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Stability Analysis of a Mathematical Model for Examination Malpractice Dynamics

Musah Konlan∗ , Razak Gbemmie Chuaya
Department of Mathematics and Statistics, University of Energy and Natural Resources, Sunyani, Ghana

musah.konlan@uenr.edu.gh, chuayarazk@gmail.com
∗Correspondence: musah.konlan@uenr.edu.gh

Abstract. Examination malpractice is one of the key challenges endangering the quality of educa-tion in Ghana. This negative act refers to any form of dishonesty or irregularity that compromisesthe integrity of any examination. In this paper, we proposed a mathematical model for exploringthe dynamics of examination malpractice at the West African Senior School Certificate Examination(WASSCE) level in Ghana. The examination malpractice-free equilibrium is computed and shown tobe both locally and globally stable if the examination malpractice reproductive number (R0) is lessthan one. The examination malpractice endemic equilibrium is also derived and found to be globallyasymptotically stable whenever R0 is greater than one. Local sensitivity analysis is performed onthe basic examination malpractice reproduction number to explore the contribution of the model pa-rameters to the evil act of examination malpractice. Finally, numerical simulations are performed toillustrate the behavior of the model sub-classes.

1. Introduction
Examination remains a prominent tool for assessing and measuring students’ academic perfor-mance throughout our educational system. It is used to determine the transition of students fromone lower level to the next higher level. On the job market, examination also serves as a means forpredicting a job seeker’s knowledge level, skills, and competence in a given domain. Unfortunately,this valuable measurement tool is being compromised at all levels given room to what is popularlycalled examination malpractice [1, 2]. Examination malpractice can be described as any deliberateact against the official rules and regulations of an examination, with the intention of giving undueadvantage to a candidate [3–6]. Examination malpractice has been constantly recorded at all levelsof our educational institutions. Data on examination malpractice at the Senior High School levelare alarming. In Ghana for example, the cases of examination malpractice at the Senior High SchoolCertificate Examination from 2020 to 2024 are shown in table 1. Also, it has been reported that inNigeria 1,767 out of 13,595 and 842 out of 12,030 candidates who took part in the Senior HighSchool Certificate Examination were involved in malpractices in 2022 and 2023 respectively [2].
Received: 24 Mar 2025.
Key words and phrases. examination malpractice; mathematical model; stability analysis; sensitivity analysis.1

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Eur. J. Math. Anal. 10.28924/ada/ma.5.16 2Examination malpractice has become a major concern in Ethiopia and Somalia [7]. According tostudies conducted in [3, 6], people involved at the school level examination malpractice includeacademic authorities, students, teachers, and parents. This bad phenomenon may take place be-fore, during and after an examination with the aim of achieving academic excellence for both thestudent and the school for a given examination. Specific forms of malpractice during examinationsinclude, but are not limited to: tattooing, examination paper leakage, support from invigilators,impersonation, smuggling of unauthorized materials into the examination room, and giraffing in theexamination room [3,4,6,8]. It is clear from the above-mentioned that examination malpractice posesa significant risk to the educational system of a nation. In a nutshell, examination malpractice low-ers educational standards, discredits certificates obtained through examination, discourages hardwork, leads to academic dishonesty and above all reduces productivity. Thus, contributing factorsto this malfeasance need to be identified and tackled [1,2]. Mathematical modelling has become aneffective toolbox for understanding the occurrence of dynamical phenomena [9,10]. However, math-ematical models for studying examination malpractice are very scarce in the literature. The authorsin [11] formulated and analyzed a dynamical model of examination malpractice taking into accountkey players in Nigeria. Their analysis indicated that leakages of examination question papers hasthe highest influence on the spread of examination malpractice. Ayoade and Farayola [11] devel-oped a mathematical model for the mechanisms of examination malpractice with control strategiessuch as: social reengineering and orientation with proportional punishment/disciplinary action forexamination malpractice victims. To fill this research gap, this work aims to propose a deterministiccompartmental model to study exam malpractice at the West African Senior High School CertificateExamination (WASSCE) in Ghana.
Table 1. WASSCE Provisional Data For Ghana (2020 - 2024)

Year N(t) M(t) C(t) Pending
2020 375, 763 2, 863 480 384

2021 446352 1, 513 174 3, 667

2022 422, 883 4, 363 518 117

2023 448, 674 4, 486 839 5, 285

2024 460, 611 4, 591 483 990

Source: https://www.myjoyonline.com/?s=Release+of+WASSCE+Results
2. Model Construction

To construct a compartmental model for examining the dynamics of examination malpractice, thetotal number of candidates sitting for the examination at time t (N(t)) is stratified into five sub-classes. Namely: S(t), M(t), C(t), R(t) and H(t) (see table 2). Candidates are recruited into the

https://doi.org/10.28924/ada/ma.5.16
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Eur. J. Math. Anal. 10.28924/ada/ma.5.16 3susceptible class (S(t)) at rate Λ. Some susceptible candidates move to honest candidates class(H(t)) at rate β. Other susceptible candidates engage into malpractices and move to malpracticeclass (M(t)) at rate θ. Candidates in the malpractice class either recover from malpractice and moveto recovered class (R(t)) at rate τ or have their entire results cancelled at rate δ. Some recoveredcandidates progress to honest candidates class at rate α. Each model sub-class is reduced at arate µ due to natural death. The model assumes that there is no examination malpractice induceddeaths. The variables and parameters used to describe the model are presented in Table 2 andTable 3 respectively.
Table 2. Model State Variables Definition

Symbol Definition

N(t) Total number of candidates sitting for the examination
S(t) Number of candidates susceptible to examination malpractice
M(t) Number of candidates who engage in any form of irregularities
C(t) Number of candidates who have their entire results cancelled following misconduct
R(t) Number of candidates who recover from malpractice
H(t) Number of law abiding/honest candidates

Table 3. Model Parameter Description
Symbol Description

Λ Candidates recruitment rate 50.0 Estimated
µ Natural human removal rate 0.0005 Assumed
β Rate at which candidates adhere to examination rules 0.5 Assumed
θ Rate at which candidates engage into irregularities 0.0044 Assumed
τ Rate at which candidates recover from malpractice 0.2 Assumed
δ Rate at which candidate entire results are canceled 0.15 Estimated
α Rate at which recovered candidates become honest 0.3 Assumed
γ Rate at which honest candidate revert to susceptible 0.9 Assumed

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Eur. J. Math. Anal. 10.28924/ada/ma.5.16 4

Figure 1. Flow Chart for Examination Malpractice
From the flow chart above, we obtain the following system of differential equations:

dS

dt
= Λ + γH − θSM − (β + µ)S

dM

dt
= θSM − (τ + δ + µ)M

dC

dt
= δM − µC

dR

dt
= τM − (α+ µ)R

dH

dt
= βS + αR − (γ + µ)H

(1)

The following notation will be used in the rest of the study:
q0 = (β + µ), q1 = (τ + δ + µ), q2 = (α+ µ) and q3 = (γ + µ)

2.1. Well-Posedness of the Model. Under this section, we establish that the system of differentialequations representing model (1) admits only non-negative solutions. Furthermore, the set overwhich the model system of equations is contextually meaningful is also determined.
Theorem 1. Given the non-negative initial value set: {S(0), M(0), C(0), R(0), H(0)} of the
dynamical system (1), it follows that the solution set {S(t), M(t), C(t), R(t), H(t)} is non-
negative and bounded ∀ t ≥ 0.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.16 5

Proof. First, we consider the differential equation for the susceptible sub-class:
dS

dt
= Λ + γH − (θM + q0)S

=⇒
dS

dt
≥ −(θM + q0)S

=⇒
∫

1

S
dS ≥ −

∫
(θM + q0)dt

=⇒ S(t) ≥ S(0)e−(qot+θ
∫ t

0 M(x)dx) ≥ 0Similarly, the following results can be obtained:
M(t) ≥ M(0)e−q1t ≥ 0

C(t) ≥ C(0)e−µt ≥ 0

R(t) ≥ R(0)e−q2t ≥ 0

H(t) ≥ H(0)e−q3t ≥ 0

Therefore, for ∀ t ≥ 0, the state variables of the model have non-negative solutions. �

Theorem 2. The feasible positive invariant region in which the solution set of the model system of
equations is meaningful is the set:

D =

{
(S,M,C,R,H) ∈ R5

+ : S +M + C + R +H ≤
Λ

µ

} (2)
Proof. The total population N at any given time t is:

N = S +M + C + R +H

=⇒
dN

dt
=
dS

dt
+
dM

dt
+
dC

dt
+
dR

dt
+
dH

dt

=⇒
dN

dt
= Λ− µN

=⇒
dN

dt
= −µ(N −

Λ

µ
)

=⇒
dN

N − Λ
µ

= −µdt

=⇒
∫

dN

N − Λ
µ

= −
∫
µdt

=⇒ N −
Λ

µ
= 0 as t → +∞

Thus, we deduce that N ≤ Λ

µ

(3)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.16 6Therefore:
D =

{
(S,M,C,R,H) ∈ R5

+ : S +M + C + R +H ≤
Λ

µ

} (4)
�

2.2. The Examination Malpractice-Free Equilibrium (EMFE). Equating the individual equationsof system (1) to zero with the condition M = C = R = 0, we obtain the EMFE (ξ∗) given by
ξ∗ = (S∗,M∗, C∗, R∗, H∗) =

(
Λq3

q0q3−βγ , 0, 0, 0, βΛ
q0q3−βγ

)
2.2.1. The Basic Reproductive Number (R0) of Examination Malpractice. In this context, the basicreproductive number defines the average number of secondary candidates that will be influencedby just one candidate who engages into examination malpractice. The method of next generatingmatrix which is mostly used is also adopted to compute the model R0. To achieve this, we expressedthe malpractice sub-system of model (1) in the form dX

dt = (F − V)XT where XT is the transposeof X = (M, C), F and V are the rates of generation of new misconducting candidates and transferrespectively. Using this malpractice/infected sub-system

dM

dt
= θSM − q1M

dC

dt
= δM − µC

(5)
we have:

F =

θSM
0

 and V =

 q1M

−δM + µC

 (6)
Evaluating the Jacobian matrices F and V of F and V at the DFE gives respectively:

F =

θS
∗ 0

0 0

 and V =

q1 0

−δ µ

 (7)
Using F and V from (7), we obtain FV −1 given by:

FV −1 =


θS∗

q1
0

0 0

 (8)
Consequently, we obtain R0 as the spectral radius of FV −1 given by:

R0 =
Λθq3

q1(q0q3 − βγ)
=

Λθ(γ + µ)

µ(τ + δ + µ)(β + γ + µ)
(9)

2.3. Stability Analysis of the Examination Malpractice-Free Equilibrium.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.16 72.3.1. Local Stability. To study the local stability of the EMFE, we adopt the linearization ap-proach.
Theorem 3. The equilibrium point

(
ξ∗ =

(
Λq3

q0q3−βγ , 0, 0, 0, βΛ
q0q3−βγ

))
is locally asymptotically sta-

ble (LAS) if R0 < 1 and unstable if R0 > 1

Proof. Let J0 be the Jacobian matrix of system (1) evaluated at ξ∗, that is:

J0 =


−q0 −θS∗ 0 0 γ

0 θS∗ − q1 0 0 0

0 δ −µ 0 0

0 τ 0 −q2 0

β 0 0 α −q3

 (10)
Clearly, the matrix in (10) admits one negative eigenvalue, namely λ1 = −µ. Furthermore, thenature of the remaining eigenvalues of the matrix in (10) can be obtained from the sub-matrix in(11) below:

J1 =


−q0 −θS∗ 0 γ

0 θS∗ − q1 0 0

0 τ −q2 0

β 0 α −q3

 (11)
According to the Routh-Hurwitz stability theorem, matrix in (11) will be stable if its trace anddeterminant are negative and positive respectively [12,13]. Now:

T race(J1) = − (q0 + q2 + q3 + q1(1− R0)) < 0 if R0 < 1 (12)and
det(J1) = q1(q0q1q3 − βγ)(1− R0) > 0 if R0 < 1 (13)Thus, the EMFE state is locally asymtotically stable whenever R0 < 1. �

Next, we examine the global stability of the examination malpractice-free state.
2.3.2. Global Stability of the Examination Malpractice-Free State. To establish the long termstability behavior of the examination malpractice-free equilibrium state, we consider the followingLyapunov function:

V (t) =
1

q1
M (14)Differentiating V(t) gives

dV (t)

dt
=

1

q1

dM

dt

=− (1− R0)M

(15)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.16 8

It is clear from (15) that dV (t)
dt ≤ 0 if R0 ≤ 1. Hence, the EMFE point is globally asymptoticallystable if R0 ≤ 1 and unstable otherwise

2.4. Existence of Examination Malpractice Endemic Equilibrium Point (EMEEP). Solving system(1) for the state variables at the EMEEP gives the following system of solutions:

S∗∗ = Λq2q3+αγτM∗∗

q2q3(θM∗∗+q2(q0q3−βγ)

M∗∗ = q1q2(q0q3−βγ)(R0−1)
θ(q1q2q3−αγτ)

C∗∗ = δq1q2(q0q3−βγ)(R0−1)
θµ(q1q2q3−αγτ)

R∗∗ = τq1(q0q3−βγ)(R0−1)
θ(q1q2q3−αγτ)

H∗∗ = βΛq2q3+αβγτM∗∗

q2q3(θq3M∗∗+(q0q3−βγ)) + ατq1(q0q3−βγ)(R0−1)
θq3(q1q2q3−αγτ)

(16)

It is clear from (16), that S∗∗, M∗∗, C∗∗, R∗∗ and H∗∗ exist if and only if R0 > 1, this gives thecondition for the existence of the EMEEP. In what follows, we examine the global stability of thisendemic equilibrium point using Liapunov function.
2.4.1. Global Stability of the Examination Malpractice Endemic Equilibrium Point.

Theorem 4. The model represented by system (1) admits a globally asymptotically stable non-
trivial endemic equilibrium ( ξ∗∗) whenever R0 > 1

Proof. Consider a positive definite function L defined by:
L (ξ∗∗) =

(
(S − S∗∗)− S∗∗ ln

S

S∗∗

)
+

(
(M −M∗∗)−M∗∗ ln

M

M∗∗

)
+

(
(C − C∗∗)− C∗∗ ln

C

C∗∗

)
+

(
(R − R∗∗)− R∗∗ ln

R

R∗∗

)
+

(
(H −H∗∗)−H∗∗ ln

H

H∗∗

)
Taking the time derivative of L (ξ∗∗) gives:
dL (ξ∗∗)

dt
=

(
1−

S∗∗

S

)
dS

dt
+

(
1−

M∗∗

M

)
dM

dt
+

(
1−

C∗∗

C

)
dC

dt
+

(
1−

R∗∗

R

)
dR

dt
+

(
1−

H∗∗

H

)
dH

dt

=

(
S − S∗∗

S

)
[Λ + γH − θSM − q0S] +

(
M −M∗∗

M

)
(θSM − q1M)

+

(
C − C∗∗

C

)
(δM − µC) +

(
R − R∗∗

R

)
(τM − q2R) +

(
H −H∗∗

H

)
(βS + αR − q3H)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.16 9

= Λ + γH + θMS∗∗ + q0S
∗∗ − q0S − (Λ + γH)

S∗∗

S
+ q1M

∗∗ − q1M − θM∗∗S

+ δM + µC∗∗ − µC − δM
C∗∗

C
+ τM + q2R

∗∗ − q2R − τM
R∗∗

R
+ βS + αR + q3H

∗∗ − q3H

− (βS + αR)
H∗∗

H

= L+ − L−where
L+ = Λ + γH + θMS∗∗ + q0S

∗∗ + q1M
∗∗ + δM + µC∗∗ + τM + q2R

∗∗ + βS

+ αR + q3H
∗∗

L− = q0S + (Λ + γH)
S∗∗

S
+ q1M + θM∗∗S + µC + δM

C∗∗

C
+ q2R + τM

R∗∗

R

+ q3H + (βS + αR)
H∗∗

H

(17)

If we now assume L+ ≤ L−then, it follows from (17) that dL(ξ∗∗)
dt ≤ 0 with the equality holding if and only if S∗∗ =

S, M∗∗ = M, C∗∗ = C, R∗∗ = R and H∗∗ = HTherefore, the largest compact invariant set within D (the model’s invariant region) is the single-ton {ξ∗∗} = {S∗∗, M∗∗, C∗∗, R∗∗, H∗∗} . Hence, following [14], the unique endemic equilibriumof system (1) is globally asymptotically stable whenever it exists. �

3. Local Sensitivity Analysis
To investigate the contribution of each model parameter to the occurrence of examination mal-practice, we calculated the sensitivity indices of the parameters of the basic examination malpracticereproductive number using the forward sensitivity index expression. The output is tabulated in Table4 below.

Table 4. Sensitivity Indices of R0 Parameters
Parameter Sensitivity index
Λ +1.00000

θ +1.00000

β −0.35702

γ +0.35682

τ −0.57061

δ −0.42796

µ −1.00123

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Eur. J. Math. Anal. 10.28924/ada/ma.5.16 104. Numerical Simulations
To examine the dynamical evolution of the model sub-classes, we simulated the proposed modelsystem of equations using Matlab ode45. We used the parameter values given in Table 3 with thefollowing assumed initial values: S(0) = 120, M(0) = 25, C(0) = 5, R(0) = 5 and H(0) = 15.The simulation graphs are shown from Figure 2 to Figure 7

5. Discussion and Conclusion
A five-compartmental model is constructed to study the phenomenon of examination malpracticeat the West African Senior School Certificate Examination (WASSCE) level in Ghana. The well-posedness of the model is established. The examination malpractice free equilibrium is shownto possess both a local and global asymptotic stability whenever the examination malpracticereproductive number (R0) is less than one. Furthermore, the examination malpractice persistentequilibrium is derived and shown to be globally stable if R0 > 1. We carried out local sensitivityanalysis on the parameters of R0 and established that candidates recruitment rate (Λ), the rateat which candidates engage into irregularities (θ) and the susceptibility rate (γ) of honest orlaw abiding candidates have positive impact on the examination malpractice reproductive number(R0). On the other hand, the rate at which susceptible candidates progress to honest or lawabiding candidates (β), the rate at which candidates recover from malpractice (τ ) and the rate ofcancellation of examination results (δ) have a negative impact on R0. In other words, increasing thevalue of any parameter with negative impact on R0 will lead to minimizing examination malpracticeat the WASSCE level in Ghana. Thus, we recommend that to control the evil act of examinationmalpractice during WASSCE, the examination bodies should intensify the orientation programsfor candidates, recruit honest and principle invigilators who can strictly monitor the candidatesin the examination rooms, strictly enforce punishment or disciplinary actions against examinationmalpractice perpetrators.

Figure 2. Plot of sus-ceptible sub-population Figure 3. Plot of mal-practice sub-population

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Eur. J. Math. Anal. 10.28924/ada/ma.5.16 11

Figure 4. Plot of can-celled sub-population Figure 5. Plot of recov-ered sub-population

Figure 6. Plot of hon-est sub-population Figure 7. Plot of allsub-population
Acknowledgment. Authors are much thankful to other faculty members and some Senior HighSchool teachers for their guidance during the development of this manuscript. Much appreciationalso goes to the respected reviewers for their valuable comments and suggestions.
Conflict of interest. Authors declare that there is no conflict of interest regarding the publicationof this work.

References
[1] M. Frempong, E. B. A. Arloo, C. K. Amissah, Factor analysis of issues contributing to examination malpractice in atertiary institution in Ghana, Math. Theory Model. 6 (2016), 1–6.[2] J. O. Okunlola, Unmasking the unseen in examination malpractices in high schools: A phenomenological study ofquality assurance leaders’ lived experiences in public examinations, J. Educ. Teach. Train. 15 (2024), 17–31.[3] L. R. Ekpenyong, N. R. George, Examination malpractices in senior secondary school mathematics: Types, causes,and solution, Int. J. Res. Educ. Sustain. Dev. 3 (2023), 62–72.

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[4] T. I. Chinebu, C. U. Okonkwo, I. L. Okoliegbe, E. A. Jooda, Mathematical analysis of the basic malpractice reproductionnumber and the dynamics of examination malpractice in Nigeria using epidemiological model, Int. J. Mechatron.Electr. Comput. Technol. 12 (2022), 5118–5123.[5] A. A. Olaoye, S. O. Asesanya, Examination malpractices in mathematics examinations as a critical factor for prolif-eration of higher education moral, Educ. Perspect. 12 (2024), 1–9.[6] M. B. Abdullahi, A. Sule, Fractional order and numerical solution of application of modeling to manage examinationmisconducts, Niger. J. Sci. Eng. Infrastruct. 2 (2024), 28–45.[7] L. Chala, M. Agago, Exploring national examination malpractice mechanisms and countermeasures: An Ethiopianperspective, Int. J. Instr. 15 (2022), 413–428.[8] K. Asante-Kyei, K. Nduro, Inclining factors towards examination malpractice among students in Takoradi Polytechnic,Ghana, J. Educ. Pract. 5 (2014), 66–73.[9] M. Konlan, B. A. Danquah, E. Okyere, S. Osman, J. A. Kessie, E. K. Donkoh, Global stability analysis and modellingonchocerciasis transmission dynamics with control measures, Infect. Ecol. Epidemiol. 14 (2024), 2347941.[10] M. Konlan, Modeling the inflow of exposed and infected migrants on the dynamics of malaria, Eur. J. Math. Anal. 4(2024), 7.[11] A. A. Ayoade, P. I. Farayola, Dynamics of examination malpractice among the key players in Nigeria, Daffodil Int.Univ. J. Sci. Technol. 15 (2020), 25–32.[12] S. Nana-Kyere, B. Seidu, K. Nantomah, Mathematical analysis of malaria epidemic: Asymptotic stability withcost-effectiveness study, J. Appl. Math. 2024 (2024), 5533885.[13] M. M. Ojo, E. F. Doungmo Goufo, Assessing the impact of control interventions and awareness on malaria: amathematical modeling approach, Commun. Math. Biol. Neurosci. 2021 (2021), 93.[14] J. P. LaSalle, Stability theory and invariance principles, in: Dynamical Systems, Academic Press, New York, 1976.

https://doi.org/10.28924/ada/ma.5.16

	1. Introduction
	2.  Model Construction
	2.1. Well-Posedness of the Model
	2.2. The Examination Malpractice-Free Equilibrium (EMFE) 
	2.3. Stability Analysis of the Examination Malpractice-Free Equilibrium
	2.4.  Existence of Examination Malpractice Endemic Equilibrium Point (EMEEP)

	3. Local Sensitivity Analysis
	4. Numerical Simulations
	5. Discussion and Conclusion
	Acknowledgment
	Conflict of interest

	References

