practice has spread in the Western countries [17, 18] and has become a global problem [19].Chewing Khat is now a public health problem, including heart failure and high blood pressure. At present, Khat chewing is a means for disseminating COVID 19 for the reason that, during harvesting, transporting, distributing, selling, and selecting a package, of course, the communal and individual act of Khat is exposed to human contact, and most of the consumers are chewing unwashed (contaminated) leaves of Khat in public places [20].The prevalence of chewing Khat is high among high school and college students due to peer pressure, family chewing habits, and living conditions. Many researchers say that a complete ban on the production and trade of Khat could harm Ethiopia’s economy in the short term [21]; some say the Ethiopian government has imposed two taxes at the local and federal levels. According to the review, chewing habits have affected the majority of productive age groups, confirmed by many students [7].According to Dhiraj Kumar Das et al. [22], media awareness is a very effective tool to generate preventive control measures of communicable diseases. To account for the effect of media awareness, there are two approaches: updating the disease transmission rate to accumulate the significant fall in transmission due to preventive measures and by incorporating a mass media compartment to represent the public interaction with mass media.Mathematical modeling plays an important role in understanding the complexities of communicable diseases such as TB, ebola, rabies, influenza, West Nile virus, zika, measles, pertussis, and HIV/AIDS. The primary purpose of studying communicable disease modeling is to design better health measures to control and ultimately to exterminate the disease. In the last few decades, both mathematicians and epidemiologists have developed many mathematical models to analyze the transmission procedure of several infectious diseases. To determine appropriate public health measures, a threshold quantity, called the basic reproduction numberR0, plays a significant role. In general, if R0<1, the disease will die out in time whereas if R0>1, the disease will persist and thus transmit through the individuals. Such type of characterization is familiar in most of the epidemiological models. [23–25].As stated by Samui et al. [26], mathematical modeling can aid in understanding the following: how transmissible the disease is, when the infectivity becomes high during the course of an epidemic, how acute the disease is, and how effective interventions has been and ought to be.Makembo, Karandzha, and David [5] developed a mathematical model of Khat (Miraa) chewing, by dividing the entire population into three parts: the number of susceptible persons St, the number of light users Lt, and the number of addicts At, and they used the SIS compartment model.We proposed here a deterministic ordinary differential equation model that can represent the dynamics of Khat-chewing communicable diseases, and its control mechanisms in public health and socio-economic crises of a country.It may be useful to assess the effectiveness of control strategies and to study the Khat-chewing methods used to predict Khat-chewing patterns. We stratified the total human population into five compartments, namely, a group of people who do not chew Khat, designated byNt, a group of people who are surrounded by Khat chewers but do not chew at present; they may chew Khat in the future, and we denote this as Σt, a group of people who chew Khat, which we represent in Ct, a group of people containing individuals who consumed Khat quite temporarily for social, spiritual, and recreational purposes, and we describe this group in Tt, and a group of people those who constantly chew Khat, and they are denoted by Ht, and the general population is denoted by Zt and is expressed as Zt=Nt+Σt+Ct+Tt+Ηt. We also note that people who chew Khat may return to vulnerable groups due to social, economic, environmental, and religious factors with a rate α2 and constant and diverse population. In this study, we used a compartment model similar to SEIR [22–29]. In this deterministic model, we have considered a system of first-order nonlinear ordinary differential equations, which is derived and analyzed analytically and analyzed only numerically using Matlab ode45. From our dynamical system, we got the free equilibrium point of Khat chewing, that is, the point where Khat chewing is not practiced in a given population, and Khat chewing presents equilibrium points and the Khat-chewing generation number, denoted by Rc0. The generation number Rc0 is intended to determine whether the chewing habits of the population have disappeared or increased.The simulations of our model were performed using parameters with their estimated values. Through simulations, the model is analyzed to obtained different situations that produce interesting results among specific classes. The article is organized as follows; in the second section, we formulate a Khat-chewing model, by classifying a constant and heterogeneous entire population into five groups; in the third section, we analyzed our model, checked its biological completeness and mathematically correct significance, determined equilibrium points and the Khat-chewing generation number, and tested existence and stability of equilibrium points of the dynamical system. In section four, we have presented numerical simulations that support our results. Finally, we conclude the manuscript with a conclusion and recommendation in the fifth section.


## 2. Model Formulation for Khat Chewing

Our model contains five first-order nonlinear ordinary differential equations. The parameters we used in our study are all positive and ranges between0and1 and are listed and described in Table 1, and our models’ diagram representation is shown in Figure 1.Table 1
Parameters and their descriptions.
No.ParametersDescription1ϵNatural birth rate2εNatural death rate3ρRate of conversion from non-Khat chewer to the exposed group4α1Rate of conversion from the exposed group to Khat chewer5α2Rate of return back from Khat chewer to exposed because of socio-economic, environmental, and religious factors6α3Rate of removal from Khat chewing to temporary Khat chewers and regular Khat chewers.7α4Rate of return back from temporarily Khat chewers to the Khat chewers group8ωThe probability of conversion from Khat chewing to temporarily Khat chewers91−ωThe probability of conversion from Khat chewing to regular Khat chewers.10δRate of conversion from temporarily chewers to permanent chewers11vKhat-chewing induced disease death rateFigure 1
Flow chart of the Khat-chewing model.We symbolized the entire population using a flow chart as follows.The mathematical representation of the above flow chart is described as follows:(1)dNtdt=εZt−εNt−ρΣtNtZt,dΣtdt=ρΣtNtZt−εΣt−α1Σt+α2Ct,dCtdt=α1Σt+α4Tt−εCt−α3Ct−α2Ct−vCt,dTtdt=α3ωCt−α4Tt−δTt−εTt,dHtdt=α31−ωCt+δTt−εΗt.Subject to the initial conditions,N0=N0, Σ0=Σ0, C0=C0, T0=T0, and H0=H0.


## 3. Model Analysis

### 3.1. Positivity of Model Solutions

Theorem 1.
If initial conditions of (1) are all positive, then it has a positive solution.Proof.
SupposeN0=N0>0, Σ0=Σ0>0, C0=C0>0, T0=T0>0, and H0=H0>0, then(i)
dNt/dt=ϵZt−εNt−ρΣtNt/Zt≥−εNt−ρΣtNt/ZtdNt/dt≥−Ntε+ρΣt/ZtdNt/Nt≥−ε+ρΣt/ZtdtlnNt≥−∫0tε+ρΣt/Ztdt+c1Nt≥N0e−∫0tε+ρΣt/Ztdt>0Hence, the solution ofNt is positive.(ii)
dΣt/dt=ρΣtNt/Zt−εΣt−α1Σt+α2Ct≥−εΣt+α1ΣtdΣt/Σt≥−ε+α1dt⇒lnΣt≥−∫0tε+α1dt+c2⇒Σt≥e−∫0tε+α1dt+c2Σt≥Σ0e−ε+α1t>0Hence, the solution ofΣt is positive.(iii)
dCt/dt=α1Σt+α4Tt−εCt−α3ωCt−α31−ωCt−α2Ct−vCtdCt/dt≥Ct−ε−α3−α2−vdCt/Ct≥−ε−α3−α2−vdtlnCt≥−∫0tε+α3+α2+vdt+c3⇒Ct≥e−∫0tε+α3+α2+vdt+c3Ct≥C0e−ε+α3+α2+vt>0
Hence, the solution ofCt is positive.
Using a similar method, solutions ofTt and Ht are positive.
Since all solutions of the dynamical system are positive, the feasible solution containing all solutions of (1) has the form:(2)ℋ=Nt,Σt,Ct,Tt,Ht∈ℝ+5.


### 3.2. Boundedness of Solutions

To say that the dynamical system (1) is biologically complete and mathematically correct, all its solutions must be bounded.Theorem 2.
For allt>0, a region ℋ exists in which all solutions of (1) are contained and bounded.Proof.
The rate of change of total population sizeZt is defined as follow:(3)dZtdt=dNtdt+dΣtdt+dCtdt+dTtdt+dHtdt.
By substituting all values of left-hand sides of (1) in the above rate of change of total solution, we obtained dZt/dt=Ztϵ−ε−vHt; assuming that disease-induced death rate v=0, initial population size Z0=Z0, and ϵ<ε,(4)dZtdt≤Ztϵ−ε=−Ztτ,dZtdt≤−Ztτ.
0≤Zt≤Z0e−τt for all t>0; hence, all solutions of (1) are bounded.


### 3.3. Existence and Stability of Equilibrium Points

#### 3.3.1. Scaling

To reduce the complexity of our model, we resize the fraction by scaling it as follows. Assuming thatet=Σt/Zt, ct=Ct/Zt, χt=Tt/Zt, nt=Nt/Zt, and ht=Ht/Zt, our dynamical system (1) is reduced to the following:(5)dntdt=ϵ−εnt−ρetnt,detdt=ρetnt−εet−α1et+α2ct,dctdt=α1et+α4χt−εct−α3ct−vct−α2ct,dχtdt=α3ωct−α4χt−δχt−εχt,dhtdt=α31−ωct+δχt−εht,subject to the initial conditions, n0=n0, e0=e0, c0=c0, χ0=χ0, and H0=H0.Since the compartmentht does not appear in the first four equations of the model (5), the dynamics of (5) is the same as the following one:(6)dntdt=ϵ−εnt−ρetnt,detdt=ρetnt−εet−α1et+α2ct,dctdt=α1et+α4χt−εct−α3ct−vct−α2ct,dχtdt=α3ωct−α4χt−δχt−εχt.


#### 3.3.2. Equilibrium Points

Khat chewing-free equilibrium points: Khat chewing-free equilibrium pointEc∗ of (1) is a balance point that exists when Khat chewing is not practiced in a given population, that is, if there are no people chewing Khat. We determine Khatchewing-free equilibrium point Ec∗∗ of (6) by assuming et=χt=ct=0 and substituting it into the right side of (6), and finally, equate the result to zero.Hence,Ec∗∗=n,e,c,χ=1,0,0,0 and Ec∗=n,e,c,χ,h=1,0,0,0,0.Khat chewing-present equilibrium points: Khat chewing-present equilibrium pointEc∗p of (1) is the equilibrium point that exists when Khat chewing is practiced in a given population, that is, if there are people who chew Khat. We determine Khat chewing-present equilibrium point Ec∗p∗ of (6) by assuming at least one of the variables, et,χt,orct≠0.Hence,(7)Ec∗p∗n1,e1,c1,χ1=1−c1α4+δ+ϵϵ+α3+v+α2ϵ+α1−α2+ϵ+α1−α2α4α3ωc1ϵα4+δ+ε,ϵ+α3+v+α2−α3α4ωα4+δ+ϵc1,c1,α3ωc1α4+δ+ϵ,Ec∗pn1,e1,c1,χ1,h1=1−c1α4+δ+ϵϵ+α3+v+α2ϵ+α1−α2+ϵ+α1−α2α4α3ωc1ϵα4+δ+ϵ,ϵ+α3+v+α2−α3α4ωα4+δ+ϵc1,c1,α3ωc1α4+δ+ϵ,c1α31−ωα4+δ+ϵ+α3ωc1ϵα4+δ+ϵ.


#### 3.3.3. Khat-Chewing Generation NumberRc0

The Khat-chewing generation numberRc0 is used to determine secondary Khat chewers, i.e., one Khat chewer individual would create another chewer over the duration of the Khat-chewing period, as long as everyone else is exposed to Khat chewers. The Khat-chewing generation number plays an important role in controlling the significant increase in the number of Khat chewers. In a more general way, the Khat-chewing generation number Rc0 can be stated as the number of new infections (new Khat chewers) created by a typical infective population at a disease-free equilibrium. Rc0 < 1 determines, on average, an infected population creates less than one new infected population during the course of its infective period, and the infection can die out but for Rc0 = 1, an infected individual can infect on an average 1 person, that is, the spread of the disease is stable. In the reverse way, Rc0 > 1 determines each infected population creates, on average, more than one new infection, and the disease can spread over the population [26, 29, 30]. To determine the Khat-chewing generation number, we used the production matrix method described in [26, 29–33].Afterwards, we determine the square matrixℳ, which is the emergence rate of new Khat-chewing devices (new infections), and ℧ which is the transmission rate of individuals. The order of both matrices is nxn, where, n is the number of infected compartments. The new infected compartment and transfer compartments are et and ct, respectively, and we get a new subsystem.(8)detdt=ρetnt−ϵet−α1et+α2ct,dctdt=α1et+α4χt−ϵct−α3ct−vct−α2ct,detdtdctdt=ρetnt0−ϵet+α1et−α2ct−α1et−α4χt+εct+α3ct+vct+α2ct.Then,ℳ=ρnt000, and ℧=ϵ+α1−α2−α1ϵ+α3+α2+v.Because℧ is nonsingular, ℧−1 is also nonsingular.ℳ℧−1 is called the next-generation matrix of our model, and its maximum positive Eigen value is known as the Khat-chewing generation number:(9)℧−1=1ϵ+α1ϵ+α3+α2+v−α2α1ϵ+α3+α2+vα2α1ϵ+α1,ℳ℧−1=1ϵ+α1ϵ+α3+α2+v−α2α1ϵ+α3+α2+vρntα2ρnt00.The eigenvalues are0 and ϵ+α3+α2+vρnt/ϵ+α1ϵ+α3+α2+v−α2α1.The maximum positive eigenvalue ofℳ℧−1 which is the Khat-chewing generation number is(10)Rc0=ϵ+α3+α2+vρntϵ+α1ϵ+α3+α2+v−α2α1.Ifϵ+α1ϵ+α3+α2+v−α2α1>ϵ+α3+α2+vρnt, then Rc0<1, Rc0<1 implies no Khat chewers or the number of Khat chewers reduced to zero [26, 29, 30].Ifϵ+α1ϵ+α3+α2+v−α2α1<ϵ+α3+α2+vρnt, then Rc0>1,Rc0>1 which indicates that the insignificant inflow of Khat chewers into the community will produce a significant number of Khat chewers [26, 29, 30].Stability analysis of the Khat chewing-free equilibrium point (analytically): the Jacobian matrix of (6) is defined as follows:(11)Jn,e,c,χ=−∈−ρet−ρnt00ρetρnt−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈.The Jacobian matrix of (6) at Khat chewing-free equilibrium point Ec∗∗=1,0,0,0 is as follows:(12)J1,0,0,0=−∈−ρ000ρ−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈.Theorem 3.
The Khat chewing-free equilibrium pointEc∗ of (1) is stable if Rc0<1, and it is unstable if Rc0>1.Proof.
Proving local stability ofEc∗∗ leads us to proving local stability of Ec∗.
The characteristics equationJ1,0,0,0 is determined using I4=0:(13)−∈−θ−ρ000ρ−∈−α1−θα200α1−∈−α3−ν−α2−θα400α3ω−α4−δ−∈−θ=0;−θ−θ3+ℵ1θ2+ℵ2θ+ℵ3=0, where(14)ℵ1=ρ−α1−∈+−α3−ν−α2−∈+−α4−δ−∈,ℵ2=−ρ−α1−∈−α3−ν−α2−∈−ρ−α1−∈−α4−δ−∈−−α3−ν−α2−∈−α4−δ−∈+α1α2,ℵ3=−ρ−α1−∈−α4α3ω+ρ−α1−∈−α3−ν−α2−∈−α4−δ−∈−−α4−δ−∈α1α2+α4α3ω.
The eigenvalues areθ1=−ε, and we can find roots of −θ3+ℵ1θ2+ℵ2θ+ℵ3=0 using the Routh–Herwitz stability criterion.Definition 1.
The Routh–Herwitz stability criterion [34] states the following:(a)
If there is no sign change in the first column of the Routh–Herwitz table, the system is stable.(b)
The number of roots of a polynomial lying in the right half-plane is equal to the number of sign changes in the first column. Consequently, the system is unstable if the poles are on the right.
Ifℵ1<0,ℵ1ℵ2+ℵ3/ℵ3<0andℵ3<0, then there are no sign changes in the first column of the Routh–Herwitz table. Using the above definition, the system is stable; this implies that Rc0<1 (the Jacobian matrix has a negative eigenvalue if Rc0<1), and Ec∗∗ is stable. Hence, Ec∗ is stable.
If at least one ofℵ1,ℵ1ℵ2+ℵ3/ℵ3orℵ3 are positive, there is a sign change in the first column of the Routh–Herwitz table. Then, the system is unstable; this implies that Rc0>1, and Ec∗∗ is unstable. Hence, Ec∗ is unstable.
Stability analysis of the Khat chewing-present equilibrium point (analytically): the Jacobian matrix of (6) is defined as follows:(15)Jn,e,c,χ=−∈−ρet−ρnt00ρetρnt−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈.
The Jacobian matrix of (6) at Khat chewing-present equilibrium point Ec∗p∗ is(16)Jn1,e1,c1,χ1=−∈−ρe1−ρn100ρe1ρn1−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈,where(17)n1=1−c1α4+δ+ϵϵ+α3+v+α2ϵ+α1−α2+ϵ+α1−α2α4α3ωc1ϵα4+δ+ϵ,e1=ϵ+α3+v+α2−α3α4ωα4+δ+ϵc1.Theorem 4.
The Khat chewing-present equilibrium pointEc∗p of (1) is stable if Rc0>1 and unstable if Rc0<1.Proof.
Proving stability ofEc∗p∗ leads us to proving local stability of Ec∗p.
The characteristics equation of the Jacobian matrix of (6) at Ec∗p∗ is(18)−∈−ρe1−θ−ρn100ρe1ρn1−∈−α1−θα200α1−∈−α3−ν−α2−θα400α3ω−α4−δ−∈−θ=0;θ4+ℵ4θ3+ℵ5θ2+ℵ6θ+ℵ7=0, where(19)ℵ4=−ρn1+α1−α3−v−α2+α4−ϵ−ρe1,ℵ5=ρe12+ρn1−ϵ−α1−ϵ−ρe1+−ϵ−ρe1−ϵ−α3−v−α2+ρn1−ϵ−α1−ϵ−α3−v−α2+−ϵ−ρe1−ϵ−α4−δ+ρn1−ϵ−α1−ϵ−α4−δ+−ϵ−α3−v−α2−ϵ−α4−δ+−ϵ−ρe1α4−α1α2+ρn1−ϵ−α1α4,ℵ6=−ρe12−ϵ−α3−v−α2−α4ρe12−ρe12−ϵ−α4−δ+−ϵ−α4−δα2α1−−ϵ−ρe1ρn1−ϵ−α1−ϵ−α3−v−α2−−ϵ−ρe1ρn1−ϵ−α1−ϵ−α4−δ−−ϵ−ρe1−ϵ−α4−δ−ϵ−α4−δ−−ϵ−ρe1−ϵ−α4−δ−ϵ−α4−δ−ρn1−ϵ−α1−ϵ−α4−δ−ϵ−α3−v−α2+−ϵ−ρe1α1α2−−ϵ−ρe1ρn1−ϵ−α1α4,ℵ7=ρe12−ϵ−α3−v−α2−ϵ−α4−δ+−ϵ−ρe1−ϵ−α3−v−α2ρn1−ϵ−α1−ϵ−α4−δ−−ϵ−ρe1−ϵ−α4−δα1α2.
Ifℵ4>0,ℵ4ℵ5−ℵ6/ℵ4>0,ℵ7>0,andℵ4ℵ5ℵ6−ℵ62−ℵ7ℵ6ℵ4/ℵ4ℵ5−ℵ6>0, or all are negative, then there are no sign changes in the first column of the Routh–Herwitz table; hence, the system is stable; this implies that Rc0<1, and Ec∗p∗ is unstable. Therefore, Ec∗p is unstable.
If at least one ofℵ4,ℵ4ℵ5−ℵ6/ℵ4,ℵ7,andℵ4ℵ5ℵ6−ℵ62−ℵ7ℵ6ℵ4/ℵ4ℵ5−ℵ6 are negative or positive, there are sign changes in the first column of the Routh–Herwitz table. Then, the system is unstable; this implies that Rc0>1, and Ec∗p∗ is stable. Hence, Ec∗p is stable.


## 3.1. Positivity of Model Solutions

Theorem 1.
If initial conditions of (1) are all positive, then it has a positive solution.Proof.
SupposeN0=N0>0, Σ0=Σ0>0, C0=C0>0, T0=T0>0, and H0=H0>0, then(i)
dNt/dt=ϵZt−εNt−ρΣtNt/Zt≥−εNt−ρΣtNt/ZtdNt/dt≥−Ntε+ρΣt/ZtdNt/Nt≥−ε+ρΣt/ZtdtlnNt≥−∫0tε+ρΣt/Ztdt+c1Nt≥N0e−∫0tε+ρΣt/Ztdt>0Hence, the solution ofNt is positive.(ii)
dΣt/dt=ρΣtNt/Zt−εΣt−α1Σt+α2Ct≥−εΣt+α1ΣtdΣt/Σt≥−ε+α1dt⇒lnΣt≥−∫0tε+α1dt+c2⇒Σt≥e−∫0tε+α1dt+c2Σt≥Σ0e−ε+α1t>0Hence, the solution ofΣt is positive.(iii)
dCt/dt=α1Σt+α4Tt−εCt−α3ωCt−α31−ωCt−α2Ct−vCtdCt/dt≥Ct−ε−α3−α2−vdCt/Ct≥−ε−α3−α2−vdtlnCt≥−∫0tε+α3+α2+vdt+c3⇒Ct≥e−∫0tε+α3+α2+vdt+c3Ct≥C0e−ε+α3+α2+vt>0
Hence, the solution ofCt is positive.
Using a similar method, solutions ofTt and Ht are positive.
Since all solutions of the dynamical system are positive, the feasible solution containing all solutions of (1) has the form:(2)ℋ=Nt,Σt,Ct,Tt,Ht∈ℝ+5.


## 3.2. Boundedness of Solutions

To say that the dynamical system (1) is biologically complete and mathematically correct, all its solutions must be bounded.Theorem 2.
For allt>0, a region ℋ exists in which all solutions of (1) are contained and bounded.Proof.
The rate of change of total population sizeZt is defined as follow:(3)dZtdt=dNtdt+dΣtdt+dCtdt+dTtdt+dHtdt.
By substituting all values of left-hand sides of (1) in the above rate of change of total solution, we obtained dZt/dt=Ztϵ−ε−vHt; assuming that disease-induced death rate v=0, initial population size Z0=Z0, and ϵ<ε,(4)dZtdt≤Ztϵ−ε=−Ztτ,dZtdt≤−Ztτ.
0≤Zt≤Z0e−τt for all t>0; hence, all solutions of (1) are bounded.


## 3.3. Existence and Stability of Equilibrium Points

### 3.3.1. Scaling

To reduce the complexity of our model, we resize the fraction by scaling it as follows. Assuming thatet=Σt/Zt, ct=Ct/Zt, χt=Tt/Zt, nt=Nt/Zt, and ht=Ht/Zt, our dynamical system (1) is reduced to the following:(5)dntdt=ϵ−εnt−ρetnt,detdt=ρetnt−εet−α1et+α2ct,dctdt=α1et+α4χt−εct−α3ct−vct−α2ct,dχtdt=α3ωct−α4χt−δχt−εχt,dhtdt=α31−ωct+δχt−εht,subject to the initial conditions, n0=n0, e0=e0, c0=c0, χ0=χ0, and H0=H0.Since the compartmentht does not appear in the first four equations of the model (5), the dynamics of (5) is the same as the following one:(6)dntdt=ϵ−εnt−ρetnt,detdt=ρetnt−εet−α1et+α2ct,dctdt=α1et+α4χt−εct−α3ct−vct−α2ct,dχtdt=α3ωct−α4χt−δχt−εχt.


### 3.3.2. Equilibrium Points

Khat chewing-free equilibrium points: Khat chewing-free equilibrium pointEc∗ of (1) is a balance point that exists when Khat chewing is not practiced in a given population, that is, if there are no people chewing Khat. We determine Khatchewing-free equilibrium point Ec∗∗ of (6) by assuming et=χt=ct=0 and substituting it into the right side of (6), and finally, equate the result to zero.Hence,Ec∗∗=n,e,c,χ=1,0,0,0 and Ec∗=n,e,c,χ,h=1,0,0,0,0.Khat chewing-present equilibrium points: Khat chewing-present equilibrium pointEc∗p of (1) is the equilibrium point that exists when Khat chewing is practiced in a given population, that is, if there are people who chew Khat. We determine Khat chewing-present equilibrium point Ec∗p∗ of (6) by assuming at least one of the variables, et,χt,orct≠0.Hence,(7)Ec∗p∗n1,e1,c1,χ1=1−c1α4+δ+ϵϵ+α3+v+α2ϵ+α1−α2+ϵ+α1−α2α4α3ωc1ϵα4+δ+ε,ϵ+α3+v+α2−α3α4ωα4+δ+ϵc1,c1,α3ωc1α4+δ+ϵ,Ec∗pn1,e1,c1,χ1,h1=1−c1α4+δ+ϵϵ+α3+v+α2ϵ+α1−α2+ϵ+α1−α2α4α3ωc1ϵα4+δ+ϵ,ϵ+α3+v+α2−α3α4ωα4+δ+ϵc1,c1,α3ωc1α4+δ+ϵ,c1α31−ωα4+δ+ϵ+α3ωc1ϵα4+δ+ϵ.


### 3.3.3. Khat-Chewing Generation NumberRc0

The Khat-chewing generation numberRc0 is used to determine secondary Khat chewers, i.e., one Khat chewer individual would create another chewer over the duration of the Khat-chewing period, as long as everyone else is exposed to Khat chewers. The Khat-chewing generation number plays an important role in controlling the significant increase in the number of Khat chewers. In a more general way, the Khat-chewing generation number Rc0 can be stated as the number of new infections (new Khat chewers) created by a typical infective population at a disease-free equilibrium. Rc0 < 1 determines, on average, an infected population creates less than one new infected population during the course of its infective period, and the infection can die out but for Rc0 = 1, an infected individual can infect on an average 1 person, that is, the spread of the disease is stable. In the reverse way, Rc0 > 1 determines each infected population creates, on average, more than one new infection, and the disease can spread over the population [26, 29, 30]. To determine the Khat-chewing generation number, we used the production matrix method described in [26, 29–33].Afterwards, we determine the square matrixℳ, which is the emergence rate of new Khat-chewing devices (new infections), and ℧ which is the transmission rate of individuals. The order of both matrices is nxn, where, n is the number of infected compartments. The new infected compartment and transfer compartments are et and ct, respectively, and we get a new subsystem.(8)detdt=ρetnt−ϵet−α1et+α2ct,dctdt=α1et+α4χt−ϵct−α3ct−vct−α2ct,detdtdctdt=ρetnt0−ϵet+α1et−α2ct−α1et−α4χt+εct+α3ct+vct+α2ct.Then,ℳ=ρnt000, and ℧=ϵ+α1−α2−α1ϵ+α3+α2+v.Because℧ is nonsingular, ℧−1 is also nonsingular.ℳ℧−1 is called the next-generation matrix of our model, and its maximum positive Eigen value is known as the Khat-chewing generation number:(9)℧−1=1ϵ+α1ϵ+α3+α2+v−α2α1ϵ+α3+α2+vα2α1ϵ+α1,ℳ℧−1=1ϵ+α1ϵ+α3+α2+v−α2α1ϵ+α3+α2+vρntα2ρnt00.The eigenvalues are0 and ϵ+α3+α2+vρnt/ϵ+α1ϵ+α3+α2+v−α2α1.The maximum positive eigenvalue ofℳ℧−1 which is the Khat-chewing generation number is(10)Rc0=ϵ+α3+α2+vρntϵ+α1ϵ+α3+α2+v−α2α1.Ifϵ+α1ϵ+α3+α2+v−α2α1>ϵ+α3+α2+vρnt, then Rc0<1, Rc0<1 implies no Khat chewers or the number of Khat chewers reduced to zero [26, 29, 30].Ifϵ+α1ϵ+α3+α2+v−α2α1<ϵ+α3+α2+vρnt, then Rc0>1,Rc0>1 which indicates that the insignificant inflow of Khat chewers into the community will produce a significant number of Khat chewers [26, 29, 30].Stability analysis of the Khat chewing-free equilibrium point (analytically): the Jacobian matrix of (6) is defined as follows:(11)Jn,e,c,χ=−∈−ρet−ρnt00ρetρnt−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈.The Jacobian matrix of (6) at Khat chewing-free equilibrium point Ec∗∗=1,0,0,0 is as follows:(12)J1,0,0,0=−∈−ρ000ρ−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈.Theorem 3.
The Khat chewing-free equilibrium pointEc∗ of (1) is stable if Rc0<1, and it is unstable if Rc0>1.Proof.
Proving local stability ofEc∗∗ leads us to proving local stability of Ec∗.
The characteristics equationJ1,0,0,0 is determined using I4=0:(13)−∈−θ−ρ000ρ−∈−α1−θα200α1−∈−α3−ν−α2−θα400α3ω−α4−δ−∈−θ=0;−θ−θ3+ℵ1θ2+ℵ2θ+ℵ3=0, where(14)ℵ1=ρ−α1−∈+−α3−ν−α2−∈+−α4−δ−∈,ℵ2=−ρ−α1−∈−α3−ν−α2−∈−ρ−α1−∈−α4−δ−∈−−α3−ν−α2−∈−α4−δ−∈+α1α2,ℵ3=−ρ−α1−∈−α4α3ω+ρ−α1−∈−α3−ν−α2−∈−α4−δ−∈−−α4−δ−∈α1α2+α4α3ω.
The eigenvalues areθ1=−ε, and we can find roots of −θ3+ℵ1θ2+ℵ2θ+ℵ3=0 using the Routh–Herwitz stability criterion.Definition 1.
The Routh–Herwitz stability criterion [34] states the following:(a)
If there is no sign change in the first column of the Routh–Herwitz table, the system is stable.(b)
The number of roots of a polynomial lying in the right half-plane is equal to the number of sign changes in the first column. Consequently, the system is unstable if the poles are on the right.
Ifℵ1<0,ℵ1ℵ2+ℵ3/ℵ3<0andℵ3<0, then there are no sign changes in the first column of the Routh–Herwitz table. Using the above definition, the system is stable; this implies that Rc0<1 (the Jacobian matrix has a negative eigenvalue if Rc0<1), and Ec∗∗ is stable. Hence, Ec∗ is stable.
If at least one ofℵ1,ℵ1ℵ2+ℵ3/ℵ3orℵ3 are positive, there is a sign change in the first column of the Routh–Herwitz table. Then, the system is unstable; this implies that Rc0>1, and Ec∗∗ is unstable. Hence, Ec∗ is unstable.
Stability analysis of the Khat chewing-present equilibrium point (analytically): the Jacobian matrix of (6) is defined as follows:(15)Jn,e,c,χ=−∈−ρet−ρnt00ρetρnt−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈.
The Jacobian matrix of (6) at Khat chewing-present equilibrium point Ec∗p∗ is(16)Jn1,e1,c1,χ1=−∈−ρe1−ρn100ρe1ρn1−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈,where(17)n1=1−c1α4+δ+ϵϵ+α3+v+α2ϵ+α1−α2+ϵ+α1−α2α4α3ωc1ϵα4+δ+ϵ,e1=ϵ+α3+v+α2−α3α4ωα4+δ+ϵc1.Theorem 4.
The Khat chewing-present equilibrium pointEc∗p of (1) is stable if Rc0>1 and unstable if Rc0<1.Proof.
Proving stability ofEc∗p∗ leads us to proving local stability of Ec∗p.
The characteristics equation of the Jacobian matrix of (6) at Ec∗p∗ is(18)−∈−ρe1−θ−ρn100ρe1ρn1−∈−α1−θα200α1−∈−α3−ν−α2−θα400α3ω−α4−δ−∈−θ=0;θ4+ℵ4θ3+ℵ5θ2+ℵ6θ+ℵ7=0, where(19)ℵ4=−ρn1+α1−α3−v−α2+α4−ϵ−ρe1,ℵ5=ρe12+ρn1−ϵ−α1−ϵ−ρe1+−ϵ−ρe1−ϵ−α3−v−α2+ρn1−ϵ−α1−ϵ−α3−v−α2+−ϵ−ρe1−ϵ−α4−δ+ρn1−ϵ−α1−ϵ−α4−δ+−ϵ−α3−v−α2−ϵ−α4−δ+−ϵ−ρe1α4−α1α2+ρn1−ϵ−α1α4,ℵ6=−ρe12−ϵ−α3−v−α2−α4ρe12−ρe12−ϵ−α4−δ+−ϵ−α4−δα2α1−−ϵ−ρe1ρn1−ϵ−α1−ϵ−α3−v−α2−−ϵ−ρe1ρn1−ϵ−α1−ϵ−α4−δ−−ϵ−ρe1−ϵ−α4−δ−ϵ−α4−δ−−ϵ−ρe1−ϵ−α4−δ−ϵ−α4−δ−ρn1−ϵ−α1−ϵ−α4−δ−ϵ−α3−v−α2+−ϵ−ρe1α1α2−−ϵ−ρe1ρn1−ϵ−α1α4,ℵ7=ρe12−ϵ−α3−v−α2−ϵ−α4−δ+−ϵ−ρe1−ϵ−α3−v−α2ρn1−ϵ−α1−ϵ−α4−δ−−ϵ−ρe1−ϵ−α4−δα1α2.
Ifℵ4>0,ℵ4ℵ5−ℵ6/ℵ4>0,ℵ7>0,andℵ4ℵ5ℵ6−ℵ62−ℵ7ℵ6ℵ4/ℵ4ℵ5−ℵ6>0, or all are negative, then there are no sign changes in the first column of the Routh–Herwitz table; hence, the system is stable; this implies that Rc0<1, and Ec∗p∗ is unstable. Therefore, Ec∗p is unstable.
If at least one ofℵ4,ℵ4ℵ5−ℵ6/ℵ4,ℵ7,andℵ4ℵ5ℵ6−ℵ62−ℵ7ℵ6ℵ4/ℵ4ℵ5−ℵ6 are negative or positive, there are sign changes in the first column of the Routh–Herwitz table. Then, the system is unstable; this implies that Rc0>1, and Ec∗p∗ is stable. Hence, Ec∗p is stable.


## 3.3.1. Scaling

To reduce the complexity of our model, we resize the fraction by scaling it as follows. Assuming thatet=Σt/Zt, ct=Ct/Zt, χt=Tt/Zt, nt=Nt/Zt, and ht=Ht/Zt, our dynamical system (1) is reduced to the following:(5)dntdt=ϵ−εnt−ρetnt,detdt=ρetnt−εet−α1et+α2ct,dctdt=α1et+α4χt−εct−α3ct−vct−α2ct,dχtdt=α3ωct−α4χt−δχt−εχt,dhtdt=α31−ωct+δχt−εht,subject to the initial conditions, n0=n0, e0=e0, c0=c0, χ0=χ0, and H0=H0.Since the compartmentht does not appear in the first four equations of the model (5), the dynamics of (5) is the same as the following one:(6)dntdt=ϵ−εnt−ρetnt,detdt=ρetnt−εet−α1et+α2ct,dctdt=α1et+α4χt−εct−α3ct−vct−α2ct,dχtdt=α3ωct−α4χt−δχt−εχt.


## 3.3.2. Equilibrium Points

Khat chewing-free equilibrium points: Khat chewing-free equilibrium pointEc∗ of (1) is a balance point that exists when Khat chewing is not practiced in a given population, that is, if there are no people chewing Khat. We determine Khatchewing-free equilibrium point Ec∗∗ of (6) by assuming et=χt=ct=0 and substituting it into the right side of (6), and finally, equate the result to zero.Hence,Ec∗∗=n,e,c,χ=1,0,0,0 and Ec∗=n,e,c,χ,h=1,0,0,0,0.Khat chewing-present equilibrium points: Khat chewing-present equilibrium pointEc∗p of (1) is the equilibrium point that exists when Khat chewing is practiced in a given population, that is, if there are people who chew Khat. We determine Khat chewing-present equilibrium point Ec∗p∗ of (6) by assuming at least one of the variables, et,χt,orct≠0.Hence,(7)Ec∗p∗n1,e1,c1,χ1=1−c1α4+δ+ϵϵ+α3+v+α2ϵ+α1−α2+ϵ+α1−α2α4α3ωc1ϵα4+δ+ε,ϵ+α3+v+α2−α3α4ωα4+δ+ϵc1,c1,α3ωc1α4+δ+ϵ,Ec∗pn1,e1,c1,χ1,h1=1−c1α4+δ+ϵϵ+α3+v+α2ϵ+α1−α2+ϵ+α1−α2α4α3ωc1ϵα4+δ+ϵ,ϵ+α3+v+α2−α3α4ωα4+δ+ϵc1,c1,α3ωc1α4+δ+ϵ,c1α31−ωα4+δ+ϵ+α3ωc1ϵα4+δ+ϵ.


## 3.3.3. Khat-Chewing Generation NumberRc0

The Khat-chewing generation numberRc0 is used to determine secondary Khat chewers, i.e., one Khat chewer individual would create another chewer over the duration of the Khat-chewing period, as long as everyone else is exposed to Khat chewers. The Khat-chewing generation number plays an important role in controlling the significant increase in the number of Khat chewers. In a more general way, the Khat-chewing generation number Rc0 can be stated as the number of new infections (new Khat chewers) created by a typical infective population at a disease-free equilibrium. Rc0 < 1 determines, on average, an infected population creates less than one new infected population during the course of its infective period, and the infection can die out but for Rc0 = 1, an infected individual can infect on an average 1 person, that is, the spread of the disease is stable. In the reverse way, Rc0 > 1 determines each infected population creates, on average, more than one new infection, and the disease can spread over the population [26, 29, 30]. To determine the Khat-chewing generation number, we used the production matrix method described in [26, 29–33].Afterwards, we determine the square matrixℳ, which is the emergence rate of new Khat-chewing devices (new infections), and ℧ which is the transmission rate of individuals. The order of both matrices is nxn, where, n is the number of infected compartments. The new infected compartment and transfer compartments are et and ct, respectively, and we get a new subsystem.(8)detdt=ρetnt−ϵet−α1et+α2ct,dctdt=α1et+α4χt−ϵct−α3ct−vct−α2ct,detdtdctdt=ρetnt0−ϵet+α1et−α2ct−α1et−α4χt+εct+α3ct+vct+α2ct.Then,ℳ=ρnt000, and ℧=ϵ+α1−α2−α1ϵ+α3+α2+v.Because℧ is nonsingular, ℧−1 is also nonsingular.ℳ℧−1 is called the next-generation matrix of our model, and its maximum positive Eigen value is known as the Khat-chewing generation number:(9)℧−1=1ϵ+α1ϵ+α3+α2+v−α2α1ϵ+α3+α2+vα2α1ϵ+α1,ℳ℧−1=1ϵ+α1ϵ+α3+α2+v−α2α1ϵ+α3+α2+vρntα2ρnt00.The eigenvalues are0 and ϵ+α3+α2+vρnt/ϵ+α1ϵ+α3+α2+v−α2α1.The maximum positive eigenvalue ofℳ℧−1 which is the Khat-chewing generation number is(10)Rc0=ϵ+α3+α2+vρntϵ+α1ϵ+α3+α2+v−α2α1.Ifϵ+α1ϵ+α3+α2+v−α2α1>ϵ+α3+α2+vρnt, then Rc0<1, Rc0<1 implies no Khat chewers or the number of Khat chewers reduced to zero [26, 29, 30].Ifϵ+α1ϵ+α3+α2+v−α2α1<ϵ+α3+α2+vρnt, then Rc0>1,Rc0>1 which indicates that the insignificant inflow of Khat chewers into the community will produce a significant number of Khat chewers [26, 29, 30].Stability analysis of the Khat chewing-free equilibrium point (analytically): the Jacobian matrix of (6) is defined as follows:(11)Jn,e,c,χ=−∈−ρet−ρnt00ρetρnt−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈.The Jacobian matrix of (6) at Khat chewing-free equilibrium point Ec∗∗=1,0,0,0 is as follows:(12)J1,0,0,0=−∈−ρ000ρ−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈.Theorem 3.
The Khat chewing-free equilibrium pointEc∗ of (1) is stable if Rc0<1, and it is unstable if Rc0>1.Proof.
Proving local stability ofEc∗∗ leads us to proving local stability of Ec∗.
The characteristics equationJ1,0,0,0 is determined using I4=0:(13)−∈−θ−ρ000ρ−∈−α1−θα200α1−∈−α3−ν−α2−θα400α3ω−α4−δ−∈−θ=0;−θ−θ3+ℵ1θ2+ℵ2θ+ℵ3=0, where(14)ℵ1=ρ−α1−∈+−α3−ν−α2−∈+−α4−δ−∈,ℵ2=−ρ−α1−∈−α3−ν−α2−∈−ρ−α1−∈−α4−δ−∈−−α3−ν−α2−∈−α4−δ−∈+α1α2,ℵ3=−ρ−α1−∈−α4α3ω+ρ−α1−∈−α3−ν−α2−∈−α4−δ−∈−−α4−δ−∈α1α2+α4α3ω.
The eigenvalues areθ1=−ε, and we can find roots of −θ3+ℵ1θ2+ℵ2θ+ℵ3=0 using the Routh–Herwitz stability criterion.Definition 1.
The Routh–Herwitz stability criterion [34] states the following:(a)
If there is no sign change in the first column of the Routh–Herwitz table, the system is stable.(b)
The number of roots of a polynomial lying in the right half-plane is equal to the number of sign changes in the first column. Consequently, the system is unstable if the poles are on the right.
Ifℵ1<0,ℵ1ℵ2+ℵ3/ℵ3<0andℵ3<0, then there are no sign changes in the first column of the Routh–Herwitz table. Using the above definition, the system is stable; this implies that Rc0<1 (the Jacobian matrix has a negative eigenvalue if Rc0<1), and Ec∗∗ is stable. Hence, Ec∗ is stable.
If at least one ofℵ1,ℵ1ℵ2+ℵ3/ℵ3orℵ3 are positive, there is a sign change in the first column of the Routh–Herwitz table. Then, the system is unstable; this implies that Rc0>1, and Ec∗∗ is unstable. Hence, Ec∗ is unstable.
Stability analysis of the Khat chewing-present equilibrium point (analytically): the Jacobian matrix of (6) is defined as follows:(15)Jn,e,c,χ=−∈−ρet−ρnt00ρetρnt−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈.
The Jacobian matrix of (6) at Khat chewing-present equilibrium point Ec∗p∗ is(16)Jn1,e1,c1,χ1=−∈−ρe1−ρn100ρe1ρn1−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈,where(17)n1=1−c1α4+δ+ϵϵ+α3+v+α2ϵ+α1−α2+ϵ+α1−α2α4α3ωc1ϵα4+δ+ϵ,e1=ϵ+α3+v+α2−α3α4ωα4+δ+ϵc1.Theorem 4.
The Khat chewing-present equilibrium pointEc∗p of (1) is stable if Rc0>1 and unstable if Rc0<1.Proof.
Proving stability ofEc∗p∗ leads us to proving local stability of Ec∗p.
The characteristics equation of the Jacobian matrix of (6) at Ec∗p∗ is(18)−∈−ρe1−θ−ρn100ρe1ρn1−∈−α1−θα200α1−∈−α3−ν−α2−θα400α3ω−α4−δ−∈−θ=0;θ4+ℵ4θ3+ℵ5θ2+ℵ6θ+ℵ7=0, where(19)ℵ4=−ρn1+α1−α3−v−α2+α4−ϵ−ρe1,ℵ5=ρe12+ρn1−ϵ−α1−ϵ−ρe1+−ϵ−ρe1−ϵ−α3−v−α2+ρn1−ϵ−α1−ϵ−α3−v−α2+−ϵ−ρe1−ϵ−α4−δ+ρn1−ϵ−α1−ϵ−α4−δ+−ϵ−α3−v−α2−ϵ−α4−δ+−ϵ−ρe1α4−α1α2+ρn1−ϵ−α1α4,ℵ6=−ρe12−ϵ−α3−v−α2−α4ρe12−ρe12−ϵ−α4−δ+−ϵ−α4−δα2α1−−ϵ−ρe1ρn1−ϵ−α1−ϵ−α3−v−α2−−ϵ−ρe1ρn1−ϵ−α1−ϵ−α4−δ−−ϵ−ρe1−ϵ−α4−δ−ϵ−α4−δ−−ϵ−ρe1−ϵ−α4−δ−ϵ−α4−δ−ρn1−ϵ−α1−ϵ−α4−δ−ϵ−α3−v−α2+−ϵ−ρe1α1α2−−ϵ−ρe1ρn1−ϵ−α1α4,ℵ7=ρe12−ϵ−α3−v−α2−ϵ−α4−δ+−ϵ−ρe1−ϵ−α3−v−α2ρn1−ϵ−α1−ϵ−α4−δ−−ϵ−ρe1−ϵ−α4−δα1α2.
Ifℵ4>0,ℵ4ℵ5−ℵ6/ℵ4>0,ℵ7>0,andℵ4ℵ5ℵ6−ℵ62−ℵ7ℵ6ℵ4/ℵ4ℵ5−ℵ6>0, or all are negative, then there are no sign changes in the first column of the Routh–Herwitz table; hence, the system is stable; this implies that Rc0<1, and Ec∗p∗ is unstable. Therefore, Ec∗p is unstable.
If at least one ofℵ4,ℵ4ℵ5−ℵ6/ℵ4,ℵ7,andℵ4ℵ5ℵ6−ℵ62−ℵ7ℵ6ℵ4/ℵ4ℵ5−ℵ6 are negative or positive, there are sign changes in the first column of the Routh–Herwitz table. Then, the system is unstable; this implies that Rc0>1, and Ec∗p∗ is stable. Hence, Ec∗p is stable.


## 4. Model Numerical Simulation

### 4.1. Parameter Estimation and Stability Analysis of Equilibrium Points Numerically

Table2 shows the estimated and recommended parameter values and their corresponding Khat-chewing generation numbers. Using the results, we analyzed our model numerically.Table 2
Estimated values of parameters and their corresponding Khat-chewing generation numbers.
ParametersValue 1Value 2Value 3Value 4Sourceϵ0.32340.32340.32340.3234WBε0.065510.065510.065510.06551[5, 35]ρ0.000150. 000380. 0550.9Estimatedα10.0650.070.650.68Estimatedα20.00750.00850.250.65Estimatedα30.350.350.350.35Estimatedα40.790.790.790.79Estimatedω0.450.450.450.45Estimated1−ω0.650.650.650.65Estimatedδ0. 590.590.590.59Estimatedv0.450.450.450.45Estimatedn1000100010001000EstimatedRc0 = 0.39Rc0 = 0.97Rc0 = 64.32Rc0 = 1193.76AtEc∗Rc0(1, 0, 0, 0, 0) = 0.00039Rc0(1, 0, 0, 0, 0) = 0.00097Rc0(1, 0, 0, 0, 0) = 0.064Rc0(1, 0, 0, 0, 0) = 1.194WB= World Bank (2017, 2018, and 2019).


### 4.2. Stability Analysis of Khat Chewing-Free Equilibrium PointEc∗∗=1,0,0,0

We analyzed the stability of Khat chewing-free equilibrium pointEc∗∗=1,0,0,0 using estimated values of column 1 and 4 of Table 2 as follows.For column 1,(20)J1,0,0,0=−0.3234−0.00015000−0.388250.0075000.065−1.13090.79000.1575−1.7034.The characteristics equation ofJ1,0,0,0 is determined using I4=0; then, its characteristic’s equation is(21)θ4+3.55θ3+3.94θ2+1.64θ+0.23=0.The eigenvalues are−1.8912−0.9460−0.3564+0.0392iand−0.3564−0.0392i.Since all Eigen values are negative (no sign change in the first column of the Routh–Herwitz table), then the system is stable which impliesRc0<1; then, Khat chewing-free equilibrium point Ec∗∗ is stable. Hence, Khat chewing-free equilibrium point Ec∗ is also stable. Therefore, the number of Khat chewers is reduced to zero.For column 4,(22)Jn,e,c,χ=−∈−ρ000ρ−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈,J1,0,0,0=−0.3234−0.9000−0.10.65000.68−1.770.79000.16−1.7.Its characteristic equation isθ4+3.89θ3+3.94θ2+0.44v−0.15=0, and its roots are −2.2140,−1.4918,−0.3242,and0.1401. There is a sign change in the first column of the Routh–Herwitz table or the roots have no same signs; then, the system is unstable which implies Rc0>1; then, Khat chewing-free equilibrium point Ec∗∗ is unstable. Hence, Khat chewing-free equilibrium point Ec∗ is also unstable. Ec∗ is unstable or Rc0>1, which indicates an insignificant inflow of Khat chewers into the community will produce a large number of Khat chewers.


### 4.3. Stability Analysis of Khat Chewing-Present Equilibrium PointEc∗p∗

We analyzed the stability of Khat chewing-present equilibrium pointEc∗p∗=−286,212,200,18.5, using estimated values of column 2 of Table 2 as follows.Jn1,e1,c1,χ1=−∈−ρe1−ρn100ρe1ρn1−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈, where(23)J−286,212,200,18.5=−0.40.11000.08.−0.50.0085000.07−1.130.79000.16−1.7.Its characteristics equation isθ4+3.78θ3+4.53θ2+2.15θ+0.34=0:(24)−2.03,−0.71+0.06i,−0.71−0.06i,and−0.33.There is no sign change in the first column of the Routh–Herwitz table or the roots have the same sign; then, the system is stable; this implies thatRc0<1, and Ec∗p∗ is unstable. Hence, Ec∗p is unstable.


### 4.4. Numerical Simulation

In this section, we cherry-pick which parameter affects the model in the event of an uncontrolled spread of Khat chewers in the community and which parameter affects the model when the number of Khat chewers in the community decreases. For initial conditionsn0=1000, e0=800, c0=600, χ0=400, and h0=300 and using column 1 of Table 2, we obtained Figure 2 which illustrates that the number of non-Khat chewers, the number of exposed, number of Khat chewers, and the number of temporary Khat chewers reduced or approaches to zero as time rises, and the number of permanent Khat chewers increases as time rises.Figure 2
Phase portrait of our model forRc0=0.39 and Rc0Ec∗=0.00039, and Ec∗ is stable.For initial conditionsn0=1000, e0=800, c0=600, χ0=400, and h0=300 and using column 1 of Table 2, we obtained Figure 3 which demonstrates that the Khat-chewing generation number increases as the value of ρ increases. Hence, if ρ<0.00075, Rc0<1, and the number of Khat chewers reduces to zero, and if ρ>0.00075, Rc0>1, andthe inflow of an insignificant number of Khat chewers into the community produces a significant number of Khat chewers.Figure 3
The Khat-chewing generation number increases as the value ofρ increases.For initial conditionsn0=1000, e0=800, c0=600, χ0=400, and h0=300 and using column 1 of Table 2, we obtained Figure 4 which demonstrates that the Khat-chewing generation number decreases as the value of α2 increases; then, the spreading of Khat chewers declines as α2 increases.Figure 4
Khat-chewing generation number decreases as the value ofα2 increases.For initial conditionsn0=1000, e0=800, c0=600, χ0=400, and h0=300 and using column 4 of Table 2, we obtained Figure 5 which represents that the Khat-chewing generation number increases as the value of ρ increases. Therefore, for ρ>0, Rc0>1, andthe inflow of an insignificant number of Khat chewers into the community produces a significant number of Khat chewers.Figure 5
Khat-chewing generation number increases as the value ofρ increases.For initial conditionsn0=1000, e0=800, c0=600, χ0=400, and h0=300 and using column 4 of Table 2, we obtained Figure 6 which symbolizes that the Khat-chewing generation number increases as the value of α2 increases in column 4 of Table 2 for 0≤α2≤1. From this graph we conclude that the inflow of a insignificant number of Khat chewers into the community produces a significant number of Khat chewers even the rate of return back from Khat chewing to exposed because of socio-economic, environmental, and religious factors increases. Thus, to turn down the spreading of Khat chewers to the community, the rate of return from Khat chewing to an exposed group because of socio-economic, environmental and religious influences must grow exponentially.Figure 6
The Khat-chewing generation number increases as the value ofα2 increases.


## 4.1. Parameter Estimation and Stability Analysis of Equilibrium Points Numerically

Table2 shows the estimated and recommended parameter values and their corresponding Khat-chewing generation numbers. Using the results, we analyzed our model numerically.Table 2
Estimated values of parameters and their corresponding Khat-chewing generation numbers.
ParametersValue 1Value 2Value 3Value 4Sourceϵ0.32340.32340.32340.3234WBε0.065510.065510.065510.06551[5, 35]ρ0.000150. 000380. 0550.9Estimatedα10.0650.070.650.68Estimatedα20.00750.00850.250.65Estimatedα30.350.350.350.35Estimatedα40.790.790.790.79Estimatedω0.450.450.450.45Estimated1−ω0.650.650.650.65Estimatedδ0. 590.590.590.59Estimatedv0.450.450.450.45Estimatedn1000100010001000EstimatedRc0 = 0.39Rc0 = 0.97Rc0 = 64.32Rc0 = 1193.76AtEc∗Rc0(1, 0, 0, 0, 0) = 0.00039Rc0(1, 0, 0, 0, 0) = 0.00097Rc0(1, 0, 0, 0, 0) = 0.064Rc0(1, 0, 0, 0, 0) = 1.194WB= World Bank (2017, 2018, and 2019).


## 4.2. Stability Analysis of Khat Chewing-Free Equilibrium PointEc∗∗=1,0,0,0

We analyzed the stability of Khat chewing-free equilibrium pointEc∗∗=1,0,0,0 using estimated values of column 1 and 4 of Table 2 as follows.For column 1,(20)J1,0,0,0=−0.3234−0.00015000−0.388250.0075000.065−1.13090.79000.1575−1.7034.The characteristics equation ofJ1,0,0,0 is determined using I4=0; then, its characteristic’s equation is(21)θ4+3.55θ3+3.94θ2+1.64θ+0.23=0.The eigenvalues are−1.8912−0.9460−0.3564+0.0392iand−0.3564−0.0392i.Since all Eigen values are negative (no sign change in the first column of the Routh–Herwitz table), then the system is stable which impliesRc0<1; then, Khat chewing-free equilibrium point Ec∗∗ is stable. Hence, Khat chewing-free equilibrium point Ec∗ is also stable. Therefore, the number of Khat chewers is reduced to zero.For column 4,(22)Jn,e,c,χ=−∈−ρ000ρ−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈,J1,0,0,0=−0.3234−0.9000−0.10.65000.68−1.770.79000.16−1.7.Its characteristic equation isθ4+3.89θ3+3.94θ2+0.44v−0.15=0, and its roots are −2.2140,−1.4918,−0.3242,and0.1401. There is a sign change in the first column of the Routh–Herwitz table or the roots have no same signs; then, the system is unstable which implies Rc0>1; then, Khat chewing-free equilibrium point Ec∗∗ is unstable. Hence, Khat chewing-free equilibrium point Ec∗ is also unstable. Ec∗ is unstable or Rc0>1, which indicates an insignificant inflow of Khat chewers into the community will produce a large number of Khat chewers.


## 4.3. Stability Analysis of Khat Chewing-Present Equilibrium PointEc∗p∗

We analyzed the stability of Khat chewing-present equilibrium pointEc∗p∗=−286,212,200,18.5, using estimated values of column 2 of Table 2 as follows.Jn1,e1,c1,χ1=−∈−ρe1−ρn100ρe1ρn1−∈−α1α200α1−∈−α3−ν−α2α400α3ω−α4−δ−∈, where(23)J−286,212,200,18.5=−0.40.11000.08.−0.50.0085000.07−1.130.79000.16−1.7.Its characteristics equation isθ4+3.78θ3+4.53θ2+2.15θ+0.34=0:(24)−2.03,−0.71+0.06i,−0.71−0.06i,and−0.33.There is no sign change in the first column of the Routh–Herwitz table or the roots have the same sign; then, the system is stable; this implies thatRc0<1, and Ec∗p∗ is unstable. Hence, Ec∗p is unstable.


## 4.4. Numerical Simulation

In this section, we cherry-pick which parameter affects the model in the event of an uncontrolled spread of Khat chewers in the community and which parameter affects the model when the number of Khat chewers in the community decreases. For initial conditionsn0=1000, e0=800, c0=600, χ0=400, and h0=300 and using column 1 of Table 2, we obtained Figure 2 which illustrates that the number of non-Khat chewers, the number of exposed, number of Khat chewers, and the number of temporary Khat chewers reduced or approaches to zero as time rises, and the number of permanent Khat chewers increases as time rises.Figure 2
Phase portrait of our model forRc0=0.39 and Rc0Ec∗=0.00039, and Ec∗ is stable.For initial conditionsn0=1000, e0=800, c0=600, χ0=400, and h0=300 and using column 1 of Table 2, we obtained Figure 3 which demonstrates that the Khat-chewing generation number increases as the value of ρ increases. Hence, if ρ<0.00075, Rc0<1, and the number of Khat chewers reduces to zero, and if ρ>0.00075, Rc0>1, andthe inflow of an insignificant number of Khat chewers into the community produces a significant number of Khat chewers.Figure 3
The Khat-chewing generation number increases as the value ofρ increases.For initial conditionsn0=1000, e0=800, c0=600, χ0=400, and h0=300 and using column 1 of Table 2, we obtained Figure 4 which demonstrates that the Khat-chewing generation number decreases as the value of α2 increases; then, the spreading of Khat chewers declines as α2 increases.Figure 4
Khat-chewing generation number decreases as the value ofα2 increases.For initial conditionsn0=1000, e0=800, c0=600, χ0=400, and h0=300 and using column 4 of Table 2, we obtained Figure 5 which represents that the Khat-chewing generation number increases as the value of ρ increases. Therefore, for ρ>0, Rc0>1, andthe inflow of an insignificant number of Khat chewers into the community produces a significant number of Khat chewers.Figure 5
Khat-chewing generation number increases as the value ofρ increases.For initial conditionsn0=1000, e0=800, c0=600, χ0=400, and h0=300 and using column 4 of Table 2, we obtained Figure 6 which symbolizes that the Khat-chewing generation number increases as the value of α2 increases in column 4 of Table 2 for 0≤α2≤1. From this graph we conclude that the inflow of a insignificant number of Khat chewers into the community produces a significant number of Khat chewers even the rate of return back from Khat chewing to exposed because of socio-economic, environmental, and religious factors increases. Thus, to turn down the spreading of Khat chewers to the community, the rate of return from Khat chewing to an exposed group because of socio-economic, environmental and religious influences must grow exponentially.Figure 6
The Khat-chewing generation number increases as the value ofα2 increases.


## 5. Conclusion and Recommendation

In this study, we examined the return from Khat chewing to the exposed group due to socio-economic, environmental, and religious factors, as well as the rate of conversion from Khat chewing to the exposed group of consequences in the Khat chewing dynamics’ model. We have derived the Khat chewing generations’ numberRc0 and Khat chewing-free and Khat chewing-present equilibrium points. For Rc0<1, the Khat chewing-free equilibrium point is stable, and it is unstable for Rc0>1, and the Khat chewing-present equilibrium point is stable if Rc0>1, and it is unstable if Rc0<1.The stability of the two equilibrium points implies that the influx of the insignificant number of Khat chewers into the community produces a significant number of Khat chewers if the conversion rate of nonchewers to vulnerable groups is high, and the inflow of Khat chewers to the community produces an insignificant number of Khat chewers if they return back from Khat chewing to the exposed group because socio-economic, environmental, and religious influences raise rapidly. We recommend to the community to stop or minimize chewing Khat habits because the practice of chewing Khat leads to economic crises, social crises (turn into isolated), and health crises, and since media awareness are a very effective tool to generate precautionary control measures of communicable diseases [22], so both governmental and nongovernmental health office stakeholders must alert their people about the impact of Khat chewing through media. In addition, to future researchers, we recommend them to extend the model by grouping the entire population based on sex, religion, and geographic locations.



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*Source: 6659551-2021-02-13.xml*