Chewing Khat Transmission Dynamics: A Mathematical Model and Stability Analysis

Fantaye, Abayneh Kebede; Birhanu, Zerihun Kinfe

doi:https://doi.org/10.1155/2022/3844885

Journal of Applied Mathematics

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Abstract Introduction Model Formulation Numerical Simulation Conclusion Data Availability Disclosure Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 3844885 | https://doi.org/10.1155/2022/3844885

Chewing Khat Transmission Dynamics: A Mathematical Model and Stability Analysis

Abayneh Kebede Fantaye¹and Zerihun Kinfe Birhanu²

Academic Editor: Ferenc Hartung

Received07 Feb 2022

Revised04 Dec 2022

Accepted12 Dec 2022

Published19 Dec 2022

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 () 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 , the chewing khat-free equilibrium point is locally and globally asymptotically stable, whereas when , the endemic equilibrium point is locally and globally asymptotically stable. The simulation results demonstrate the analytical results.

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 , a group of people who are surrounded by khat chewers but do not chew at the moment but may chew khat in the future , a group of people who chew khat , a group of people includes people who only consumed khat temporarily for social, spiritual ,and recreational purposes , and a group of people who chew khat constantly . 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 () 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 , moderate , addicted , and quitter 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 of to describe the interaction between khat chewer classes. The total population is divided into four compartments: susceptible , moderate , addicted and quitters of khat chewer 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 rate into the susceptible class. It increased with the rate and decreased by effective contact with the moderate chewers at a rate of (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 at rate and decreased when moderate chewers become addicted and quit at rate and , 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 rate and decreases when they join to quitter class at rate of (iv)Quitters of khat chewers are individuals who are quitting of chewing khat. It is increased with the rates of and and while it decreased by the rate of .(v)There is a positive natural death rate for all classes and disease-induced death rate in addicted class. The parameters of the model are presented in Table 1

Based on the assumption and the flow chart shown in Figure 1 for the khat chewing dynamics, we have the following system of nonlinear ordinary differential equations: with

3. Model Analysis

This section studies the solution of Equation (1) in the epidemiologically feasible region.

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. Let and be the initial solution of model Equation (1); then, and are positive in for all .

Proof. Consider Since , and are continuous, we deduce that . If , then positivity holds, but, if , or or or . Now, from the first equation of system (1), we have where . So that, by integrating and simplifying, Equation (8) becomes where , and from the definition of , we have , and then, the solution , and hence, . Again, from the second equation of system (1), we can obtain that Then, by integrating and simplifying Equation (10), we get where , and from the above ; then, the solution , and hence, . Similarly, ; hence, and ; hence, . Thus, based on definition, is not finite which means , 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 . So, we will establish the following lemma:

Lemma 2. All feasible solutions , and of the system of Equation (1) are bounded for the system.

Proof. Differentiating the total human population, with respect to time of our model Equation (1), we have Substituting all state equations from the system of Equation (1) in Equation (13), we get implies that By rearranging and then multiplying both sides of Equation (15) by integrating factor and after some simplification, we have where is the initial values of the total human population,
From Equation (16), we have . Thus, is positively invariant (all solutions in remain in for all time).

Remark 1. In the region , 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, 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 and . Then, we obtain

3.3.1. The Basic Reproduction Number ()

The basic reproduction number measures the expected number of secondary khat chewers that result from one newly khat chewer individuals introduced into a susceptible population. We can obtain 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 , which represents the rate of new khat-chewing devices (new infections) and which represents the rate of individual transmission. Both matrices have a order of , where is the total number of infected compartments. We obtain a new subsystem and a new infected compartment and transfer compartment, designated as and , respectively.

That is

Now, let , and then, the above system of equation can be rewritten as where

The Jacobian matrices of and at disease free equilibrium point, , is given by

As a result, becomes

So that the eigenvalues of matrix are and . Using the method of next-generation matrix, the basic reproduction number, is the spectral radius of or the dominant eigenvalue of and thus, the basic reproduction number is given by

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 by . Now, a system of Equation (1) can be rewritten as:

From the second and third equation of system Equation (25), we have

Then, by dividing Equation (29) by Equation (30), we get

Again by adding the first and second equation of system Equation (25), we have

From Equation (30), we obtain

Substituting Equation (33) in the last equation of Equation (33), we get

Equating Equations (32) and (34), we get

Substituting Equation (35) in Equation (33), we obtain

Again substituting Equation (35) in the last equation of system Equation (34), we obtain

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 solution of the nonlinear system of Equation (1) is locally asymptotically stable, if

Proof. The Jacobian matrix of system Equation (1) at chewing khat-free equilibrium point is The characteristic equation of Equation (29) at chewing khat-free equilibrium point, Equation (39) can be rearranged and simplified as Here, from Equation (40), we obtain Here, and . Furthermore, using Routh-Hurwitz criteria, the last equation of Equation (40) has strictly negative real part since and if . Hence, with , our system of Equation (1) at the chewing khat-free equilibrium point 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: where which represents the number of uninfected individuals, while which represents the number of infected individuals, and represent a vector at disease free equilibrium point pf the same length as . Based on [20], the chewing khat-free equilibrium point, , is globally asymptotically stable if the following two conditions are fulfilled: (i)(): should be a matrix with real negative eigenvalues.(ii)(): should be Metzler matrix.

Theorem 5. The chewing khat-free equilibrium point, , is globally asymptotically stable if .

Proof. From system of Equation (1), we have The system of Equation (1) together with Equation (42) can be written as Using uninfected entry of Jacobian matrix of model Equation (1) and the representation in Equation (42), the matrices are This implies that the sufficient conditions are satisfied, Therefore, chewing khat-free equilibrium point, , is globally asymptotically stable if .

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, , of the model is locally asymptotically stable when .

Proof. The Jacobian matrix of system Equation (1) at chewing khat endemic equilibrium point is The characterstic equation of Equation (47) at is Equation (48) can be simplified as where Using the Routh-Hurwitz criterion, all roots of characteristic polynomial have negative real parts if and only if for . Thus, the chewing khat endemic equilibrium point, , 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 Table 1. The simulations and analysis made are based on these initial conditions: .

The time series plot of state variables for and 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 which is less than one. This supports theorem that the stability of khat chewer free equilibrium point exists when , that is, if , 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 which is greater than one. This supports theorem that the stability of khat chewers endemic equilibrium point exists when , that is, if , 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.

From Figure 4, we observe that as the effective contact rate, , 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.

Moreover, from Figure 5, we observe that as the rate at which joins to and , , increases, moderate chewer individuals decreases, while addicted individuals are increased. From Figure 6, we observe that as the rate at which recovery rate, , 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 becomes to , , increases, the susceptible individuals are increased, while the quitter individuals are decreased.

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 number , chewing khat-free , and endemic equilibrium point are computed. Furthermore, we proved that for , then the chewing khat-free equilibrium point is locally and globally asymptotically stable. We also showed that, chewing khat of an endemic equilibrium point exists and locally and globally asymptotically stable for . Finally, the simulation result shows that, as the contact rate, , as well as the rate at which joins to , , decreases, and as the recovery rate, , increases, the addicted individuals decreased from the society. Designing optimal control would be considered in a future study.

Data Availability

The data used to support the findings of this study are included in the article. Actually, we used data from other papers for the simulation. The papers are properly cited.

Disclosure

The research did not receive specific funding but was performed as part of the employment of the authors at Hawassa University and Debre Tabor University.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

M. El-Setouhy, R. Alsanosy, A. Makeen, K. Ghailan, A. Alsharqi, and K. Sheikh, “Impact of khat price increases on consumption behavior - price elasticity analysis,” Substance Abuse Treatment, Prevention, and Policy, vol. 14, no. 1, p. 19, 2019.
View at: Publisher Site | Google Scholar
E. Gebissa, “Khat in the Horn of Africa: Historical perspectives and current trends,” Journal of Ethnopharmacology, vol. 132, no. 3, pp. 607–614, 2010.
View at: Publisher Site | Google Scholar
A. Alem, D. Kebede, and G. Kullgren, “The prevalence and socio-demographic correlates of khat chewing in butajira, Ethiopia,” Acta Psychiatrica Scandinavica, vol. 397, pp. 84–91, 1999.
View at: Google Scholar
M. Odenwald, A. Klein, and N. Warfa, “Introduction to the special issue: the changing use and misuse of khat (Catha edulis)--tradition, trade and tragedy,” Journal of Ethnopharmacology, vol. 132, no. 3, pp. 537–539, 2010.
View at: Publisher Site | Google Scholar
R. A. Rather, S. Berhanu, L. Abaynah, and M. Sultan, “Prevalence of khat (Catha edulis) chewing and its determinants: a respondent-driven survey from Hossana, Ethiopia,” Substance Abuse and Rehabilitation, vol. 12, pp. 41–48, 2021.
View at: Publisher Site | Google Scholar
K. G. Wubneh, F. M. Desta, and H. A. Kahsay, “Mathematical modeling and analysis of khat-chewing dynamics,” Journal of Mathematics, vol. 2021, Article ID 6659551, 12 pages, 2021.
View at: Publisher Site | Google Scholar
T. Abdeta, D. Tolessa, K. Adorjan, and M. Abera, “Prevalence, withdrawal symptoms and associated factors of khat chewing among students at Jimma University in Ethiopia,” BMC Psychiatry, vol. 17, no. 1, p. 142, 2017.
View at: Publisher Site | Google Scholar
T. Y. Akalu, A. G. Baraki, H. F. Wolde, A. M. Lakew, and K. A. Gonete, “Factors affecting current khat chewing among male adults 15–59 years in Ethiopia, 2016: a multi-level analysis from Ethiopian demographic health survey,” BMC Psychiatry, vol. 20, no. 1, pp. 1–8, 2020.
View at: Publisher Site | Google Scholar
H. Douglas, M. Boyle, and N. Lintzeris, “The health impacts of khat: a qualitative study among Somali-Australians,” The Medical Journal of Australia, vol. 195, no. 11-12, pp. 666–669, 2011.
View at: Publisher Site | Google Scholar
F. Teni, A. Surur, A. Hailemariam et al., “Prevalence, reasons, and perceived effects of khat chewing among students of a college in Gondar town, northwestern Ethiopia: a cross-sectional study,” Annals of Medical and Health Sciences Research, vol. 5, no. 6, pp. 454–460, 2015.
View at: Publisher Site | Google Scholar
S. A. Adeoya-Osiguwa and L. R. Fraser, “Cathine and norephedrine, both phenylpropanolamines, accelerate capacitation and then inhibit spontaneous acrosome loss,” Human Reproduction, vol. 20, no. 1, pp. 198–207, 2005.
View at: Publisher Site | Google Scholar
A. Yahya, Y. Rajeshwar, T. Eticha et al., “Socio-Economic and Health Effects of Khat Chewing in Mekelle, Tigray Region, Ethiopia,” Internation Journal of Pharmacy and Pharmaceutical Research, 2016.
View at: Google Scholar
M. Al-Habori, “The potential adverse effects of habitual use ofCatha edulis(khat),” Expert Opinion on Drug Safety, vol. 4, no. 6, pp. 1145–1154, 2005.
View at: Publisher Site | Google Scholar
Y. Yitayih and J. van Os, “Prevalence and determinants of chewing khat among women in Ethiopia: data from Ethiopian demographic and health survey 2016,” BMC Psychiatry, vol. 21, no. 1, pp. 1–8, 2021.
View at: Publisher Site | Google Scholar
W. M. Ali, M. Zubaid, A. Al-Motarreb et al., “Association of khat chewing with increased risk of stroke and death in patients presenting with acute coronary syndrome,” Mayo Clinic Proceedings, vol. 85, pp. 974–980, 2010.
View at: Google Scholar
K. A. Sheikh, M. El-Setouhy, U. Yagoub, R. Alsanosy, and Z. Ahmed, “Khat chewing and health related quality of life: cross-sectional study in Jazan Region, Kingdom of Saudi Arabia,” Health and quality of life outcomes, vol. 12, no. 1, pp. 44–7525, 2014.
View at: Publisher Site | Google Scholar
B. Milanovic, “Qat expenditures in Yemen and Djibouti: an empirical analysis,” Journal of African Economies, vol. 17, no. 5, pp. 661–687, 2008.
View at: Publisher Site | Google Scholar
J. Havlicek, A. Garcia, and D. Smith, “Mental health and substance use disorders among foster youth transitioning to adulthood: past research and future directions,” Children and Youth Services Review, vol. 35, no. 1, pp. 194–203, 2013.
View at: Publisher Site | Google Scholar
P. Sunthrayuth, M. A. Khan, and F. S. Alshammari, “Mathematical modeling to determine the fifth wave of COVID-19 in South Africa,” BioMed Research international, vol. 2022, Article ID 9932483, 14 pages, 2022.
View at: Publisher Site | Google Scholar
A. K. Fantaye and Z. K. Birhanu, “Mathematical model and analysis of corruption dynamics with optimal control,” Journal of Applied Mathematics, vol. 2022, Article ID 8073877, 16 pages, 2022.
View at: Publisher Site | Google Scholar
X. P. Li, Y. Wang, M. A. Khan, M. Y. Alshahrani, and T. Muhammad, “A dynamical study of SARS-CoV-2: a study of third wave,” Results in Physics, vol. 29, article 104705, 2021.
View at: Publisher Site | Google Scholar
M. S. Alqarni, M. Alghamdi, T. Muhammad, A. S. Alshomrani, and M. A. Khan, “Mathematical modeling for novel coronavirus (COVID-19) and control,” Numerical Methods for Partial Differential Equations, vol. 38, no. 4, pp. 760–776, 2022.
View at: Publisher Site | Google Scholar
A. K. Fantaye, “Modelling and stability analysis of cotton leaf curl virus (CLCuV) transmission dynamics in cotton plant,” Journal of Applied Mathematics, vol. 2022, Article ID 6988197, 12 pages, 2022.
View at: Publisher Site | Google Scholar
A. K. Fantaye and M. Muchie, “Eco-epidemiological model and stability analysis of cotton leaf curl virus (CLCuV) transmission dynamics,” Communications in Mathematical Biology and Neuroscience, vol. 2022, 2022.
View at: Google Scholar
P. K. Tiwari, R. K. Rai, S. Khajanchi, R. K. Gupta, and A. K. Misra, “Dynamics of coronavirus pandemic: effects of community awareness and global information campaigns,” The European Physical Journal Plus, vol. 136, no. 10, p. 994, 2021.
View at: Publisher Site | Google Scholar
M. Sardar, S. Khajanchi, S. Biswas, S. F. Abdelwahab, and K. S. Nisar, “Exploring the dynamics of a tumor-immune interplay with time delay,” Alexandria Engineering Journal, vol. 60, no. 5, pp. 4875–4888, 2021.
View at: Publisher Site | Google Scholar
S. Khajanchi, S. Bera, and T. K. Roy, “Mathematical analysis of the global dynamics of a HTLV-I infection model, considering the role of cytotoxic t-lymphocytes,” Mathematics and Computers in Simulation, vol. 180, pp. 354–378, 2021.
View at: Publisher Site | Google Scholar
P. K. Tiwari, R. K. Singh, S. Khajanchi, Y. Kang, and A. K. Misra, “A mathematical model to restore water quality in urban lakes using Phoslock,” Discrete & Continuous Dynamical Systems-B, vol. 26, no. 6, p. 3143, 2021.
View at: Publisher Site | Google Scholar
D. K. Das, S. Khajanchi, and T. K. Kar, “The impact of the media awareness and optimal strategy on the prevalence of tuberculosis,” Applied Mathematics and Computation, vol. 366, article 124732, 2020.
View at: Publisher Site | Google Scholar
Z. K. Birhanu and A. K. Fantaye, “Modeling and analysis of corruption dynamics incorporating media coverage,” Communications in Mathematical Biology and Neuroscience, vol. 2022, 2022.
View at: Google Scholar
D. K. Das, S. Khajanchi, and T. Kar, “Transmission dynamics of tuberculosis with multiple re-infections,” Chaos, Solitons & Fractals, vol. 130, article 109450, 2020.
View at: Publisher Site | Google Scholar
A. K. Misra, R. K. Singh, P. K. Tiwari, S. Khajanchi, and Y. Kang, “Dynamics of algae blooming: effects of budget allocation and time delay,” Nonlinear Dynamics, vol. 100, no. 2, pp. 1779–1807, 2020.
View at: Publisher Site | Google Scholar
L. Wong, K. Lee, I. Russell, and C. Chen, “Endogenous controls for real-time quantitation of miRNA using TaqMan microRNA assays,” Applied Biosystems Application Note, p. P11, 2007.
View at: Google Scholar
M. George, S. Karanja, and D. Theuri, “Modelling Miraa addiction like a disease incorporating voluntary quitting,” Journal of Advances in Mathematics and Computer Science, vol. 24, no. 4, pp. 1–19, 2017.
View at: Publisher Site | Google Scholar
P. Van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, no. 1-2, pp. 29–48, 2002.
View at: Publisher Site | Google Scholar
A. Iggidr, J. Mbang, G. Sallet, and J. J. Tewa, “Multi-compartment models,” Conference Publications, vol. 2007, pp. 506–519, 2007.
View at: Google Scholar

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Copyright © 2022 Abayneh Kebede Fantaye and Zerihun Kinfe Birhanu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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