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Qualitative properties of mathematical model of English language education
Advances in Difference Equations volumeÂ 2021, ArticleÂ number:Â 15 (2021)
Abstract
This paper presents a mathematical model that examines the impacts of traditional and modern educational programs. We calculate two reproduction numbers. By using the Chavez and Song theorem, we show that backward bifurcation occurs. In addition, we investigate the existence and local and global stability of boundary equilibria and coexistence equilibrium point and the global stability of the coexistence equilibrium point using compound matrices.
1 Introduction
Learning a foreign language has always been one main issue for individuals who are living in a globalized world. To achieve this aim, the individuals have spent expense, energy, and time.
The main problem in foreign language education relates to how to teach it: traditional or modern education. The classes are still traditional i.e., teachercentered, and the methods used for teaching do not attend to the individualâ€™s personal needs. Moreover, these methods do not truly support the ambition to develop team working abilities among the students. Not surprisingly, a redesign of the situation is necessary to improve it. Modern education (computeraided procedures) can function as a supplement to traditional education. Thanks to advancements in educational technology such as computers and social media, learning a language does not seem the same painstaking task as before, since as time passes it has turned out to be quite enhanced moving from teachercentered to learnercentered methods.
Firstly, some do not succumb to the reality of the benefits they can have from emerging technology, and secondly, there are others who fascinatingly hurry to use technology with no learning discipline. Modern education is not devoid of problems. The major problem refers to the point that normally sufficient computers are not available. The learners may also be short of time to use computers for learning. Moreover, in this system of education, the learner requires some skills in the appropriate use of computers.
The majority of educational programs can be applied in certain conditions. For example, the software which works in the Windows environment cannot be used in the IOS environment. The opposite case is also true.
Another problem with modern education is that the software and hardware applications and systems are usually expensive and having access to them is difficult. Some learners do not easily accept technology and prefer to be taught through traditional methods.
Levine [5] and other scholars state that although in both methods of education, i.e. traditional and modern, the learners can enjoy the same level of language learning, this level can be achieved in less time in modern education.
We can consider individuals who tend to learn a foreign language either through traditional methods or modern methods according to their abilities and interests. In addition, they can switch between traditional and modern education. Individuals who fail to learn English as a foreign language (EFL) can participate in training again. The present study aims to compare the two educational systems of traditional and modern by using mathematical modeling to examine which system is more effective. This modeling becomes the development of epidemic model with two strains and superinfection [7, 8]. We use the mathematical modeling of LotkaVolterra model see [3, 4].
At first in Sect.Â 2, the researchers present the model and define two basic reproduction numbers. They examine some preliminaries such as boundedness and the positivity of the solutions of the model. In Sect.Â 3, the existence of equilibria and local and global asymptotical stable is investigated. In Sect.Â 4, the occurrence of backward bifurcation is shown by using the theorem of CastilloChavez and Song [2]. In Sect.Â 5, they use a geometric approach, introduced by Muldowney and Li [6], and show that the nontrivial positive equilibrium is globally stable. Finally, a numerical simulation is carried out to support the analytical results.
2 Mathematical model
In this paper, the model of learning the English language by university students has been considered. It has the following compartments: \(S(t)\) is the number of susceptible individuals who tend to participate in language learning classes, \(C(t)\) is the number of students who have selected modern education (computeraided procedures). \(T(t)\) is the number of students who have selected traditional education. \(F(t)\) is the number of students who have successfully completed the course.
Class T selected traditional education due to several reasons including not having easy access to computer within the class, not being skilled enough in working with computer, the high expense of educational software, and the teachersâ€™ unwillingness or their less proficiency to work with computer.
Class C selected modern education due to the problems of traditional education. With the passage of time, an individual who has been present in the traditional class of T may prefer to learn English through modern education maybe because he/she has learned the necessary skills for working with a computer, and is now able to overcome the problems of modern education. Hence, he/she may be transferred to class C with the rate of \(\delta _{1}\). Another case may happen too. An individual in the modern class may come to the conclusion that the traditional class is more suitable for him/her. So, he/she is transferred to class T with the rate of \(\delta _{2}\). In mathematical modeling, the rate of death is considered as Î¼. It is hypothesized that in modern education the individuals with the rate of \(\alpha _{1}\) and in the traditional education the individuals with the rate of \(\alpha _{2}\) are successful in learning English. With the rates of \(1 \alpha _{1}\) and \(1 \alpha _{2}\), the individuals who have not been successful in learning English are transferred to the class of susceptible individuals. We also assume that susceptible individuals enter the classes of modern and traditional education at the rates of \(\frac{\beta _{1} C}{N}\) and \(\frac{\beta _{2} T}{N}\), respectively.
Based on the above assumptions and model diagram in the Fig.Â 1, the proposed model is formulated as the following model:
The parameters in the model (2.1) are as follows:

N: Size of the total population.

S: The number of susceptible individuals who tend to participate in English language class.

T: The number of individuals who tend to learn English language through traditional methods not computeraided procedures.

C: The number of individuals who tend to learn English language via computeraided procedures (modern education).

F: Individuals who have finished the course successfully.

Î›: The rate at which recruits enter the susceptible population.

\(\beta _{1}\): The probability that a susceptible individual uses traditional education to learn English language.

\(\beta _{2}\): The probability that a susceptible individual uses modern education (computeraided procedures) to learn English language.

Î¼: The natural death rate of the general population.

\(\delta _{1}\): The rate at which a language learner who uses traditional education turns to a language learner who uses modern education (computeraided procedures).

\(\delta _{2}\): The rate at which a language learner who uses modern education (computeraided procedures) turns to a language learner who uses traditional education.

\(\alpha _{1}\): The rate of ending the traditional course by a language learner successfully.

\(\alpha _{2}\): The rate of ending the modern course by a language learner successfully.
The total population \(N(t) = S(t)+C(t)+T(t)+F(t)\) satisfies the following relation:
hence \(\limsup_{t \to \infty } N(t)=\frac{\Lambda }{\mu }\). Therefore the set
is a positively invariant set for (2.1). Therefore the dynamics of the model can be studied only in Î©.
We define two reproduction numbers:
Let
where \(s = 1  c  g  f\). Hence system (2.1) is converted to the following system:
Now, we prove the positivity of the solutions.
Proposition 2.1
If the initial conditions are positive, i.e., \(c(0)>0\), \(g(0) >0\) and \(f(0) > 0\), then the solution \((c(t), g(t), f(t))\) of the system is positive for all \(t \geq 0\).
Proof
Let \((c(t), g(t), f(t))\) be a solution of the system with \(c(0) > 0\), \(g(0) > 0\) and \(f(0) > 0\). Assume the conclusion is not true, then there is a \(t^{*} > 0\) such that
and
If \(\min \{c(t^{*}), g(t^{*}), f(t^{*})\} = f(t^{*})\), then we have
Therefore \(0=f(t^{*}) \geq f(0) \exp (  \mu t) >0\), hence it is a contradiction. Similarly, if \(\min \{c(t^{*}), g(t^{*}), f(t^{*})\} \) is equal to \(c(t^{*})\) or \(g(t^{*})\), then this leads to a contradiction.â€ƒâ–¡
3 Existence and stability of equilibria \(E_{0}\), \(E_{1}\) and \(E_{2}\)
System (2.5) has the following equilibria:
The equilibria \(E_{1}\) and \(E_{2}\) exist provided that \(R_{0} > 1\).
Theorem 3.1
The educationfree equilibrium \(E_{0}\) is asymptotically stable provided that \(R_{0} < 1\).
Proof
We need to be careful in computing the Jacobian. In particular, we need to keep in mind that \(N(t) = S(t)+C(t)+T(t)+F(t)\). The Jacobian matrix of the system has the following form:
where
The Jacobian matrix at the equilibrium \(E_{0} \) is
\(J_{0}\) has the eigenvalues
Since \(R_{0} < 1\), all eigenvalues are negative. Therefore \(E_{0}\) is asymptotically stable.â€ƒâ–¡
We use Lasalleâ€™s invariance principle and we show that the educationfree \(E_{0}\) is globally asymptotically stable.
Theorem 3.2
(Lasalleâ€™s invariance principle)
Let \(X^{*}\) be an equilibrium point for \(X'=F(X)\) and let \(L:U \to \mathbb{R}\) be a Lyapunov function for \(X^{*}\), where U is an open set containing \(X^{*}\). Let \(P \subset U\) be a neighborhood of \(X^{*}\) that is closed and bounded. Suppose that P is positively invariant and that there is no entire solution in \(PX^{*}\) on which L is constant. Then \(X^{*}\) is asymptotically stable, and P is contained in the basin of attraction of \(X^{*}\).
Theorem 3.3
The educationfree equilibrium \(E_{0}\) is globally asymptotically stable provided that \(R_{0} < 1\).
Proof
Consider the following Lyapunov function on \(\mathbb{R}^{3}_{+}\):
It is easy to see that \(V=0 \) at the diseasefree equilibrium. We compute the derivative of V with respect to t:
Since \(R_{0}<1\), then dV is nonpositive. Hence the educationfree equilibrium \(E_{0}\) is globally asymptotically stable by the Lassalle invariance principle.â€ƒâ–¡
In the following theorem, we get the conditions that the equilibrium \(E_{1} = (\hat{c}, 0, \hat{f})\) is locally asymptotically stable.
Theorem 3.4
Let
The equilibrium \(E_{1} = (\hat{c}, 0, \hat{f})\) is locally asymptotically stable if and only if \(\widehat{R_{2}^{1}} < 1\).
Proof
The local stability of the equilibrium \(E_{1}\) is given by the Jacobian matrix at \(E_{1}\):
where
An eigenvalue of matrix \(J_{1}\) is
The other eigenvalues are the eigenvalues of the following submatrix:
Matrix \(J_{11}\) has eigenvalues with the negative real parts if \(\operatorname{Tr} (J_{11}) <0\) and \(\det (J_{11})>0\). We can easily see that
By replacing
in Eq.Â (3.2), since \(R_{1}>1\), we get
Hence, stability \(E_{1}\) is determined by the sign of the eigenvalue \(\lambda _{11}\),
The equilibrium \(E_{1}\) will be locally asymptotically stable if and only if
\(\lambda _{11}<0\), that is,
Therefore, the equilibrium \(E_{1}= (\hat{c}, 0 , \hat{f})\) is locally asymptotically stable.â€ƒâ–¡
Theorem 3.5
Suppose \(2\alpha _{1}=2\alpha _{1}=\beta _{1}=\beta _{2}\). The equilibrium \(E_{1}= (\hat{c}, 0 , \hat{f})\) is globally asymptotically stable provided that \(\widehat{R_{2}^{1}} < 1\).
Proof
We consider the following Lyapunov function on \(\mathbb{R}_{+}^{3}\):
It is not hard to see that \(V=0\) at \(E_{1}\). We take the derivative of V with respect to t:
We evaluate the coefficients \(K_{1}\), \(K_{2}\), \(K_{3}\), such that the coefficients of \(g (f \hat{f})\) and \((c \hat{c})(f \hat{f})\) are equal to zero:
The following inequality has been obtained from the equation for \(dV/ dt\) and by using the inequalities \(a^{2} + b^{2} \geq 2ab\) and \(a^{2} + b^{2} \geq 2ab\):
If \(\widehat{R_{2}^{1}} <1\), then \(\frac{dV}{dt}\) is negative. Therefore, by the Lasalle invariance principle, the equilibrium \(E_{1}\) is globally asymptotically stable.â€ƒâ–¡
The proof of locally and globally asymptotically stability of \(E_{2}\) is similar to the one discussed above. For this purpose, we define
Now, we get an equilibrium in which both types of educational methods are present. This equilibrium is a nontrivial solution of the following system:
System (3.5) gives the following values:
where \(\delta = \delta _{1} \beta _{1}  \delta _{2} \beta _{2}\). Therefore the following theorem holds.
Theorem 3.6
Suppose \(\widehat{R_{2}^{1}} < 1\) and \(\widehat{R_{1}^{2}} < 1\). The equilibrium \(E_{3} = (c^{*}, t^{*}, f^{*})\) exists provided that one of the following conditions holds:
1) \(\delta >0\), \(\beta _{1} < \beta _{2}\) and \(\alpha _{2}< \alpha _{1}\),
2) \(\delta <0\), \(\beta _{2} < \beta _{1}\) and \(\alpha _{1}< \alpha _{2}\).
4 Backward bifurcation
Initially, we will state CastilloChavez and Song theorem, then we show that backward bifurcation occurs.
Theorem 4.1
(CastilloChavez and Song theorem)
Assume

A1:
\(A = D_{x}f(0,0) = ( \frac{\partial f_{i}}{\partial x_{j}}(0,0) )\) is the linearization matrix of the following system around the equilibrium 0 with Ï• evaluated at 0 (zero is a simple eigenvalue of A and all other eigenvalues of A have negative real parts):
$$\begin{aligned} & \frac{dx}{dt}=f(x,\phi ), \qquad f: \mathbb{R}^{n}\times \mathbb{R} \to \mathbb{R}\textit{ and } f \in \mathbb{C}^{2}\bigl( \mathbb{R}^{n}\times \mathbb{R}\bigr) \end{aligned}$$(4.1) 
A2:
Matrix A has a nonnegative right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue.
Let \(f_{k}\) be the kth component of f and
$$\begin{aligned} & \textbf{a}=\sum_{k,i,j=1}^{3} v_{k} w_{i} w_{j} \frac{\partial ^{2} F_{k}}{\partial x_{i} \partial x_{j}}(0,0), \\ &\textbf{b}= \sum_{k,i=1}^{3} v_{k} w_{i} \frac{\partial ^{2} F_{k}}{\partial x_{i} \partial \varphi } (0,0). \end{aligned}$$The local dynamics of (4.1) around 0 are totally determined by a and b.

i.
\(a > 0\), \(b > 0\). When \(\phi <0\) with \(\phi \ll 1\), 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when \(0< \phi \ll 1\), 0 is unstable and there exists a negative and a locally asymptotically stable equilibrium.

ii.
\(a< 0\), \(b > 0\). When \(\phi <0\) with \(\phi \ll 1\), 0 is unstable; when \(0< \phi \ll 1\), 0 is locally asymptotically stable, and there exists a positive unstable equilibrium.

iii.
\(a > 0\), \(b < 0\). When \(\phi <0\) with \(\phi \ll 1\), 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when \(0< \phi \ll 1\), 0 is stable, and a positive unstable equilibrium appears.

iv.
\(a < 0\), \(b > 0\). When Ï• changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable.

i.
We apply the theorem of CastilloChavez and Song to show that in system backward bifurcation occurs in the following cases:
Case 1. \(R_{1} = 1\) and \(R_{2} < 1\).
Since \(R_{1} = 1\), we can compute \(\beta _{1}\) as
The eigenvalues of \(J(E_{0}, \beta _{1}^{*})\) are 0, âˆ’Î¼, \(\beta _{2} (1 \frac{1}{R_{2}} )\). Therefore, zero is simple and other eigenvalues are negative real numbers. If \(R_{1} = 1\), then \(\lambda _{1}=0\) and corresponding right and left eigenvalues are \(w = (1, 0, \frac{\alpha _{1}}{\mu } )\) and \(v = (1, 0, 0)\). We rewrite the system (2.5) as follows:
Now, we compute the quantities a and b in the following form:
Case 2. \(R_{2}=1\) and \(R_{1}<1\).
From \(R_{2}=1\), we conclude that
The Jacobian matrix \(J(E_{0}, \beta _{2}^{*})\) has eigenvalues 0, âˆ’Î¼, \(\beta _{1} (1 \frac{1}{R_{1}} )\). Therefore zero is simple, since \(R_{1}<1\) other eigenvalues are negative real numbers. The right and left eigenvectors of \(J(E_{0}, \beta _{2}^{*})\) associated with \(\lambda _{1}=0\) are \(w = (0,1, \frac{\alpha _{2}}{\mu } )\) and \(v = (0,1, 0)\), respectively. Hence
Since \(\textbf{a} <0\) and \(\textbf{b} >0\), model (2.1) undergoes a backward bifurcation.
5 Global stability of the nontrivial positive equilibrium
The second additive compound of the Jacobian matrix J is the \(3 \times 3\) matrix given by \(J^{[2]}\):
where
Now, we use the geometric method for the global stability problem and prove sufficient conditions for the global stability of the nontrivial positive equilibrium(see [6]). We denote unit ball of \(\mathbb{R}^{2}\) and its closure and boundary by \(\mathcal{B}\), \(\overline{\mathcal{B}}\) and \(\partial \mathcal{B}\), respectively. The collection of all Lipschitzian functions from X to Y is denoted by \(\operatorname{Lip}(X \to Y )\). A function \(\phi \in \operatorname{Lip}(\overline{\mathcal{B} } \to \Omega )\) is considered as a simply connected and rectifiable surface in \(\Omega \subseteq \mathbb{R}^{n}\). A rectifiable and closed curve in Î© is a function \(\phi \in \operatorname{Lip}( \partial \mathcal{B} \to \Omega )\) and it is called simple if it is onetoone. Assume \(\sum (\psi , \Omega )= \{ \psi \in \operatorname{Lip}(\overline{ \mathcal{B}} \to \Omega ):\phi _{ \partial \mathcal{B} } =\psi \}\). For any simple, closed and rectifiable curve in Î©, \(\sigma (\psi ,\Omega )\) is a nonvoid set provided that Î© is an open domain which is simply connected.
We consider a norm \(\Vert \cdot \Vert \) on . We define a functional \(\mathcal{S}\) on the surfaces in Î© by the following relation:
In which the mapping \(u \to \phi (u)\) is Lipschitzian on \(\overline{\mathcal{B}}\), and \(\frac{\partial \phi }{\partial u_{1}} \wedge \frac{\partial \phi }{\partial u_{2}}\) is the wedge product in . Furthermore, the matrix function P is invertible and \(\Vert P^{1} \Vert \) is a bounded function on \(\phi (\overline{\mathcal{B}})\). The following result is stated in [6].
Consider a \(C^{1} \) function on the set \(\Omega \subseteq \mathbb{R}^{n}\) such as \(x \to f(x) \in \mathbb{R}^{n}\) and the following ODE system:
We consider the function \(\phi _{t}(u)=x(t, \phi (u))\) as the solution of (5.2), passing through \((0; \phi (u))\), for any Ï•. We define the righthand derivative of \(\mathcal{S} \phi _{t}\), by the following relation:
In which \(Q = P_{f} P^{1} + P \frac{\partial f^{[2]}}{\partial x} P^{1}\), \(P_{f}\) represents the directional derivative of P in the direction of the vector field f, and \(\frac{\partial f^{[2]}}{\partial x}\) denotes the second additive compound matrix of \(\frac{\partial f}{\partial x}\). Furthermore, we consider the following differential equation:
For which the solution is of the form \(z=P.(\frac{\partial \phi }{\partial u_{1}} \wedge \frac{\partial \phi }{\partial u_{2}})\). The formula \(D_{+} \mathcal{S} \phi _{t}\) can be expressed as,
Let P be the following matrix:
Therefore, we have the matrix
hence the matrix \(P J^{[2]} P^{1}= J^{[2]}\) and
where
We use the following norm for \(z=(z_{1}, z_{2},z_{3})^{T}\):
Lemma 5.1
There exists a constant \(\tau >0\) such that \(D_{+} \Vert z \Vert \leq  \tau \Vert z \Vert \) for all \(z \in \mathbb{R}^{3}\) and all \(c,g, f \geq 0\), where z is a solution of \(\frac{dz}{dt} = Q(\phi _{t}(u)) z \) provided
Proof
We prove the existence of some \(\tau > 0\) such that \(D_{+} \Vert z\Vert \leq \tau \Vert z\Vert \) and z is a solution of \(\frac{dz}{dt} = Q(\phi _{t}(u)) z \). The proof contains eight cases based on the different octants and the definition of the norm in (5.6). We consider \(\delta = \delta _{1} \beta _{1} \delta _{2} \beta _{2} >0\). By using Theorem 3.6, we have \(\beta _{1} < \beta _{2}\) and \(\alpha _{2} < \alpha _{1}\). For \(\delta <0\), the proof is similar.
Case 1: \(z_{1}, z_{2}, z_{3} > 0\) and \(\vert z_{1} \vert +\vert z_{3} \vert > \vert z_{2} \vert +\vert z_{3} \vert \). In this case \(\Vert z \Vert =\vert z_{1} \vert + \vert z_{3}\vert \)
We consider \(K_{1} = 2\beta _{2} cg+2\beta _{2} g^{2}+2\beta _{2} gf +2\delta cg\), since \(\vert z_{2}\vert < \vert z_{1} \vert \), therefore, \(K_{1} \vert z_{2} \vert < K_{1} \vert z_{1} \vert \). We have
where
\(D_{+} \Vert z \Vert \) be bounded away from zero on the negative side for all z and \(c, g, f>0\), therefore, we require
Case 2: \(z_{1}, z_{2}, z_{3} > 0\) and \(z_{1}+z_{3}< z_{2}+z_{3}\). In this case \(\Vert z \Vert =z_{2}+z_{3}\) and
Since the coefficient \(z_{1}\) is smaller than \(\alpha _{2}\) and \(z_{1}< z_{2}\),
Here
\(D_{+} \Vert z \Vert \) be bounded away from zero on the negative side for all z and \(c, g, f>0\), therefore, we require
Case 3: \(z_{1}<0< z_{2}, z_{3}\) and \(z_{1}+z_{3}> z_{2}+z_{3}\), therefore \(\Vert z \Vert =z_{1}+z_{3}\) and
Since \(\vert z_{2} \vert < \vert z_{1} \vert \), we have
Here
Therefore, we require
Case 4: \(z_{1}< 0< z_{2},z_{3}\) and \(z_{1}+z_{3}< z_{2}+z_{3}\). Hence \(\Vert z\Vert =z_{2}+z_{3}\) and \(D_{+} \Vert z\Vert =D_{+} (z_{2}+z_{3})\). We have
Since \(\vert z_{1} \vert < \vert z_{2}\vert \), \(\alpha _{1} \vert z_{1} \vert < \alpha _{1} \vert z_{2}\vert \), we have
Here
Therefore, we require
Case 5: \(z_{2}< 0< z_{1},z_{3}\) and \(z_{1}+z_{3}> z_{2}\). Hence \(\Vert z\Vert =z_{1}+z_{3}\) and
We consider \(K_{5} = 2\beta _{2} g+ \delta g\). Therefore \(K_{5} z_{2} < K_{5}z_{1} + K_{5}z_{3}\). Hence we have
Here
Therefore, we require
Case 6: \(z_{2}< 0< z_{1},z_{3}\) and \(z_{1}+z_{3}< z_{2}\). Hence \(\Vert z\Vert =z_{2}\) and
We consider \(K_{6}= \beta _{1} c + \delta g c\). Since \(z_{1}< z_{2}\) and \(z_{3}< z_{2}\), we have \(K_{6} z_{3}< K_{6} z_{2}\), therefore
Therefore we require
Case 7: \(z_{3}< 0< z_{1},z_{2}\) and \(z_{1}+z_{3}> z_{2}\). Therefore \(\Vert z\Vert =z_{1}+z_{3}\)
Since \(z_{1}+z_{3}> z_{2}\), the relation \(\delta g z_{2} < \delta g z_{1}+ \delta g z_{3}\) holds, therefore
Here
Therefore, we require
Case 8: \(z_{3}< 0< z_{1},z_{2}\) and \(z_{1}+z_{3}< z_{2}\). Therefore \(\Vert z\Vert =z_{2}\) and
We define \(K_{8} = \beta _{1} c + \delta g c\). Since \(z_{3} <0\), \(\delta > 0\), we have \(\alpha _{2} \vert z_{1} \vert < \alpha _{2} \vert z_{2} \vert \) and \(K_{8} \vert z_{3} \vert < K_{8} \vert z_{2} \vert \) and
Therefore, we require
In [6], the global stability of a unique steady state is investigated by the geometric method. In such cases, a compact absorbing set exists. therefore surfaces remain in Î© for all times. But there is no such set, when a model such as model (2.5) has backward bifurcation. Hence, the following lemma proves the existence of the sequence \(\varphi ^{k}\) of surfaces (see [1]).â€ƒâ–¡
Lemma 5.2
Suppose that Ïˆ be a simple and closed curve in Î©. There exist \(\epsilon >0\) and a surface \(\varphi ^{k}\) that minimizes S with respect to \(\Sigma (\psi , \Omega )\) such that \(\varphi _{t}^{k} \subseteq \Omega \) for all \(k=1,2,3,\ldots \) and for all \(t \in [0,\epsilon ]\).
Proof
Suppose \(\xi = \frac{1}{2} \min \{ c : (c, g , f) \in \psi \}\). We consider \(\delta >0\), if \(\delta <0\) then the proof is similar. In this case
therefore, there exists \(\epsilon >0\) such that if \(c(0) \geq \xi \) then the solution remains in Î© for \(t \in [0,\epsilon ]\). Hence we must show that there exists a sequence \(\{\varphi ^{k} \}\) which minimizes S relative to \(\Sigma (\psi , \overline{\Omega })\), where \(\overline{\Omega } = \{ (c , g, f) \in \Omega : c \geq \xi \}\). Let \(\varphi (u)=(c(u), g(u), f(u)) \in \Sigma (\psi , \Omega )\), we define a new surface \(\tilde{\varphi }(u)=(\tilde{c}(u), \tilde{g}(u), \tilde{f}(u)) \), by the following relation:
\(\tilde{\varphi }(u) \in \Sigma (\psi , \overline{\Omega })\). Now, we demonstrate that \(\mathcal{S} \tilde{\varphi } \leq \mathcal{S} \varphi \):
is a vector in \(\mathbb{R}^{3}\). We denote \(\frac{\partial \tilde{\varphi }}{\partial u_{1}} \wedge \frac{\partial \tilde{\varphi }}{\partial u_{2}} =(\tilde{x}_{1} , \tilde{x}_{2}, \tilde{x}_{3})^{T}\) and \(\frac{\partial \varphi }{\partial u_{1}} \wedge \frac{\partial \varphi }{\partial u_{2}} =(x_{1} , x_{2}, x_{3})^{T}\) and show \(\Vert \frac{\partial \tilde{\varphi }}{\partial u_{1}} \wedge \frac{\partial \tilde{\varphi }}{\partial u_{1}} \Vert \leq \Vert \frac{\partial \varphi }{\partial u_{1}} \wedge \frac{\partial \varphi }{\partial u_{2}} \Vert \).
Case 1. If \(c \geq \xi \) then \(\tilde{\varphi } =\varphi \) and \(\vert \tilde{x}_{i} \vert = \vert x_{i} \vert \) \((i=1,2,3)\), therefore \(\Vert \frac{\partial \tilde{\varphi }}{\partial u_{1}} \wedge \frac{\partial \tilde{\varphi }}{\partial u_{1}} \Vert = \Vert \frac{\partial \varphi }{\partial u_{1}} \wedge \frac{\partial \varphi }{\partial u_{2}} \Vert \).
Case 2. If \(c < \xi \) and \(\xi + g(u)+f(u) \leq 1\), then \(\tilde{\varphi } (v)= (\xi , g(v), f(v))\). Therefore
almost everywhere. Hence it follows that \(\vert \tilde{x}_{i} \vert \leq \vert x_{i} \vert \), thus \(\Vert \frac{\partial \tilde{\varphi }}{\partial u_{1}} \wedge \frac{\partial \tilde{\varphi }}{\partial u_{1}} \Vert \leq \Vert \frac{\partial \varphi }{\partial u_{1}} \wedge \frac{\partial \varphi }{\partial u_{2}} \Vert \).
Case 3. If \(c< \xi \) and \(\xi + g(u)+f(u) > 1\), then \(\tilde{\varphi } (v)= ( \xi , \frac{g}{g+f}(1\xi ) , \frac{f}{g+f}(1\xi ) )\). Therefore
Thus, \(\frac{\partial \tilde{\varphi }}{\partial u_{1}}\) and \(\frac{\partial \tilde{\varphi }}{\partial u_{2}}\) are linearly dependent, and hence their wedge product is \(\frac{\partial \tilde{\varphi }}{\partial u_{1}} \wedge \frac{\partial \tilde{\varphi }}{\partial u_{2}} =0\). Therefore \(\Vert \frac{\partial \tilde{\varphi }}{\partial u_{1}} \wedge \frac{\partial \tilde{\varphi }}{\partial u_{1}} \Vert =0 \leq \Vert \frac{\partial \varphi }{\partial u_{1}} \wedge \frac{\partial \varphi }{\partial u_{2}} \Vert \). We denote \(\tilde{i}_{2}(u)= \max \{ c , \xi \}\) and hence \(\frac{1}{\tilde{c}} < \frac{1}{c}\). Therefore
Suppose that \(\{\varphi ^{k}\}\) is a sequence of surfaces that minimizes \(\mathcal{S}\) respect to \(\Sigma (\psi ,\Omega )\). Let \(\{\tilde{\varphi }^{k}\}\) be a sequence of surfaces in \(\Sigma (\psi , \tilde{\Omega })\) defined by the above definition. Since \(\mathcal{S} \tilde{\varphi } \leq \mathcal{S} \varphi \) for every k, and \(\Sigma (\psi , \tilde{\Omega })\) is a subset of \(\Sigma (\psi ,\Omega )\), one concludes that \(\tilde{\varphi }^{k}\) minimizes S relative to \(\Sigma (\psi ,\tilde{\Omega })\).â€ƒâ–¡
Lemmas 6.1 and 6.2 and Corollary 5.4 in [1] lead one to conclude to the following theorem.
Theorem 5.3
Positive semitrajectories of system converges to an equilibrium point, i.e., any Ï‰limit point of 2.1 in Î© is an equilibrium point.
Finally, the above theorem implies the following result.
Theorem 5.4
Suppose the inequalities in LemmaÂ 6.2 hold, if \(R_{0}>1\), then all solutions of (2.5) tends to the unique positive equilibrium point.
6 Example
In the present study, 30 EFL students at Semnan University participated. They chose modern and traditional education equally. Hence \(\beta _{1} = \beta _{2}\). There were 15 students in each group. Before the treatment, all the participants took a reading comprehension test as pretest. TableÂ 1 shows the results of pretest.
An independent samples ttest was run at the 0.05 level of significance to determine whether the difference between the means was significant or not. The results of the ttest presented in TableÂ 2 show that there was no significant difference between the groups as far as their reading comprehension proficiency was concerned before the treatment.
After the treatment, another reading comprehension test was given to the participants. This test served as a posttest. The results of this test are shown in the following tables. As the results in TableÂ 3 show, the modern education group outperformed the traditional education group in posttest. To see if the difference in means was significant, another independent samples ttest was run. TableÂ 4 demonstrates the results of the ttest for the posttest.
The analysis of the data in TableÂ 4 shows that at the 0.05 level of significance there was a significant difference between the posttest reading mean scores of the subjects from the modern education group and the posttest reading mean scores of the subjects from the traditional education group (t observed (3.717) is bigger than t critical (0.001)).
7 Conclusion
In this paper, English learning in a university is modeled. This modeling becomes the development of an epidemic model with two strains and superinfection. In most superinfection models, individuals infected with strain two can superinfect individuals infected with strain one. We consider the case in which superinfection goes in both directions.
In this model, individuals can participate in two groups of modern and traditional education based on their abilities and interests. We carried out a qualitative study of this model including the computation of basic reproduction and the existence and locally and globally stable of boundary equilibria and coexistence equilibrium point. The educationfree \(E_{0}\) is shown; it is local and global stable under suitable conditions.
By using compound matrices, local and global stability of the coexistence equilibrium point was proven. The global stability conditions of this point indicate that initially, individuals chose the traditional education class more than the modern education class. But eventually the number of modern education graduates was higher.
The analysis of the model showed that the occurrence of backward bifurcation and more complexity can occur when we attempt to eliminate the education problems. Therefore, when \(R_{0} <1\), education problems may be persistent.
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We would like to thank all who participated in this study and generously gave their time toward this research. We are also grateful for both helpful comments by the anonymous reviewers and guidance from the editorial team.
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Ghasemabadi, A., Soltanian, N. Qualitative properties of mathematical model of English language education. Adv Differ Equ 2021, 15 (2021). https://doi.org/10.1186/s13662020031800
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DOI: https://doi.org/10.1186/s13662020031800