EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6345
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Hamiltonian and Neural Network-Based Framework for
Modeling Conjunctivitis Transmission with Medical

Intervention

Nadeem Abbas1,∗, Wasfi Shatanawi1,2, Syeda Alishwa Zanib3,∗

1 Department of Mathematics and Sciences, College of Humanities and Sciences,
Prince Sultan University, Riyadh, 11586, Saudi Arabia
2 Department of Mathematics, Faculty of Science, The Hashemite University,
P.O Box 330127, Zarqa 13133, Jordan
3Department of Mathematics, Riphah International University, Main Satyana Road,
Faisalabad 44000, Pakistan

Abstract. This study introduces a novel integrated mathematical and machine learning frame-
work to optimize control strategies for conjunctivitis (pink eye). We develop a dynamic compart-
mental model that explicitly incorporates key interventions, including self-isolation, medication,
and treatment, to simulate and curb disease transmission. The model’s well-posedness is rigor-
ously established through invariant region and boundedness analysis. Analytical derivation of
the basic reproduction number (R0) quantifies the epidemic threshold, while sensitivity analy-
sis identifies critical parameters, incubation rate (ρ), transmission rate of conjunctivitis (κ), and
natural birth rate (δ), as primary drivers of disease dynamics. Stability analysis of equilibrium
points informs the design of optimal, time-dependent intervention strategies. Employing Pontrya-
gin’s Maximum Principle, we derive and numerically solve the optimality system, demonstrating
that a combined strategy involving self-isolation, medication, and treatment control can reduce
conjunctivitis incidence by 38–62% compared to baseline measures. To further enhance predic-
tive capability, Artificial Neural Networks (ANNs) are trained on simulated datasets with noise
perturbation, achieving mean squared errors ranging from 0.19 to 0.98 across test scenarios and
confirming robust forecasting accuracy. This work bridges mechanistic modeling with data-driven
prediction, offering actionable insights for public health policy and resource allocation in managing
conjunctivitis outbreaks.

2020 Mathematics Subject Classifications: 26A33,34A08, 03C65

Key Words and Phrases: Conjunctivitis, Optimal Control Theory, Self-Isolation Strategies,
Medication Compliance, Artificial Neural Networks, Hamiltonian Approach

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6345

Email addresses: nabbas@psu.edu.sa (N.Abbas),
wshatanawi@psu.edu.sa (W. Shatanawi), 19907@riphahfsd.edu.pk (S.A.Zanib)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 2 of 33

1. Introduction

Conjunctivitis, commonly called ”pink eye,” is when the clear tissue covering the white part
of the eye and inner eyelid becomes inflamed. This inflammation can be caused by bacteria,
viruses, allergies (like pollen or dust), or irritants such as dirt and shampoos. In this study,
we are specifically looking at conjunctivitis caused by viruses. Symptoms may encompass
redness in the white of the eye and inner eyelid, itching, blurred vision, and increased tear
production shown in figure 1. Typically, these signs persist for approximately 5-7 days
and resolve within about 2 weeks [1]. Moreover, infectious conjunctivitis, encompassing
bacterial, viral, and neonatal conjunctivitis, arises from the infection of the conjunctiva and
the white part of the eye. Symptoms of conjunctivitis include itching, eye soreness, tear
production, pus discharge, and light sensitivity. Utilizing antibiotic eye drops, maintaining
hygiene, isolation, and allowing the natural course of the disease over 2 to 3 weeks are
effective measures to halt the spread of conjunctivitis. The prevalence of conjunctivitis is
observed to be higher in tropical regions [2, 3].

Conjunctivitis Dynamics

Medication Self-Isolation

Isolation through self-
isolation and 

medication strategies 
for infected 
individuals

Effectiveness of combining 
isolation through sick leave, 
medication strategies, and 

treatment to reduce overall 
conjunctivitis cases

Eye Protection

Mathematical Model

𝒄𝟏 using drops and 
wearing glasses 
during medication.

𝒄𝟐 awareness of 
self-isolation with 
medical campaign.

Figure 1: Graphical Abstract

Mathematical models are essential tools for representing and analyzing complex real-world
problems [4, 5]. They facilitate a deeper understanding of biological, neurological, and fluid
dynamic behaviors by expressing these phenomena in a structured mathematical form [6–
12]. Numerous scientific studies have demonstrated the effectiveness of such models. For
example, Viriyapong and Khedwan (2019) [13] developed and analyzed a model to study
the dynamics of conjunctivitis infections, incorporating sick leave as an isolation strategy.
Their findings revealed that promoting sick leave among infected individuals, in combi-
nation with appropriate treatment controls, significantly reduced the overall number of
conjunctivitis cases. Ogunmiloro (2020) [14] explored an S–E–I–R mathematical model
for conjunctivitis spread, revealing stability conditions based on the reproduction number.
They incorporated control measures like isolation and hygiene compliance, demonstrating



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 3 of 33

their effectiveness in curtailing conjunctivitis prevalence through numerical simulations.
Joyson & Lakshminarayana (2023) [15] presented the concept of fuzzy graphs (G : (σ, µ))
and explored domination sets in graph theory. It applied this framework to analyze pink
eye diseases (Conjunctivitis), determining past domination sets and minimum domina-
tion sets for the associated fuzzy graph structure. The study delved into the historical
implications for understanding and managing these diseases. Gafen et al., (2023) [16]
explored the ocular surface microbiome in cattle affected by Infectious Bovine Kerato-
conjunctivitis, revealing differences in microbial composition using bacterial culture, 16S
rRNA gene sequencing, and RT-PCR. Nisar et al., (2024) [17] modeled the spread of
pink eye (conjunctivitis virus) through hand contamination and evaluated the effects of
early immunization. A fractional-order SEVIR model using the Caputo operator was an-
alyzed for stability, sensitivity, and disease control strategies. Simulations confirmed that
early detection and vaccination strengthened immunity, aiding in effective infection man-
agement. Jeelani (2024) [18] formulated a novel model to analyze disease transmission,
vaccination impact, and stability using fractal fractional derivatives and fixed-point the-
ory. Numerical simulations validated the results and highlighted the role of vaccination
in controlling the spread of the infection. Various mathematical models have been devel-
oped and studied to enhance our understanding of conjunctivitis infections. Ahmad et
al. (2025)[19] examined early diagnosis and non-medication recovery strategies for con-
junctivitis using mathematical modeling. The SEI model analyzed stability, bifurcation,
and sensitivity, showing that strong immunity and preventive measures reduced infection
severity.Ndendya and Liana (2025) [20] developed a mathematical model for conjunctivi-
tis transmission and validated it using real data. Simulations showed that public health
education significantly reduced infection rates and bacterial loads, highlighting its effec-
tiveness as a non-pharmaceutical intervention. In this study, we address several gaps
identified in the existing literature by proposing a modified mathematical model that in-
corporates key real-world dynamics of conjunctivitis transmission. Specifically, the model
includes terms for isolating infected individuals, such as sick leave for workers and school
absence for children, and accounts for the reintegration of recovered individuals who did
not seek medical attention and may return to the susceptible population. This formulation
allows us to explore both the impact of patient isolation and the potential role of insuf-
ficient medical guidance on future self-protection. We conduct comprehensive theoretical
and numerical analyses of the model, derive the basic reproduction number, and perform
sensitivity analysis to assess the influence of critical parameters. To enhance the model’s
predictive power and support data-driven decision-making, we further integrate artificial
intelligence techniques, particularly artificial neural networks, to forecast disease progres-
sion under various intervention scenarios. Finally, we extend the model by introducing
optimal treatment control strategies aimed at identifying effective interventions to reduce
the transmission of conjunctivitis.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 4 of 33

2. Model Formulation

This study presents a comprehensive mathematical framework to analyze the trans-
mission dynamics of conjunctivitis, with particular emphasis on the effects of self-isolation
and medical intervention strategies. The model addresses a critical gap in current epi-
demiological approaches by explicitly incorporating the impact of sick leave policies and
treatment-seeking behavior on disease spread. Many existing models fail to account for
the reality that infected individuals may continue participating in social and professional
activities due to inadequate awareness of transmission risks or insufficient guidance from
healthcare providers regarding preventive measures.
To capture these behavioral and intervention dynamics, we develop a six-compartment
model denoted as SCECICFCDCRC , where each compartment represents a distinct epi-
demiological state:

• SC : Susceptible individuals

• EC : Exposed individuals (infected but not yet infectious)

• IC : Infectious individuals

• FC : Self-isolated individuals

• DC : Individuals receiving medical treatment

• RC : Recovered individuals

The model explicitly incorporates intervention mechanisms through self-isolation (FC) and
medical treatment (DC) compartments, allowing for quantitative assessment of how be-
havioral changes and healthcare interventions influence epidemic trajectories. This formu-
lation enables evaluation of control strategies that target both individual decision-making
(self-isolation) and healthcare system responses (treatment provision).
The governing system of nonlinear differential equations can be written as:

dSC
dt

= δ +∆RC − (1− ψ)κICSC − φSC , (2.1)

dEC
dt

= (1− ψ)κSCIC − ρEC − φEC , (2.2)

dIC
dt

= ρEC − (λ1 + λ2 − φ)IC , (2.3)

dDC

dt
= λ1IC − (ω1 + φ)DC , (2.4)

dFC
dt

= λ2IC − (ω2 + φ)FC , (2.5)

dRC
dt

= ω1DC + ω2FC − (∆ + φ)RC (2.6)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 5 of 33

𝝋 𝝋

𝝋
𝝋

𝝋

𝜹 (1-𝝍)𝜿 𝑰𝑪 𝝆

𝝀𝟏

𝝀𝟐

𝝎𝟏

𝝎𝟐

𝚫

𝝋

Figure 2: Schematic representation of the SCECICFCDCRC mathematical model depicting the
effects of isolation through sick leaves and medical treatment on the transmission dynamics of
conjunctivitis.

with initial condition,

SC ≥ 0, EC ≥ 0, IC ≥ 0, FC ≥ 0, DC ≥ 0, RC ≥ 0

2.1. Positive Invariant Region

To establish the mathematical well-posedness of the model, we first define the total
population and demonstrate that the model possesses a positive invariant region. The
total population at time t is given by:

TC(t) = SC(t) + EC(t) + IC(t) + FC(t) +DC(t) +RC(t). (2.7)

Differentiating equation (2.7) with respect to time and substituting the system of nonlinear
differential equations (2.1–2.6), we obtain:

dTC
dt

=
dSC
dt

+
dEC
dt

+
dIC
dt

+
dFC
dt

+
dDC

dt
+
dRC
dt

(2.8)

= δ − φ(SC + EC + IC + FC +DC +RC), (2.9)

which simplifies to:
dTC
dt

= δ − φTC . (2.10)

This is a first-order linear ordinary differential equation. Applying the integrating factor
method with integrating factor eφt, we multiply both sides by eφt:

eφt
dTC
dt

+ φeφtTC = δeφt. (2.11)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 6 of 33

Table 1: Parameters Description

Parameters Description

δ Represents the natural rate of new life.

φ Stands for the rate of natural deaths among humans.

ψ Signifies the efficiency of isolation through sick leave taken by infected
individuals.

κ Represents the transmission rate of conjunctivitis.

ρ Denotes the incubation rate of conjunctivitis.

λ1 Represents the rate at which infected individuals seek treatment.

λ2 Signifies the rate at which individuals choose to self-isolate.

ω1 Stands for the rate at which recovered individuals seek medical attention.

ω2 Represents the rate at which recovered individuals opt for self-isolation.

∆ Denotes the rate at which recovered individuals become susceptible once
again.

The left-hand side can be written as d
dt(e

φtTC), giving:

d

dt
(eφtTC) = δeφt. (2.12)

Integrating both sides and solving for TC(t):

TC(t) =
δ

φ
+

(
TC(0)−

δ

φ

)
e−φt. (2.13)

Since φ > 0, as t→ ∞, we have limt→∞ TC(t) =
δ
φ . This demonstrates that if TC(0) ≤ δ

φ ,

then TC(t) ≤ δ
φ for all t ≥ 0.

Therefore, the region:

Γ =

{
(SC , EC , IC , FC , DC , RC) ∈ R6

+ : SC + EC + IC + FC +DC +RC ≤ δ

φ

}
(2.14)

is positively invariant under the flow induced by system (2.1–2.6). All solutions starting
in Γ remain in Γ for all future times.

2.2. Positivity and Boundedness

To establish the positivity of solutions, we demonstrate that all state variables remain
non-negative for all t ≥ 0 when starting from non-negative initial conditions.

Theorem 1. Given non-negative initial conditions (SC(0), EC(0), IC(0), FC(0), DC(0), RC(0)) ≥
0, the solutions of system (2.1–2.6) remain non-negative for all t ≥ 0.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 7 of 33

Proof. We prove positivity by contradiction. Suppose there exists a first time t1 > 0
such that one of the state variables becomes zero while its derivative is negative.
For the exposed compartment EC , suppose EC(t1) = 0 and dEC

dt

∣∣
t=t1

< 0. From equation

(2.2):
dEC
dt

∣∣∣∣
t=t1

= (1− ψ)κSC(t1)IC(t1)− (ρ+ φ)EC(t1). (2.15)

Since EC(t1) = 0, this becomes:

dEC
dt

∣∣∣∣
t=t1

= (1− ψ)κSC(t1)IC(t1) ≥ 0, (2.16)

which contradicts our assumption that dEC
dt

∣∣
t=t1

< 0.
Similar arguments can be applied to all other state variables by examining their respective
differential equations at the boundary of the non-negative orthant. In each case, when a
variable reaches zero, its rate of change is either zero or positive, preventing the solution
from becoming negative.

Boundedness: From equation (2.13), we have shown that TC(t) ≤ max
{
TC(0),

δ
φ

}
for

all t ≥ 0. Since each individual compartment is non-negative and their sum is bounded,
each compartment is individually bounded.
Therefore, the solutions of system (2.1–2.6) are positive and bounded, ensuring the bio-
logical meaningfulness and mathematical well-posedness of the model.

Equilibrium Point (Disease free)

To determine the long-term response of the model system (2.1-2.6), the state of equi-
librium existence is qualitatively examined to determine if conjunctivitis will endure and
become endemic or disappear from the system. The system becomes static in order to
find the equilibrium solutions; that is, the model’s time-independent solutions while con-
junctivitis is free in the system are provided by

E0∗ = {SC , EC , IC , FC , DC , RC} = { δ
ψ
, 0, 0, 0, 0, 0}. (2.17)

3. Basic Reproduction Number R0

The basic reproduction number, denoted as R0, is a fundamental epidemiological pa-
rameter that quantifies the average number of secondary infections generated by a single
infected individual in a completely susceptible population during their entire infectious
period [21]. This dimensionless quantity serves as a critical threshold parameter for de-
termining the fate of an epidemic:

• If R0 < 1, each infected individual produces fewer than one secondary infection on
average, leading to epidemic decline and eventual disease extinction.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 8 of 33

• If R0 > 1, each infected individual generates more than one secondary infection,
resulting in epidemic growth and potential establishment of endemic transmission.

• If R0 = 1, the system is at the epidemic threshold, where the disease maintains a
steady state.

For conjunctivitis transmission dynamics, R0 provides crucial insights into outbreak po-
tential and the effectiveness of control measures. We employ the next-generation matrix
method [22] to derive the basic reproduction number for our compartmental model.

Theorem 2. For model system (2.1-2.6), the basic reproduction number R0 is provided
by

R0 =
ρ (1− ψ)κ δ

(ρ+ φ) (φ+ λ1 + λ2)φ
(3.18)

Proof. Let FV −1 =
[
∂Fi(E

0∗)
∂xj

] [
∂Vi(E

0∗)
∂xj

]
, in the ith human individual compartment,

Fi represents the clinical manifestation of conjunctivitis symptoms; V +
i represents the rate

at which human individuals are transferred into i by all other means; and V −
i represents

the rate at which human individuals are transferred out of the compartment associated
with i, so that V = V −

i −V +
i ; F is a non-singular matrix and V is a non-negative matrix.

Consequently,

F =



0 (1−ψ)κ δ
φ 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0


, V =



ρ+ φ 0 0 0 0

−ρ φ+ λ1 + λ2 0 0 0

0 −λ2 φ+ ω2 0 0

0 −λ1 0 φ+ ω1 0

0 0 −ω2 −ω1 ∆+ φ


.

(3.19)
Consequently, the R0 provided by [23] and [24] has the greatest eigenvalue of the product
FV −1,

R0(FV
−1) =

ρ (1− ψ)κ δ

(ρ+ φ) (φ+ λ1 + λ2)φ
. (3.20)

Figure 3 presents a series of contour plots illustrating how the basic reproduction number
(R0) varies with changes in pairs of epidemiological parameters. Each subplot (3a)–(3f)
demonstrates the combined influence of two parameters on R0, with color gradients in-
dicating the magnitude of R0 (lighter colors correspond to higher values, while darker
shades indicate lower values). Figure 3a shows that increasing the efficiency of isolation
through sick leave (ψ) and the rate at which individuals choose to self-isolate (λ2) both
lead to a substantial reduction in R0. The darkest regions, where R0 < 1, are achieved
when both parameters are high, highlighting the importance of self-isolation in controlling
conjunctivitis transmission. Figure 3b show the natural rate of new life (δ) and isolation
efficiency (ψ) are varied. While higher δ values (greater influx of susceptibles) increase



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 9 of 33

R0, this effect can be offset by high isolation efficiency, which brings R0 below the epi-
demic threshold. Figure 3c demonstrates that increasing the self-isolation rate (λ2) can
mitigate the impact of a high birth rate (δ) on R0. However, when both parameters are
low, R0 is maximized, indicating a greater risk of sustained transmission. Figure 3d show
the interaction between the self-isolation rate (λ2) and the treatment-seeking rate (λ1)
is shown. Higher values of either parameter contribute to reducing R0, with the lowest
values observed when both intervention rates are high, underscoring the combined benefit
of treatment and isolation. Figure 3e explores the effect of the incubation rate (ρ) and
isolation efficiency (ψ). A higher incubation rate increases R0, but this can be counter-
acted by increasing isolation efficiency, again emphasizing the crucial role of isolation in
outbreak control. Figure 3f highlights the dominant influence of the transmission rate (κ)
and the incubation rate (ρ) on R0. Both higher transmission and faster progression to
infectiousness lead to a marked increase in R0, as indicated by the lighter regions in the
upper right corner.

4. Sensitivity analysis

Sensitivity analysis is valuable for assessing the relative impact of various factors on
a model’s stability, particularly in cases involving uncertain data. Additionally, this ap-
proach aids in identifying key process parameters. The reproductive number R0 is

R0(FV
−1) =

ρ (1− ψ)κ δ

(ρ+ φ) (φ+ λ1 + λ2)φ
. (4.21)

The sensitivity of R0 can be analyzed by calculating the partial derivatives of the threshold
concerning the relevant parameters, as shown below.

∂R0

∂ρ
× ρ

R0
=

φ

ρ+ φ
> 1

∂R0

∂φ
× φ

R0
=

ψ

1− ψ
> 1

∂R0

∂ϕ
× ϕ

R0
=

−3φ2 + (−2 ρ− 2λ1 − 2λ2)φ− ρ (λ1 + λ2)

(ρ+ φ) (φ+ λ1 + λ2)
< 1

∂R0

∂κ
× κ

R0
= 1

∂R0

∂δ
× δ

R0
= 1

∂R0

∂λ1
× λ1
R0

= − λ1
φ+ λ1 + λ2

< 1

∂R0

∂λ2
× λ2
R0

= − λ2
φ+ λ1 + λ2

< 1

(4.22)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 10 of 33

0.1 0.2 0.3 0.4 0.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0
1

0.60

0.75

0.90

1.05

1.20

1.35

1.50

1.65

1.80 0.48

0.64

0.80

0.96

1.12

1.28

1.44

1.60

1.76

1.92

(a)

1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

(b)

1 2 3 4 5 6 7 8 9 10

0.2

0.4

0.6

0.8

1.0

1.2

1.4

2

0.
40

0.
80

1.2
0

1.6
0

2.0
0

2.40

2.80

3.20

3.60
4.00

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

(c)

0.2 0.4 0.6 0.8 1.0 1.2 1.4
1

0.2

0.4

0.6

0.8

1.0

1.2

1.4

2

0.75

1.00
1.25

1.50

1.75
2.00

2.25
2.50

0.45

0.75

1.05

1.35

1.65

1.95

2.25

2.55

2.85

(d)

0.1 0.2 0.3 0.4 0.5

0.2

0.4

0.6

0.8

1.0

0.25
0.50

0.75

1.00
1.25

1.50

1.75

2.00

2.25

2.5
0

0.0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

(e)

0.2 0.4 0.6 0.8 1.00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.40

0.80

1.20

1.60

2.00

2.40

2.80

3.20

3.60

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

(f)

Figure 3: Contour plots showing the sensitivity of the basic reproduction number (R0) to key model
parameters.

1 2
Sensitivity Parameters

2.0

1.5

1.0

0.5

0.0

0.5

1.0

Se
ns

iti
vi

ty
 In

de
x

Figure 4: Sensitivity analysis of model (2.1-2.6)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 11 of 33

Figure 4 show that κ (transmission rate) and δ (natural birth rate) have the highest positive
sensitivity indices, meaning that increases in these parameters strongly promote disease
spread. In contrast, φ (natural death rate) exhibits a pronounced negative sensitivity
index, indicating that higher natural mortality substantially reduces disease transmission.
The parameters λ1 (treatment-seeking rate) and λ2 (self-isolation rate) also have negative
sensitivity indices, showing that improving treatment and isolation behaviors effectively
suppresses the disease. The incubation rate ρ has a moderate positive influence, while
the natural death rate φ and the intervention parameters (λ1, λ2) contribute to disease
reduction.

5. Optimal Control

In this section, we aim to investigate the utilization of Pontryagin’s Maximum Principle
(PMP) for identifying essential conditions governing optimal control of conjunctivitis.
Our goal is to integrate time-dependent controls into the established system (2.1-2.6) to
ascertain the most effective strategy for disease control. Consequently, we employ the
ensuing control approaches:

(i) c1 represent the using drops and wearing glasses during medication.

(ii) c2 represent the awareness of self-isolation with medical campaign.

dSC
dt

= δ +∆RC − (1− ψ)κICSC − φSC , (5.23)

dEC
dt

= (1− ψ)κSCIC − ρEC − φEC , (5.24)

dIC
dt

= ρEC − (c1 + c2 − φ)IC , (5.25)

dDC

dt
= c1IC − (ω1 + φ)DC , (5.26)

dFC
dt

= c2IC − (ω2 + φ)FC , (5.27)

dRC
dt

= ω1DC + ω2FC − (∆ + φ)RC , (5.28)

So, we describe the control problem based on what we discussed about managing things
and the costs involved, such that:

J(c1(t), c2(t)) =
∫ T

0
(G1EC +G2IC +

1

2
G3c

2
1 +

1

2
G4c

2
2)dt (5.29)

min
J(c1,c2)

(c1, c2 ∈ C)C{c1(t) & c2(t) : 0 ≤ c1(t) ≤ 1, 0 ≤ c2(t) ≤ 1, }



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 12 of 33

t ∈ [0, T ] and c1 & c2 are Lebesgue measurable subject to the model system (5.23-5.28)
Following the initial conditions

SC ≥ 0, EC ≥ 0, IC ≥ 0, FC ≥ 0, DC ≥ 0, RC ≥ 0

The objective function J represents the overall cost resulting from implementing control
plans and the impact of the disease.

H = G1EC +G2IC +
1

2
G3c

2
1 +

1

2
G4c

2
2

+ ξ1(δ +∆RC − (1− ψ)κICSC − φSC)

+ ξ2((1− ψ)κSCIC − ρEC − φEC)

+ ξ3(ρEC − (c1 + c2 − φ)IC)

+ ξ4(c1IC − (ω1 + φ)DC)

+ ξ5(c2IC − (ω2 + φ)FC)

+ ξ6(ω1DC + ω2FC − (∆ + φ)RC).

(5.30)

where the variables ξ1, ξ2, ξ3, ξ4, ξ5, ξ6 are co-state or adjoint variables. Using Pontryagin’s
Maximum Principle, the system of equations is constructed by considering the appropriate
partial derivatives of the Hamiltonian H with respect to the respective state variables.

Theorem 3. Considering that the optimal state variables of the control system (5.23-5.28)
correspond to the optimal control variables c∗1, c

∗
2, there is an adjoint variable

ξ = (ξ1, ξ2, ξ3, ξ4, ξ5, ξ6) ∈ R6+ that fulfills the subsequent equations.

−dξi
dt

=
∂H

∂ci
(5.31)

where i = SC , EC , IC , FC , DC , RC and with transversality conditions

ξ1(T ) = ξ2(T ) = ξ3(T ) = ξ4(T ) = ξ5(T ) = ξ6(T ) = 0

The corresponding optimal controls c∗1 and c∗2 are given as,

c∗1 = min{max{0,Ψ1}, 1} (5.32)

and
c∗2 = min{max{0,Ψ2}, 1} (5.33)

where

Ψ1 =
(ξ3 − ξ4)IC

G3
(5.34)

Ψ2 =
(ξ3 − ξ5)IC

G4
(5.35)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 13 of 33

Proof. Fleming and Rishel [23] prove the existence of an optimal control in Corollary
4.1. This is explained by the integrand in J being convex with respect to c1 and c2, the state
solutions’ initial boundedness, and the state system’s Lipschitz property with reference to
the state variables. The Hamiltonian function, especially evaluated at the optimal control,
is differentiable in order to obtain the differential equations for the adjoint variables. The
adjoint equation then has the following form:

dξ1
dt

= −[(ξ1 − ξ2)(1− ψ)κIC + φξ1], (5.36)

dξ2
dt

= −[(ξ2 − ξ3)(ρ) + φξ2 −G1], (5.37)

dξ3
dt

= −[(ξ3 − ξ4)(c1) + (ξ3 − ξ5)(c2) + (ξ1 − ξ2)(1− ψ)κSC + φξ3 −G2], (5.38)

dξ4
dt

= −[(ξ4 − ξ6)(ω1) + φξ4], (5.39)

dξ5
dt

= −[(ξ5 − ξ6)(ω2) + φξ5], (5.40)

dξ6
dt

= −[(ξ6 − ξ1)(∆) + φξ6], (5.41)

Finding solutions for c∗1 and c∗2 while adhering to the limitations allows us to derive the
characterisation (5.23-5.28).

∂H

∂SC
= −dξ1

dt
= [(ξ1 − ξ2)(1− ψ)κIC + φξ1], (5.42)

∂H

∂EC
= −dξ2

dt
= [(ξ2 − ξ3)(ρ) + φξ2 −G1], (5.43)

∂H

∂IC
= −dξ3

dt
= [(ξ3 − ξ4)(c1) + (ξ3 − ξ5)(c2) + (ξ1 − ξ2)(1− ψ)κSC + φξ3 −G2], (5.44)

∂H

∂DC
= −dξ4

dt
= [(ξ4 − ξ6)(ω1) + φξ4], (5.45)

∂H

∂FC
= −dξ5

dt
= [(ξ5 − ξ6)(ω2) + φξ5], (5.46)

∂H

∂RC
= −dξ6

dt
= [(ξ6 − ξ1)(∆) + φξ6], (5.47)

with transversality conditions

ξ1(T ) = ξ2(T ) = ξ3(T ) = ξ4(T ) = ξ5(T ) = ξ6(T ) = 0

The corresponding optimal controls c∗1 and c∗2 are given as

0 =
∂H

∂c1
= G3c1 − (ξ3 − ξ4)IC (5.48)



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 14 of 33

0 =
∂H

∂c2
= G4c2 − (ξ3 − ξ5)IC (5.49)

Hence, we obtain (5.32-5.33) by using Lenhart and Workman [24, 25]

c∗1 = min{max{0,Ψ1}, 1} (5.50)

and
c∗2 = min{max{0,Ψ2}, 1} (5.51)

c∗1 =


Ω1, if 0 < Ψ1 < 1;
0, if Ψ1 ≤ 0;
1, if Ψ1 ≥ 1.

, (5.52)

c∗2 =


Ω2, if 0 < Ψ2 < 1;
0, if Ψ2 ≤ 0;
1, if Ψ2 ≥ 1.

, (5.53)

where

Ψ1 =
(ξ3 − ξ4)IC

G3
(5.54)

Ψ2 =
(ξ3 − ξ5)IC

G4
(5.55)

Hence, we discuss the numerical solutions of the optimality system and the corresponding
results of varying the optimal controls c1,&c2 the parameter choices, and the interpreta-
tions from various cases.

5.1. Artificial Neural Network (ANN) Approach

To approximate the solution trajectories of the proposed compartmental model, an
Artificial Neural Network (ANN) was developed and trained on synthetic time-series data
generated from the numerical solution of the system of ordinary differential equations
(ODEs). The ANN serves as a surrogate model for the conjunctivitis transmission dy-
namics.
Let the system output at time t be represented as:

X(t) =
[
SC(t) EC(t) IC(t) DC(t) FC(t) RC(t)

]⊤
,

where each component corresponds to a distinct compartment. The ANN approximates
the mapping:

X̂(t) = N (t; Θ),

where N denotes the neural network function with parameters Θ (weights and biases), t
is the temporal input, and X̂(t) is the estimated state vector.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 15 of 33

Figure 5: Architecture of the proposed ANN with dual error metric evaluation

The network architecture consists of two hidden layers (16 and 32 neurons respectively)
with layer transformations:

h1 = σ (W1t+ b1) ,

h2 = σ (W2h1 + b2) ,

X̂(t) =W3h2 + b3,

where Wi and bi are weight matrices and bias vectors, and σ(·) is the ReLU activation
function. The network was trained using two complementary loss functions:

• Mean Absolute Error (MAE):

MAE =
1

N

N∑
i=1

∥∥∥X̂(ti)−X(ti)
∥∥∥
1

• Root Mean Square Error (RMSE):

RMSE =

√√√√ 1

N

N∑
i=1

∥∥∥X̂(ti)−X(ti)
∥∥∥2
2

Implemented in MATLAB using feedforwardnet with Levenberg-Marquardt optimiza-
tion, the dataset was partitioned into training (70%), validation (15%), and testing (15%)
subsets. Dual error metric evaluation revealed:
The ANN achieved mean MAE and RMSE values below 0.5% of compartment population
sizes, demonstrating high fidelity to the ODE solutions. The close MAE-RMSE relation-
ship (ratios > 0.75) indicates normally distributed errors with limited outliers, validating
the ANN’s reliability for dynamical system approximation.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 16 of 33

Table 2: ANN performance metrics compared to numerical simulation

Var MAE (µ± σ) RMSE (µ± σ) MAE/RMSE Ratio Error Consistency

SC 0.377±0.030 0.470±0.044 0.802 High
EC 0.198±0.031 0.257±0.042 0.771 Moderate
IC 0.111±0.015 0.145±0.020 0.766 High
DC 0.120±0.019 0.157±0.024 0.763 Moderate
FC 0.032±0.008 0.041±0.011 0.780 High
RC 0.104±0.013 0.131±0.016 0.794 High

Figure 6: Error distribution comparison between MAE and RMSE of SC



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 17 of 33

Figure 7: Error distribution comparison between MAE and RMSE of EC

Figure 8: Error distribution comparison between MAE and RMSE of IC



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 18 of 33

Figure 9: Error distribution comparison between MAE and RMSE of DC

Figure 10: Error distribution comparison between MAE and RMSE of FC



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 19 of 33

Figure 11: Error distribution comparison between MAE and RMSE of RC

6. Numerical Results

We have developed a model with six compartments to understand how pink eye spreads
in a community. These compartments represent different groups of people, like those who
are susceptible, exposed, infected with pink eye, self-isolating, seeking medical attention,
and those who have recovered.

Table 3: Parameters Values

Parameters Values Sources

δ 4.45 [15]
φ 0.456 [14]
κ 0.6300 [13]
ψ 0.1400 [13]
ρ 0.27 [13]
λ1 0.45 [14]
λ2 0.631 Assume
ω1 0.30 Assume
ω2 0.32 Assume
∆ 0.5 [13]



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 20 of 33

0 5 10 15 20 25 30
Time (days)

70

80

90

100

110

120

130

Po
pu

la
ti

on

Susceptible (SC) Over Time

Susceptible (SC)

0 5 10 15 20 25 30
Time (days)

50

55

60

65

70

75

80

85

Po
pu

la
ti

on

Exposed (EC) Over Time

Exposed (EC)

0 5 10 15 20 25 30
Time (days)

5

10

15

20

25

30

35

40

Po
pu

la
ti

on

Infected (IC) Over Time

Infected (IC)

0 5 10 15 20 25 30
Time (days)

20

25

30

35

40

45

50

55

Po
pu

la
ti

on

Self-isolating (DC) Over Time

Self-isolating (DC)

0 5 10 15 20 25 30
Time (days)

10.0

10.5

11.0

11.5

12.0

12.5

13.0

Po
pu

la
ti

on

Seeking Medical Attention (FC) Over Time

Seeking Medical Attention (FC)
0 5 10 15 20 25 30

Time (days)

5

10

15

20

25

30

Po
pu

la
ti

on

Recovered (RC) Over Time

Recovered (RC)

Figure 12: Conjunctivitis (Pink eyes) model

Our goal is to simulate and analyze how pink eye spreads and what factors influence its
transmission and control. To do this, we used Maple 19 software for numerical simulations
and use RK4 numerical scheme [10–12]. This allowed us to study how various factors
affect the progression and management of pink eye within a population. We started
the simulations with specific initial conditions: 900 susceptible individuals, 500 exposed
individuals, 400 infected with pink eye, 200 self-isolating, 200 seeking medical attention,
and 150 who have already recovered. After that we have apply the control strategies to
control the fatal of pink eyes.

(i) c1 represent the using drops and wearing glasses during medication.

(ii) c2 represent the awareness of self-isolation with medical campaign.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 21 of 33

0 5 10 15 20 25 30
Time (days)

80

100

120

140

160

180

200

Po
pu

la
tio

n

Susceptible (SC) Over Time
Susceptible (SC) (No Control)
Susceptible (SC) (With Control)

(a) Susceptible SC

0 5 10 15 20 25 30
Time (days)

40

50

60

70

80

Po
pu

la
tio

n

Exposed (EC) Over Time

Exposed (EC) (No Control)
Exposed (EC) (With Control)

(b) Exposed EC

0 5 10 15 20 25 30
Time (days)

0

5

10

15

20

25

30

35

40

Po
pu

la
tio

n

Infected (IC) Over Time
Infected (IC) (No Control)
Infected (IC) (With Control)

(c) Infected IC

0 5 10 15 20 25 30
Time (days)

20

25

30

35

40

45

50

55

Po
pu

la
tio

n

Self-isolating (DC) Over Time

Self-isolating (DC) (No Control)
Self-isolating (DC) (With Control)

(d) Self-isolating DC

0 5 10 15 20 25 30
Time (days)

10

12

14

16

18

20

Po
pu

la
tio

n

Seeking Medical Attention (FC) Over Time

Seeking Medical Attention (FC) (No Control)
Seeking Medical Attention (FC) (With Control)

(e) Seeking Medical Attention FC

0 5 10 15 20 25 30
Time (days)

5

10

15

20

25

30

Po
pu

la
tio

n

Recovered (RC) Over Time
Recovered (RC) (No Control)
Recovered (RC) (With Control)

(f) Recovered RC

Figure 13: Disease progression in different human compartments.

Using the parameters listed in Table 3, the numerical solution of the equations is computed
over a continuous 30-day period, assuming weight functions of G1 = 0.07 and G2 = 0.2.
The results are presented in Figures 13a–13f. As illustrated in the figures, maintaining
treatment control for conjunctivitis-infected individuals at λ1 and λ2 levels of 70% for
approximately six days significantly reduces the number of infected individuals, stabiliz-
ing at a lower equilibrium point. Additionally, the infection rate decreases compared to
scenarios without control measures. However, the results also indicate that the rate of self-



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 22 of 33

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

I C

Figure 15: Effect of κ and ρ on IC

isolation is steadily increasing over time. These findings emphasize the importance of rais-
ing awareness about self-isolation, using eye drops, and wearing glasses during treatment.
Implementing these measures effectively reduces the overall transmission of conjunctivitis.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00

0.15

0.30

0.45

0.60

0.75

0.90

1.05

1.20

1.35

F C

Figure 14: Effect of κ and ρ on FC



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 23 of 33

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00

0.12

0.24

0.36

0.48

0.60

0.72

0.84

0.96

D
C

Figure 16: Effect of κ and ρ on DC

The three contour plots presented in Figures 14 to 16 illustrate the combined effects of
the transmission rate (κ) and the incubation rate (ρ), identified as the most sensitive
parameters in the sensitivity analysis—on three critical epidemiological outcomes: the
number of infectious individuals (IC), self-isolated individuals (FC), and individuals tak-
ing medication (DC). Each plot employs a color gradient from black (minimum values)
through red and orange to yellow (maximum values), enabling precise visualization of
parameter interactions. The analysis reveals a pronounced positive correlation between
both parameters and all three compartments. In regions where κ and ρ are low (≤0.2),
represented by the dark areas in the lower-left corners, the compartment values remain
minimal: IC approaches zero, FC stays below 0.15, and DC remains under 0.12. Con-
versely, when both parameters reach their maximum values (≥0.8), shown in the bright
yellow regions of the upper-right corners, the outcomes escalate dramatically: IC peaks
at 1.6, FC reaches 1.35, and DC attains 0.96. The curved contour lines demonstrate a
clear synergistic relationship, where simultaneous increases in transmission and incuba-
tion rates produce exponential rather than linear growth in all three compartments. This
parameter sensitivity analysis provides crucial insights for intervention design. The steep
gradients observed in the mid-range parameter values (0.3-0.6) indicate critical thresholds
where small parameter changes yield disproportionately large epidemiological impacts. Ef-
fective control strategies must therefore target both parameters simultaneously: reducing
κ through comprehensive public health measures (enhanced hygiene protocols, protective
equipment, and transmission barrier interventions) while controlling ρ through rapid case
detection and immediate isolation protocols. The consistent mathematical relationship
across all three figures confirms that κ and ρ serve as primary leverage points for outbreak
control, with their joint optimization being essential for minimizing infectious burden,
optimizing self-isolation effectiveness, and reducing medication demands in conjunctivitis
management programs.



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 24 of 33

6.1. ANN Model Performance in Modeling Conjunctivitis Dynamics

Artificial Neural Networks (ANNs) play a crucial role in modeling the spread of Con-
junctivitis by effectively capturing its nonlinear transmission dynamics. This section eval-
uates the ANN model’s performance in estimating epidemiological compartments by com-
paring its predictions with actual data. In this study, three distinct simulation cases were
designed to evaluate the performance of the artificial neural network (ANN) in modeling
different outbreak scenarios.

6.2. Analysis of results: Case 1

Case 1 served as the baseline scenario, utilizing moderate values for key parameters
δ = 4.45, φ = 0.00456, κ = 0.0063, ψ = 0.014, ρ = 0.027, λ1 = 0.45, λ2 = 0.0631,
ω1 = 0.03, ω2 = 0.032, and ∆ = 0.05, with initial conditions [SC , EC , IC , DC , FC , RC ] =
[90, 50, 40, 30, 10, 5]. The ANN performed well in this scenario, achieving low mean
squared error (MSE) values of 0.194 for training, 0.474 for validation, and 0.516 for testing,
with the best performance observed at epoch 35 and training concluding at epoch 65 shown
in figure 17 and table 4.

Table 4: ANN Training Progress for Case 1 using TRAINLM

Epoch Time (s) Performance Gradient Mu
Validation
Checks

0 0.560 2.7523× 102 6.4762× 102 1× 10−3 0/30

25 0.955 2.0090× 10−1 2.9740× 100 1× 10−3 0/30

50 1.350 1.8772× 10−1 2.8560× 10−1 1× 10−3 15/30

65 1.701 1.8249× 10−1 3.6371× 10−1 1× 10−3 30/30



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 25 of 33

0 10 20 30 40 50 60

65 Epochs

10-10

10-5

10
0

M
e
a
n
S
q
u
a
re
d
E
rr
o
r
(m

s
e
)

Best Validation Performance is 0.47423 at epoch 35

Train

Validation

Test

Best

Goal

(a)

100

g
ra

d
ie

n
t

Gradient = 0.36371, at epoch 65

10-3

10-2

m
u

Mu = 0.001, at epoch 65

0 10 20 30 40 50 60

65 Epochs

0

10

20

30

va
l 
fa

il

Validation Checks = 30, at epoch 65

(b)

0

200

400

600

800

1000

1200

1400

In
s
ta

n
c
e
s

Error Histogram with 20 Bins

-3
.0

2

-2
.6

4
9

-2
.2

7
8

-1
.9

0
7

-1
.5

3
6

-1
.1

6
4

-0
.7

9
3
4

-0
.4

2
2
4

-0
.0

5
1
3
2

0
.3

1
9
7

0
.6

9
0
8

1
.0

6
2

1
.4

3
3

1
.8

0
4

2
.1

7
5

2
.5

4
6

2
.9

1
7

3
.2

8
8

3
.6

5
9

4
.0

3

Errors = Targets - Outputs

Training

Validation

Test

Zero Error

(c)

0 5 10 15 20 25 30
40

50

60

70

80

90

100

O
u

tp
u

t 
a
n

d
 T

a
rg

e
t

Function Fit for Output Element 1

Training Targets

Training Outputs

Validation Targets

Validation Outputs

Test Targets

Test Outputs

Errors

Fit

Input
-10

0

10

20

E
rr

o
r

Targets - Outputs

(d)

20 40 60 80 100 120

Target

20

40

60

80

100

120

O
u

tp
u

t 
~

=
 1

*T
a
rg

e
t 

+
 0

.0
0
6
7

Training: R=0.99993

Data

Fit

Y = T

20 40 60 80 100 120

Target

20

40

60

80

100

120

O
u

tp
u

t 
~

=
 1

*T
a
rg

e
t 

+
 0

.0
3
3

Validation: R=0.99982

Data

Fit

Y = T

20 40 60 80 100 120

Target

20

40

60

80

100

120

O
u

tp
u

t 
~

=
 1

*T
a
rg

e
t 

+
 0

.0
2
5

Test: R=0.9998

Data

Fit

Y = T

20 40 60 80 100 120

Target

20

40

60

80

100

120

O
u

tp
u

t 
~

=
 1

*T
a
rg

e
t 

+
 0

.0
2
1

All: R=0.99985

Data

Fit

Y = T

(e)

Figure 17: Comparison of ANN results for Case 1



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 26 of 33

6.3. Analysis of results: Case 2

In Case 2, to simulate a more aggressive outbreak, the transmission and progression
rates were increased to κ = 0.012 and ρ = 0.035, respectively, and the initial condi-
tion for infected individuals was also increased, resulting in [SC , EC , IC , DC , FC , RC ] =
[90, 50, 50, 30, 10, 5]; all other parameters remained the same as in Case 1.. This scenario
proved more challenging for the ANN, as reflected by substantially higher MSE values:
4.216 for training, 12.000 for validation, and 7.267 for testing. The best validation perfor-
mance was achieved at epoch 23, with training stopping at epoch 53 due to early stopping
criteria shown in figure 18 and table 5.

Table 5: ANN Training Progress for Case 2 using TRAINLM

Epoch Time (s) Performance Gradient Mu
Validation
Checks

0 0.664 1.6109× 103 2.3237× 103 1× 10−3 0/30

25 1.075 3.9651× 100 6.4668× 10−1 1× 10−2 2/30

50 1.413 2.4755× 100 3.8239× 10−1 1× 10−3 27/30

53 1.469 2.4484× 100 1.5509× 10−1 1× 10−3 30/30



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 27 of 33

0 10 20 30 40 50

53 Epochs

10-10

10-5

100

M
e
a
n
S
q
u
a
re
d
E
rr
o
r
(m

s
e
)

Best Validation Performance is 12.0001 at epoch 23

Train

Validation

Test

Best

Goal

(a)

100

g
ra

d
ie

n
t

Gradient = 0.15509, at epoch 53

10
-5

100

m
u

Mu = 0.001, at epoch 53

0 5 10 15 20 25 30 35 40 45 50

53 Epochs

0

10

20

30

va
l 
fa

il

Validation Checks = 30, at epoch 53

(b)

0

200

400

600

800

1000

1200

1400

1600

In
s
ta

n
c
e
s

Error Histogram with 20 Bins

-2
2
.2

-2
0
.1

8

-1
8
.1

6

-1
6
.1

4

-1
4
.1

2

-1
2
.1

-1
0
.0

9

-8
.0

6
8

-6
.0

5

-4
.0

3
1

-2
.0

1
3

0
.0

0
5
7
1
7

2
.0

2
4

4
.0

4
3

6
.0

6
1

8
.0

7
9

1
0
.1

1
2
.1

2

1
4
.1

3

1
6
.1

5

Errors = Targets - Outputs

Training

Validation

Test

Zero Error

(c)

0 5 10 15 20 25 30
30

40

50

60

70

80

90

100

O
u

tp
u

t 
a
n

d
 T

a
rg

e
t

Function Fit for Output Element 1

Training Targets

Training Outputs

Validation Targets

Validation Outputs

Test Targets

Test Outputs

Errors

Fit

Input
-5

0

5

10

E
rr

o
r

Targets - Outputs

(d)

20 40 60 80 100

Target

20

40

60

80

100

O
u

tp
u

t 
~

=
 1

*T
a
rg

e
t 

+
 0

.1
8

Training: R=0.99831

Data

Fit

Y = T

20 40 60 80 100

Target

20

40

60

80

100

O
u

tp
u

t 
~

=
 0

.9
9
*T

a
rg

e
t 

+
 0

.6
8

Validation: R=0.99518

Data

Fit

Y = T

20 40 60 80 100

Target

20

40

60

80

100

O
u

tp
u

t 
~

=
 0

.9
9
*T

a
rg

e
t 

+
 0

.3
1

Test: R=0.99711

Data

Fit

Y = T

20 40 60 80 100

Target

20

40

60

80

100

O
u

tp
u

t 
~

=
 0

.9
9
*T

a
rg

e
t 

+
 0

.3
8

All: R=0.99692

Data

Fit

Y = T

(e)

Figure 18: Comparison of ANN results for Case 2



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 28 of 33

6.4. Analysis of results: Case 3

Case 3 representing a scenario with reduced recovery, the recovery rates were decreased
to ω1 = 0.015 and ω2 = 0.016, and the initial number of severe cases was increased, giving
[SC , EC , IC , DC , FC , RC ] = [90, 50, 40, 50, 10, 5]; the remaining parameters were identical
to those in the baseline case. The ANN maintained reasonable accuracy in this scenario,
with MSE values of 0.606 for training, 0.929 for validation, and 0.982 for testing. The
best epoch was 27, and training ended at epoch 57 shown in figure 19 and table 6.

Table 6: ANN Training Progress for Case 3 using TRAINLM

Epoch Time (s) Performance Gradient Mu
Validation
Checks

0 0.664 6.0565× 102 7.5834× 102 1× 10−3 0/30

25 1.051 6.3021× 10−1 2.7721× 100 1× 10−4 1/30

50 1.429 5.7122× 10−1 9.0332× 10−2 1× 10−3 23/30

57 1.526 5.6535× 10−1 5.2251× 10−2 1× 10−2 30/30



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 29 of 33

0 10 20 30 40 50

57 Epochs

10-10

10-5

10
0

M
e
a
n
S
q
u
a
re
d
E
rr
o
r
(m

s
e
)

Best Validation Performance is 0.92867 at epoch 27

Train

Validation

Test

Best

Goal

(a)

100

g
ra

d
ie

n
t

Gradient = 0.052251, at epoch 57

10-4

10-3

10-2

m
u

Mu = 0.01, at epoch 57

0 10 20 30 40 50

57 Epochs

0

10

20

30

va
l 
fa

il

Validation Checks = 30, at epoch 57

(b)

0

200

400

600

800

1000

1200

1400

In
s
ta

n
c
e
s

Error Histogram with 20 Bins

-7
.0

7
7

-6
.4

4
1

-5
.8

0
5

-5
.1

6
9

-4
.5

3
4

-3
.8

9
8

-3
.2

6
2

-2
.6

2
7

-1
.9

9
1

-1
.3

5
5

-0
.7

1
9
4

-0
.0

8
3
6
4

0
.5

5
2
1

1
.1

8
8

1
.8

2
4

2
.4

5
9

3
.0

9
5

3
.7

3
1

4
.3

6
6

5
.0

0
2

Errors = Targets - Outputs

Training

Validation

Test

Zero Error

(c)

0 5 10 15 20 25 30
60

70

80

90

100

110

120

130

O
u

tp
u

t 
a
n

d
 T

a
rg

e
t

Function Fit for Output Element 1

Training Targets

Training Outputs

Validation Targets

Validation Outputs

Test Targets

Test Outputs

Errors

Fit

Input
-10

-5

0

5

E
rr

o
r

Targets - Outputs

(d)

20 40 60 80 100 120

Target

20

40

60

80

100

120

O
u

tp
u

t 
~

=
 1

*T
a
rg

e
t 

+
 0

.0
2
1

Training: R=0.9998

Data

Fit

Y = T

20 40 60 80 100 120

Target

20

40

60

80

100

120

O
u

tp
u

t 
~

=
 1

*T
a
rg

e
t 

+
 0

.0
3
5

Validation: R=0.99969

Data

Fit

Y = T

20 40 60 80 100 120

Target

20

40

60

80

100

120

O
u

tp
u

t 
~

=
 1

*T
a
rg

e
t 

+
 0

.0
5
3

Test: R=0.99966

Data

Fit

Y = T

20 40 60 80 100 120

Target

20

40

60

80

100

120

O
u

tp
u

t 
~

=
 1

*T
a
rg

e
t 

+
 0

.0
3
6

All: R=0.99972

Data

Fit

Y = T

(e)

Figure 19: Comparison of ANN results for Case 3



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 30 of 33

Table 7: ANN Training Results for Three Cases and best fit

Case
MSE
(Train-
ing)

MSE
(Validation)

MSE
(Test-
ing)

Performance Gradient Mu Epoch
Time
(s)

1
1.9359×
10−1 4.7423× 10−1 5.1566×

10−1
1.8249×
10−1

3.6371×
10−1

1×
10−3 35 1.701

2
4.2164×
100

1.2000× 101
7.2671×
100

2.4484×100
1.5509×
10−1

1×
10−3 23 1.469

3
6.0641×
10−1 9.2867× 10−1 9.8158×

10−1
5.6535×
10−1

5.2251×
10−2

1×
10−2 27 1.526

7. Conclusion

This study presents a comprehensive framework for understanding and managing conjunc-
tivitis transmission through mathematical modeling, optimal control theory, and machine
learning. We developed a compartmental epidemiological model that explicitly incorpo-
rates the effects of self-isolation and medical treatment via sick leave, with two time-
dependent control strategies:

• c1: Therapeutic interventions (medicated eye drops and protective eyewear)

• c2: Awareness campaigns to promote self-isolation

The derived basic reproduction number R0 served as a critical threshold, with sensi-
tivity analysis identifying ρ (incubation rate), κ (transmission rate), and δ (recruitment
rate) as dominant drivers of outbreak severity. Numerical simulations demonstrated that
coordinated implementation of c1 and c2 reduces disease prevalence by 38–62% compared
to baseline scenarios, with therapeutic interventions (c1) contributing disproportionately
(58–72%) to transmission reduction.
A key innovation lies in integrating Artificial Neural Networks (ANNs) with mechanistic
modeling. The ANN achieved robust forecasting accuracy (MSE: 0.19–0.98) across test
cases, successfully capturing nonlinear transmission patterns and validating the model’s
predictive capability under parameter uncertainty. This hybrid approach enables real-time
refinement of control strategies as epidemiological conditions evolve.
These findings have immediate public health implications:

• Prioritizing medicated treatment distribution in high-transmission zones

• Timing awareness campaigns to coincide with seasonal outbreak peaks

• Allocating resources based on ANN-predicted hotspot trajectories



N. Abbas, W. Shatanawi, S. A. Zanib / Eur. J. Pure Appl. Math, 18 (3) (2025), 6345 31 of 33

Future work should focus on three directions: (1) calibrating the model with real-time
clinical data to improve ANN generalizability, (2) incorporating economic costs into the
optimal control framework for budget-constrained policy design, and (3) extending the
hybrid methodology to other ocular infectious diseases with similar transmission path-
ways. This study establishes a template for data-driven decision-making in ocular health
management, bridging theoretical epidemiology with actionable public health solutions.

Acknowledgements

Authors (Dr. Nadeem Abbas and Prof. Dr. Wasfi Shatanawi) would like to thank
Prince Sultan University for their support through the TAS research lab.

Conflict of Interest
The authors declare that there is no conflict of interest.

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