Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2022.019781

ARTICLE

Dynamical Model to Optimize Student’s Academic Performance

Evren Hincal and Amna Hashim Alzadjali*

Department of Mathematics, Near East University, Mersin, 10 KKTC, Cyprus
*Corresponding Author: Amna Hashim Alzadjali. Email: gtr-y2006@hotmail.com
Received: 14 October 2021; Accepted: 17 January 2022

1  Introduction

The human capital theory perceives education and learning activities as an investment in people to increase the productivity of goods and services [1]. The industrial and technological development of a country depends on literacy as a requirement for its success. This is especially when a literate member of society engages in an active and effective role in the development process. There is no doubt that combining the skills of improving income generation with knowledge of sustainable development assist mankind in improving his material condition of living through the use of resources available to him.

The academic performance of a student serves as the bedrock for knowledge acquisition and the development of skills that directly impact the socio-economic development of a country [2]. It determines the success or failure of any academic institution [3].

There are many factors that enhance and impede students’ academic performance attributed to students, parents, teachers and environments. The student’s factors include self- motivation, interest in a subject, punctuality in class, regular studying and access to learning materials. Class attendance and students’ attitudes toward their learning have an impact on academic performance. In [4], it is confirmed that in the case of mathematics, students’ attitude towards the subject has a direct impact on their academic performance.

Qualified teachers and facilitators render effective facilitation which enhances academic performance. However, performance target, completion of syllabus, paying attention to weak students, assignment and student evaluation have significant impact too [5].

Parental background and status have significant impact on student’s academic performance. Educated parent provide home school tutorial to their ward and are more encouraging as well. In [6], it was shown that, students with high level of parental involvement in their academics excel their counterparts with no such involvement.

Environmental factors that influence academic performance are enabling environment, infrastructure, adequate facilities and learning materials, well-equipped laboratories, etc. In [7], it is revealed that the availability of physical resources such as the library, textbooks, adequacy of classroom and spacious playing ground affect the performance of the students. Reference [8] emphasized that the use of instructional equipment facilitates effective service delivery and enhances teaching and learning. Distanced school also affects students’ performance in the sence that the more school distance, the more tired students become [9,10].

Also, fairly disciplined schools perform better than less or no disciplined schools. Effective discipline is used to control students’ behavior, which has a direct impact on their academic performance [11]. Furthermore, the student-to-teacher ratio or class size also affects performance. Effective teaching in a moderate class ratio enhances performance [12].

Age has a significant impact on academic performance; older students are likely to drop out than younger ones. Reference [10] showed a significant positive impact of age on academic performance in mathematics and science but the degree of the association is weak.

Mathematical models play a significant role in solving real-life problems [13–15]. Moreover, a mathematical model is an important tool used to optimize a real-life problem for the quickest and effective resolution [16–20]. In this paper, we study the uppermost priority and goal of educators and facilitators. The dubious marginal rate between admission and graduation rates unveils the rates of dropout and withdrawal from school. To improve the academic performance of students, we optimize the performance indices to the dynamics describing the academic performance in the form of nonlinear ODE.

The paper is arranged as follows: Introduction is given in chapter one, followed by definitions of terms and important theorems in Chapter two. Chapter three gives the detail of model formulation while Chapter four gives stability analysis of the solutions of the model. Chapter five is the detailed formation of optimal control and numerical simulations.

2  Definition and Theorems

Definition 1: [21] (Optimal Control)

A fairly general continuous time optimal control problem can be defined as follows:

Problem i: To find the control vector trajectory u:[t0,tf]∈R→Rn minimize the performance index:

J(u)=φ(x(tf))+∫t0tfL(x(t),u(t),t)dt,(1)

subject to:

x˙(t)=f(x(t),u(t),t),x(t0)=x0,(2)

where

x=(x1,x2,x3,…,xn)T, f=(f1,f2,f3,…,fn)T and φ:Rn×R→R is a terminal cost function.

Problem ii: Find tf and u(t) to minimize:

J=∫t0tf1dt=tf−t0,(3)

subject to:

x˙(t)=f(x(t),u(t),t),x(t0)=x0.(4)

This special type of optimal control problem is called the minimum time problem.

Definition 2: [21] (Hamiltonian)

With a time varying Largrange’s multiplier function λ:[t0,tf]→R, also known as the co-state define Hamiltonian function H as:

H(x(t),u(t),λ(t),t)=L(x(t),u(t),t)+λ(t)Tf(x(t),u(t),t),(5)

such that

J(u)=φ(x(tf))+∫t0tf{H(x(t),u(t),λ(t),t)−λT(t)x}dt.(6)

Theorem 1: [22] (Banach’s Fixed Point Theorem)

Let (X,d) be a complete metric space and let T:X→X be contractive operator, i.e., there is a constant α∈(0,1) such that

||Tx−Ty||≤α||x−y||,∀x,y∈X,(7)

then there exists a unique fixed point x∈X such that Tx=x.

The above theorem states the conditions sufficient for the existence and uniqueness of a fixed point, which we will see in a point that is mapped to itself.

Theorem 2: [23] (Lyapunov Function Theorem)

Let x∗ be an equilibrium point and D⊆Rn be a neighborhood of x∗. Let V:D⊆Rn→R be continuously differentiable function such that

     i)  V(x∗)=0andV(x)>0forx∈D−{x∗}.

    ii)  d(v(x(t))dt≤0,x∈D then the equilibrium point x∗ is stable.

Further, x∗ is asymptotically stable if

d(v(x(t))dt<0,x∈D−{x∗}.

Theorem 3: [24] (Pontryagin Maximum Principle)

If u∗(t),x∗(t)(t∈[t0,tf]) is a solution of the optimal control problem (1), (2) then there exist a non-zero absolutely continuous function λ(t) such that λ(t),x∗(t),u∗(t) satisfy the system

dxdt=∂H∂λ,dλdt=−∂H∂x,(8)

such that, for almost all t∈[t0,tf] the function in Eq. (5) attains its maximum:

H(λ(t),x∗(t),u∗(t))=M(λ,x),

M(λ(tf),x∗(tf))=sup{H(λ,x,u):u∈U},(9)

and such that at terminal time tf the conditions:

M(λ(tf),x∗(tf))=0,λ0(tf)≤0 are satisfied.

If the functions λ(t),x(t),u(t) satisfy the relation (8), (9) (i.e., x(t),u(t) are Portryagin extremals), then the condition,

M(t)=M(λ(t),x(t))=const.,λ0(t)=const, hold.

This theorem is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls.

Remark 1: [25] For a minimum, it is necessary for the stationary (optimality) condition to give:

∂HT∂u=0.(10)

3  Model Formulation

The model is constructed based on the assumption that the new intake is admitted into the average class at the rate λ. The average student A may then become weak, excellent or graduate at the rates ϕ,β and θ2, respectively. Below average student B may become weak, average or graduate at the rates γ,α and θ1, respectively. Excellent student E graduate at rate θ3. But upon mingling with weak student W, the excellent student E may be influenced to become weak at the rate η. The rate at which student leaves school either through death or expelled is assumed to be the same in all the compartments. The schematic diagram of the model is given in Fig. 1.

Figure 1: Schematic diagram describing the dynamics of student academic performance

The performance dynamics is described by the nonlinear system of ODE

dBdt=pηWE−θ1B−αBA−μB−γBW,(11)

dAdt=αBA+λ−θ2A−μA+(1−p)WE−ϕWA−βAE,(12)

dEdt=βAE−ηWE−θ3E−μE,(13)

dWdt=γBW+ϕWA−μW,(14)

dGdt=θ1B+θ2A+θ3E−μG.(15)

3.1 Uniform Boundedness

Theorem 4: All the solutions of model are confined within bounded subset

Ψ={(B,A,E,W,G)∈R5:B,A,E,W,G≤λμ}.

Proof

Let the population size be

N=B+A+E+W+G,(16)

then

dNdt=λ−μN.(17)

Solving the linear ODE (17) gives

N(t)=λμ+ke−μt.(18)

The long-term behavior of (18) yield

limt→∞N(t)=λμ.(19)

Meanwhile, as time increases without bound all the solutions converge to the equilibrium Ne=λμ, of (18), hence the equilibrium is globally asymptotically stable.

3.2 Existence and Uniqueness

Theorem 5: The system (11)–(15) is Lipschitz continuous

Proof

Let the system (11)–(15) be of the form

f1(t,B)=pηWE−θ1B−αBA−μB−γBW,(20)

f2(t,A)=αBA+λ−θ2A−μA+(1−p)WE−ϕWA−βAE,(21)

f3(t,E)=βAE−ηWE−θ3E−μE,(22)

f4(t,W)=γBW+ϕWA−μW,(23)

f5(t,G)=θ1B+θ2A+θ3E−μG,(24)

|f1(t,B)−f1(t,B∗)|=|θ1+αA+μ+γW|||B−B∗||

≤(|θ1|+|α|maxt∈[0,T]||A||+|μ|+|γ|maxt∈[0,T]||W||)||B−B∗||.

∴||f1(t,B)−f1(t,B∗)||≤L1||B−B∗||,

where,

L1=|θ1|+|α|maxt∈[0,T]||A||+|μ|+|γ|maxt∈[0,T]||W||.(25)

Analogously, we obtained

||f2(t,A)−f2(t,A∗)||≤L2||A−A∗||,(26)

||f3(t,E)−f3(t,E∗)||≤L3||E−E∗||,(27)

||f4(t,W)−f4(t,W∗)||≤L4||W−W∗||,(28)

||f5(t,G)−f5(t,G∗)||≤L5||G−G∗||,(29)

where

L2=|α|maxt∈[0,T]||B||+|θ2|+|μ|+|ϕ|maxt∈[0,T]||W||+|β|maxt∈[0,T]||E||<∞,

L3=|β|maxt∈[0,T]||A||+|η|maxt∈[0,T]||W||+|θ3|+|μ|<∞,

L4=|γ|maxt∈[0,T]||B||+|ϕ|maxt∈[0,T]||A||+|μ|<∞,

and

L5=|μ|<∞.

Composing the system (11)–(15) in vector form as

x′=f(t,x),x(t0)=x0.(30)

f(t,x)=Px+g(x)+Λ,x=(B,A,E,W,G)T,

P=[−θ1−μ000θ10−θ2−μ00θ200−θ3−μ0θ3000−μ00000−μ],g(x)=[pηWE−αBA−γBWαBA+(1−p)ηWE−ϕWA−βAEβAE−ηWEγBW+ϕWA0],

Λ=(λ,0,0,0,0)T.

Since f∈C([t0−ϵ,t0+ϵ]×U⊆R5), then (30) has the equivalent integral equation

x(t)=x0+∫t0tf(τ,x(τ))dτ,(31)

Theorem 6: Let f:[t0−ϵ,t0+ϵ]×U⊆R5→R5 be uniformly continuous in t and Lipschitz in the second variable. Suppose f is bounded, i.e., M=maxt∈[t0−ϵ,t0+ϵ]||f(t,x)|| such that||f(t,x)||≤M. If L0ϵ<1 then there exist unique x that solves (30).

Proof

Let X={x∈C[t0−ϵ,t0+ϵ]:||x(t)−x0||≤r}

Since X is closed, for any Cauchy sequence (xn)n≥1∈X and that limn→∞xn=x∗ then x∗≤r. Equivalently, x∗∈Br¯(x0)⊆U;

Hence X is complete metric space induced by norm.

Define the fixed point operator T:X→X by

Tx(t)=x0+∫t0tf(τ,x(τ))dτ,(32)

T is well defined

||Tx(t)−x0||=||∫t0tf(τ,x(τ))dτ||

≤∫t0t||f(τ,x(τ))||dτ

≤M∫t0tdτ

=M|t−t0|

≤Mϵ.

||Tx(t)−x0||≤r,ϵ≤rM.(33)

So, Tx(t)∈Br¯(x0)∀t∈[t0−ϵ,t0+ϵ].

For contraction

||Tx(t)−Ty(t)||=||∫t0t[f(τ,x(τ))−f(τ,y(τ))]dτ||

≤∫t0t||f(τ,x(τ))−f(τ,y(τ))||dτ

≤∫t0tL(τ)||x(τ)−y(τ)||dτ

≤∫t0tL(τ)maxs∈[t0−ϵ,t0+ϵ]||x(s)−y(s)||dτ

≤L0||x−y|||t−t0|

||Tx(t)−Ty(t)||≤L0ϵ||x−y||(34)

Since L0ϵ<1 then T is contraction.

4  Stability Analysis

Since the state Eq. (15) depends on the previous states, it suffices to analyze (11)–(15).

4.1 Equilibrium Solution

The system (11)–(15) has the following equilibrium solutions

The independence equilibrium E∗=(B0,A0,E0,W0)=(0,λμ+θ2,0,0), and the interdependence equilibrium E∗∗=(B∗,A∗,E∗,W∗), where,

B∗=μ−ϕA∗γ,(35)

W∗=βA∗−θ3−μη,(36)

E∗=[η(θ1+μ)−γ(θ3+μ)+(ηα+γβ)A∗](μ−ϕA∗)γpη(βA∗−θ3−μ),(37)

and A∗ depend on the solution of

a0A3+a1A2+a2A+a3=0,(38)

where,

a0=pβϕ(ηα+γβ),

a1=pϕβ(γβ−ηα)−(ηα+γβ)[pβμ−ϕ(1−p)(θ3+μ)]+pβϕ[η(θ1+μ)−γ(θ3+μ)],

a2=pηα[μβ+ϕ(ϕβ+μ)]+pγβ[η(λ−θ2−μ)−2ϕ(θ3+μ)]−μ(1−p)(θ3+μ)(ηα+γβ)−[pβμ−ϕ(1−p)(θ3+μ)][η(θ1+μ)−γ(θ3+μ)],

a3=−pη[αμ(ϕβ+μ)+γ(θ3+μ)(λ−θ2−μ)]+pϕγ(θ3+μ)2−μ(1−p)(θ3+μ)[η(θ1+μ)−γ(θ3+μ)].

4.2 Local Stability

From system (11)–(15) we formulate the following Jacobian matrix, and then we test the equilibrium solution in the Jacobian matrix, if all the eigenvalues are negative, then the solution is locally stable, otherwise it is unstable [26,27].

J=[θ1−αA−μ−γWαA0γW−αBαB−θ2−μ−ϕW−βEβEϕWpηW(1−p)ηW−βAβA−ηW−θ3−μ0pηE(1−p)ηE−ϕA−ηEϕA+γB−μ].

Theorem 7: The independence equilibrium (E0) is locally asymptotically stable.

Proof

JE∗=[−θ1−αλθ2+μ−μαλθ2+μ000−θ2−μ000−βλθ2+μβλθ2+μ−θ3−μ00−ϕλθ2+μ0ϕλθ2+μ−μ].

The eigenvalues of JE∗ are

K1=−θ1−αλθ2+μ−μ,(39)

K2=−θ2−μ,(40)

K3=βλθ2+μ−θ3−μ,(41)

K4=ϕλθ2+μ−μ.(42)

For E∗ to be stable we require K3,K4<0, which implies that

βλ(θ2+μ)(θ3+μ)−1<0,

and

ϕλμ(θ2+μ)−1<0.

4.3 Global Stability

Theorem 8: The independence equilibrium E∗ is globally asymptotically stable in the interior of Ψ.

Proof

Define L:{(B,A,E,W)∈Ψ:A>0}→R, by

L=12[B+A−A0+E+W]2.(43)

Since,

L(B0,A0,E0,W0)=0

L(B,A,E,W)>0,B≠B0,A≠A0,E≠E0,W≠W0.

Then L is positive definite.

The time derivative of L∈C′

dLdt=[B+A−A0+E+W][dBdt+dAdt+dEdt+dWdt]

=[B+A−A0+E+W][−(θ1+μ)B+(θ2+μ)A−(θ3+μ)E−μW+λ]

=−(θ1+μ)B2−(θ3+μ)E2−μW2−(θ2+μ)AB−(θ3+μ)BE−μWE−(θ1+μ)(A−A0)B−(θ2+μ)(A−A0)A−(θ3+μ)(A−A0)E−μW(A−A0)−(θ1+μ)BE−(θ2+μ)AE−μWE−(θ1+μ)BW−(θ2+μ)AW−(θ3+μ)EW+λ(B+A−A0+E+W).(44)

Applying the relation between arithmetic and geometric means: ∀x,y∈R,xy≤12x2+12y2

dLdt≤−(θ1+μ)B2−(θ3+μ)E2−μW2−(θ2+μ)(12A2+12B2)−(θ3+μ)(12B2+12E2)−μ(12W2+12B2)−(θ1+μ)(12(A−A0)2+12B2)−(θ2+μ)(12A2+12(A−A0)2)−(θ3+μ)(12(A−A0)2+12E2)−μ(12W2+12(A−A0)2)−(θ1+μ)(12B2+12E2)−(θ3,thmeticandgeometricmeans()2+μ)(12A2+12E2)−μ(12W2+12E2)−(θ1+μ)(12B2+12W2)−(θ2+μ)(12A2+12W2)−(θ3+μ)(12E2+12W2)+λ(B+A−A0+E+W)

=−[(4μ+52θ1+12θ2+12θ3)B2+(4μ+12θ1+12θ2+52θ3)E2+2(θ2+μ)A2+(4μ+12θ1+12θ2+12θ3)W2+(2μ+12θ1+12θ2+12θ3)(A−A0)2−λ(B+A−A0+E+W)].(45)

Since,

(4μ+52θ1+12θ2+12θ3)B2+(4μ+12θ1+12θ2+52θ3)E2+2(θ2+μ)A2+(4μ+12θ1+12θ2+12θ3)W2+(2μ+12θ1+12θ2+12θ3)(A−A0)2>λ(B+A−A0+E+W),

then

dLdt<0.(46)

Hence E∗ is globally asymptotically stable.

Theorem 9: The interdependence equilibrium E∗∗ is globally asymptotically stable in the interior of Ψ.

Proof

Define L:{(B,A,E,W)∈Ψ:B,A,E,W>0}→R by

L=12[B−B∗+A−A∗+E−E∗+W−W∗]2,(47)

since,

L(B∗,A∗,E∗,W∗)=0,

L(B,A,E,R)>0,B≠B∗,A≠A∗,E≠E∗,W≠W∗,

then L is positive definite.

The time derivative of L∈C′

dLdt=[B−B∗+A−A∗+E−E∗+W−W∗][dBdt+dAdt+dEdt+dWdt]

=−(θ1+μ)(B−B∗)B−(θ2+μ)(B−B∗)A−(θ3+μ)(B−B∗)E−μW(B−B∗)−(θ1+μ)(A−A∗)B−(θ2+μ)(A−A∗)A−(θ3+μ)(A−A∗)E−μW(A−A∗)−(θ1+μ)(E−E∗)B−(θ2+μ)(E−E∗)A−(θ3+μ)(E−E∗)E−μW(E−E∗)−(θ1+μ)(W−W∗)B−(θ2+μ)(W−W∗)A−(θ3+μ)(W−W∗)E−μ(W−W∗)W+λ(B−B∗+A−A∗+E−E∗+W−W∗).(48)

Applying the relation between arithmetic and geometric means: ∀x,y∈R,xy≤12x2+12y2

≤−(θ1+μ)(12B2+12(B−B∗)2)−(θ2+μ)(12A2+12(B−B∗)2)−(θ3+μ)(12E2+12(B−B∗)2)−μ(12W2+12(B−B∗)2)−(θ1+μ)(12B2+12(A−A∗)2)−(θ2+μ)(12A2+12(A−A∗)2)−(θ3+μ)(12E2+12(A−A∗)2)−μ(12W2+12(A−A∗)2)−(θ1+μ)(12B2+12(E−E∗)2)−(θ2+μ)(12A2+12(E−E∗)2)−(θ3+μ)(12E2+12(E−E∗)2)−μ(12W2+12(E−E∗)2)−(θ1+μ)(12B2+12(W−W∗)2)−(θ2+μ)(12A2+12(W−W∗)2)−(θ3+μ)(12E2+12(W−W∗)2)−μ(12W2+12(W−W∗)2)+λ(B−B∗+A−A∗+E−E∗+W−W∗)

=−[2(θ1+μ)B2+2(θ2+μ)A2+2(θ3+μ)E2+2μW2+(2μ+12θ1+12θ2+12θ3)(B−B∗)2+(2μ+12θ1+12θ2+12θ3)(A−A∗)2+(2μ+12θ1+12θ2+12θ3)(E−E∗)2+(2μ+12θ1+12θ2+12θ3)(W−W∗)2−λ(B−B∗+A−A∗+E−E∗+W−W∗)],

Since,

2(θ1+μ)B2+2(θ2+μ)A2+2(θ3+μ)E2+2μW2+(2μ+12θ1+12θ2+12θ3)(B−B∗)2+(2μ+12θ1+12θ2+12θ3)(A−A∗)2+(2μ+12θ1+12θ2+12θ3)(E−E∗)2+(2μ+12θ1+12θ2+12θ3)(W−W∗)2>λ(B−B∗+A−A∗+E−E∗+W−W∗),

Then

dLdt<0.(49)

Hence E∗∗ is globally asymptotically stable.

5  Formation of Optimal Control

The optimal control strategy is aimed at optimizing student’s academic performance which reflects in the increase of number of graduating students.

Let the control rates:

u1(t)∈[0,u1(t)max] be the self-motivation that makes weak student to become below average student.

u2(t)∈[0,u2(t)max] be punctuality in class that makes weak student to become average.

u3(t)∈[0,u3(t)max] be the interest in the subject that makes below average student to become average.

u4(t)∈[0,u4(t)max] be regular studying that makes average student to become excellent.

u5(t)∈[0,u5(t)max] be examination performance and character that make below average, average and excellent students to graduate.

Then the control dynamics is described by the nonlinear system of ODE below:

dBdt=pηWE−(θ1+u5)B−(α+u3)BA−μB−(γ+u1)BW,(50)

dAdt=(α+u3)BA+λ−(θ2+u5)A−μA+(1−p)WE−(ϕ+u2)WA−(β+u4)AE,(51)