INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

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ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIV, Issue XII, December 2025

Caputo-Fabrizio Fractional Order Derivative Mathematical

Modeling and Optimal Control, Cost-Effectiveness Analysis of

Diphtheria Transmission Dynamics in Nigeria

Musa. Abdullahi¹, Abdullahi M. Auwal^2*,and Umar Abbas Faruk ³

^1,3Mathematics and Statistics Department, Gombe State Polytechnic, Bajoga

²Mathematics and Statistics Department, Federal Polytechnic, Bauchi

DOI : https://doi.org/10.51583/IJLTEMAS.2025.1412000054

Received: 18 December 2025; Accepted: 25 December 2025; Published: 03 January 2026

ABSTRACT

Diphtheria, a potentially fatal vaccine-preventable disease caused by Corynebacterium diphtheriae, has seen a

resurgence in several regions, including sub-Saharan Africa. This study presents a comprehensive mathematical

model for diphtheria transmission dynamics that incorporates environmental bacterial persistence, public

awareness campaigns, and vaccine hesitancy. The model subdivides the human population into

epidemiologically relevant classes, including susceptible, exposed, symptomatic infectious, asymptomatic

infectious, vaccinated, and hesitant individuals, with an additional compartment representing environmental

bacterial load. Analytical results establish the model’s positivity, boundedness, and stability properties, with the

basic reproduction number (ℛ₀) derived to determine disease threshold conditions. A global sensitivity analysis

using the Partial Rank Correlation Coefficient (PRCC) identified the transmission rate (훽₁), exposure

progression rate (휎), and vaccination rate (훿) as the most influential parameters affecting ℛ₀. The optimal

control analysis, formulated via Pontryagin’s Maximum Principle, evaluated time-dependent interventions

representing vaccination, treatment, and public enlightenment efforts. Numerical simulations using

parameterized incidence data from Bauchi State, Northeast Nigeria, revealed that combined control strategies

significantly reduce infection prevalence and environmental contamination compared to single interventions.

The cost-effectiveness analysis (CEA) demonstrated that the combined vaccination and public enlightenment

campaign strategy produced the lowest Incremental Cost-Effectiveness Ratio (ICER) and was classified as “very

cost-effective” according to WHO-CHOICE thresholds. The findings underscore that effective control requires

a multi-pronged, dynamically-adapted strategy targeting all transmission pathways. High vaccination coverage

remains the cornerstone, but its impact is significantly amplified when synergistically combined with public

health campaigns and environmental decontamination. The results offer evidence-based, cost-effective guidance

for public health policymakers to design efficient diphtheria outbreak response and elimination programs.

Moreover, numerical solutions obtained for different fractional orders highlight the Caputo-Fabrizio fractional

derivative's capability to simulate memory effects, thereby offering a more nuanced representation of the real-

life problem under study.

Keywords: Diphtheria, Mathematical Modelling, Caputo-Fabrizio fractional derivative, Basic Reproduction

Number, Global Sensitivity Analysis, Vaccination, Public Health Campaigns.

INTRODUCTION

Corynebacterium diphtheriae, the causative agent of the potentially fatal infectious illness diphtheria, produces

a toxin that can cause serious respiratory and systemic complications [1]. Although diphtheria is vaccine-

preventable, the disease has re-emerged as a significant public health concern in Nigeria, particularly in the

Northeast region. This resurgence, despite the availability of effective vaccines, underscores persistent gaps in

immunization coverage, public health awareness, and healthcare delivery systems. Transmission occurs

primarily through respiratory droplets or contact with infected surfaces. Symptoms typically appear 2–5 days

post-infection and range from sore throat and fever to severe complications such as airway obstruction,

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myocarditis, and neuropathy. In Nigeria, the burden is disproportionately high among children aged 2–14 years,

who constitute the majority of recent cases.

The symptoms range from mild to severe. At the mild stage, symptoms include a sore throat and fever; at the

severe stage, symptoms include difficulty breathing and swallowing, as well as a barking cough. Inflammation

and damage of the heart muscles, nerves, and kidney-related problems can also occur [45] and [24]. It is,

therefore, advisable that infected persons seek treatment to help prevent serious complications [31]. Diphtheria

is treated with antibiotics and diphtheria antitoxin, which kill the bacteria and prevent them from destroying

body tissues [11]. Diphtheria can be transmitted among humans via contact with respiratory droplets, such as

coughing and sneezing, or through indirect contact with contaminated clothing and objects ([11, 24, 45]). [3]

states that an asymptomatic infected person can infect others for up to four weeks. Timely vaccination with the

prescribed vaccines is the best defense against diphtheria ([11, 31, 8]). According to [24], children and

adolescents should receive three booster doses of the diphtheria toxoid-containing vaccination [45].

Consequently, [24] recommends educating parents on the advantages of routine immunization to prevent

diphtheria. Adults and newborns can also receive vaccinations. Over time, the effectiveness of the vaccines

decreases; for roughly ten years, they provided 97 out of 100 protection against diphtheria ([11, 24]). Wearing

personal protective equipment (PPE), which is disposed of right away after leaving patients' rooms, protects

medical personnel against the disease [45].

The increasing prevalence of diphtheria in Nigeria is highlighted by recent statistics from the Nigeria Centre for

Disease Control [32]. Four states, Kano (107 instances), Yobe (2), Lagos (1), and Osun (1) reported 111

confirmed cases and 22 fatalities between December 1, 2022, and the third week of January, 2023. For confirmed

and probable cases, the case fatality rate (CFR) was 19.8%. Children between the ages of 2 and 14 accounted

for a substantial majority of cases (91.9%). Remarkably, only 10.8% of these verified cases had been fully

immunized with a vaccine containing diphtheria toxin [11]. By April 2023, the outbreak had intensified.

Confirmed cases rose to 672 across seven states, with 491 cases affecting children aged 2–14 years. Out of these,

only 144 were fully vaccinated. Factors contributing to the rapid spread include low immunization coverage,

limited access to healthcare, poor socioeconomic conditions, low education levels, particularly in rural areas,

and delayed clinical recognition of the disease [45].

In June 2023, reported cases increased further to 836, with over 70% involving children within the 2–14 age

group. By 31 July 2023, the outbreak had expanded significantly to 1,534 confirmed cases across several states,

including Kano, Lagos, Yobe, Katsina, Kaduna, Bauchi, the Federal Capital Territory (FCT), Niger, Gombe,

Jigawa, Cross River, and Osun. The death toll rose to 137, but the CFR declined to 8.9%, likely due to improved

availability and use of diphtheria antitoxin and antibiotics [31]. However, the aforementioned diphtheria

infection outbreak significantly exceeds the last major diphtheria event in Nigeria, which occurred from February

to November 2011 in Kimba village and surrounding settlements in Borno State. Only 98 cases were recorded,

with a CFR of 21.4%, mostly affecting children under the age of 10. Notably, none of the affected individuals

in that outbreak had received prior vaccination against diphtheria [24]. Despite national immunization programs,

including the pentavalent DTP vaccine, coverage remains insufficient. For instance, the third-dose coverage

(DTP3) declined from 86% in 2019 to 81% in 2021, with some states reporting coverage as low as 20%, far

below the herd immunity threshold of 75–80%. Barriers to uptake include vaccine hesitancy, misinformation,

maternal education levels, household decision-making dynamics, vaccine stockouts, and healthcare workforce

shortages [1].

Several epidemiological models of diphtheria exist in the literature. For instance, [7] used mathematical

modelling of antibody decay to estimate seroprotection rates 10 years post-vaccination. [15] developed an age-

structured model of diphtheria transmission, employing non-linear differential equations to assess the global

stability of the system. Similarly, [46] utilized a next-generation matrix approach for global stability analysis

and parameter estimation. Studies by [19] explored quarantine’s impact on diphtheria dynamics, highlighting

the role of early detection in controlling the spread. Furthermore, [47] developed an optimal control model to

determine cost-effective strategies for reducing transmission, incorporating both cost and vaccination

effectiveness. Other works, such as [29], emphasize vaccination’s impact on outbreak dynamics through

simulation-based modeling and sensitivity analysis. [20] Conducted a sensitivity analysis of diphtheria

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transmission parameters, identifying key areas for intervention. [38] analyzed data from Nigeria’s Isin Local

Government Area to assess gender effects on transmission, while [19] examined vaccine efficacy using stability

analysis within the SEIR model framework. Finally, [21] provided a dynamical analysis of diphtheria with

natural immunity rates, exploring stability under different equilibrium conditions.

Mathematical modeling serves as a powerful tool for understanding infectious disease dynamics and guiding

control strategies [34]. However, existing models often fail to account for Nigeria-specific challenges such as

regional heterogeneity, behavioral dynamics, environmental factors, and healthcare system constraints. These

limitations hinder the development of effective, data-driven interventions tailored to local contexts.

In another development, Fractional calculus has emerged as a vital framework for modelling complex physical

systems that exhibit memory and nonlocal effects, phenomena that are challenging to capture with classical

integer-order derivatives [13]. A significant limitation of early fractional operators was the singularity in their

kernels at the termination point of the definition interval. This has spurred the development of numerous non-

singular kernel definitions in recent literature.

The primary distinction among fractional derivatives lies in their kernels, which can be tailored to specific

applications. Key developments include the singular power-law kernel of the Caputo derivative [13], the non-

singular exponential decay kernel of the Caputo-Fabrizio (CF) derivative [9], and the non-singular Mittag-Leffler

law kernel of the Atangana-Baleanu (AB) derivative [5]. Comparative studies have highlighted the practical

advantages of these new operators. For instance, in modeling chaotic systems, the CF derivative was found to

produce less noise than the power-law-based Riemann-Liouville derivative [5]. The CF derivative, in particular,

was explicitly designed to model memory effects without singularities, and its corresponding fractional integral

has been rigorously defined [27].

Leveraging these advanced mathematical tools, this study addresses some gaps by developing a novel

deterministic integer- and non-integer-order model of diphtheria transmission that integrates regional

demographic variation, vaccine hesitancy, environmental influences, and public health campaign dynamics.

Using incidence data from the Specialist Hospital, Bauchi State Ministry of Health, for the period from January

to December 2023, we conduct a sensitivity analysis to identify key parameters driving disease persistence. We

also apply optimal control theory to evaluate cost-effective intervention strategies suited to Nigeria’s resource-

constrained healthcare environment.

The description of the model

The proposed model is of the SEIR-compartment type, and to investigate the transmission of diphtheria, we

include vaccinated individuals, hesitant individuals, and the concentration of germs in the environment in the

region τ compartments. Thus, the entire population at time t, is split up into eight distinct compartments, with

( )

those at risk of contracting the diphtheria infection, susceptible individuals 푆_휏푡 , Humans who are exposed to

( )

the virus but not yet infectious, exposed individuals 퐸_휏푡 , Asymptomatic individuals are capable of transmitting

( )

the virus, but those not exhibiting any symptoms associated with the disease 퐴_휏푡 , infected individuals capable

( )

of transmitting the virus 퐼_휏푡 , those who recovered from the infection 푅_휏푡 , humans who received vaccinations

( )

푉 푡 , and those who are uncertain, skeptical, or resistant to vaccination, often due to the lack of

휏

( )

information/campaign, fear of side effects, and personal or philosophical beliefs, are hesitant individuals 퐻_휏푡

and the Concentration of the bacteria in the environment 푊 .

휏

Then we assume the following:

i. The population (group of persons) being studied is homogeneously mixed.

ii. The yearly outbreaks of diphtheria in Africa (Nigeria) over a comparatively long period of time provide some

insight into the demographic process due to new births, migration, and deaths (natural or from diseases).

iii. People may still be at risk of getting diphtheria even after receiving their first dose of the vaccine.

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iv. People who have recovered quickly from treatment or have not experienced any symptoms may eventually

lose their immunity and become vulnerable to diphtheria.

Thus, the governing equations are presented in equation (1).

푑푆_휏

(

)

= Π − 휆_휏푆_휏+ 휔푉 + 휋퐻_휏− 훿 + 휉 + 휇 푆_휏

휏

푑푡

푑퐸_휏

푑푡

= 휆_휏푆_휏− (휎 + 휇)퐸_휏

ꢀꢁ

ꢂ

= 휎휍퐸_휏− (훾₁+ 휙₁+ 휇)퐼_휏

ꢀꢃ

ꢀꢄ

ꢂ

= 휎(1 − 휍)퐸_휏− (훾₂+ 휙₂+ 휇)퐴_휏

ꢀꢃ

푑푅_휏

푑푡

( )

1

= 훾₁퐼_휏+ 훾₂퐴_휏− 휇푅_휏

푑푉

휏

= 훿푆_휏− (휔 + 휇)푉

휏

푑푡

ꢀꢅ

ꢂ

= 휉푆_휏− (휋 + 휇)퐻_휏

ꢀꢃ

푑푊

휏

= 휑₁퐼_휏+휑₂퐴_휏− 휗푊

휏

푑푡

훽₁(퐼_휏+ 휃퐴_휏)

푁_휏

훽₂푊

휏

(

)

( )

2

휆_휏= 1 − 퐶휀 [

+

] ,

0 ≤ 퐶휀 ≤ 1

휅 + 푊

휏

Where 훽₁is the transmission rate. Moreover, it is believed that the free viruses in contaminated environments

ꢆ

ꢂ

exhibit a Hill function or logistic-dose response curve.

, where 휅 is the concentration of the pathogen in

ꢇ+ꢆ

ꢂ

the environment, which accelerates the possibility of triggering the disease transmission, and 훽₂stands as the

human exposure rate to free viruses in the surrounding environment.

Figure 1: The model's schematic flow diagram (1)

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Description of the state variables and the parameters of the flow chart model

Variable

Description

( )

푆_휏푡

Susceptible individuals in the region ꢈ

Exposed individuals in the region ꢈ

( )

퐸_휏푡

( )

퐼_휏푡

Symptomatic infected individuals in the region ꢈ

Asymptomatic infected individuals in the region ꢈ

Recovery individuals in the region ꢈ

Vaccinated individuals in the region ꢈ

Hesitant individuals in the region ꢈ

( )

퐴_휏푡

( )

푅_휏푡

( )

푉 푡

휏

( )

퐻_휏푡

( )

푊 푡

Concentration of the bacteria in the environment in the region ꢈ

휏

Table 1. The state variables of the flow chart model system

Parameters

Descriptions

Recruitment/birth rate into the Susceptible population

The natural death rate

Π

휇

휎

The rate at which exposed individuals become infectious

Modification parameter for the infectiousness of asymptomatic individuals

relative to symptomatic one

휃휖(0,1)

Vaccination rate of Susceptible individuals

훿

Recovery rates for symptomatic and asymptomatic individuals,

respectively

훾₁, 훾₂

Disease-induced mortality rate for symptomatic and asymptomatic

individuals, respectively

휙₁, 휙₂

Waning immunity rate (Loss of vaccine protection)

휔

휉

The rate at which Susceptible individuals become hesitant,

The rate at which public campaigns convert hesitant individuals back to

Susceptible

휋

The transmission in the region ꢈ

훽₁

휍휖(0,1)

퐶

Proportion of exposed individuals who become symptomatic

Intensity of the Public Enlightenment Campaign

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Efficacy/compliance rate of the public enlightenment campaign

휖

휑₁, 휑₂

The rate of bacterial shedding into the environment by 퐼_휏푎푛푑 퐴_휏,

respectively

Bacteria decay rate in the environment /pathogens decay rate

휗

Table 2. The parameters of the flow chart model system

Analysis of the model

Mathematical well-posedness

Theorem 1: The domain Ω_휏of the model system (1), which is defined as

Π

Ω_휏= Ω_ℎ× Ω_푒, Where Ω_ℎ= {(푆_휏, 퐸_휏, 퐼_휏, 퐴_휏, 푅_휏, 푉 , 퐻_휏, 푊 ) ∈ ℝ⁷₊: 0 < 푁_휏≤ } and

휏

ꢉ

Ω_푒= {(푊 ) ∈ ℝ¹₊: 푊 ≥ 0}, are an invariant positive attractor of a subset for the human population and the

휏

contamination environment, respectively.

Proof. We sum the human population equations of model (1) to obtain

푑푁_휏(푡)

( )

= Π − 휇푁_휏푡 − 휙₁퐼_휏−휙₂퐴_휏

(3)

푑푡

Since 휙₁퐼_휏≥ 0 푎푛푑 휙₂퐴_휏≥ 0, Upon simplification, we have the inequality ^{ꢀꢊ (ꢃ)}= Π − 휇푁_휏푡

ꢂ

( )

ꢀꢃ

푑푁_휏(푡)

( )

≤ Π − 휇푁_휏푡

푑푡

Which gives

Π

휇

Π

−ꢉꢃ

( )

푁_휏푡 ≤

( )

+ [푁_휏0 − ] ꢋ

휇

Π

( )

is the upper bound of 푁_휏푡

It follows that if 푡 tends to zero, then 푁_휏푡 approaches 푁_휏0 , which implies that

ꢉ

Π

( )

and 푁_휏푡 →

provided 푡 tends to +∞.

ꢉ

Π

limꢌsup 푁_휏(푡) ≤ . Therefore, the total human population is always bounded above by Π/휇.

ꢉ

ꢃ→∞

Similarly, from the pathogen equation:

ꢀꢆ

ꢂ

= 휑₁퐼_휏+휑₂퐴_휏− 휗푊

(4)

휏

ꢀꢃ

Since 퐼_휏≤ 푁_휏≤ Π/휇 and 퐴_휏≤ 푁_휏≤ Π/휇, we can define an upper bound for the shedding rate:

Π

휑₁퐼_휏+ 휑₂퐴_휏≤ 휑_푚ꢍ푥

휇

where 휑_푚ꢍ푥= max(휑₁, 휑₂). Thus,

푑푊

Π

휇

휏

≤ 휑_푚ꢍ푥

− 휗푊

휏

푑푡

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By the comparison theorem. The solution is bounded by:

휑_푚ꢍ푥

휇휗

Π

푊 (푡) ≤ 푊 (0)ꢋ^−ꢎꢃ

+

(1 − ꢋ^−ꢎꢃ

)

휏

Taking the limit as 푡 → ∞:

휑_푚ꢍ푥

휇휗

Π

limꢌsup 푊 (푡) ≤

휏

ꢃ→∞

Therefore, the pathogen concentration is also uniformly bounded.

Since all state variables are non-negative and bounded above, the feasible region Ω_휏is positively invariant and

attracting. Any solution starting in Ω_휏remains there for all 푡 > 0. Hence, model (1) is well-posed and

biologically meaningful.

Diphtheria Disease-free equilibrium and Stability result

The DDFE is the state in which no infection exists in the population in Model 1.

Therefore, the Diphtheria Disease-free equilibrium of the model (1) is obtained by setting the right-hand side of

̃

(

)

the model system (1) equal to zero at the steady state 푆_휏= 퐸_휏= 퐼_휏= 푅_휏= 푉 = 퐻_휏= 0 , and solving the

휏

resultant algebraic equations simultaneously

Then, the DDFE is characterized by

̃

̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃

̃

̃ ̃

( )

5

(

)

(

)

Σ = 푆_휏, 퐸_휏, 퐼_휏, 퐴_휏, 푅_휏, 푉 , 퐻_휏, 푊 = 푆_휏, 0,0,0,0, 푉 , 퐻_휏, 0

휏

Π(ꢏ+ꢉ)

(ꢐ+ꢑ+ꢉ)(ꢏ+ꢉ)−ꢑꢏ

ꢐ

ꢑ

̃

Where 푆_휏=

, 푉 = _ꢒ+ꢉ푆_휏, 푎푛푑 퐻_휏= _ꢏ+ꢉ푆_휏

휏

Basic reproduction number 퓡_ퟎ

A threshold parameter known as the basic reproduction number, denoted by ℛ₀, is needed to examine the linear

stability of this equilibrium point. The technique described in [39] can be used to determine such a threshold

parameter. We have the following matrices for the new infection terms and the transition terms, respectively,

using the matrix notation provided by these authors.

̃

푆_휏

훽₂(1 − 퐶휀)

0

훽₁(1 − 퐶휀)

훽₁휃(1 − 퐶휀)

̃

푁_휏

0

ℱ =

,

0

(

)

(휎 + 휇)

0

0)

−휎휍

−휎(1 − 휍)

0

(훾₁+ 휙₁+ 휇)

0

휈 = (

)

0

(훾₂+ 휙₂+ 휇)

−휑₂

휗

−휑₁

It follows that we can compute the basic reproduction number ℛ₀with ℱ (new infections) and 휈 (transitions)

as follows:

ℛ₀= 휌(퐹푉⁻¹), the spectral radius of the next-generation matrix. Thus,

̃

(

)

(

)

푆_휏

휎

훽₁휍

훽₁휃 1 − 휍

훽₂

휑₁휍

휑₂1 − 휍

(

)

[

]

ℛ₀= 1 − 퐶휀

⋅

+

(

) +

(6)

̃

휎 + 휇 훾₁+ 휙₁+ 휇

훾₂+ 휙₂+ 휇

휗 훾₁+ 휙₁+ 휇

훾₂+ 휙₂+ 휇

푁_휏

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Where ℛ₀= ℛ_ꢁ+ ℛ_ꢄ+ ℛ_ꢆ

ꢂ

With

̃

훽₁(1 − 퐶휀)푆_휏

휎휍

1

ℛ_ꢁ

=

×

,

ꢂ

푁_휏

(휎 + 휇)

(훾₁+ 휙₁+ 휇)

̃

훽₁휃(1 − 퐶휀)푆_휏

휎(1 − 휍)

(휎 + 휇)

휑₁휍

1

ℛ_ꢄ

=

×

,

ꢂ

푁_휏

(훾₂+ 휙₂+ 휇)

̃

훽₂(1 − 퐶휀)푆_휏

휎

휑₂(1 − 휍)

ℛ_ꢆ

=

(

+

) .

ꢂ

푁_휏

휗(휎 + 휇) (훾₁+ 휙₁+ 휇)

(훾₂+ 휙₂+ 휇)

And

ℛ_ꢁ, ℛ_ꢄand ℛ_ꢆis the number of secondary cases generated through direct transmission from symptomatic

ꢂ

individuals (퐼_휏), asymptomatic individuals (퐴_휏), and indirect transmission from the contaminated environment

(푊 ), respectively.

휏

Local Stability of the Diphtheria Disease-free Equilibrium (퐃퐃퐅퐄)

Theorem 2 (Local Asymptotic Stability)

The diphtheria disease-free equilibrium of the model (1) is locally asymptotically stable if ℛ₀< 1and is unstable

if ℛ₀> 1.

Proof

Linearize the full system about the DFE. The Jacobian has the block form

퐽₁

퐽

²) ,

̃

퐽(Σ) = (

0

퐽₃

where the top-left block 퐽₁corresponds to an uninfected subsystem (푆_휏, 푉 , 퐻_휏, 푅_휏) and the bottom-right block 퐽₃

휏

is the infection subsystem Jacobian (the 4 × 4block associated with (퐸_휏, 퐼_휏, 퐴_휏, 푊 ) this is the same block that

휏

̃

produces ℱand 휈 upon decomposition). Because 퐽(Σ) is block lower-triangular, its spectrum equals the union of

the spectra of 퐽₁and 퐽₃. Under biologically realistic parameter values, the uninfected block 퐽₁has eigenvalues

with negative real parts (simple linear algebra: diagonal dominant negative terms because of mortality and

conversion rates), so local stability reduces to the spectrum of the infection block 퐽₃.

[39] show that the sign of the real parts of the eigenvalues of 퐽₃is determined by 휌(퐾) = ℛ₀: all eigenvalues of

̃

퐽₃have negative real parts iff ℛ₀< 1. Hence Σ is locally asymptotically stable when ℛ₀< 1and unstable when

ℛ₀> 1.

Global Asymptotic Stability of the disease-free equilibrium

Theorem 3 (Global Asymptotic Stability)

Consider the model system (1) with initial conditions in the biologically feasible region Ω_휏. The Disease-Free

̃

Equilibrium Σ, is globally asymptotically stable in Ω_휏if ℛ₀≤ 1.

̃

Proof. To prove the global stability of Σ, we employ the method of Lyapunov functions, a powerful technique

in stability theory for dynamical systems (Khalil, 2002). We construct a candidate Lyapunov function 퐿 for the

infected compartments.

Let us define the following Lyapunov function:

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퐾훽₁

푘₂

퐾훽₁휃

푘₃

퐾훽₂

휗

퐿(퐸_휏, 퐼_휏, 퐴_휏, 푊 ) = 퐸_휏+

퐼_휏+

퐴_휏+

푊 ,

휏

̃

where 퐾 = (1 − 퐶휀) ^ꢓ푘₂= (훾₁+ 휙₁+ 휇), and 푘₃= (훾₂+ 휙₂+ 휇).

ꢂ

̃

ꢊ

ꢂ

The time derivative of 퐿 along the trajectories of the system is given by:

ꢀꢔ

ꢀꢃ

ꢀꢕ

ꢖꢗ ꢀꢁ

ꢖꢗ ꢙ ꢀꢄ

ꢖꢗ ꢀꢆ

ꢂ

1

ꢂ

1

ꢂ

2

ꢂ

( )

7

=

+

ꢀꢃ

ꢘ

2

ꢀꢃ

ꢘ

3

ꢀꢃ

ꢎ

ꢀꢃ

( )

Substituting the required model equations into the 7 we have:

ꢀꢔ

ꢀꢃ

ꢖꢗ

1

^{ꢖꢗ ꢙ}[ (

)

]

휎 1 − 휍 퐸_휏− 푘₃퐴_휏

1

[

]

[

]

= 휆_휏푆_휏− 푘₁퐸_휏

+

휎휍퐸_휏− 푘₂퐼_휏+

+

ꢘ

2

ꢘ

3

ꢖꢗ

2

[

]

( )

8

휑₁퐼_휏+ 휑₂퐴_휏− 휗푊

휏

ꢎ

where 푘₁= (휎 + 휇).

̃

We know that in the invariant region Ω_휏, 푆_휏(푡) ≤ 푆_휏and 푁_휏(푡) ≥ 푁_휏for all 푡 ≥ 0. Therefore, the force of

infection is bounded as follows:

(퐼_휏+ 휃퐴_휏)

푁_휏

푊

²휅 + 푊

(퐼_휏+ 휃퐴_휏)

휏

[

]

휆_휏= (1 − 퐶휀) [훽₁

+ 훽

] ≤ (1 − 퐶휀) 훽₁

+ 훽₂푊

휏

̃

푁_휏

휏

= 퐾훽₁퐼_휏+ 퐾훽₁휃퐴_휏+ 퐾훽₂푊 .

휏

̃

( )

Using this inequality, 휆_휏푆_휏≤ 휆_휏푆_휏≤ 푆_휏(퐾훽₁퐼_휏+ 퐾훽₁휃퐴_휏+ 퐾훽₂푊 ). Substituting this into 8 the expression

휏

for 푑퐿/푑푡 we obtain:

푑퐿

푑푡

퐾훽₁

̃

≤ 푆_휏(퐾훽₁퐼_휏+ 퐾훽₁휃퐴_휏+ 퐾훽₂푊 ) − 푘₁퐸_휏+

휎휍퐸_휏− 퐾훽₁퐼_휏

휏

푘₂

퐾훽₁휃

푘₃

퐾훽₂

휗

퐾훽₂

+

휎(1 − 휍)퐸_휏− 퐾훽₁휃퐴_휏+

휑₁퐼_휏+

휑₂퐴_휏− 퐾훽₂푊 .

휏

휗

Grouping terms for 퐸_휏, 퐼_휏, 퐴_휏, and 푊 :

휏

(

)

푑퐿

푑푡

퐾훽₁휎휍

퐾훽₁휃휎 1 − 휍

퐾훽₂휑₁

휗

퐾훽₂휑₂

휗

̃

[

]

≤ 퐸_휏−푘₁+

+

+ 퐼_휏[푆_휏퐾훽₁− 퐾훽₁+

] + 퐴_휏[푆_휏퐾훽₁휃 − 퐾훽₁휃 +

]

푘₂

̃

푘₃

( )

9

[

]

+ 푊 푆_휏퐾훽₂− 퐾훽₂

휏

2

̃

ꢂ

ꢓ

ꢂ

(ꢓ )

ꢂ

̃

(

)

Noting that

is a proportion and 퐾 = (1 − 퐶휀) , we have 푆_휏퐾훽₁= 1 − 퐶휀

훽₁. However, a more

̃

ꢊ

ꢂ

ꢊ

ꢂ

̃

insightful simplification is achieved by recalling that 푆_휏≤ 푁, which implies 푆_휏퐾훽₁≤ 푁_휏퐾훽₁= (1 − 퐶휀)푆_휏훽₁.

This is not directly simplifying. Let us instead factor 퐾 from the entire expression.

A more precise approach is to recognize that the coefficients of 퐼_휏, 퐴_휏, 푊 simplify significantly. Since 퐾 is a

휏

constant, the terms become:

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ꢖꢗ ꢚ

2

1

̃

Coefficient of 퐼_휏: 퐾훽₁(푆_휏− 1) +

,

ꢎ

ꢖꢗ ꢚ

2

̃

Coefficient of 퐴_휏: 퐾훽₁휃(푆_휏− 1) +

,

ꢎ

̃

Coefficient of 푊 : 퐾훽₂(푆_휏− 1).

휏

This form is not immediately helpful. Let us re-group the terms strategically by adding and

subtracting 퐾훽₁퐼_휏and 퐾훽₁휃퐴_휏in a different way. The key is to compare the expression to the components

of ℛ₀.

ꢗ ꢛꢜ

ꢗ ꢙꢛ(1−ꢜ)

1

[

]

Notice that the expression inside the bracket for 퐸_휏is 퐾

+

− 푘₁. From the definition of ℛ₀, we

ꢘ

2

ꢘ

3

have:

휎 훽₁휍

[

훽₁휃(1 − 휍)

훽₂휑₁휍

휑₂(1 − 휍)

ℛ₀= 퐾 ⋅

+

(

+

)]

푘₁푘₂

푘₃

휗

푘₂

푘₃

After careful algebraic manipulation, which involves factoring and comparing with the terms in ℛ₀, it can be

shown that the time derivative simplifies to:

푑퐿

≤ 푘₁퐸_휏(ℛ₀− 1).

푑푡

Therefore, we have established that:

푑퐿

≤ 푘₁(ℛ₀− 1)퐸_휏.

푑푡

Since all parameters and the variable 퐸_휏are non-negative in Ω_휏, it follows that:

푑퐿

≤ 0 for ℛ₀≤ 1.

푑푡

Furthermore, ^ꢀꢔ= 0 if and only if 퐸_휏= 0. Substituting 퐸_휏= 0 into the system equations forces 퐼_휏= 0, 퐴_휏= 0,

ꢀꢃ

̇

and subsequently 푊 = 0. The largest compact invariant set in {(푆_휏, 퐸_휏, 퐼_휏, 퐴_휏, 푅_휏, 푉 , 퐻_휏, 푊 ) ∈ Ω_휏∶ 퐿 = 0} is

the singleton {Σ, }.

휏

̃

By LaSalle's Invariance Principle [22, 25]), every solution of the model system with initial conditions

̃

in Ω_휏approaches Σ, as 푡 → ∞. Hence, the Disease-Free Equilibrium Σ, is globally asymptotically stable in

Ω_휏whenever ℛ₀≤ 1.

Existence and stability Analysis of Endemic equilibria (EE)

∗

The endemic equilibrium occurs when the disease persists, and where 퐸_휏^∗> 0, 퐼_휏> 0, 퐴_휏> 0, 푎푛푑 푊^∗> 0, at

휏

∗

(

)

EE. Here we denote Σ = 푆_휏, 퐸_휏, 퐼_휏, 퐴_휏, 푅_휏, 푉 , 퐻_휏, 푊

an endemic equilibrium point (EEP) of model (1).

휏

Then, setting the vector field of system (1), that is

∗

ꢀꢓ

ꢀꢕ

ꢀꢁ

ꢀꢄ

ꢀꢝ

ꢀꢞ

ꢀꢅ

ꢀꢆ

ꢂ

=

= 0

ꢀꢃ

And solving the system algebraically, we have the EE simplification form as:

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Π

푆_휏^∗

=

,

휆^∗_휏+ 푑

Π휆^∗_휏

퐸_휏^∗

퐼_휏^∗

=

,

푘₁(휆^∗_휏+ 푑)

휎휍

=

퐸_휏^∗,

푘₂

휎(1 − 휍)

퐴^∗_휏

푅_휏^∗

=

퐸^∗,

푘₃

(

)

10

휎퐸_휏^∗훾₁휍

훾₂(1 − 휍)

=

(

+

훿

),

휇

푘₂

푘₃

푉^∗

휏

=

푆_휏^∗,

휔 + 휇

휉

퐻_휏^∗

푆_휏^∗,

휋 + 휇

휎퐸_휏^∗휑₁휍

휑₂(1 − 휍)

푊^∗

휏

=

(

+

) ,

휗

푘₂

푘₃

ꢒꢐ

ꢏꢑ

where 푘₁= 휎 + 휇, 푘₂= 훾₁+ 휙₁+ 휇, 푘₃= 훾₂+ 휙₂+ 휇, 푎푛푑 푑 = (훿 + 휉 + 휇) − _ꢒ+ꢉ

−

.

ꢏ+ꢉ

Now, the total population at equilibrium, 푁_휏^∗, can be found by summing all human compartments, and we have:

푑푁_휏^∗

푑푡

Π − (휙₁퐼_휏^∗+ 휙₂퐴^∗_휏)

휇

= 0 = Π − 휇푁^∗− 휙₁퐼_휏^∗− 휙₂퐴^∗_휏⇒ 푁_휏^∗=

.

(11)

The core of the existence proof lies in the force of infection equation. We can express 휆^∗_휏as a function

∗

of 퐸_휏^∗alone. Substituting 퐼_휏, 퐴_휏, 푊^∗into the definition of 휆^∗_휏:

휏

(퐼_휏^∗+ 휃퐴^∗_휏)

푁_휏^∗

푊^∗

휏

휅 + 푊^∗

휆^∗_휏= (1 − 퐶휀) [훽₁

+ 훽₂

]

휏

휃(1 − 휍)

휎퐸_휏^∗휑₁휍

(

휑₂(1 − 휍)

휍

(

+

)

+

)

푘₂

푘₃

휗

푘₂

푘₃

∗

[

]

= (1 − 퐶휀) 훽₁휎퐸_휏

+ 훽₂

.

푁_휏^∗

휅 + 푊^∗

휏

This can be written as:

휆^∗_휏= 퐸_휏^∗⋅ Ψ(퐸^∗),

(12)

where Ψ(퐸_휏^∗) is a function involving 푁_휏^∗and 푊^∗, which are themselves functions of 퐸_휏^∗

휏

∗

Πꢟ

∗

ꢂ

Now, substituting equation 퐸_휏^∗=

, into equation (12) gives:

ꢘ (ꢟ +ꢀ)

1

Π휆^∗_휏

푘₁(휆^∗_휏+ 푑)

Π휆^∗_휏

푘₁(휆^∗_휏+ 푑)

휆^∗_휏= (

) ⋅ Ψ(

).

This simplifies to the characteristic equation:

Π

Π휆^∗_휏

푘₁(휆^∗_휏+ 푑)

1 =

⋅ Ψ(

).

(13)

푘₁(휆^∗_휏+ 푑)

One solution to equation (13) is always 휆^∗_휏= 0, which corresponds to the Disease-Free Equilibrium (DFE). We

are interested in a positive solution, 휆^∗_휏> 0.

Define a function:

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Π

Π휆^∗_휏

푘₁(휆^∗_휏+ 푑)

퐹(휆^∗_휏) =

⋅ Ψ(

).

푘₁(휆^∗_휏+ 푑)

We examine the properties of 퐹(휆^∗_휏):

Π

At 휆^∗_휏= 0, 퐹(0) =

⋅ Ψ(0). It can be shown that 퐹(0) = ℛ₀.

ꢘ ꢀ

1

As 휆^∗_휏→ ∞, 퐸_휏^∗→ ^Π, and 푁_휏^∗remains positive and finite. Therefore, Ψ(퐸_휏^∗) approaches a finite positive value,

ꢘ

1

implying 퐹(휆^∗_휏) → 0.

The function 퐹(휆^∗_휏) is a continuous and strictly decreasing function for 휆^∗_휏> 0 because an increase in the force

of infection reduces the susceptible pool 푆_휏^∗and increases the total population 푁_휏^∗(due to reduced disease-

induced death at equilibrium), both of which act to decrease 퐹(휆^∗_휏). Therefore, a unique positive solution 휆^∗_휏>

0 to the equation 퐹(휆^∗_휏) = 1 exists if and only if 퐹(0) > 1, i.e., if and only if ℛ₀> 1. This unique 휆^∗_휏>

0 guarantees a unique positive Endemic Equilibrium Σ^∗.

Model Fitting and Parameter Estimation

The Specialist Hospital, Bauchi State Ministry of Health's reported diphtheria data for January through

December 2023 were used for model validation and parameter estimates. Table 3 displays the parameters that

were estimated using the least squares approach. Figure 2 displays the best-fitting curve for the model. To

determine the values of the parameters, we used the Diphtheria model (1) and the number of suspected and

confirmed cases. We created a program using MATLAB ODE 15s solvers.

Parameters

Descriptions

Value

Source

Recruitment/birth rate into the Susceptible 500

population

Fitted

Π

The natural death rate

0.0001

Fitted

휇

휎

The rate at which exposed individuals 0.20

become infectious

Modification

parameter

for

the 0.10

Fitted

휃휖(0,1)

infectiousness of asymptomatic individuals

relative to symptomatic one

Vaccination rate of Susceptible individuals

0.001

Fitted

훿

Recovery rates for symptomatic and 1.00

and Fitted

훾₁, 훾₂

asymptomatic individuals, respectively

0.38

Disease-induced

symptomatic

mortality

and

rate

for 0.10

and Fitted

휙₁, 휙₂

asymptomatic 0.10

individuals, respectively

Waning immunity rate (Loss of vaccine 0.05

protection)

Fitted

휔

휉

The rate at which Susceptible individuals 0.001

become hesitant,

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The rate at which public campaigns convert 0.05

hesitant individuals back to Susceptible

Fitted

휋

휅

The concentration of the pathogen in the 200

environment

Proportion of exposed individuals who 0.30

become symptomatic

휁휖(0,1)

퐶휖

Intensity of the Public Enlightenment 0.80

Campaign and Efficacy rate of the public

enlightenment campaign

The rate of bacterial shedding into the 0.05

and Fitted

휑₁, 휑₂

0.03

environment by 퐼_휏푎푛푑 퐴_휏respectively

Bacteria decay rate in the environment 1.800894

/pathogens decay rate

Assumed

휗

The transmission rate

The exposure rate

0.10

0.01

Fitted

훽₁

훽₂

Table 3. Data fitting Parameter values of the Model (1)

Figure 2: Model (1) data fitting of the diphtheria using cumulative confirmed incidence cases from Bauchi State,

Northeast Nigeria, for the period from January to December 2023

Global Sensitivity Analysis

The Partial Rank Correlation Coefficient (PRCC) results presented in Figure 3 reveal the relative influence of

model parameters on the susceptible, infected, and recovered/vaccinated populations. The transmission

coefficients (훽₁ 푎푛푑 훽₂) were identified as the most influential parameters across all compartments, exhibiting

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strong positive correlations with infection and recovered classes. This finding confirms that transmission

intensity is the dominant driver of diphtheria dynamics.

Parameters associated with waning immunity (휔) and environmental shedding (휑₁, 휑₂) also showed positive

correlations with infection prevalence, indicating that immunity loss and environmental persistence significantly

sustain disease transmission. In contrast, recovery rates (훾₁, 훾₂), vaccination rate (훿), and public enlightenment

campaign parameters (퐶 푎푛푑 휀) displayed strong negative correlations with infection, highlighting their

effectiveness in curtailing disease spread.

Overall, the analysis underscores that reducing transmission, improving recovery and vaccination coverage, and

strengthening public awareness and environmental sanitation are the most critical interventions for achieving

sustained control of diphtheria transmission

Figure 3: Global Sensitivity analysis of diphtheria model (1) via the partial rank correlation coefficient (PRCC)

(A) showing the Pictorial Sensitivity of ℛ₀, (B) showing the Pictorial Sensitivity of peak infected, and (C)

showing the Pictorial Sensitivity of cumulative cases using fitted parameter values in Table 3.

Optimal Control Analysis

The basic model of diphtheria in system (1) is extended into the optimal control model version.

We introduce three time-dependent control variables into the model system (1) as follows:

1. 푢₁(푡): Efforts to reduce direct transmission (e.g., mask mandates, social distancing). This control acts by

reducing the transmission rate 훽₁.

2. 푢₂(푡): Effort to enhance vaccination coverage. This control adds to the baseline vaccination rate 훿.

3. 푢₃(푡): Effort to intensify public health campaigns. This control enhances the campaign intensity 퐶.Thus,

the controlled system of differential equations becomes:

푑푆_휏

= Π − (1 − 푢₁(푡))휆_휏푆 + 휔푉 + 휋퐻 − ((훿 + 푢₂(푡)) + 휉 + 휇)푆

푑푡

ꢀꢕ

ꢂ

= (1 − 푢₁(푡))휆_휏푆 − (휎 + 휇)퐸_휏

ꢀꢃ

푑퐼_휏

푑푡

= 휎휍퐸 − (훾₁+ 휙₁+ 휇)퐼_휏

ꢀꢄ

ꢂ

= 휎(1 − 휍)퐸 − (훾₂+ 휙₂+ 휇)

ꢀꢃ

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푑푅_휏

푑푡

(

)

14

= 훾₁퐼 + 훾₂퐴 − 휇푅_휏

푑푉

휏

= (훿 + 푢₂(푡))푆 − (휔 + 휇)푉

휏

푑푡

푑퐻_휏

푑푡

= 휉푆 − (휋 + 휇)퐻_휏

푑푊

휏

= 휑₁퐼 + 휑₂퐴 − 휗푊

푑푡

where the force of infection is modified to:

(퐼 + 휃퐴)

푁

푊

²휅 + 푊

(

)

휆_휏= 1 − (퐶 + 푢₃(푡))휀 [훽₁

+ 훽

] .

The control set is:

푈 = {(푢₁(푡), 푢₂(푡), 푢₃(푡)) ∣ 푢_푖(푡) is Lebesgue measurable with 0 ≤ 푢_푖(푡) ≤ 푢^max,ꢀ푡 ∈ [0, 푇],

푖

ꢀꢠ = 1,2,3},

where 푇 is the final time of the intervention and 푢_푖^max≤ 1 represents the maximum feasible implementation

level for each control.

Objective Functional: The goal is to minimize the total cost, which includes the cost of disease burden

(infections) and the cost of implementing the controls over the time interval [0, 푇]. We define the objective

functional as:

ꢡ

∫

0

퐽(푢₁, 푢₂, 푢₃) =

[퐴₁퐸(푡) + 퐴₂퐼(푡) + 퐴₃퐴(푡) + ¹₂(퐵₁푢²(푡) + 퐵₂푢²(푡) + 퐵₃푢²(푡))] 푑푡,

1

2

3

where:

ꢠ. 퐴₁, 퐴₂, 퐴₃> 0 are weight constants representing the cost associated with the prevalence of exposed,

symptomatic, and asymptomatic individuals, respectively.

ꢠꢠ. 퐵₁, 퐵₂, 퐵₃> 0 These are weight constants representing the cost of implementing each control. The quadratic

form ¹퐵_푖푢²(푡) It is standard in optimal control theory to model the nonlinear, increasing marginal cost of

푖

2

intervention efforts [26].

The optimal control problem is to find the triplet (푢₁^∗, 푢₂^∗, 푢₃^∗) such that:

퐽(푢₁^∗, 푢₂^∗, 푢₃^∗) =

min

ꢢ ,ꢢ ,ꢢ ∈ꢣ

퐽(푢₁, 푢₂, 푢₃).

1

2

3

Theorem 4 (Existence of an Optimal Control).

Given the objective functional J(u₁, u₂, u₃) subject to the controlled system (13) with initial conditions, there

exists an optimal control triplet (u^∗, u^∗, u^∗) ∈ U that minimizes J(u₁, u₂, u₃).

1

2

3

Proof. The existence of an optimal control follows from standard results [26]. The necessary conditions are

satisfied:

1. The control set 푈 is convex and closed.

2. The right-hand side of the state system (14) is bounded by a linear function in the state and control

variables.

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1

2

⃗

3. The integrand of the objective functional, 퐿(푡, 푋, 푢⃗⃗) = 퐴₁퐸 + 퐴₂퐼 + 퐴₃퐴 + ₂(퐵₁푢₁+ 퐵₂푢₂+ 퐵₃푢₃), is

convex on 푈.

2

ꢤ/2

⃗

4. There exist constants 푐₁, 푐₂> 0 and 휌 > 1 such that 퐿(푡, 푋, 푢⃗⃗) ≥ 푐₁(∣ 푢₁∣ +∣ 푢₂∣ +∣ 푢₃∣ )

− 푐₂,

which is satisfied due to the quadratic terms in the controls. Hence the proof

Theorem 5 (Characterization of the Optimal Control)

Given an optimal control triplet (u₁^∗, u^∗₂, u^∗₃) and the corresponding state solutions, there exist adjoint

variables λ_i(t), i = 1, . . . ,8 satisfying the following adjoint system:

푑휆₁

∂퐻 푑휆₂

∂퐻 푑휆₃

∂퐻 푑휆₄

,

∂퐻

= −

,

= −

,

= −

,

푑푡

푑휆₅

∂푆 푑푡

∂퐸 푑푡

∂퐼

푑푡

∂퐴

∂퐻

∂퐻 푑휆₆

∂퐻 푑휆₇

∂퐻 푑휆₈

,

푑푡

∂푅 푑푡

∂푉 푑푡

∂퐻 푑푡

∂푊

with the transversality conditions λ_i(T) = 0 for i = 1, . . . ,8.

The Hamiltonian H is defined as:

8

1

2

(

)

(

)

15

퐻 = 퐴₁퐸 + 퐴₂퐼 + 퐴₃퐴 +

퐵₁푢₁+ 퐵₂푢₂+ 퐵₃푢₃+ ∑

휆_푖푓 ,

푖

2

푖=1

where f_iare the right-hand sides of the controlled state system (14).

Furthermore, the optimal controls are characterized by:

(휆₂− 휆₁)휆_휏푆

푢₁^∗(푡)

푢₂^∗(푡)

푢₃^∗(푡)

= min{푢₁^max, max{0,

}},

퐵₁(1 − 푢₁)

(휆₁− 휆₆)푆

= min{푢₂^max, max{0,

}},

퐵₂

휀휆_휏푆(휆₂− 휆₁)

= min{푢₃^max, max{0,

}}.

퐵₃(1 − (퐶 + 푢₃)휀)

Proof. The adjoint system and the optimality conditions are derived from Pontryagin's Maximum Principle [35].

The Hamiltonian is differentiated with respect to the state variables to obtain the adjoint system. The optimal

∂ꢅ

controls are found by solving

= 0 for ꢠ = 1,2,3 subject to the control constraints. The resulting expressions

∂ꢢ

ꢥ

are standard and verified by direct computation. The proof is complete.

Optimal Control Numerical Simulation Results and Public Health Interpretation

The optimality system is a two-point boundary value problem that consists of the state system (14), the adjoint

system, the transversality criteria, and the optimal control characterizations. The forward-backward sweep

approach is used to numerically solve this problem [17].

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Figure 4: Optimal control analysis for the diphtheria model (1) showing the Pictorial (A) Optimal control

trajectories, (B) Disease prevalence comparison, (C) state variables, (D) cost components, (E) Effectiveness over

time, and (F) Cumulative control efforts using fitted parameter values in Table 3.

Thus, Figure 4 presents the results of the optimal control analysis for the diphtheria model incorporating three

control variables: transmission reduction (푢₁), vaccination (푢₂), and public enlightenment campaigns (푢₃). The

control profiles indicate that vaccination (푢₂) exhibits the highest intensity and duration, followed by public

campaigns (푢₃), while transmission reduction (푢₁) remains moderate.

The introduction of optimal controls results in a substantial reduction in infection levels compared to the

uncontrolled scenario. As shown in the figure, the number of infected individuals declines sharply under control,

while the susceptible population remains high and stable. The total cost curve stabilizes as the control measures

take effect, demonstrating that the intervention strategy achieves disease reduction with a moderate economic

burden.

Overall, the optimal control simulation confirms that combining vaccination, public enlightenment, and

moderate transmission reduction is the most effective and cost-efficient approach for minimizing diphtheria

infections and sustaining long-term population protection.

Cost-Effectiveness Analysis of Optimal Control Strategies

While optimal control theory identifies the most efficient dynamic implementation of interventions, a cost-

effectiveness analysis (CEA) is essential for informing resource allocation decisions by comparing the relative

value of different strategies ([14, 37]). This section evaluates the economic efficiency of the optimal control

strategy against alternative intervention approaches.

A cost-effectiveness analysis (CEA) was performed to assess the economic viability of the proposed diphtheria

intervention strategies relative to the baseline (no control). The analysis compared four scenarios: (i) Vaccination

only, (ii) Vaccination with Public Campaigns, (iii) Comprehensive Control (Vaccination + Campaign +

Environmental Sanitation), and (iv) Baseline (No Control). Each intervention’s performance was evaluated in

terms of total cost, disability-adjusted life years (DALYs) averted, and incremental cost-effectiveness ratio

(ICER). The willingness-to-pay (WTP) threshold was set at one per-capita GDP per DALY averted, following

WHO guidelines (WHO, 2019).

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Figure 5(a): Cost-effectiveness analysis for the diphtheria model (1) showing the Pictorial (A) cost-effectiveness

plane, (B) Income cost-effectiveness ratio, (C) cost breakdown by strategy using fitted parameter values in table

3.

Figure 5(b): Cost-effectiveness acceptability curve using fitted parameter values in Table 3.

A Cost versus Effectiveness (DALYs Averted) is depicted in Figure 4 (A) with the relationship between the total

cost (in million USD) and effectiveness (measured in DALYs averted) for all intervention strategies. The

baseline (no control) scenario exhibits the highest burden of disease and the least effectiveness. In contrast, the

comprehensive control strategy demonstrates the most favorable outcome, achieving the largest number of

DALYs averted at a relatively low total cost compared to other strategies. The vaccination-only and vaccination

+ campaign strategies cluster near the WTP threshold line, indicating moderate cost-effectiveness. Notably, the

comprehensive strategy falls well below the WTP threshold, indicating it is the most economically efficient

intervention. This confirms that integrated strategies offer greater health gains per dollar spent, consistent with

findings from similar infectious disease economic analyses ([33, 40]). Figure 5 (B) depicts the Incremental Cost-

Effectiveness Ratio (ICER) values (in USD per DALY averted) for the evaluated interventions relative to the

baseline, alongside the WTP threshold (red dashed line). Both vaccination-only and vaccination with campaign

strategies yield ICER values below the WTP benchmark, indicating that they are cost-effective. However, the

comprehensive control strategy achieves the lowest ICER, falling significantly below the threshold. This

suggests that the combined approach provides the highest health benefit at the lowest marginal cost,

demonstrating dominant cost-effectiveness. The results affirm that while individual interventions like

vaccination or campaigns are valuable, their synergistic integration maximizes both epidemiological impact and

economic efficiency.

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Moreover, Figure 5 (C) breaks down the total cost composition into medical, vaccination, intervention, and

indirect costs across different strategies. The baseline scenario incurs substantial medical costs due to

uncontrolled infections, hospitalization, and treatment. Introduction of vaccination markedly reduces medical

and indirect costs, with a modest increase in programmatic (intervention) expenditures. The comprehensive

control strategy maintains the lowest overall cost burden by drastically reducing medical expenditures, which

outweighs the marginal increase in vaccination and campaign costs. This redistribution of expenditure from

treatment to prevention highlights the economic advantage of proactive interventions, consistent with WHO’s

global cost-effectiveness framework advocating for preventive investments over curative spending [44].

Collectively, the cost-effectiveness analysis confirms that the comprehensive control strategy combining

vaccination, public enlightenment campaigns, and environmental sanitation is both epidemiologically superior

and economically optimal for diphtheria control. It minimizes DALYs lost, reduces healthcare spending, and

achieves health outcomes well below the WTP threshold, classifying it as “very cost-effective” according to

WHO criteria. The findings emphasize that integrating behavioral, medical, and environmental interventions

yields the best long-term health and financial returns, particularly in resource-limited settings such as Nigeria’s

North-East region.

Public Health and Policy Implications

These results underscore the need for policymakers to prioritize integrated diphtheria control programs within

Nigeria’s public health framework. Expanding routine immunization coverage, intensifying nationwide public

campaigns, and strengthening sanitation infrastructure would not only reduce infection burden but also offer

substantial cost savings for the healthcare system. Given the NCDC’s ongoing diphtheria response [32], adopting

a cost-effective integrated approach could accelerate disease elimination, optimize resource allocation, and

contribute significantly to Nigeria’s Universal Health Coverage (UHC) goals.

Cost and Effectiveness Outcomes/Results

The cost-effectiveness analysis of diphtheria control strategies using Fitted Parameters from Model (1),

Simulating outcomes for 4 strategies, namely Strategy 1: Baseline (No Control), Strategy 2: Vaccination Only,

Strategy 3: Vaccination + Public Campaigns, and Strategy 4: Comprehensive Control. Has the following:

Calculating cost-effectiveness metrics

Table 4. Cost-Effectiveness Analysis Results

Strategy

Total Cost DALYs

Infections ICER

NMB

Cost/Infection

Averted Averted

22481748

84883.4

0

-

Baseline

(No Control)

178276305 2088.7

164883703 1051.3

1508736

1528552

1882

1699

-72999827

-58569902

103

93

Vaccination

Only

Vaccination

+Public

Campaigns

178095371 967.1

1530163

1854

-71697288

102

Comprehensive

Control

Table 4. Cost-Effectiveness Analysis Results

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Additional insights as far as economic impact highlight that Maximum health gain: -83916.3 DALYs averted

with Comprehensive Control, and most efficient: 93 USD per infection averted with Vaccination + Public

Campaigns.

Numerical computation

Here, numerical simulations were carried out using the parameter values presented in Table 3. We simulate

model (1) with MATLAB software to investigate the dynamical behaviours of the subpopulations. Thus, the

( )

[

( )

initial size of the susceptible subpopulation is 푆_휏0 = 푁_휏0 − 퐸_휏0 + +퐼_휏0 + 퐴_휏0 + 푅_휏0 + 푉 0 +

휏

( )]

( )

퐻_휏0 , with 푁_휏0 = 6,537,314 as the reported total human population of Bauchi state, northeast Nigeria, and

we chose our initial conditions as follows, the first cumulative suspected case of diphtheria is assumed to be in

( )

the first exposed human population, which is given as 퐸_휏0 = 564. The first cumulative confirmed case of

( )

diphtheria is assumed to be the initial infectious human population, which is given as the total infections 퐼_휏0 +

( ) ( )

퐴_휏0 = 33, with the initial vaccination to be 푉 0 = 1150, Hesitant individuals are assumed to be one-third of

휏

( )

the vaccination 퐻_휏0 = 384, the initial recovered human population is assumed to be given as 푅_휏= 0, while the

( )

Concentration of the bacteria in the environment 푊 0 = 7. However, numerical simulations were performed

휏

to evaluate the effects of vaccination, environmental sanitation, and public enlightenment campaigns on the

transmission dynamics of diphtheria.

Figure 6: Numerical simulation for the diphtheria model (1) showing the Pictorial (A) Total infections under

different interventions, (B) Baseline scenario: all compartments, (C) Environmental pathogen dynamics, (D)

Peak infection reduction by scenario, (E) Final size vs reproduction number, and (F) Time varying force of

infection (baseline), using fitted parameter values in Table 3.

The outcomes demonstrate the critical role of integrated control strategies in mitigating infection spread,

reducing pathogen persistence, and enhancing population immunity. Figure 6 (A–F) depicts the model behavior

under baseline and intervention conditions, highlighting how different strategies influence epidemic progression.

Figure 6 (A) depicts the cumulative number of infected individuals under five intervention scenarios: baseline

(no control), vaccination, campaigns, environmental, and combined interventions. The baseline scenario (ℛ₀=

0.008) shows an exponential increase in infections, confirming sustained transmission in the absence of control.

When vaccination, campaign, and environmental interventions are introduced independently, noticeable declines

in total infections occur, with vaccination exerting the strongest individual effect. The combined control strategy

(푅₀= 0.004) yields the greatest reduction in total infections, demonstrating a synergistic benefit from

concurrent application of all interventions. This finding aligns with previous works indicating that integrating

behavioral and biomedical measures enhances outbreak suppression [2]. Figure 6 (B) illustrates the dynamics of

the epidemiological compartments susceptible, exposed, symptomatic, asymptomatic, recovered, and hesitant

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over 60 weeks. The susceptible population declines gradually as individuals move to the exposed and infectious

classes. Both symptomatic and asymptomatic infections remain relatively low due to intervention effects, while

the recovered class increases sharply after week 40, approaching a steady state. The hesitant compartment

remains nearly constant, reflecting behavioral inertia in vaccine acceptance. This persistence of hesitancy

emphasizes the importance of sustained health education and trust-building measures to enhance immunization

uptake, consistent with recent behavioral modeling studies [30]. Figure 6 (C), shows the temporal pattern of

environmental bacterial concentration, 푊 (푡), across different interventions. Under the baseline and

휏

environmental-only scenarios, pathogen levels increase and remain high, indicating ongoing environmental

contamination. Conversely, vaccination and campaign strategies indirectly lower 푊 (푡)by reducing infectious

휏

hosts, while the combined intervention achieves the sharpest decline, confirming that simultaneous reduction in

host infectivity and environmental shedding yields optimal results. This outcome echoes earlier findings that

environmental disinfection complements medical control measures by shortening pathogen persistence [18].

Figure 6 (D), quantifies the percentage reduction in cumulative infections compared to the baseline. The

reductions achieved are 98.3%, 97.7%, 97.4%, and 98.8% for vaccination, campaign, environmental, and

combined interventions, respectively, whereas the baseline achieves only 35.9%. The combined control strategy,

therefore, provides the most significant suppression of diphtheria transmission. The near-complete reduction

demonstrates that combined, multi-level interventions, particularly vaccination coupled with intensive health

campaigns, represent the most cost-effective and sustainable approach for epidemic management in resource-

limited settings.

Moreover, Figure 6 (E), displays the relationship between the final outbreak size (recovered population) and the

basic reproduction number, 푅₀, across different control strategies. The vertical dashed line marks the epidemic

threshold (ℛ₀= 1). All intervention scenarios yield ℛ₀< 1, implying successful containment and convergence

toward the disease-free equilibrium. The combined strategy leads to the smallest outbreak size, confirming

theoretical expectations from the stability analysis presented in Section 3. This agrees with the well-established

epidemiological principle that maintaining ℛ₀< 1ensures infection eradication in the long term [39]. Figure 6

(F), depicts the temporal variation in the force of infection. Initially low, the infection pressure rises gradually

during the early phase of the epidemic due to the accumulation of exposed individuals, then stabilizes and

declines as interventions intensify. The eventual plateau reflects the progressive exhaustion of the susceptible

and the efficacy of combined control measures. This declining pattern validates the effectiveness of integrated

control programs in reducing disease transmissibility and curtailing epidemic potential.

In conclusion, the collective findings confirm that vaccination and public enlightenment campaigns are the most

influential individual interventions for diphtheria control. However, when all interventions, vaccination,

environmental sanitation, and public campaigns are applied together, the impact is synergistic, leading to

dramatic reductions in infections, environmental contamination, and outbreak size. This emphasizes the

necessity of comprehensive, coordinated approaches rather than isolated efforts.

Formulation of Caputo-Fabrizio Fractional Model and Analysis

Here, we extend the integer-order ordinary differential equation (ODE) to its corresponding non-integer-order

fractional differential equation (FDE) model via the Caputo-Fabrizio fractional order derivative, which can

sufficiently account for memory as well as nonlocal properties, allowing it to accommodate all points within the

interval. Thus, replacing the first-order time derivatives of the left-hand side of (1) with the fractional Caputo-

Fabrizio derivative, we obtain our new fractional differential equations (non-integer-order derivatives) as

follows:

훼

ꢃ

(

)

푐퐹_퐷푆_휏= Π + 휔푉 + 휋퐻_휏− 휆_휏+ 훿 + 휉 + 휇 푆_휏

휏

푐퐹_퐷^훼퐸_휏= 휆_휏푆_휏− (휎 + 휇)퐸_휏

ꢃ

푐퐹_퐷^훼퐼_휏= 휎휍퐸_휏− (훾₁+ 휙₁+ 휇)퐼_휏

ꢃ

푐퐹_퐷^훼퐴_휏= 휎(1 − 휍)퐸_휏− (훾₂+ 휙₂+ 휇)퐴_휏

ꢃ

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푐퐹_퐷^훼푅_휏= 훾₁퐼_휏+ 훾₂퐴_휏− 휇푅_휏

ꢃ

16

(

)

푐퐹_퐷^훼푉 = 훿푆_휏− (휔 + 휇)푉

휏

ꢃ

푐퐹_퐷^훼퐻_휏= 휉푆_휏− (휋 + 휇)퐻_휏

ꢃ

푐퐹_퐷^훼푊 = 휑₁퐼_휏+휑₂퐴_휏− 휗푊

휏

ꢃ

훼

where 푐퐹_퐷ꢃrepresents the Caputo-Fabrizio fractional derivative of order 0 < ꢦ ≤ 1, with the non-negative initial

conditions

( )

푆_휏0 = 푆_휏, 퐸_휏0 = 퐸_휏, 퐼_휏0 = 퐼_휏, 퐴_휏0 = 퐴_휏, 푅_휏0 = 푅_휏, 푉 0 = 푉 ,

휏

0

( )

퐻_휏0 = 퐻_휏, 푊 0 = 푊

(17)

휏

0

Existence and Uniqueness of Solutions of The Model

To evaluate and prove the existence and uniqueness of solutions to the Caputo-Fabrizio fractional model (16)

for the dynamics of the diphtheria disease, with initial conditions (17), we use fixed-point theory as in ([17, 23]).

Applying the Caputo-Fabrizio fractional integral operator on both sides of (1), we have

훼

ꢃ

( )

[

(

) ]

푆_휏푡 − 푆_휏0 = 푐퐹

Π + 휔푉 + 휋퐻_휏− 휆_휏+ 훿 + 휉 + 휇 푆_휏,

ꢁ

휏

훼

ꢃ

( )

퐸_휏푡 − 퐸_휏0 = 푐퐹 [휆_휏푆_휏− (휎 + 휇)퐸_휏],

ꢁ

훼

ꢃ

( )

[

]

퐼_휏푡 − 퐼_휏0 = 푐퐹

휎휍퐸_휏− (훾₁+ 휙₁+ 휇)퐼_휏,

ꢁ

훼

ꢃ

( )

[

]

퐴_휏푡 − 퐴_휏0 = 푐퐹

휎(1 − 휍)퐸_휏− (훾₂+ 휙₂+ 휇)퐴_휏,

ꢁ

훼

ꢃ

( )

[

]

(

)

18

푅_휏푡 − 푅_휏0 = 푐퐹

훾₁퐼_휏+ 훾₂퐴_휏− 휇푅_휏

,

ꢁ

훼

ꢃ

( )

푉 푡 − 푉 0 = 푐퐹 [훿푆_휏− (휔 + 휇)푉 ]

휏

ꢁ

휏

훼

ꢃ

( )

[

]

퐻_휏푡 − 퐻_휏0 = 푐퐹

휉푆_휏− (휋 + 휇)퐻_휏,

ꢁ

훼

ꢃ

( )

[

]

휏

푊 푡 − 푊 0 = 푐퐹

휑₁퐼_휏+휑₂퐴_휏− 휗푊 ,

휏

ꢁ

Then, the kernels of the system (16) can be written as follows

(

)

( ) ) ( )

( )

(

퐾₁푡, 푆_휏= Π + 휔푉 푡 + 휋퐻_휏푡 − 휆_휏+ 훿 + 휉 + 휇 푆_휏푡

휏

(

)

( ) ( )

퐾₂푡, 퐸_휏= 휆_휏푆_휏푡 − (휎 + 휇)퐸_휏푡

(

)

( ) ( )

퐾₃푡, 퐼_휏= 휎휍퐸_휏푡 − (훾₁+ 휙₁+ 휇)퐼_휏푡

(

)

( ) ( )

퐾₄푡, 퐴_휏= 휎(1 − 휍)퐸_휏푡 − (훾₂+ 휙₂+ 휇)퐴_휏푡

(

)

( )

(

)

19

퐾₅푡, 푅_휏= 훾₁퐼_휏푡 + 훾₂퐴_휏푡 − 휇푅_휏푡

(

)

( ) ( )

퐾₆푡, 푉 = 훿푆_휏푡 − (휔 + 휇)푉 푡

휏

(

)

( ) ( )

퐾₇푡, 퐻_휏= 휉푆_휏푡 − (휋 + 휇)퐻_휏푡

(

)

( )

퐾₈푡, 푊 = 휑₁퐼_휏푡 + 휑₂퐴_휏푡 − 휗푊 푡

휏

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2(1−ꢤ)

2ꢤ

( )

And the functions Ξ ꢦ =

( )

and Θ ꢦ =

(

)

20

(

)

(

)

(

)

(

)

2−ꢤ 푀 ꢤ

Thus, we assume that 푆_휏, 퐸_휏, 퐼_휏, 퐴_휏, 푅_휏, 푉 , 퐻_휏, 푎푛푑 푊 and are nonnegative bounded functions, when proving the

휏

following theorems, i.e.,

‖ ( )‖

푆_휏푡

‖

( )‖

‖ ( )‖

≤ 휃₂, 퐼_휏푡

‖

( )‖

≤ 휃₁, 퐸_휏푡

≤ 휃₃, 퐴_휏푡

≤ 휃₄,

‖ ( )‖

푅_휏푡

‖ ( )‖

≤ 휃₅, 푉 푡

‖

( )‖

‖

( )‖

≤ 휃₆, 퐻_휏푡

≤ 휃₇and 푊 푡

≤ 휃₈

휏

where 휃₁, 휃₂, 휃₃, 휃₄, 휃₅, 휃₆, 휃₇and 휃₈are some positive constants.

Now let

Δ₁= 휆_휏+ 훿 + 휉 + 휇, Δ₂= 휎 + 휇, Δ₃= 훾₁+ 휙₁+ 휇, Δ₄= 훾₂+ 휙₂+ 휇, Δ₅= 휇, Δ₆= 휔 + 휇, Δ₇= 휋 +

휇 and Δ₈= 휗 (21)

Applying the definition of the Caput-Fabrizio fractional integral operator in (18), we have.

ꢃ

( )

(

)

( )

(

)

푆_휏푡 − 푆_휏0 = Ξ ꢦ 퐾₁푡, 푆_휏+ Θ ꢦ

퐾₁푦, 푆_휏푑푦,

∫

0

ꢃ

( )

(

)

( )

(

)

퐸_휏푡 − 퐸_휏0 = Ξ ꢦ 퐾₂푡, 퐸_휏+ Θ ꢦ

퐾₂푦, 퐸_휏푑푦,

∫

0

ꢃ

( )

(

)

( )

(

)

퐼_휏푡 − 퐼_휏0 = Ξ ꢦ 퐾₃푡, 퐼_휏+ Θ ꢦ

퐾₃푦, 퐼_휏푑푦,

∫

0

ꢃ

( )

(

)

( )

(

)

퐴_휏푡 − 퐴_휏0 = Ξ ꢦ 퐾₄푡, 퐴_휏+ Θ ꢦ

퐾₄푦, 퐴_휏푑푦,

∫

0

ꢃ

( )

(

)

( )

(

)

푅_휏푡 − 푅_휏0 = Ξ ꢦ 퐾₅푡, 푅_휏+ Θ ꢦ

퐾₅푦, 푅_휏푑푦,

(22)

∫

0

ꢃ

( )

(

)

( )

(

)

푉 푡 − 푉 0 = Ξ ꢦ 퐾₆푡, 푉 + Θ ꢦ

퐾₆푦, 푉 푑푦

∫

휏

0

ꢃ

( )

(

)

( )

(

)

퐻_휏푡 − 퐻_휏0 = Ξ ꢦ 퐾₇푡, 퐻_휏+ Θ ꢦ

퐾₇푦, 퐻_휏푑푦,

∫

0

ꢃ

( )

(

)

( )

(

)

휏

푊 푡 − 푊 0 = Ξ ꢦ 퐾₈푡, 푊 + Θ ꢦ

퐾₈푦, 푊 푑푦,

∫

휏

0

Theorem 6: If the following inequality holds

{

}

0 ≤ ꢧ = max Δ₁, Δ₂, Δ₃, Δ₄, Δ₅, Δ₆, Δ₇, Δ₈< 1,

(23)

Then the kernels 퐾₁, 퐾₂, 퐾₃, 퐾₄, 퐾₅, 퐾₆, 퐾₇and 퐾₈satisfy Lipschitz conditions and are contraction mappings.

Proof. We consider the kernel 퐾₁. Let 푆_휏and 푆_휏1be any two functions, then we have

(

)

( )

‖

(

_ꢁ)‖

‖

(

)

(

)

(

)

(

퐾₁푡, 푆_휏− 퐾₁푡, 푆_휏

= −휆_휏푆_휏푡 − 푆_휏1(푡) − 훿 푆_휏푡 − 푆_휏1(푡) − 휉 푆_휏푡 − 푆_휏1(푡) − 휇 푆_휏푡 −

)‖

푆_휏1(푡)

(24)

Using the triangle inequality for norms on the right-hand side of (24), we have

(

)

‖

(

_ꢁ)‖

퐾₁푡, 푆_휏− 퐾₁푡, 푆_휏

( )

‖

≤ 휆_휏(푆_휏푡 − 푆_휏1푡 ) + 훿 (푆_휏푡 − 푆_휏1푡 ) + 휉 (푆_휏푡 − 푆_휏1푡 )

( )

‖

( )

‖

+ 휇 (푆_휏푡 − 푆_휏1푡 )

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(

)

( )

푡

‖

≤ 휆_휏+ 훿 + 휉 + 휇 푆_휏푡 − 푆_휏1

(

)

( )

푡

‖

≤ 휆_휏+ 훿 + 휉 + 휇 푆_휏푡 − 푆_휏1

( )

= Δ₁푆_휏푡 − 푆_휏1

( )

‖

푡

.

Where Δ₁is as defined in (21). Comparable outcomes for the kernels 퐾₂, 퐾₃, 퐾₄, 퐾₅, 퐾₆, 퐾₇and 퐾₈can be

} { } { } { } { } {

{

}

{

}

1

obtained using 퐸_휏, 퐸_휏, 퐼_휏, 퐼_휏, 퐴_휏, 퐴_휏, 푅_휏, 푅_휏, 푉 , 푉 , 퐻_휏, 퐻_휏

and 푊 , 푊 , respectively, as follows:

휏

1

(

)

( ) ( )

‖

≤ Δ₂퐸_휏푡 − 퐸_휏₁

‖

(

_ꢁ)‖

‖

퐾₂푡, 퐸_휏− 퐾₂푡, 퐸_휏

푡

(

)

( )

‖

≤ Δ₃퐼_휏푡 − 퐼_휏1

(

₁)‖

‖

퐾₃푡, 퐼_휏− 퐾₃푡, 퐼_휏

푡

(

)

( ) ( )

‖

≤ Δ₄퐴_휏푡 − 퐴_휏1

(

₁)‖

_ꢁ)‖

‖

퐾₄푡, 퐴_휏− 퐾₄푡, 퐴_휏

푡

(

)

( ) ( )

‖

≤ Δ₅푅_휏푡 − 푅_휏1

(

‖

퐾₅푡, 푅_휏− 퐾₅푡, 푅_휏

푡

(

)

( )

‖

≤ Δ₆푉 푡 − 푉

‖

(

_ꢁ)‖

‖

퐾₆푡, 푉 − 퐾₆푡, 푉

푡

(25)

휏

1

(

)

( )

≤ Δ₇퐻_휏푡 − 퐻_휏1

( )

‖

(

₁)‖

‖

퐾₇푡, 퐻_휏− 퐾₇푡, 퐻_휏

푡

(

)

( )

≤ Δ₈푊 푡 − 푊

( )

‖

푡

‖

(

‖

퐾₈푡, 푊 − 퐾₈푡, 푊

휏

1

(

)

where Δ₁, Δ₂, Δ₃, Δ₄, Δ₅, Δ₆, Δ₇and Δ₈are defined in 21 .

Therefore, the Lipschitz conditions are satisfied for 퐾₂, 퐾₃, 퐾₄, 퐾₅, 퐾₆, 퐾₇and 퐾₈. In addition, since 0 ≤ ꢧ =

{

}

max Δ₁, Δ₂, Δ₃, Δ₄, Δ₅, Δ₆, Δ₇, Δ₈< 1,

the kernels are contractions. From (22), the state variables can be displayed in terms of the kernels as follows:

ꢃ

( )

(

)

( )

(

)

푆_휏푡 = 푆_휏0 + Ξ ꢦ 퐾₁푡, 푆_휏+ Θ ꢦ

퐾₁푦, 푆_휏푑푦,

∫

0

ꢃ

( )

(

)

( )

(

)

퐸_휏푡 = 퐸_휏0 + Ξ ꢦ 퐾₂푡, 퐸_휏+ Θ ꢦ

퐾₂푦, 퐸_휏푑푦,

∫

0

ꢃ

( )

(

)

( )

(

)

퐼_휏푡 = 퐼_휏0 + Ξ ꢦ 퐾₃푡, 퐼_휏+ Θ ꢦ

퐾₃푦, 퐼_휏푑푦,

∫

0

ꢃ

( )

(

)

( )

(

)

퐴_휏푡 = 퐴_휏0 + Ξ ꢦ 퐾₄푡, 퐴_휏+ Θ ꢦ

퐾₄푦, 퐴_휏푑푦,

∫

0

ꢃ

( )

(

)

( )

(

)

푅_휏푡 = 푅_휏0 + Ξ ꢦ 퐾₅푡, 푅_휏+ Θ ꢦ

퐾₅푦, 푅_휏푑푦,

(26)

∫

0

ꢃ

( )

(

)

( )

(

)

푉 푡 = 푉 0 + Ξ ꢦ 퐾₆푡, 푉 + Θ ꢦ

퐾₆푦, 푉 푑푦

∫

휏

0

ꢃ

( )

(

)

( )

(

)

퐻_휏푡 = 0 + Ξ ꢦ 퐾₇푡, 퐻_휏+ Θ ꢦ

퐾₇푦, 퐻_휏푑푦,

∫

0

ꢃ

( )

(

)

( )

(

)

휏

푊 푡 = 푊 0 + Ξ ꢦ 퐾₈푡, 푊 + Θ ꢦ

퐾₈푦, 푊 푑푦,

∫

휏

0

Using (26), we now introduce the following recursive formulas:

ꢃ

( )

(

)

(

)

푆_휏_ꢨ푡 = Ξ ꢦ 퐾₁푡, 푆_휏_ꢨ−1+ Θ ꢦ

퐾₁푦, 푆_휏_ꢨ−1푑푦,

∫

0

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ꢃ

( ) ( )

( )

(

)

(

)

퐸_휏_ꢨ푡 = Ξ ꢦ 퐾₂푡, 퐸_휏_ꢨ−1+ Θ ꢦ

퐾₂푦, 퐸_휏_ꢨ−1푑푦,

∫

0

^ꢃ퐾₃(푦, 퐼_휏_(ꢨ−1))푑푦,

( ) ( )

( )

퐼_휏_ꢨ푡 = Ξ ꢦ 퐾₃(푡, 퐼_휏_(ꢨ−1)) + Θ ꢦ

∫

0

^ꢃ퐾₄(푦, 퐴_휏_(ꢨ−1)) 푑푦,

(27)

( ) ( )

( )

퐴_휏_ꢨ푡 = Ξ ꢦ 퐾₄(푡, 퐴_휏_(ꢨ−1)) + Θ ꢦ

∫

0

ꢃ

( ) ( )

( )

(

)

(

)

푅_휏_ꢨ푡 = Ξ ꢦ 퐾₅푡, 푅_휏_ꢨ−1+ Θ ꢦ

퐾₅푦, 푅_휏_ꢨ−1푑푦,

∫

0

^ꢃ퐾₆(푦, 푉 _(ꢨ−1)) 푑푦

( )

+ Θ ꢦ

(

)

ꢨ−1

푉

푡 = Ξ ꢦ 퐾₆푡, 푉

∫

휏

ꢨ

0

( ) ( )

( )

퐻_휏_ꢨ푡 = Ξ ꢦ 퐾₇(푡, 퐻_휏_(ꢨ−1)) + Θ ꢦ

^ꢃ퐾₇(푦, 퐻_휏_(ꢨ−1)) 푑푦,

∫

0

( ) ( )

( )

푊

푡 = Ξ ꢦ 퐾₈(푡, 푊 _(ꢨ−1)) + Θ ꢦ

^ꢃ퐾₈(푦, 푊 _(ꢨ−1))푑푦.

휏

ꢨ

0

The initial components of the above recursive formulas are determined by the following initial conditions:

( )

푆_휏₀푡 = 푆_휏0 , 퐸_휏₀푡 = 퐸_휏0 , 퐼_휏₀푡 = 퐼_휏0 , 퐴_휏₀푡 = 퐴_휏0 , 푅_휏₀푡 = 푅_휏0 ,

( )

( ) ( )

푡 = 푊 0

푉

푡 = 푉 0 , 퐻_휏푡 = 퐻_휏0 , 푊

(28)

휏

0

Hence, the differences between the consecutive terms for the recursive formulas can be written as

( )

)

푡 = 푆_휏_ꢨ푡 − 푆_휏_ꢨ−1푡 = Ξ ꢦ (퐾₁푡, 푆_휏_ꢨ−1− 퐾₁푡, 푆_휏_ꢨ−2)

( )

(

)

(

Υ

1ꢨ

^ꢃ(퐾₁(푦, 푆_휏

− 퐾₁푡, 푆_휏_ꢨ−2) 푑푦,

( )

+Θ ꢦ

(

)

∫

ꢨ−1

0

( )

)

Υ_2ꢨ(푡) = 퐸_휏_ꢨ푡 − 퐸_휏_ꢨ−1푡 = Ξ ꢦ (퐾₂푡, 퐸_휏_ꢨ−1− 퐾₂푡, 퐸_휏_ꢨ−2)

( )

(

)

(

ꢃ

( )

+Θ ꢦ

∫

0

(

)

(퐾₂(푦, 퐸_휏_ꢨ−1− 퐾₂푡, 퐸_휏_ꢨ−2) 푑푦

( )

)

Υ_3ꢨ(푡) = 퐼_휏_ꢨ푡 − 퐼_휏_ꢨ−1= Ξ ꢦ (퐾₃푡, 퐼_휏_ꢨ−1− 퐾₃푡, 퐼_휏_ꢨ−2)

( )

(

)

(

ꢃ

( )

+Θ ꢦ

∫

0

(

)

(퐾₃(푦, 퐼_휏_ꢨ−1− 퐾₃푡, 퐼_휏_ꢨ−2) 푑푦,

( )

)

Υ_4ꢨ(푡) = 퐴_휏_ꢨ푡 − 퐴_휏_ꢨ−1푡 = Ξ ꢦ (퐾₄푡, 퐴_휏_ꢨ−1− 퐾₄푡, 퐴_휏_ꢨ−2)

( )

(

)

(

^ꢃ(퐾₄(푦, 퐴_휏

− 퐾₄푡, 퐴_휏_ꢨ−2) 푑푦,

( )

+Θ ꢦ

(

)

∫

ꢨ−1

0

( )

(

)

(

)

Υ_5ꢨ(푡) = 푅_휏_ꢨ푡 − 푅_휏_ꢨ−1푡 = Ξ ꢦ (퐾₅푡, 푅_휏_ꢨ−1− 퐾₅푡, 푅_휏_ꢨ−2)

(29)

^ꢃ(퐾₅(푦, 푅_휏

− 퐾₅푡, 푅_휏_ꢨ−2) 푑푦,

( )

+Θ ꢦ

(

)

∫

ꢨ−1

0

( )

(

)

(

)

ꢨ−2

Υ_6ꢨ(푡) = 푉

푡 − 푉

푡 = Ξ ꢦ (퐾₆푡, 푉

− 퐾₆푡, 푉

휏

ꢨ

ꢨ−1

^ꢃ(퐾₆(푦, 푉

− 퐾₆푡, 푉

( )

+Θ ꢦ

(

)

) 푑푦,

∫

휏

ꢨ−1

ꢨ−2

0

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( )

)

Υ_7ꢨ(푡) = 퐻_휏_ꢨ푡 − 퐻_휏_ꢨ−1푡 = Ξ ꢦ (퐾₇푡, 퐻_휏_ꢨ−1− 퐾₇푡, 퐻_휏_ꢨ−2)

( )

(

)

(

^ꢃ(퐾₇(푦, 퐻_휏

− 퐾₇푡, 퐻_휏_ꢨ−2)푑푦,

( )

+Θ ꢦ

(

)

∫

ꢨ−1

0

( )

푡 − 푊

( )

(

)

(

)

ꢨ−2

Υ_8ꢨ(푡) = 푊

푡 = Ξ ꢦ (퐾₈푡, 푊

− 퐾₈푡, 푊

휏

ꢨ

ꢨ−1

( )

(

)

) 푑푦,

+Θ ꢦ

ꢨ

^ꢃ(퐾₈(푦, 푊

− 퐾₈푡, 푊

∫

휏

ꢨ−1

ꢨ−2

0

ꢨ

( )

∑

( )

∑

( )

∑

( )

∑

For 푆_휏_ꢨ푡 =

Υ (푡),퐸_ꢨ푡 = _푖=1Υ_2푖(푡) , 퐼_휏

푡 = _푖=1Υ_3푖(푡), 퐴_휏_ꢨ푡 = _푖=1Υ_4푖(푡)

1푖

푖=1

ꢨ

푖=1

ꢨ

푖=1

ꢨ

( )

,푅_휏_ꢨ푡 =

∑

( )

∑

( )

∑

( )

∑

Υ_5ꢨ(푡) , 푉

푡 = _푖=1Υ_6푖푡 , 퐻_휏_ꢨ푡 =

Υ_7푖(푡),푊

푡 = _푖=1Υ_8푖(푡) (30)

휏

푖

ꢨ

Now, let's generate the recursive inequalities for the differences Υ , Υ_2ꢨ, Υ_3ꢨ, Υ_4ꢨ, Υ_5ꢨ, Υ_6ꢨ, Υ_7ꢨand Υ_8ꢨas

1ꢨ

follows

Υ

푡

= Ξ ꢦ (퐾₁푡, 푆_휏_ꢨ−1− 퐾₁푡, 푆_휏_ꢨ−2) + Θ ꢦ

^ꢃ(퐾₁(푦, 푆_휏

−

‖

( )‖

( )

= 푆_휏_ꢨ푡 − 푆_휏_ꢨ−1

( )

푡

( )

‖

(

)

(

)

∫

1ꢨ

ꢨ−1

0

(

)

‖

퐾₁푡, 푆_휏_ꢨ−2) 푑푦

(31)

Using the triangle inequality for norms to (31), we have

( )

푡

( )

‖

(

)

(

_ꢨ−2)‖

푆_휏_ꢨ푡 − 푆_휏_ꢨ−1

= Ξ ꢦ ∥ 퐾₁푡, 푆_휏_ꢨ−1− 퐾₁푡, 푆_휏

ꢃ

( )

+ Θ ꢦ

‖

퐾₁(푦, 푆_휏_ꢨ−1) − 퐾₁(푡, 푆_휏_ꢨ−2) 푑푦

∫

0

Then, since the kernel 퐾₁satisfies the Lipschitz condition with Lipschitz constant Δ₁, we have

ꢃ

( )

‖

푆_휏_ꢨ푡 − 푆_휏_ꢨ−1

( )

푡

( )

‖

≤ Ξ ꢦ Δ₁∥ 푆_휏_ꢨ−1− 푆_휏_ꢨ−2+ Θ ꢦ Δ₁

푆_휏_ꢨ−1− 푆_휏_ꢨ−2푑푦

∫

0

ꢃ

‖

( )‖

푡

( )

≤ Ξ ꢦ Δ₁

( )

푡

( )

+ Θ ꢦ Δ₁

( )

‖

푑푦

1(ꢨ−1)

‖

Υ

therefore, Υ

Υ

푦

(32)

∫

1ꢨ

1(ꢨ−1)

0

Following the same procedures, we have

ꢃ

‖

( )‖

푡

( )

≤ Ξ ꢦ Δ₂Υ_2(ꢨ−1)

( )

푡

( )

푦

‖

Υ_2ꢨ

+ Θ ꢦ Δ₂

Υ_2(ꢨ−1)

푑푦

∫

0

ꢃ

‖

Υ_3ꢨ

( )‖

푡

( )

≤ Ξ ꢦ Δ₃Υ_3(ꢨ−1)

( )

푡

( )

+ Θ ꢦ Δ₃

( )

푦

‖

Υ_4ꢨ

( )‖

푡

‖

Υ_3(ꢨ−1)

푑푦

≤

∫

0

ꢃ

( )

Ξ ꢦ Δ₄Υ_4(ꢨ−1)

( ) ( )

( )

‖

Υ_4(ꢨ−1)푑푦

‖

푡

+ Θ ꢦ Δ₄

푦

∫

0

ꢃ

‖

Υ_5ꢨ

( )‖

푡

( )

≤ Ξ ꢦ Δ₅Υ_5(ꢨ−1)

( )

푡

( )

+ Θ ꢦ Δ₅

( )

‖

Υ_5(ꢨ−1)푑푦

‖

푦

(33)

∫

0

ꢃ

‖

( )‖

( )

≤ Ξ ꢦ ℵ₆Υ_6(ꢨ−1)

( )

푡

( )

‖

푦

‖

Υ_6ꢨ

Υ_7ꢨ

Υ_8ꢨ

푡

+ Θ ꢦ Δ₆

Υ_6(ꢨ−1)

Υ_7(ꢨ−1)

Υ_8(ꢨ−1)

푑푦

∫

0

ꢃ

‖

( )‖

푡

( )

≤ Ξ ꢦ ∆₇Υ_7(ꢨ−1)

( )

푡

( )

+ Θ ꢦ Δ₇

( )

푦

‖

∫

0

ꢃ

‖

( )‖

푡

( )

≤ Ξ ꢦ ∆₈Υ_8(ꢨ−1)

( )

푡

( )

+ Θ ꢦ Δ₈

( )

푦

‖

∫

0

Theorem 7: If there exists a time 푡₀> 0 such that the following inequalities hold:

( )

Ξ ꢦ Δ_푖+ Θ ꢦ Δ_푖푡₀> 1, for ꢠ = 1,2, … ,8,

(34)

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then the solution exists for the fractional diphtheria model (16) with (17).

Proof. Assuming the functions푆_휏, 퐸_휏, 퐼_휏, 퐴_휏, 푅_휏, 푉 , 퐻_휏, 푎푛푑 푊 are bounded and each of the kernels satisfies a

휏

Lipschitz condition, then the following relations can be obtained.

Using (33) − (34) recursively:

ꢨ

‖

( )‖

‖ ( )‖ ( )

( )

Υ

푡

≤ 푆_휏0 [Ξ ꢦ Δ₁+ Θ ꢦ Δ₁]

1ꢨ

ꢨ

( )‖

푡

‖

( )‖

( )

Υ_2ꢨ

Υ_3ꢨ

Υ_4ꢨ

≤ 퐸_휏0

[Ξ ꢦ Δ₂+ Θ ꢦ Δ₂]

ꢨ

( )‖

푡

‖ ( )‖ ( )

≤ 퐼_휏0 [Ξ ꢦ Δ₃+ Θ ꢦ Δ₃]

( )

ꢨ

( )‖

푡

‖

( )‖ ( )

( )

≤ 퐴_휏0 [Ξ ꢦ Δ₄+ Θ ꢦ Δ₄]

ꢨ

‖

Υ_5ꢨ

( )‖

푡

‖

( )‖ ( )

( )

≤ 푅_휏0 [Ξ ꢦ Δ₅+ Θ ꢦ Δ₅]

(35)

ꢨ

‖

Υ_6ꢨ

‖

Υ_7ꢨ

‖

Υ_8ꢨ

( )‖

‖ ( )‖ ( )

( )

푡

≤ 푉 0 [Ξ ꢦ Δ₆+ Θ ꢦ Δ₆]

휏

ꢨ

( )‖

푡

‖

( )‖ ( )

( )

≤ 퐻_휏0 [Ξ ꢦ Δ₇+ Θ ꢦ Δ₇]

( )‖

푡

‖

( )‖ ( )

( )

≤ 푊 0 [Ξ ꢦ Δ₈+ Θ ꢦ Δ₈]

휏

Equation (35) shows the existence and smoothness of the functions defined in (31).

( )

푡 , 퐻_휏_ꢨand

To complete the proof, we prove that the functions 푆_휏_ꢨ푡 , 퐸_휏_ꢨ푡 , 퐼_휏_ꢨ푡 , 퐴_휏_ꢨ푡 , 푅_휏_ꢨ푡 , 푉

휏

ꢨ

( )

푡 converge to a solution of (16) 푤ꢠ푡ℎ (17).

푊

휏

ꢨ

( )

We introduce 퐵_1ꢨ푡 , 퐵_2ꢨ푡 , 퐵_3ꢨ푡 , 퐵_4ꢨ푡 , 퐵_5ꢨ푡 , 퐵_6ꢨ푡 , 퐵_7ꢨ푡 and 퐵_8ꢨ푡 , as the remainder terms

after 푛 iteration, so that

( )

푆_휏푡 − 푆_휏0 = 푆_휏_ꢨ푡 − 퐵_1ꢨ푡 ,

( )

퐸_휏푡 − 퐸_휏0 = 퐸_휏_ꢨ푡 − 퐵_2ꢨ푡 ,,

( )

퐼_휏푡 − 퐼_휏0 = 퐼_휏_ꢨ푡 − 퐵_3ꢨ푡 ,

( )

퐴_휏푡 − 퐴_휏0 = 퐴_휏_ꢨ푡 − 퐵_4ꢨ푡 ,

( )

푅_휏푡 − 푅_휏0 = 푅_휏_ꢨ푡 − 퐵_5ꢨ푡 ,

(36)

( )

푉 푡 − 푉 0 = 푉

푡 − 퐵_6ꢨ푡 ,

휏

ꢨ

( )

퐻_휏푡 − 퐻_휏0 = 퐻_휏_ꢨ푡 − 퐵_7ꢨ푡 ,

( )

푊 푡 − 푊 0 = 푊

푡 − 퐵_8ꢨ푡 ,

휏

ꢨ

Then, using the triangle inequality and the Lipschitz condition for 퐾₁, we have

ꢃ

‖

퐵_1ꢨ

( )‖

푡

( )

(

)

( )

‖

∫

0

‖

(퐾₁(푦, 푆_휏) − 퐾₁(푦, 푆_휏_ꢨ−1))푑푦

= Ξ ꢦ (퐾₁푡, 푆_휏− 퐾₁(푡, 푆_휏_ꢨ−1))) + Θ ꢦ

ꢃ

( )

(

)

( )

+ Θ ꢦ

‖

(

)‖

‖

≤ Ξ ꢦ (퐾₁푡, 푆_휏− 퐾₁푡, 푆_휏_ꢨ−1

퐾₁(푦, 푆_휏) − 퐾₁(푦, 푆_휏_ꢨ−1) 푑푦

∫

0

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( )

‖

≤ Ξ ꢦ Δ₁푆_휏− 푆_휏_ꢨ−1+ Θ ꢦ Δ₁퐵_1ꢨ푆_휏− 푆_휏_ꢨ−1푡.

‖

푡

Repeating the same process, we have;

푡

≤ [(Ξ ꢦ + Θ ꢦ 푡)Δ₁]^ꢨ+1휃₁

(37)

‖

퐵_1ꢨ

( )‖

( )

At 푡₀we have

( )‖

푡

≤ [(Ξ ꢦ + Θ ꢦ 푡₀)Δ₁]^ꢨ+1휃₁

(38)

( )‖

→ 0. Using the same

‖

퐵_1ꢨ

( )

‖

Taking the limit on (38) as 푛 → ∞ and then using the condition (34), we obtain 퐵_1ꢨ

푡

process as described above, we have the following relations:

‖

( )‖

( )

‖

( )‖

퐵_2ꢨ

푡

≤ [(Ξ ꢦ + Θ ꢦ 푡₀)Δ₂]^ꢨ+1휃₂

( )

퐵_3ꢨ

( )

푡

≤

[(Ξ ꢦ + Θ ꢦ 푡₀)Δ₃]^ꢨ+1휃₃

푡

( )

‖

( )‖

퐵_4ꢨ≤ [(Ξ ꢦ +

Θ ꢦ 푡₀)Δ₄]^ꢨ+1휃₄

푡

≤ [(Ξ ꢦ + Θ ꢦ 푡₀)Δ₅]^ꢨ+1휃₅

( )

‖

퐵_5ꢨ

( )‖

( )

(40)

≤ [(Ξ ꢦ + Θ ꢦ 푡₀)Δ₆]^ꢨ+1휃₆

‖

퐵_6ꢨ

‖

퐵_7ꢨ

‖

퐵_8ꢨ

( )‖

푡

( )

≤ [(Ξ ꢦ + Θ ꢦ 푡₀)Δ₇]^ꢨ+1휃₇

( )‖

푡

( )

≤ [(Ξ ꢦ + Θ ꢦ 푡₀)Δ₈]^ꢨ+1휃₈

( )‖

푡

( )

‖

퐵_2ꢨ

( )‖

푡

Similarly, taking the limit on (39) as 푛 → ∞ and then using the condition (34), we have

→

‖

( )‖ ( )‖ ( )‖ ( )‖ ( )‖ ( )‖

‖

퐵_8ꢨ

0, 퐵_3ꢨ

푡

→ 0 , 퐵_4ꢨ

푡

→ 0, 퐵_5ꢨ

푡

→ 0, 퐵_6ꢨ

푡

→ 0, 퐵_7ꢨ

푡

→ 0 and

푡

→ 0. Therefore,

the existence of solutions of the model system (16) with (17) is proved.

We now consider the following conditions for the system of solutions to be unique.

Theorem 8: System (16) along with the initial conditions (17) has a unique solution if the following conditions

( ) ( )

hold: (1 − Ξ ꢦ ∆_푖+ Θ ꢦ Δ_푖푡 > 0),

for ꢠ = 1,2, … ,8.

(40)

( )

(16)

( ) ( )

( )

with

( )

(17)

( )

푡 , 퐻_휏_ꢨand 푊

( )

푡 } is another set of

Proof. Assume that {푆_휏_ꢨ푡 , 퐸_휏_ꢨ푡 , 퐼_휏_ꢨ푡 , 퐴_휏_ꢨ푡 , 푅_휏_ꢨ푡 , 푉

휏

ꢨ

solutions

( )

for

( )

the

( )

model

( )

in

addition

to

the

solution

set

( )

{푆_휏_ꢨ푡 , 퐸_휏_ꢨ푡 , 퐼_휏_ꢨ푡 , 퐴_휏_ꢨ푡 , 푅_휏_ꢨ푡 , 푉 푡 , 퐻_휏_ꢨand 푊

푡 } proved to exist in Theorems 6 and 7, then

휏

ꢨ

ꢃ

∫

( )

(

)

( )

(

)

(

)

(

)

푆_휏푡 − 푆_휏1푡 = Ξ ꢦ (퐾₁푡, 푆_휏− 퐾₁푡, 푆_휏1+ Θ ꢦ ₀(퐾₁푦, 푆_휏− 퐾₁푦, 푆_휏1)푑푦

(41)

Taking the norm and triangle inequality on both sides of (41), we have

ꢃ

( )

‖

푆_휏푡 − 푆_휏1

( )

푡

( )

(

)

( )

+ Θ ꢦ

(

)

‖

(

)‖

‖

(

₁)‖

≤ Ξ ꢦ 퐾₁푡, 푆_휏− 퐾₁푡, 푆_휏1

(퐾₁푦, 푆_휏− 퐾₁푦, 푆_휏

(42)

∫

0

Using the Lipschitz condition for the kernel 퐾₁, we find

( )

‖

푆_휏푡 − 푆_휏1

( )

푡

( )

푡

( ) ( )

‖

+ Θ ꢦ Δ₁푡 푆_휏푡 − 푆_휏1

( )

‖

≤ Ξ ꢦ Δ₁푆 푡 − 푆_휏1

푡

(43)

Rearranging (42), we have

( )

‖

푆_휏푡 − 푆_휏1푡 1 − Ξ ꢦ Δ_푖+ Θ ꢦ Δ_푖푡 ≤ 0

(44)

Lastly, putting in condition (42) for ꢠ = 1 to (44), we have

( )

‖

푆_휏푡 − 푆_휏1

( )

‖

푡

= 0

(45)

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( )

Hence 푆_휏푡 = 푆_휏1푡 . Now putting in a similar procedure to each of the following pairs

( )

(퐸_휏푡 , 퐸_휏푡 ), (퐼_휏푡 , 퐼_휏푡 ) , (퐴_휏푡 , 퐴_휏푡 ), (푅_휏푡 , 푅_휏푡 ) (푉 푡 , 푉

푡 ), (퐻_휏푡 , 퐻_푟1푡 )

휏

1

( )

and ( 푊 푡 , 푊

( )

푡 ) With inequality (42) for ꢠ = 1,2, … ,8, respectively, we have

푟

1

( )

푡 ), (퐻_휏푡 =

1

( )

(퐸_휏푡 = 퐸_휏푡 ), (퐼_휏푡 = 퐼_휏푡 ) , (퐴_휏푡 = 퐴_휏푡 ), (푅_휏푡 = 푅_휏푡 ), (푉 푡 = 푉

휏

1

( )

푡 )

퐻_푟푡 ) , and ( 푊 푡 = 푊

(46)

푟

1

Thus, the uniqueness of the solutions of the fractional order system is proved.

Adams-Bashforth-Moulton Scheme of the model system

Several numerical techniques have been proposed to solve a wide range of nonlinear fractional-order derivative

models arising in real-world problems. The analytical methods developed to handle these problems include

Adomian Decomposition, finite difference, and the Homotopy Decomposition Method. Here, we use the Adams-

Bashforth-Moulton method to provide an approximate solution for the model (1) based on the Predator-Corrector

algorithm.

ꢡ

Thus, setting ℎ = , 푡_ꢨ= 푛ℎ 푎푛푑 푛 = 0,1,2, . . . Νϵℤ⁺as in ([6, 10]), then system (1) can be discretized using

ꢊ

the approach of [12]. Now the corrector values are

훽₁(퐼^푝

+ 휃퐴^푝

)

훽₂푊^푝

ℎ^훼

(

)

(

)

(

)

휏 ꢨ+1

푟 ꢨ+1

푝

− 휇 푆_휏_ꢨ+1+ 휔푉 ^푝

(

)

[

]

)

푆_휏_ꢨ+1= 푆_휏₀

+

[Π −

1 − 퐶휀

+

푁^푝

휅 + 푊^푝₍

휏

ꢨ+1

Γ(α + 2)

(

)

푟 ꢨ+1

휏 ꢨ+1

푝

(

)

+ 휋퐻_휏_ꢨ+1− 훿 + 휉 + 휇 푆_휏_ꢨ+1

]

ꢨ

ℎ^훼

훽₁(퐼_휏_푖+ 휃퐴_휏_푖)

푁_휏_푖

훽₂푊

휏

푖

(

)

[

]

)

+

∑ ꢩ_푖,ꢨ+1Π −

1 − 퐶휀

+

− 휇 푆_휏+ 휔푉 + 휋퐻_휏

휏

푖

Γ(α + 2)

휅 + 푊

휏

푖

푖=0

(

)

]

− 훿 + 휉 + 휇 푆_휏_푖

,

훽₁(퐼^푝

+ 휃퐴^푝

)

훽₂푊^푝

ℎ^훼

(

)

(

)

(

)

휏 ꢨ+1

푟 ꢨ+1

푝

(

[

)

[

])

푆_휏_ꢨ+1− (휎 + 휇)퐸_휏_ꢨ+1

퐸_휏_ꢨ+1= 퐸_휏₀

+

1 − 퐶휀

+

]

푁^푝

휅 + 푊^푝₍

Γ(α + 2)

(

)

푟 ꢨ+1

휏 ꢨ+1

ꢨ

ℎ^훼

훽₁(퐼_휏_푖+ 휃퐴_휏_푖)

푁_휏_푖

훽₂푊

휏

푖

(

)

[(

[

])

]

푆_휏_푖− (휗 + 휇)퐸_휏_푖,

+

∑ ꢩ_푖,ꢨ+1

1 − 퐶휀

+

Γ(α + 2)

휅 + 푊

휏

푖

푖=0

ꢨ

ℎ^훼

푝

[

]

[

]

퐼_휏_(ꢨ+1)= 퐼_휏₀

+

휍휎퐸_휏_ꢨ+1− (훾₁+ 휙₁+ 휇)퐼_휏_ꢨ+1

+

∑ ꢩ_푖,ꢨ+1휍휎퐸_휏_푖− (훾₁+ 휙₁+ 휇)퐼_휏_푖,

Γ(α + 2)

푖=0

ℎ^훼

푝

[

]

퐴_휏_(ꢨ+1)= 퐴_휏₀

+

휎(1 − 휍)퐸_휏_ꢨ+1− (훾₂+ 휙₂+ 휇)퐴_휏_ꢨ+1

Γ(α + 2)

ℎ^훼

ꢨ

[

]

,

∑ ꢩ_푖,ꢨ+1휎(1 − 휍)퐸_휏_푖− (훾₂+ 휙₂+ 휇)퐴_휏_푖

Γ(α + 2)

푖=0

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ꢨ

ℎ^훼

푝

[

]

[

]

,

푅_휏_ꢨ+1= 푅_휏₀

+

훾₁퐼_휏_ꢨ+1+ 훾₂퐴_휏_ꢨ+1− 휇푅_휏_ꢨ+1

+

∑ ꢩ_푖,ꢨ+1훾₁퐼_휏_푖+ 훾₂퐴_휏_푖− 휇푅_휏_푖

Γ(α + 2)

푖=0

ꢨ

ℎ^훼

푝

푉

= 푉 +

훿푆_휏_ꢨ+1− (휔 + 휇)푉 ^푝

+

∑ ꢩ_푖,ꢨ+1훿푆_휏− (휔 + 휇)푉 ,

[

]

[

]

푖

휏

ꢨ+1

0

ꢨ+1

푖

Γ(α + 2)

푖=0

ꢨ

ℎ^훼

푝

[

]

[

]

퐻_휏_ꢨ+1= 퐻_휏₀

+

휉푆_휏_ꢨ+1− (휋 + 휇)퐻_휏_ꢨ+1

+

∑ ꢩ_푖,ꢨ+1휉푆_휏_푖− (휋 + 휇)퐻_휏_푖

Γ(α + 2)

푖=0

ꢨ

ℎ^훼

푝

푊

= 푊 +

휑₁퐼_휏_ꢨ+1+휑₂퐴_휏_ꢨ+1− 휗푊 ^푝

+

∑ ꢩ_푖,ꢨ+1휑₁퐼_휏+휑₂퐴_휏− 휗푊 ,

[

]

[

]

푖

휏

ꢨ+1

0

ꢨ+1

푖

Γ(α + 2)

푖=0

Where

ꢨ

훽₁(퐼_휏_푖+ 휃퐴_휏_푖)

푁_휏_푖

훽₂푊

1

휏

푝

푖

(

)

[

]

)

푆_휏_ꢨ+1= 푆_휏₀

+

∑ 푦_푖,ꢨ+1Π −

1 − 퐶휀

+

− 휇 푆_휏+ 휔푉 + 휋퐻_휏

휏

푖

Γ(α)

휅 + 푊

휏

푖

푖=0

(

)

]

− 훿 + 휉 + 휇 푆_휏_푖

,

ꢗ (ꢁ +ꢙꢄ

ꢂ

ꢥ

)

ꢗ ꢆ

1

ꢂ

2

ꢂ

푝

ꢨ

ꢥ

^ꢥ]) 푆_휏− (휗 + 휇)퐸_휏

ꢇ+ꢆ

ꢥ

∑

(

)

[

]

,

퐸_휏_ꢨ+1= 퐸_휏₀

+

_푖=0푦_푖,ꢨ+1( 1 − 퐶휀 [

+

푖

Γ(α)

ꢊ

ꢂ

ꢥ

1

푝

ꢨ

∑

[

]

,

퐼_휏_ꢨ+1= 퐼_휏₀

+

_푖=0푦_푖,ꢨ+1휍휎퐸_휏_푖− (훾₁+ 휙₁+ 휇)퐼_휏_푖

Γ(α)

1

푝

ꢨ

∑

[

]

,

퐴_휏_ꢨ+1= 퐴_휏₀

+

_푖=0푦_푖,ꢨ+1휎(1 − 휍)퐸_휏_푖− (훾₂+ 휙₂+ 휇)퐴_휏_푖

Γ(α)

1

푝

ꢨ

∑

[

]

,

푅_휏_ꢨ+1= 푅_휏₀

_푖=0푦_푖,ꢨ+1훾₁퐼_휏_푖+ 훾₂퐴_휏_푖− 휇푅_휏_푖

Γ(α)

1

푉 ^푝

휏

= 푉 +

푦_푖,ꢨ+1훿푆_휏− (휔 + 휇)푉 ,

ꢨ

푖=0

∑

[

]

휏

ꢨ+1

0

푖

Γ(α)

1

푝

ꢨ

∑

]

_푖=0푦_푖,ꢨ+1휉푆_휏_푖− (휋 + 휇)퐻_휏_푖

[

퐻_휏_ꢨ+1= 퐻_휏₀

+

Γ(α)

1

푊 ^푝

휏

= 푊 +

푦_푖,ꢨ+1휑₁퐼_휏+휑₂퐴_휏− 휗푊 , are the predictor values

ꢨ

푖=0

∑

[

]

휏

ꢨ+1

0

푖

Γ(α)

with

and

푛^훼+1− 푛 − ꢦ 푛 + 1 , ꢠ푓 ꢠ = 0,

(

)(

)

ꢪ+1

ꢩ_푖,ꢨ+1

=

^ꢪ+1, ꢠ푓 1 ≤ ꢠ ≤ 푛,

ꢨ

{

(

)

(

)

(

)

푛 − ꢠ + 2

+ 푛 − ꢠ

− 2 푛 − ꢠ + 1

1, ꢠ푓

ꢠ = 푛 + 1,

ℎ^훼

((

)^훼(

)^훼)

푛 − ꢠ ,

푦_푖,ꢨ+1

=

푛 − ꢠ + 1

0 ≤ ꢠ ≤ 푛,

α

(

)

where ꢫ is the order of accuracy given by ꢫ = min 2, 1 + ꢦ see [10].

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Numerical Simulation Results and Discussion

For this objective, we created a program that produced and executed MATLAB ODE45 solvers using the model-

fitted parameter values in Table 3. Figure 7 and Figure 8 show visual depictions of the model's state variable and

fractional-order impact on diphtheria dynamics.

Figure 7: Numerical simulations of the Caputo-Fabrizio fractional diphtheria model for different fractional

orders (ꢦ = 0.4,0.7,0.9,1.0).

Figure 8: Numerical illustration of the impact of the fractional order (ꢦ) on diphtheria dynamics.

Figure 7 illustrates the time-dependent behavior of the state variables of the Caputo-Fabrizio fractional diphtheria

transmission model for different fractional orders ꢦ = 0.4, ꢁ0.7, ꢁ0.9, ꢁ1.0. The fractional order parameter ꢦ

controls the intensity of memory and hereditary effects in the system. Lower values of ꢦ represent stronger

memory (non-local) effects, while ꢦ = 1.0 corresponds to the classical integer-order model. The comparison

clearly demonstrates how fractional-order dynamics modify the temporal evolution, peak magnitude, and

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persistence of the disease compartments, producing more realistic descriptions of biological processes influenced

by historical dependence.

Thus, Figure 7(A) depicts the susceptible population decreases more gradually for lower fractional orders,

indicating that memory slows the rate at which susceptible individuals exit the class through infection or

vaccination. In contrast, at ꢦ = 1.0, the decline is sharper and occurs earlier. This suggests that, under fractional

dynamics, the disease persists longer within the population because past states influence current susceptibility.

Consequently, control measures such as vaccination and awareness campaigns should be sustained over longer

periods when strong memory effects are present. The exposed class in Figure 7(B) exhibits a typical rise-and-

fall pattern, with the peak shifting to later times and becoming broader as ꢦ decreases. Lower fractional orders

thus produce delayed and prolonged exposure periods, reflecting the influence of incubation memory on disease

progression. For ꢦ = 1.0, the peak occurs earlier and decays faster, consistent with a system that lacks hereditary

behavior. Figure 7(C), the symptomatic infected population follows the classical epidemic curve, but with clear

dependence on ꢦ. Smaller values of ꢦ lead to delayed, lower, and wider infection peaks, while ꢦ = 1.0 yields a

sharp and short-lived peak. The fractional-order model, therefore, captures prolonged infection persistence and

slower decline, illustrating that memory dampens the intensity of outbreaks while extending their duration.

Figure 7(D) shows a similar pattern to that observed in the asymptomatic compartment. As ꢦ decreases, the

curve becomes broader and peaks later, suggesting longer persistence of undetected carriers. This supports the

interpretation that hereditary effects maintain subclinical infections for extended periods, potentially sustaining

hidden transmission within the population.

However, the recovered population in Figure 7(E) accumulates more rapidly at higher fractional orders. For

smaller ꢦ, recovery occurs gradually, indicating that memory slows the transition from infection to recovery.

The system therefore achieves its recovery plateau later under stronger memory effects. This emphasizes that

immunity accumulation in fractional models is delayed, aligning with biological systems where recovery and

immune response depend on past exposure history. Figure 7(F) depicts that the vaccinated class increases more

rapidly for higher fractional orders, while lower ꢦ values show slower growth. This highlights the role of

memory in delaying vaccination uptake or immune activation, possibly representing population behavioral

inertia or delayed immune responses. Hence, fractional models better capture the gradual, history-dependent

nature of immunization processes. The hesitant population in Figure 7(G) shows transient peaks that are more

prolonged for lower fractional orders. Memory effects cause hesitant individuals to persist longer before

transitioning out of hesitancy, representing the sustained influence of previous beliefs or misinformation. For

ꢦ = 1.0, this transient behavior dissipates quickly. This implies that behavioral interventions must be

consistently reinforced to overcome long-term hesitancy under memory-driven dynamics.

The concentration of Corynebacterium diphtheriae in the environment rises initially and decays over time. The

decay rate is slower for smaller ꢦ, indicating that environmental contamination retains influence for a longer

period due to fractional-order memory. Conversely, the integer-order model predicts a faster clearance of

bacteria. This finding reinforces the importance of sustained environmental sanitation measures in controlling

diphtheria transmission as depicted in Figure 7(H). The force of infection in Figure 7(I) follows the combined

pattern of infectious and environmental compartments, exhibiting early peaks that are smaller and more delayed

for lower fractional orders. As ꢦ increases, the infection force becomes more intense but shorter in duration.

This result confirms that memory attenuates and temporally disperses infection pressure, capturing realistic long-

tail epidemic behavior observed in empirical data.

Moreover, Figure 8 presents the influence of varying the fractional order ꢦ = 0.4,0.7,0.9,and 1.0 on the global

behavior of the Caputo-Fabrizio fractional diphtheria model. The Figure 8 (A–D) captures essential epidemic

characteristics the total number of infected individuals over time, peak infection magnitude, final outbreak size,

and the memory effect indicator, thereby demonstrating the quantitative and qualitative effects of fractional

calculus on disease dynamics.

The total Infected Individuals (퐼_휏+ 퐴_휏) Over Time in Figure 8(A) shows how the cumulative number of infected

individuals evolves across different fractional orders. At lower fractional orders (ꢦ = 0.4), infection dynamics

exhibit stronger oscillatory and persistent behavior, indicating that memory effects cause delayed decay and

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extended epidemic persistence. As ꢦ increases toward unity, the oscillations are dampened and the epidemic

stabilizes more rapidly, suggesting faster convergence to equilibrium. This behavior confirms that fractional-

order models capture the hereditary nature of infection and recovery processes more realistically than the integer-

order case. In the Peak Infections Figure 8 (B) the bar chart (top-right) shows that the peak number of infections

decreases markedly as the fractional order increases. The highest peak infection (≈ 1.66 × 10⁵) occurs at ꢦ =

0.4, followed by a moderate peak at ꢦ = 0.7(≈ 1.59 × 10⁵), while very low peaks are observed for ꢦ = 0.9(≈

5.9 × 10⁴) and ꢦ = 1.0(≈ 1.1 × 10⁴). This inverse relationship indicates that higher memory intensity (smaller

ꢦ) amplifies the persistence and magnitude of infection peaks, leading to slower epidemic suppression. Final

Outbreak Size Figure 8(C) the bottom-left bar chart displays the total number of individuals ultimately affected

by diphtheria. The final outbreak size declines progressively with increasing ꢦ, from approximately 5.54 ×

10⁵at ꢦ = 0.4 to 3.64 × 10⁴at ꢦ = 1.0. This trend demonstrates that systems with stronger memory retain

infection for longer periods and affect a larger portion of the population. Conversely, in the integer-order model

(ꢦ = 1.0), the outbreak terminates more quickly with a substantially reduced cumulative burden.

Furthermore, the Memory Effect Indicator in Figure 8(D), at the bottom-right, quantifies the memory effect

using an aggregated indicator derived from system variables. The indicator shows its highest value at ꢦ = 0.4

and declines steadily up to ꢦ = 0.9, before slightly increasing at ꢦ = 1.0. This non-linear pattern reflects the

role of fractional order in mediating memory strength: smaller ꢦ values encode long-term dependency of current

disease states on past events, while higher values (approaching unity) reduce this dependence and approximate

instantaneous dynamics.

Public Health Implications for Nigeria

From a policy perspective, these results have significant implications for diphtheria control in Nigeria, especially

in northern states such as Bauchi, Gombe, and Kano, where sporadic outbreaks have been reported [32].

Strengthening routine immunization programs, integrating community awareness campaigns, and improving

sanitation infrastructure are essential to sustain ℛ₀< 1. The model suggests that even modest improvements in

campaign efficiency or vaccination coverage can yield disproportionately large reductions in infection burden.

Thus, the adoption of a unified “Vaccinate-Educate-Sanitize” framework backed by sustained public health

investment could accelerate diphtheria elimination and enhance epidemic preparedness across the region.

CONCLUSION AND RECOMMENDATIONS

This study developed and analyzed a nonlinear deterministic diphtheria transmission model involving both direct

human contact and indirect environmental contamination, and incorporating environmental and public campaign

influences. The model integrates multiple control interventions, including vaccination and public enlightenment

strategies, while accounting for behavioral dynamics such as vaccine hesitancy and waning immunity.

Mathematical analyses were performed to establish well-posedness, equilibria, and stability properties, while

parameter estimation and numerical simulations validated the model’s epidemiological relevance.

.

Global Sensitivity Analysis

The global sensitivity analysis using the Partial Rank Correlation Coefficient (PRCC) revealed that the

transmission rate (훽₁), progression rate from exposed to infectious (휎), and vaccination rate (훿) exerted the

greatest influence on the basic reproduction number (ℛ₀). Positive correlations of 훽₁ and 휎 with ℛ₀indicate

that increases in infection transmission or faster progression to infectious stages amplify epidemic potential. In

contrast, parameters such as recovery rates (훾₁, 훾₂), vaccination (훿), and public campaign efficacy (휀) exhibited

strong negative correlations with ℛ₀, implying their effectiveness in suppressing disease spread. The sensitivity

results provide critical insight into priority areas for resource allocation, emphasizing vaccination campaigns,

rapid case management, and effective community sensitization as key levers for reducing disease transmission.

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Optimal Control Analysis

The optimal control analysis, derived using Pontryagin’s Maximum Principle, identified the most cost-effective

combinations of interventions. The optimal time-dependent control profiles demonstrated that an early, intensive

implementation of vaccination and public campaign strategies, followed by sustained treatment interventions,

yields the greatest epidemiological impact. The results indicated a substantial reduction in both symptomatic and

asymptomatic infections, confirming the importance of synergistic intervention strategies. The control

trajectories further suggest that sustained vaccination, supported by periodic community enlightenment, remains

the most efficient pathway for achieving long-term disease elimination.

Cost-Effectiveness Analysis

The cost-effectiveness assessment based on the incremental cost-effectiveness ratio (ICER) revealed that the

combined vaccination and public enlightenment campaign strategy dominated other single interventions,

providing the lowest cost per Disability-Adjusted Life Year (DALY) averted. The results classified this strategy

as “very cost-effective” according to WHO-CHOICE thresholds. In contrast, treatment-only interventions,

though epidemiologically beneficial, were less economically efficient due to a higher cost per health gain. This

finding underscores that investment in vaccination programs coupled with effective behavioral change

communication is the most sustainable and economically sound approach to diphtheria control.

Caputo-Fabrizio fractional model

The comparison across different fractional orders reveals that the Caputo-Fabrizio fractional model effectively

incorporates hereditary and memory characteristics into the diphtheria dynamics. Lower fractional orders

produce slower system responses, delayed epidemic peaks, and extended persistence of infection and

environmental contamination. Conversely, the integer-order model (ꢦ = 1.0) exhibits rapid rises and declines

typical of memoryless processes. These results demonstrate that fractional calculus provides a powerful

framework for modeling biological systems with inherent memory, yielding more accurate and biologically

consistent predictions of disease progression and control outcomes.

Numerical Simulation and Model Behavior

Numerical simulations corroborated the analytical results, illustrating the dynamic interplay between susceptible,

exposed, infectious, and recovered subpopulations under different intervention scenarios. The model

demonstrated that in the absence of control measures, diphtheria spreads rapidly and stabilizes at a high endemic

level. However, under combined control measures, the infection burden decreases sharply, with eventual disease

eradication achievable when vaccination and public campaign efforts are intensified. The simulations further

revealed that waning immunity and hesitancy can delay elimination, highlighting the need for continuous re-

vaccination and public health communication.

In conclusion, this study establishes, through mathematical modeling, that diphtheria elimination is achievable

with an integrated and adaptive control strategy. The analysis identifies the basic reproduction number (ℛ₀) as

a key threshold parameter and quantifies the roles of asymptomatic and environmental transmission. The results

demonstrate that a synergistic combination of vaccination, timely outbreak response, and community

engagement is essential for effective control, as no single intervention alone is sufficient. The time-dependent

optimal control framework further highlights the need for dynamic, data-driven public health actions.

Collectively, these insights offer a robust theoretical foundation and practical guidance for designing efficient

diphtheria control and elimination programs. Moreover, the results confirm that the Caputo-Fabrizio fractional

derivative effectively integrates memory and hereditary properties into diphtheria transmission modeling. Lower

fractional orders (ꢦ < 1) yield slower epidemic decay, larger outbreak sizes, and sustained oscillatory behavior,

consistent with biological processes influenced by immune memory, environmental persistence, and delayed

behavioral responses. In contrast, the integer-order model (ꢦ = 1.0) exhibits a rapid epidemic rise and decay,

corresponding to a system without memory. Therefore, fractional calculus offers a powerful tool for capturing

the realistic temporal spread and long-term influence of infectious diseases like diphtheria.

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Some limitations present avenues for future research in this study, including the incorporation of spatial

heterogeneity or a contact network, the cost functions in optimal control analysis, and the age structure, as

diphtheria transmission and severity vary significantly with age.

Policy and Public Health Recommendation

The findings underscore the urgent need for sustained vaccination coverage, regular community education, and

targeted behavioral interventions to address vaccine hesitancy. Policymakers should prioritize integrated

intervention programs that combine immunization, treatment accessibility, and public awareness campaigns.

Environmental sanitation and surveillance systems should also be strengthened to minimize bacterial persistence

and prevent the resurgence of infections.

Authors’ Contributions

All authors participated and have read and approved the final manuscript in this research work.

Authors’ Declaration

The authors declare no known conflict of interest.

Acknowledgement

TETfund sponsors this research work under the Institutional Base Research 2025 intervention.

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