

































IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA 

INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS 

   

  

 

Abstract: 

Diphtheria is a bacterial infectious disease that can lead to severe complications and even deaths. This work 

presents the Caputo-Fabrizio Fractional derivatives of the aged-structured deterministic model of diphtheria 

infection. The existence and the uniqueness of the solution of the model are investigated and established using 

the contraction principle. The stability of the model is investigated with the help of the well-known Ulem-Hyers 

and the generalized Ulem-Hyers theorems. Analyzing the model using the Laplace Adomian Decomposition 

Methods, the system’s analytical solution, in the form of an infinite series that converges quickly to it exact 

value is obtained. 

KeyWords: Diphtheria, Caputo-Fabrizio, Adomian Decomposition, aged-structured, contraction principle  

Introduction  

1.1 Introduction    

Diphtheria is one of the respiratory diseases raphaging the population in recent time. It is a bacterial 

(Corynebacteriumdiptheriae) infectious disease that can lead to severe complications such as respiratory failure, 

heart problems and even deaths if it is not detected early. This infection that mostly affects the throat and the 

nose can be prevented by vaccination. Case-fatality occurs only in places where there is poor sanitation 

condition and inadequate vaccination coverage as a result of low resources [1,2,3]  

Diphtheria is a highly contagious infection that spreads primarily through person-to-person contact via 

respiratory droplets(coughing, sneezing or spitting) or direct contact with infected skin lessions or any material 

(like clothe) that has been in contact with the bacteria. It is possible to get diphtheria more than once. Anyone 

who is not protected by diphtheria vaccine and comes in close contact with diphtheria is susceptible. Some of 

the symptoms/signs of diphtheria are: throat pain, weakness/ fatigue, fever, swollen neck glands, problems 

breathing due to tissues obstructing nose and throat, nerve, kidney or heart problems ( if the bacteria enters the 

blood stream). The incubation period is one to ten days after exposure( 4, 5, 6,7,8) 

  Thomas, Henry Sylvester and Udofia, Ekere Sunday

                                Akpan, Ubong Dominic Department of mathematics, AKSU

          Uwakwe, Joy Ijeoma, Department of mathematics,Alvan Ikoku University of Education, Owerri

Department of Mathematics AkwaIbom State University, IkotAkpaden, AkwaIbom State, Nigeria

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

To effectively contend the spread of Diphtheria, different control measures, such as isolation of patient, 

maintenance of  one meter between patients, keeping  patient care areas with good ventilation, the use mask that 

is medically prepared and cover any wound/lesions on patient’s body by  patient who may  have to move out of 

the isolation areas,  population subgroup such as young children under five years of age, school children, elderly 

who are at greater risk and have close contact with diphtheria infection and health workers should highly 

prioritized with treatment and vaccination, epidemiological surveillance ensuring early detection of diphtheria 

outbreak, administering antitoxin to neutralize the toxin and antibiotics to kill the bacteria, reducing 

complication and mortality should be implemented [6,7,8].  

Understanding, describing and analyzing the dynamics of infectious diseases [ 9, 10, 11, 12, 13, 14, 15,16, 

17,18] have been key in guiding decisions and policies in public health system. 

In recent time, mathematicians and epidemiologist have demonstrated great effort in understanding and 

describing the dynamics of diphtheria infection.  

[19], presented the mathematical model of diphtheria diseases that categorizes the individuals based on 

susceptibility, vaccination, infected and recovery status. The stability of the system wa confirmed, the basic 

reproductive ratio was calculated. They also converted the deterministic model into caputo-Fabrizio fractional 

order model, analyzed it for existence and uniqueness of solution using appropriate principle. Adomian 

Decomposition method was applied for numerical solution of the model. 

In the research under consideration, we seek to transform the aged-structured deterministic model of diphtheria 

infection [8,9] into the Caputo-Fractional order of aged-structured model of diphtheria infection. The existence 

and the uniqueness of the solution of the model shall be investigated and established using the contraction 

principle. The stability of the model shall be investigated with the help of the well-known Ulem-Hyers and the 

generalized Ulem-Hyers theorems. Analyzing the model using the Laplace Adomian Decomposition Methods, 

the system’s analytical solution, in the form of an infinite series that converges quickly to it exact value shall be 

obtained.  

2.0 MODEL FORMULATION 

2.1 ASSUMPTIONS 

1. The control of diphtheria is based on primary prevention of disease by ensuring high population immunity of the 

infant ((0-1 year) and school children by vaccination 
2. Isolation of detected cases (that is confirmed cases are not allowed to interact with the population freely.  

3. Epidemiological surveillance ensuring early detection through contact tracing is carried out 
4. Secondary prevention spread by the rapid investigation of close contacts to ensure prompt treatment of those 

infected 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

5. The total population of human  at time  � under consideration denoted by �� is split into mutually 

exclusive sub-population of �� Susceptible infant at time  � (0-1years), �� , Susceptible school children 

population at time  �, �, Vaccination population at time  �, �,  Exposed population at time  � , ��, 

Asymptomatic infection population at time  �, ��., Symptomatic infection population at time  �, �., 

Recovered  population at time  �. ��., Detected infectious humans at time  �(Asymptomatic and 
symptomatic)  population through testing, 

��   =  ��    +  ��  + � + � + �� +  �� + � +  �� 

2.2 STATE VARIABLES  

�� −The total population of human at time  � under consideration  

�� − Susceptible infantat time  � (0-1years), 

�� − Susceptible school children populationat time  �, 

� −Vaccination populationat time  �, 

� −  Exposed populationat time  , 

��. − Asymptomatic infection populationat time  �, 

��. − Symptomatic infection populationat time  �,  

� −Recovered  populationat time  �. 

��. − Detected infectious humansat time  �(Asymptomatic and symptomatic)  population through testing, 

 

2.3 PARAMETERS 

 

�� − �������� ���� ���� ������� �� ��, Asymptomatic infection population  

�� − �������� ���� ����� ������� �� ��, symptomatic infection population 

� − �������� �� ��� ��������� �ℎ�� ���  ��, Asymptomatic  

1 − � − �������� �� ��� ��������� �ℎ�� ���  ��, symptomatic  

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

�� − ��������� ������������ ���� ���� �� 

�� − ��������� ������������ ���� ���� �� 

�� − ����������� ��������  ���  �� 

�� − ����������� ��������  ���  �� 

� − ������� ��������  

� − �������� ���� ����  �� �� �� 

� − ��� ������ ����ℎ ���� �� ℎ�����  

��� − ������������ ���� ��  �� 

��� − ������������ ���� ��  �� 

� −  ����  �� ����������� 

�� − ��������� ���� (��� ������� ������� )  ���  �� 

�� − ��������� ���� (��� ������� ������� )  ���  �� 

� − ����ℎ ��� �� ���������  

� − ������� ����ℎ ���� 

��(�)   =  
��(���� + ����)

�� − ��
 

��(�)   =  
��(���� + ����)

�� − ��
 

 

2.4 MODEL EQUATIONS  

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

Using the above described state variables and parameters together with the schematic diagram 

in figure 1, the model of the diphtheria infection transmission dynamics results in the following 

system of deterministic non-linear first order differential equations  

���

��
  = ��� − 

��(���� + ����)

�� − ��
�� − ��1�1 − ��� −  ��1               (2.1) 

���

��
  =  ��� − 

��(���� + ����)

�� − ��
�� − ��2�2 −  ��2                    (2.2) 

��

��
  = ��1�1 +  ��2�2 −  ��                                         (2.3) 

��

��
  =   

��(���� + ����)

�� − ��
�� +  

��(���� + ����)

�� − ��
��  − �1�� − �2�1 − ��� −  ��      (2.4) 

���

��
  =  �1�� − ���1 − �1�1 − ��1 − ��1                         (2.5) 

���

��
  =  �2�1 − ��� − ���2 − �2�2 − ��2 − ��2                   (2.6) 

���

��
  =  �1�1 + �2�2 − ���� − ��� − ���                               (2.7) 

��

��
  = ���1 + ���2 + ���� −  ��                                              (2.8) 

 

2.5 FRACTIONAL ORDER MODEL 

 TheCaputo-Fabrizio order derivatives of  (2.1) –(2.8)  is given as follows 

��
�

��(�)�
�� = ��� −  

��(���� + ����)

�� − ��
�� − ����� − ��� −  ���               (3.1) 

��
�

��(�)�
��   =  ��� −  

��(���� + ����)

�� − ��
�� − ����� −  ���                    (3.2) 

��
�

�(�)�
��   = ����� +  ����� −  ��                                         (3.3) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

��
�

�(�)�
��   =   

��(���� + ����)

�� − ��
�� +   

��(���� + ����)

�� − ��
��  − ���� − ��(1 − �)� −  ��      (3.4) 

��
�

��(�)�
�   =  ���� −  ���� − ���� − ��� − ���                         (3.5) 

��
�

��(�)�
��   =  ��(1 − �)� − ���� − ���� − ��� − ���                   (3.6) 

��
�

��(�)�
��   =  ���� +  ���� −  ���� − ��� − ���                               (3.7) 

��
�

 �(�)�
��   = ���� + ���� + ���� −  ��                                              (3.8) 

ssss 

With the initial conditions,  

��(0) =  ���, ��(�)  =  ���, �(�)  =  ��, �(�) = ��,  ��(�) = ���, ��(�) = ���, , ��(�)  =  ����(�) = �� 

��(�) +    ��(�) + �(�) + �(�) + ��(�) + ��(�) + ��(�) +  �(�) = 1 

��
�

�
�� Represents the Caputo-Fabrizio fractional derivative of order � �[0, 1] 

3.0 ANALYSIS OF THE MODEL 

3.1 EXISTENCE AND UNIQUENESS OF SOLUTION 

We shall use some basic fixed point theorem to establish the existence and uniqueness of the 

solution of (3.1) –(3.8) 

��
�

�(�)�
�� = ℒ(�, �(�) 

         (3.9) 

�(0) = �� 

�(�) = ���(�), ��(�), �(�), �(�), ��(�), ��(�),   ��(�), �(�)�
�

∈ ℜ� ��� � ∈ [0. ����] denotes the state of the 

model and ℒ represent the continuous vector given below 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

ℒ =  

⎝

⎜
⎜
⎜
⎜
⎜
⎛

ℒ2

ℒ3

ℒ4

ℒ5

ℒ6

ℒ7

ℒ8
⎠

⎟
⎟
⎟
⎟
⎟
⎞

  =  

⎝

⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎛

��� −  
��(���������)

�����
�� − ����� − ��� −  ���

��� − 
��(���������)

�����
�� − ����� −  ���     

����� +  �����  − �� −  ��
��(���������)

�����
�� +   

��(���������)

�����
��  − ���� − ��(1 − �)� −  ��

���� −  ���� − ���� − ��� − ���

��(1 − �)� −  ���� − ���� − ��� − ���

���� + ���� − ���� − ��� − ���    

���� +  ���� + ���� −  �� ⎠

⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎞

 (3.10) 

The initial condition of the variable of the model is denoted by ���(0), ��(0), �(0), �(0), ��(0),

  �20,   ��0, �0� 

Where �(0) = ���(0), ��(0), �(0), �(0), ��(0), ��(0),   ��(0), �(0)�
�

 ��� , �� = (���, ���, ��, ��, ���,

  �20,   ��0, �0� 

In addition, we define ℒ: [0. ����] × ℜ� →   ℜ� is said to satisfy Lipschitz condition in the second argument, if 
we have: 

�ℒ��, ��� − ℒ��, ���� ≤ ���� − ��� ∀ � ∈ [0, ����] ∀ ���, ��  ∈ ℝ� �ℎ��� �

> 0, ���� �� ����� ���� … … … … … … (3.11) 

The existence of a unique solution to the model (3.1)    -   (3.8)is established in the following 
theorem: 
Theorem 3.1. 
There exists a unique solution to the initial value problem (3.9) on �([0, ����], ℝ�), provided 
that (3.11) and 

�
2(1 − �)�

(2 − �)ℱ(�)
+

2��

(2 − �)ℱ(�)
����� < 1 … … … … … … . . (3.12) 

are satisfied. 
Proof: 
If we apply the Caputo-Fabrizio fractional integral on each sides of (3.9), then we 
havetheCaputo-Fabrizio time-fractional integral of the function �(�) of order � is defined by 

�(�) =   �� +
2(1 − �)

(2 − �)ℱ(�)
�(�, �) +

2�

(2 − �)ℱ(�)
� ℒ(�, �(�)��

�

0

 (3.13)  

Let us defined the operator Κ: �([0, ����], ℝ�)    → �([0, ����], ℝ�)    by  

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

Κ[�](�) = �(�), �, V ∈ ��[0, ����], ℝ��                         (3.14) 
 
Where  

�(�) =   �
0

+
2(1 − �)

(2 − �)ℱ(�)
�(�, �) +

2�

(2 − �)ℱ(�)
� ℒ(�, �(�)��

�

�

 

The supremum norm on ��[0, ����], ℝ�� is given by: 

‖�(�)‖ =  ‖�(�)‖
�∈[�,����]

���
  , ∀ � ∈ �([0, ����], ℝ�)   

Clearly, �([0, ����], ℝ�) equipped with ‖. ‖ is a Banach space. 
Suppose, ℘ is the fixed point of the operator  
Κ: �([0, ����], ℝ�)    → �([0, ����], ℝ�), then ℘ becomes the solution of the initial value 
problem (3.9), and 

Κ℘(�) = ℘(�) 
Where  

℘(�) =   �
0

+
2(1 − �)

(2 − �)ℱ(�)
ℒ(�, ℘(t)) +

2�

(2 − �)ℱ(�)
� ℒ(�, ℘(�)��

�

�

 

Consider  
‖K[�](�) − K[℘](�)‖

=  ��
0

+
2(1 − �)

(2 − �)ℱ(�)
ℒ(�, V(t)) +

2�

(2 − �)ℱ(�)
� ℒ(�, �(�)��

�

�

− ��
0

+
2(1 − �)

(2 − �)ℱ(�)
ℒ(�, ℘(t)) +

2�

(2 − �)ℱ(�)
� ℒ(�, ℘(�)��

�

�

�� 

‖K[�](�) − K[℘](�)‖

≤  �
2(1 − �)

(2 − �)ℱ(�)

1

Γ(�)
�ℒ��, V(t)� − ℒ��, ℘(t)��

+
2�

(2 − �)ℱ(�)
�[ℒ(�, �(�) − ℒ(�, ℘(�)]��

�

�

� 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

≤  
2(1 − �)

(2 − �)ℱ(�)
��ℒ��, V(t)� − ℒ��, ℘(t)���

+
2�

(2 − �)ℱ(�)
��[ℒ(�, �(�) − ℒ(�, ℘(�)]��

�

�

�        (3.15) 

 
Since the operator ℒsatisfies the Lipschitz condition (eq. 3.11), we have that 

‖K[�](�) − K[℘](�)‖ ≤
2(1 − �)

(2 − �)ℱ(�)
�‖V(t) − ℘(t)‖ +

2��

(2 − �)ℱ(�)
�‖V(t) − ℘(t)‖��

�

�

 

 
‖K[�](�) − K[℘](�)‖  

≤
2(1 − �)

(2 − �)ℱ(�)
� ‖V(t) − ℘(t)‖

�∈[�,����]
���

+
2��

(2 − �)ℱ(�)
� ‖V(t) − ℘(t)‖��

�  ��� 

� �∈[�,����]

           (3.16) 

 
 

≤
2(1 − �)�

(2 − �)ℱ(�)
+

2�� ∫ ��
�  

�  

(2 − �)ℱ(�)
‖V − ℘‖ 

≤
2(1 − �)�

(2 − �)ℱ(�)
+

2������

(2 − �)ℱ(�)
‖V − ℘‖ 

Thus if the condition (3.12)holds then, 

‖K[�](�) − K[�](�)‖ ≤
2(1 − �)�

(2 − �)ℱ(�)
+

2������

(2 − �)ℱ(�)
‖V − W‖ 

 
 

Hence, the operator K becomes a contraction. Therefore K has a unique fixed point 
which is a solution to the initial value problem (3.9)and hence asolution to the 
system ((3.1)-(3.8)). 
 
 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

ULAM-HYERSSTABILITY 
TheUlam-Hyers (UH) stability and generalized UH stability [20,21]for the fractional system 
(3.1) –(3.8) using the Caputo-Fabrizio operator is discussed in this 
section.LetS = �([0, ����]: ℝ�) be the space of all continuous functions from [0, ����]toℝ�, 

endowed with the norm:‖�‖ = ‖�‖
�∈[�,����]

���
,  

Consider 

��
�

�
�� �(�) = ℒ��. �(�)�                                         (3.17) 

�(�) =  � 0 

Also let ε > 0. Consider the following inequality: 

� ��
�

�
�� ��(�) − ℒ��. ��(�)�� ≤ �, � �ℑ,   � = max(��)� , � = 1,2,3, … ,8, �� ∈ S     (3.18) 

Remark 3.1. “A function � ∈ S satisfies the inequality (3.18)if and only if there exists a 

function p ∈ S, having the following properties; 

(�)|�(�)| ≤ �, � = max���� , � �ℑ (ii) ��
�

�
� ��(�) = ℒ��. ��(�)� + �(�), � �ℑ 

Definition 3.1. The fractional model ((3.1)-(3.8))or the transformed system (3.17)is UH stable 
if for every ε > 0 there exists k > 0, such that for any solution 
φ ∈ S of the inequality (3.18), there exists a unique solution � ∈ S, of the fractional system 
(3.17)such that the following inequality is satisfied: 

‖��(�) − �(�)‖ ≤ ��, � �ℑ� = max���� , � = 1, 2, 3, … , 8            (3.19) 

 
Where  

�(�) = ���(�), ��(�), �(�), �(�), ��(�), ��(�),   ��(�), �(�)�
�

 

��(�) = ���̅
� (�), ��

� (�), ��(�), ��(�), ��
�(�), ��

�(�),   ��
� (�), ��(�)�

�

 

�(0) = ���(0), ��(0), �(0), �(0), ��(0), ��(0), ��(0), �(0)�
�
 

Definition 3.2. Themodel system (3.17) is generalized UH stable if there exists a continuous 

function�: ℝ+ → ℝ+satisfying�(0) =  0, such that for anysolution � ∈ S of system (3.18), there 

exists a unique solution � ∈ S such that the following inequality is satisfied: 

‖��(�) − �(�)‖ ≤ �(�), � �ℑ� = max ����
�

, � = 1, 2, 3, … , 8         (3.20) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

 
Theorem 3.4. If Φ ∈ Ε satisfies the system (3.17), then we have the following: 

���(�) − ��
�

(�) −
2(1 − �)

(2 − �)ℱ(�)
�(�, �) +

2�

(2 − �)ℱ(�)
� ℒ��. ��(�)���

�

0

� ≤ Ωε 

�ℎ��� Ω =
2(1 − �)

(2 − �)ℱ(�)
+

2�

(2 − �)ℱ(�)
� ��

�

0

 (3.19) 

Proof:  

Using remark 3.1(ii) ��
�

�
�� ��(�) = ℒ��. ��(�)� + �(�), � �ℑ, which on applying the Caputo-

Fabrizio integral gives 

��(�) =  ���
(�) +

2(1 − �)

(2 − �)ℱ(�)
�(�, �) +

2�

(2 − �)ℱ(�)
� ℒ��. ��(�)���

�

�

+
2(1 − �)

(2 − �)ℱ(�)
�(�)

+
2�

(2 − �)ℱ(�)
� �(�)��

�

�

 

By rearranging, applying norm on both sides and using remark 4.1 (i), it follows that; 

���(�) − ���
(�) −

2(1 − �)

(2 − �)ℱ(�)
�(�, �) −

2�

(2 − �)ℱ(�)
� ℒ��. ��(�)���

�

�

�

≤
2(1 − �)

(2 − �)ℱ(�)
�(�) +

2�

(2 − �)ℱ(�)
�|�(�)|��

�

�

≤ Ωε 

 
 
 
Theorem 3.5. Supposeℒ: ℑ × ℝ� → ℝ� satisfies the Lipschitz condition, with Lipschitz 
constant � > 0 and(1 − Ω) � > 0, then the model (3.17)is generalized UH stable. 
Proof: 
Suppose that �∈S satisfies the inequality in (3.18)and �∈S is a unique solution of (3.17). Then 
∀ε > 0;  ��[0, ����], we have 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

|��(�) − �(�)|

=  ���(�) − ���
(�) −

2(1 − �)

(2 − �)ℱ(�)
��(�, �) −

2�

(2 − �)ℱ(�)
� ℒ��. ��(�)���

�

�

�

��[�,����]

���

 

≤ ���(�) −  ���
(�) −

2(1 − �)

(2 − �)ℱ(�)
��(�, �)  −

2�

(2 − �)ℱ(�)
� ℒ��. ��(�)���

�

�

�

��[�,����]

���

+ ��
2(1 − �)

(2 − �)ℱ(�)
��(�, �) −

2(1 − �)

(2 − �)ℱ(�)
�(�, �)��

��[�,����]

���

+
2�

(2 − �)ℱ(�)
��ℒ��. ��(�)� − ℒ��. �(�)��

�

�

��

��[�,����]

���

 

≤ ���(�) −  ���
(�) −

2(1 − �)

(2 − �)ℱ(�)
��(�, �)  −

2�

(2 − �)ℱ(�)
� ℒ��. ��(�)���

�

�

�

��[�,����]

���

+
2(1 − �)

(2 − �)ℱ(�)
|��(�, �) − �(�, �)|

��[�,����]

���

+
2�

(2 − �)ℱ(�)
�|��(�) − �(�)|

�

�

��

��[�,����]

���

 

 
≤ �Ω + Ωℳ|�� − �| 

Thus, we have ‖�� − �‖ ≤ ��, �ℎ��� � =
�

���ℳ
              (3.20) 

Hence equating �(�) = ��, so that �(0) = 0we conclude that the model (3.16),is both UH and 
generalized UH stable. 
 
4.0 Iterative schemes involving the Caputo fractional operators 
This section, we shall study an iterative scheme using the Caputo-Fabrizio fractional derivative. 
We seek to derive explicit expressions for the unknown 

functions,��(�), ��(�), �(�), �(�), ��(�), ��(�), �(�), ��(�) using series representation approach based on 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

Laplace Adomian Decomposition Method. The system is transformed to algebraic equations by the 

application of Laplace transform to the Caputo-Fabrizio  fractional order derivative (3.1) – (3.8). 

Laplace Adomian Decomposition Method empowers us to construct a convergent series solution for 

��(�), ��(�), �(�), �(�), ��(�), ��(�), �(�), ��(�)which can be evaluated numerically to obtain accurate 

approximations. 

 
For the solution of themodel(3.1)-(3.8), we shall adopt the Laplace Adomian 
Decompositionmethod. Applying the Laplace transform of the Caputo-Fabrizio fractional 
operator toboth sides of the system (3.1)-(3.8), we have 
 
 
 
 
 
 
 
 

ℒ � ��
�

��(�)�
�� � = ℒ ���� −  

��(���� + ����)

�� − ��
�� − ����� − ��� −  ����               (4.1.1) 

ℒ � ��
�

��(�)�
�� �   =  ℒ ���� −  

��(���� + ����)

�� − ��
�� − ����� −  ���     �               (4.1.2) 

ℒ � ��
�

�(�)�
�� �   = ℒ{����� +  �����  − �� −  ��}                                         (4.1.3) 

ℒ � ��
�

�(�)�
�� �   =  ℒ �

��(���� + ����)

�� − ��
�� +  

��(���� + ����)

�� − ��
��  − ���� − ��(1 − �)� −  ���      (4.1.4) 

ℒ � ��
�

��(�)�
�� �   = ℒ{���� −  ���� − ���� − ��� − ���}                         (4.1.5) 

ℒ � ��
�

��(�)�
�� �  =  ℒ{��(1 − �)� − ���� − ���� − ��� − ���}                   (4.1.6) 

ℒ � ��
�

 �(�)�
� �   = ℒ{���� +  ���� +  ���� −  �� }                                             (4.1.7) 

ℒ � ��
�

��(�)�
�� �   =  ℒ{���� + ���� − ���� − ��� − ���    }                           (4.1.8) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

Using the property of Laplace transform for Caputo-Fabrizio fractional derivatives, we obtain 

Following the definition of Laplace transform for the Caputo-Fabrizio derivative, 
(Laplace transform of the Caputo-Fabrizio derivative of functions) 
 

The Laplace transform of the Caputo-Fabrizio derivative is given by 

ℒ{ ��
�

�
�� �(�, �)}(�) =  

(2 − �)ℱ(�)

2

�ℒ{�(�, �)} − �(�, 0)

� + �(1 − �)
 

 

�ℒ{�1(�)} − �1(0)

� + �(1 − �)
= 

ℒ ���� −  
��(���� + ����)

�� − ��
�� − ����� − ��� −  ����               (4.2.1) 

�ℒ{�2(�)} − �2(0)

� + �(1 − �)
=    ℒ ���1 −  

�
2
��1�1 + �2�

2
�

�ℎ − ��

�2 − ��2�2 −  ��2     �               (4.2.2) 

�ℒ{�(�)} − �(0)

� + �(1 − �)
= ℒ{��1�1 +  ��2�2  − �� −  ��}                                         (4.2.3) 

�ℒ{�(�)} − �(0)

� + �(1 − �)

=   ℒ �
�

1
(�1�1 + �2�

2
)

�ℎ − ��

�1 +   
�

2
(�1�1 + �2�

2
)

�ℎ − ��

�2  − �1�� − �2(1 − �)� −  ���      (4.2.4) 

�ℒ{�1(�)} − �1(0)

� + �(1 − �)
=  ℒ��1�� −  �

1
�1 − �1�1 − ��1 − ��1�                         (4.2.5) 

�ℒ{�2(�)} − �2(0)

� + �(1 − �)
=   ℒ��2(1 − �)� −  �

2
�2 − �2�2 − ��2 − ��2�                   (4.2.6) 

�ℒ{��(�)} − ��(0)

� + �(1 − �)
=   ℒ��1�1 +  �2�2 −  �

3
�� − ��� − ���    �                           (4,2.7) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

�ℒ{�(�)} − �(0)

� + �(1 − �)
=  ℒ��

1
�1 +  �

2
�2 +  �

3
�� −  �� �                                             (4.2. 8) 

ℒ{�1(�)} =
�1(0)

�
+

� + �(1 − �)

�
ℒ ���ℎ −  

�
1
��1�1 + �2�

2
�

�ℎ − ��

�1 − ��1�1 − ��1 −  ��1�               (4.3.1) 

ℒ{�2(�)} =
�2(0)

�
+

� + �(1 − �)

�
 ℒ ���1 −  

�
2
��1�1 + �2�

2
�

�ℎ − ��

�2 − ��2�2 −  ��2     �               (4.3.2) 

ℒ{�(�)} =
�(0)

�
+

� + �(1 − �)

�
ℒ{��1�1 +  ��2�2  − �� −  ��}                                         (4.3.3) 

ℒ{�(�)} =
�(0)

�

+
� + �(1 − �)

�
  ℒ �

�
1

(�1�1 + �2�
2

)

�ℎ − ��

�1 +   
�

2
(�1�1 + �2�

2
)

�ℎ − ��

�2  − �1�� − �2(1 − �)�

−  ���      (4.3.4) 

ℒ{�1(�)} =
�1(0)

�
+

� + �(1 − �)

�
 ℒ��1�� −  �

1
�1 − �1�1 − ��1 − ��1�                         (4.3.5) 

ℒ{�2(�)} =
�2(0)

�
+

� + �(1 − �)

�
  ℒ��2(1 − �)� −  �

2
�2 − �2�2 − ��2 − ��2�                   (4.3.6) 

ℒ{��(�)} =
��(0)

�
+

� + �(1 − �)

�
 ℒ��

1
�1 +  �

2
�2 +  �

3
�� −  �� �                                             (4.3. 7) 

 

ℒ{�(�)} =
�(0)

�
+

� + �(1 − �)

�
 ℒ��1�1 +  �2�2 −  �

3
�� − ��� − ���    �                           (4.3.8) 

According to the Adomian decomposition method, the solution will be in the following series 
type 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

��(�) = ∑ ���(�)�
��� , ��(�) = ∑ ���(�)�

��� , �(�) = ∑ ��(�)�
��� , �(�) = ∑ ��(�)�

��� , ��(�) =
∑ ���(�)�

��� , ��(�) = ∑ ���(�)�
��� ��(�) = ∑ ���(�)(�)�

��� ,             R(�) = ∑ ��(�)�
���              

(4.4) 
 

The non- linear term involved in the model are ��(�)��(�),         ��(�)��(�), ��(�)��(�) ,    ��(�)��(�). These are 

decomposed by Adomian Decomposition polynomial as 

��(�)��(�) = ∑ ���
�
��� , ��(�)��(�) = ∑ ���

�
��� , ��(�)��(�) = ∑ ���

�
��� ,   ��(�)��(�) = ∑ ���

�
���       (4.5) 

Where �� is Adomian polynomial defined as: 

�� =
�

�(���)

��

���
�∑ ℎ���

�
��� ∑ ℎ���

�
��� �

� ��
           (4.6) 

�� =  ���� ,    �� =  ���� + ����,   �� =  ���� +  ���� + ����,    �� =  ���� +  ���� + ���� + ����,.     .     .      (4.7) 

 

Applying equation (4.4)-(4.7) into the system (4.3.1)-(4.3.8), we have 

ℒ �� ���(�)

�

���

�

=
�1(0)

�

+
� + �(1 − �)

�
ℒ ���ℎ − −

�
1

�1 ∑ �1�
∞
�=0

�ℎ − ∑ ���
∞
�=0

−
�

1
�2 ∑ �2�

∞
�=0

�ℎ − ∑ ���
∞
�=0

− (��1 + � +  �) � �1�

∞

�=0

�               (4.8.1) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

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“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

ℒ �� ���(�)

�

���

�

=
�2(0)

�

+
� + �(1 − �)

�
 ℒ �� � �1�

∞

�=0

−   
�

2
�1 ∑ �3�

∞
�=0

�ℎ − ∑ ���
∞
�=0

−
�

2
�2 ∑ �4�

∞
�=0

�ℎ − ∑ ���
∞
�=0

− (��2 +  �) � �2�

∞

�=0

�               (4.8.2) 

ℒ �� ��(�)

�

���

�

=
�(0)

�

+
� + �(1 − �)

�
ℒ ���1 � �1�

∞

�=0

+ ��2 � �2�

∞

�=0

 − (�

+  �) � ��

∞

�=0

�                                         (4.8.3) 

ℒ �� ��(�)

�

���

�

=
�(0)

�

+
� + �(1 − �)

�
  ℒ �

�
1
�1 ∑ �1�

∞
�=0

�ℎ − ∑ ���
∞
�=0

+
�

1
�2 ∑ �2�

∞
�=0

�ℎ − ∑ ���
∞
�=0

+   
�

2
�1 ∑ �3�

∞
�=0

�ℎ − ∑ ���
∞
�=0

+
�

2
�2 ∑ �4�

∞
�=0

�ℎ − ∑ ���
∞
�=0

 

− (�1� + �2(1 − �) +  �) � ��

∞

�=0

�      (4.8.4) 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

ℒ �� ���(�)

�

���

�

=
�1(0)

�
+

� + �(1 − �)

�
 ℒ ��1� � ��

∞

�=0

− ��
1

+ �1 + � + �� � �1�

∞

�=0

�                         (4.8.5) 

 

 

ℒ �� ���(�)

�

���

�

=
�2(0)

�

+
� + �(1 − �)

�
  ℒ ��2(1 − �) � ��

∞

�=0

− (�
2

+ �2 + � + �) � �2�

∞

�=0

�                   (4.8.6) 

ℒ �� ���(�)

�

���

�

=
��(0)

�

+
� + �(1 − �)

�
 ℒ ��1 � �1�

∞

�=0

+  �2 � �2�

∞

�=0

− ( �
3

+ �

+ �) � ���

∞

�=0

�                                             (4.8. 7) 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

ℒ �� ��(�)

�

���

�

=
�(0)

�

+
� + �(1 − �)

�
 ℒ ��

1
� �1�

∞

�=0

+  �
2

� �2�

∞

�=0

+  �
3

� ���

∞

�=0

−  � � ��

∞

�=0

�                           (4.8.8) 

Using initial value condition, ��(0) = ���, �(0) = ��, ��(0) =  �����(0) =  ���, �(0) =  ��, �(�) =  ��, 

��(0) =  ���, ��(�) =  ���, matching the items on both sides of (4.8.1) –(4.8.8) and applying  

���(�)���(�) = ���, ���(�)���(�) = ���, ���(�)���(�) = ���,   ���(�)���(�) = ���  the general term of the 

model is given below 

� ��(���)(�)

�

���

=  ℒ�� �
� + �(1 − �)

�
ℒ ���� −

����������

�� − ���
−

����������

�� − ���
− (��� + � +  �)����� (4.9.1) 

� ��(���)(�)

�

���

=   ℒ�� �
� + �(1 − �)

�
ℒ 

1

��
ℒ ����� −  

����������

�� − ���
−

����������

�� − ���

− (��� +  �)�����        (4.9.2) 

� ����(�)

�

���

=  ℒ�� �
� + �(1 − �)

�
ℒ{������ +  ������  − (� +  �)��}�                    (4.9.3) 

� ����(�)

�

���

= ℒ�� �
� + �(1 − �)

�
ℒ �

����������

�� − ���
+

����������

�� − ���
+  

����������

�� − ���
+

����������

�� − ���
 

− (��� + ��(1 − �) +  �)����    (9.4) 

� ��(���)(�)

�

���

=   ℒ�� �
� + �(1 − �)

�
ℒ{����� − (�� + �� + � + �)���}�                        (4.9.5) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

� ��(���)(�)

�

���

=   ℒ�� �
� + �(1 − �)

�
 ℒ{��(1 − �)�� − (�� + �� + � + �)���}�                  (4.9.6) 

� ��(���)(�)

�

���

=   ℒ�� �
� + �(1 − �)

�
ℒ{����� + ����� − ( �� + � + �)���}�                 (4.9.7) 

� ����(�)

�

���

= ℒ�� �
� + �(1 − �)

�
ℒ{�����  +  ����� + ����� −  ���}�              (4.9. 8) 

 

Let � = 0 

���(�) =  ℒ�� �
� + �(1 − �)

�
ℒ ���� −

����������

�� − ���
−

����������

�� − ���
− (��� + � +  �)����� (4.10.1) 

���(�) =   ℒ�� �
� + �(1 − �)

�
ℒ 

1

��
ℒ ����� − 

����������

�� − ���
−

����������

�� − ���
− (��� +  �)�����        (4.10.2) 

��(�) =  ℒ�� �
� + �(1 − �)

�
ℒ{������ +  ������  − (� +  �)��}�                    (4.10.3) 

��(�) = ℒ�� �
� + �(1 − �)

�
ℒ �

����������

�� − ���
+

����������

�� − ���
+  

����������

�� − ���
+

����������

�� − ���
 

− (��� + ��(1 − �) +  �)����    (4.10.4) 

���(�) =   ℒ�� �
� + �(1 − �)

�
ℒ{����� − (�� + �� + � + �)���}�                        (4.10.5) 

���(�) =   ℒ�� �
� + �(1 − �)

�
 ℒ{��(1 − �)�� − (�� + �� + � + �)���}�                  (4.10.6) 

���(�) =   ℒ�� �
� + �(1 − �)

�
ℒ{����� + ����� − ( �� + � + �)���}�                 (4.10.7) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

��(�) = ℒ�� �
� + �(1 − �)

�
ℒ{�����  + ����� + ����� −  ���}�              (4.10. 8) 

���(�) =  ℒ�� �
� + �(1 − �)

�
ℒ ���� −

����������

�� − ���
−

����������

�� − ���
− (��� + � +  �)����� (4.10.1) 

���(�) =   ℒ�� �
� + �(1 − �)

�
ℒ 

1

��
ℒ ����� − 

����������

�� − ���
−

����������

�� − ���
− (��� +  �)�����        (4.10.2) 

��(�) =  ℒ�� �
� + �(1 − �)

�
ℒ{������ +  ������  − (� +  �)��}�                    (4.10.3) 

��(�) = ℒ�� �
� + �(1 − �)

�
ℒ �

����������

�� − ���
+

����������

�� − ���
+  

����������

�� − ���
+

����������

�� − ���
 

− (��� + ��(1 − �) +  �)����    (4.10.4) 

���(�) =   ℒ�� �
� + �(1 − �)

�
ℒ{����� − (�� + �� + � + �)���}�                        (4.10.5) 

���(�) =   ℒ�� �
� + �(1 − �)

�
 ℒ{��(1 − �)�� − (�� + �� + � + �)���}�                  (4.10.6) 

���(�) =   ℒ�� �
� + �(1 − �)

�
ℒ{����� + ����� − ( �� + � + �)���}�                 (4.10.7) 

��(�) = ℒ�� �
� + �(1 − �)

�
ℒ{�����  + ����� + ����� −  ���}�              (4.10. 8) 

Applying the inverse Laplace transform, we obtain 

���(�) =  ℒ�� �
� + �(1 − �)

�
ℒ ���� −

����������

�� − ���
−

����������

�� − ���
− (��� + � +  �)����� (4.10.1) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

���(�) =   ℒ�� �
� + �(1 − �)

�
ℒ 

1

��
ℒ ����� − 

����������

�� − ���
−

����������

�� − ���
− (��� +  �)�����        (4.10.2) 

��(�) =  ℒ�� �
� + �(1 − �)

�
ℒ{������ +  ������  − (� +  �)��}�                    (4.10.3) 

��(�) = ℒ�� �
� + �(1 − �)

�
ℒ �

����������

�� − ���
+

����������

�� − ���
+  

����������

�� − ���
+

����������

�� − ���
 

− (��� + ��(1 − �) +  �)����    (4.10.4) 

���(�) =   ℒ�� �
� + �(1 − �)

�
ℒ{����� − (�� + �� + � + �)���}�                        (4.10.5) 

���(�) =   ℒ�� �
� + �(1 − �)

�
 ℒ{��(1 − �)�� − (�� + �� + � + �)���}�                  (4.10.6) 

���(�) =   ℒ�� �
� + �(1 − �)

�
ℒ{����� + ����� − ( �� + � + �)���}�                 (4.10.7) 

��(�) = ℒ�� �
� + �(1 − �)

�
ℒ{�����  + ����� + ����� −  ���}�              (4.10. 8) 

���(�) =  ���� −
����������

�� − ���
−

����������

�� − ���
− (��� + � +  �)���� {1 + �(� − 1)} (4.11.1) 

���(�) =   ����� −  
����������

�� − ���
−

����������

�� − ���
− (��� +  �)���� {1 + �(� − 1)}        (4.11.2) 

��(�) =  {������ +  ������  − (� +  �)��}{1 + �(� − 1)}                    (4.11.3) 

��(�) = �
����������

�� − ���
+

����������

�� − ���
+   

����������

�� − ���
+

����������

�� − ���
 − (��� + ��(1 − �) +  �)��� {1

+ �(� − 1)}    (4.11.4) 

���(�) =   {����� − (�� + �� + � + �)���}{1 + �(� − 1)}                        (4.11.5) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

���(�) =   {��(1 − �)�� − (�� + �� + � + �)���}{1 + �(� − 1)}                  (4.11.6) 

���(�) =   {����� + ����� − ( �� + � + �)���}{1 + �(� − 1)}               (4.11.7) 

��(�) = {�����  +  ����� +  ����� −  ���}{1 + �(� − 1)}              (4.11. 8) 

 

Let � = 1  

���(�) =  ℒ�� �
� + �(1 − �)

�
ℒ ���� −

����������

�� − ���
−

����������

�� − ���
− (��� + � +  �)����� (4.12.1) 

���(�) =   ℒ�� �
� + �(1 − �)

�
ℒ ����� − 

����������

�� − ���
−

����������

�� − ���
− (��� +  �)�����        ( 4.12.2) 

��(�) =  ℒ�� �
� + �(1 − �)

�
ℒ{������ +  ������  − (� +  �)��}�                    (4.12.3) 

��(�) = ℒ�� �
� + �(1 − �)

�
ℒ �

����������

�� − ���
+

����������

�� − ���
+  

����������

�� − ���
+

����������

�� − ���
 

− (��� + ��(1 − �) +  �)����    (4.12.4) 

���(�) =   ℒ�� �
� + �(1 − �)

�
ℒ{����� − (�� + �� + � + �)���}�                        (4.12.5) 

���(�) =   ℒ�� �
� + �(1 − �)

�
 ℒ{��(1 − �)�� − (�� + �� + � + �)���}�                  (4.12.6) 

���(�) =   ℒ�� �
� + �(1 − �)

�
ℒ{����� + ����� − ( �� + � + �)���}�                 (4.12.7) 

��(�) = ℒ�� �
� + �(1 − �)

�
ℒ{�����  +  ����� +  ����� −  ���}�              (4 .12. 8) 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

 

 

 

���(�) = ���� − �
1

�� − [{����� +  ����� − ( �� + � + �)���}{1 + �(� − 1)}]
������{�����

− (�� + �� + � + �)���}{1 + �(� − 1)}�

− �����{��(1 − �)�� − (�� + �� + � + �)���}{1 + �(� − 1)}�� − (��� + � +  �)� ����

−
����������

�� − ���
−

����������

�� − ���
− (��� + � +  �)���� {1 + �(� − 1)}� {1 + �(� − 1)}   4.13.1 

���(�) =  = �� ���� −
����������

�� − ���
−

����������

�� − ���
− (��� + � +  �)����

−  
1

�� − {����� +  ����� − ( �� + � + �)���}{1 + �(� − 1)}
[����{�����

− (�� + �� + � + �)���} − ����{��(1 − �)�� − (�� + �� + � + �)���}] − (��� +  �)� �����

−  
����������

�� − ���
−

����������

�� − ���
− (��� +  �)���� {1 + �(� − 1)}{1 + �(� − 1)}(4.13.2) 

 

��(�) = ���� ���� −
����������

�� − ���
−

����������

�� − ���
− (��� + � +  �)����

+  ��� ����� −  
����������

�� − ���
−

����������

�� − ���
− (��� +  �)����  − (�

+  �){������ +  ������  − (� +  �)��}� {1 + �(� − 1)}{1 + �(� − 1)}                   (4.13.3) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

��(�) =  �
1

�� − {����� + ����� − ( �� + � + �)���}{1 + �(� − 1)}
[����{�����

− (�� + �� + � + �)���} ����� − 
����������

�� − ���
−

����������

�� − ���
− (��� +  �)����

+ ����{��(1 − �)�� − (�� + �� + � + �)���} ���� −
����������

�� − ���
−

����������

�� − ���

− (��� + � +  �)����

+   ����{����� − (�� + �� + � + �)���} ����� − 
����������

�� − ���
−

����������

�� − ���
− (��� +  �)����

+ ����{��(1 − �)�� − (�� + �� + � + �)���} ����� − 
����������

�� − ���
−

����������

�� − ���

− (��� +  �)����]{1 + �(� − 1)} 

− (��� + ��(1 − �) +  �) �
����������

�� − ���
+

����������

�� − ���
+   

����������

�� − ���
+

����������

�� − ���
 

− (��� + ��(1 − �) +  �)��� {1 + �(� − 1)}� {1 + �(� − 1)}    (4.13.4) 

���(�) =  ���� �
����������

�� − ���
+

����������

�� − ���
+   

����������

�� − ���
+

����������

�� − ���
 − (��� + ��(1 − �) +  �)���

− (�� + �� + � + �){����� − (�� + �� + � + �)���}� {1 + �(� − 1)}{1

+ �(� − 1)}                        (4.13.5)  

 

���(�) = ���(1 − �) �
����������

�� − ���
+

����������

�� − ���
+   

����������

�� − ���
+

����������

�� − ���
 − (��� + ��(1 − �) +  �)���  

− (�� + �� + � + �){����� − (�� + �� + � + �)���}� {1 + �(� − 1)}{1

+ �(� − 1)}                 (4.13.6) 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

���(�) =  ���{����� − (�� + �� + � + �)���}{1 + �(� − 1)}

+  ��{��(1 − �)�� − (�� + �� + � + �)���}{1 + �(� − 1)} − ( �� + �

+ �){����� + ����� − ( �� + � + �)���}� 

��(�) =   ���{����� − (�� + �� + � + �)���}  + ��{��(1 − �)�� − (�� + �� + � + �)���}

+  ��{����� +  ����� − ( �� + � + �)���}  −  �{�����  + ����� +  ����� −  ���}�{1

+ �(� − 1)}{1 + �(� − 1)}           (4.13. 8) 

 Hence the required solution  

 

��(�) = ���(�) + ���(�) + ���(�) + .  .  . 

��(�) = ���(�) + ���(�) + ���(�) + .  .  . 

�(�) = ��(�) + ��(�) + ��(�) + .  .  . 

�(�) = ��(�) + ��(�) + ��(�) + .  .  . 

��(�) = ���(�) + ���(�) + ���(�) + .  .  . 

��(�) = ���(�) + ���(�) + ���(�) + .  .  . 

��(�) = ���(�) + ���(�) + ���(�) + .  .  . 

�(�) = ��(�) + ��(�) + ��(�) + .  .  . 

 

Summary 

Diphtheria remains a re-emerging public health concern despite vaccination programs, with children and adults 
exhibiting different levels of susceptibility and transmission potential. Classical integer-order models often fail 
to capture the memory effects inherent in disease transmission, such as waning immunity and delayed 
intervention impact, and few models incorporate age structure, which is crucial for diphtheria dynamics. 
Motivated by these limitations, this study develops a novel age-structured fractional-order model of 
diphtheria using the Caputo–Fabrizio derivative, which accounts for non-locality and memory effects in 
disease spread. 

The model stratifies the population into susceptible children, susceptible adults, vaccinated, exposed, infectious 
children, infectious adults, isolated infectious, and recovered individuals. Rigorous analysis establishes the 
existence and uniqueness of solutions via the contraction mapping principle and proves Ulam–Hyers 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

stability, confirming the robustness of the system under perturbations. To obtain approximate analytic–
numerical solutions, the Laplace Adomian Decomposition Method (LADM) is applied, yielding a rapidly 
convergent series representation. Results show that the fractional order significantly alters outbreak 
intensity and timing, reflecting the role of memory in diphtheria persistence and control. 

This work achieves three key outcomes: (i) the formulation of a new fractional-order, age-structured 
diphtheria model; (ii) provision of rigorous mathematical guarantees for solution behavior; and (iii) 
demonstration of efficient analytic–numerical solutions through LADM. The study contributes to knowledge by 
introducing a more realistic modeling framework that integrates age heterogeneity and memory effects, offering 
deeper epidemiological insights and practical guidance for sustaining vaccination and isolation strategies in 
diphtheria control. 

 

REFERENCES 

1. Truelove SA, Keegan LT, Moss WJ, et al. Clinical and epidemiological aspects of diphtheria: a 
systematic review and pooled analysis. Clin Infect Dis 2020; 71: 89–97. [PMC free 
article] [PubMed] [Google Scholar] 

2. Oyelola A. Adegboye,  Faith O. Alele,  Anton Pak,  Maria E. Castellanos,  Malachy I. Okeke, 
Mohammed A.S., Theophilus I. Emeto,  Emma S. McBryde, A resurgence and re-emergence of 
diphtheria in Nigeria, 2023, TherAdv Infect Dis. 2023 Jan-Dec; 10: 20499361231161936, Published 
online 2023 March 28. doi: 10.1177/20499361231161936 

3.  Al-Dar AA, Al-Qassimi M, Ezzadeen FH, et al. Diphtheria resurgence in Sada’a-Yemen, 2017–
2020. BMC Infect Dis 2022; 22: 46. [PMC free article] [PubMed] [Google Scholar] 

4. Centers for disease control and preventionhttps://www.cdc.gov/diphtheria 
5. Centers for disease control and prevention. Travel-Related Infectious 

Diseases:Diptheriahttps://wwwnc.cdc.gov/travel/yellowbook/2020/travel-related-infectious-
diseases/diphtheria 

6. National Library of Medicine. Diptheria (https://medlineplus.gov/diphtheri.html 
7. Ekere S. Udofia, Ubong D. Akpan, Joy IjeomaUwakwe, Henry S. Thomas(2024)  Earthline Journal of 

Mathematical Sciences, volume 14, number 3, 2024, page 391-404  

 

8. E. S. Udofia, J. I. Uwakwe, H. S. Thomas(2024) Optimal control and sensitivity analysis of age-

structured mathematical model of Diphtheria infection. International  Journal of Mathematical Analysis 
and Modelling,( 2024),7(1):122-143  

 

9. ONUOHA JOY LJEOMA, INYAMA SIMEON CHIOMA , UDOFIA EKERE SUNDAY AND 
OMAME ANDREW (2015), Mathematical Model of the Transmission Dynamics of Swine Flu 

IJO JOURNALS

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https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7312233/
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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                                           Thomas, Henry Sylvester *  

https://ijojournals.com/                                                                                 Volume 08 || Issue 09 || September, 2025 || 

“CAPUTO-FABRIZIO FRACTIONAL DRIVATIVES OF AGE-STRUCTURED DIPPTHERIA INFECTION MODEL WITH LAPLACE ADOMIAN DECOMPOSITION ANALYSIS" 

 

 

with the Vaccination of Non Newborns,  International J. of Math.  Sci. &Engg. Appls.  (IJMSEA) 
ISSN 0973-9424, Vol. 9 No. I (March, 2015), pp. 1-17 www.ascent-journals.com 

10. Ekere S. Udofia, (2023), Mathematical Model of Male Circumcision in HIV/AIDS Preventions, 
International Journal of Innovative Science and Research Technology, Volume 8, Issue 8, August 
– 2023 ISSN No: -2456-2165 www.ijisrt.com 

11. ONUOHA JOY LJEOMA, INYAMA SIMEON CHIOMA AND UDOFIA SUNDAY EKERE(2014) 
Mathematical Model of the Transmission Dynamics of Swine Flu with the Vaccination of Newborns,  
International J. of Math. Sci. &Engg. Appls. (IJMSEA) ISSN 0973-9424, Vol. 8 No. V (September, 
2014), pp. 217-229  

12. UdofiaEkere Sunday and Inyama Simeon Chioma (2012), Application of Optimal Control to the 
Epidemiology of Fowl Pox Transmission Dynamics in Poultry,  Journal of Mathematics and Statistics 8 
(2): 248-252, ISSN 1549-3644, © 2012 Science Publications 

13. Udofia, Ekere Sunday and Inyama, SimeonChioma Mathematical model of Structural Strategy( Delayed 
First Intercourse) in HIVAIDS Prevention, Journal of the Nigerian Association of Mathematical Physics 
Volume 24 (July, 2013) pp 257-260 © J. of NAMP 

14. Udofia, Ekere Sunday, Sampson, Marshal Imeh (2014) Mathematical Model for the Epidemiology 
of Fowl Pox Infection Transmission That Incorporates Discrete Delay, IOSR Journal of 
Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 10, Issue 4 Ver. V (July 
- Aug. 2014), PP 08-16 www.iosrjournals.org 

15. IA Agwu, SC Inyama, RA Umana, A Omame, N Ukanwoke, A Ofomata, HI Mbachu, ES Udofia, JI 
Uwakwe (2018), Determining the impact of variation of Harvesting Effort on the Qualitative Behaviour 
of a Coexistence Steady State Solution and its Stability in Prey-Predator Fishery Model, Academic 
Journal of Applied Mathematical Sciences Vol. 4 Issue 10 Pages 119-128, 2018 

16. UdofiaEkere Sunday and Amos AmosIdungafa (2018), Mathematical Model of Bacteria-Nutrient 
Harvesting in A Cultured Environment, Journal of the Nigerian Association of Mathematical Physics 
Volume 46 (May, 2018 Issue), pp115 –118, 2018 © J. of NAMP 

17. Udofia, Ekere S, and EtukudoIdorenyin A (2019), Optimal Allocation of Biscuit Ingredient in the 
Production Process - An Invariant Property Based Algorithm Approach, Mathematical Theory and 
Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) DOI: 10.7176/MTM Vol.9, 
No.4, 2019  

18. I. J. Udom, E. S. Udofia, S. A. Nta, G. A. Usoh, E. O. Sam (2023)’ Prediction of Piggery Wastewater 
Nutrient Attenuation by Constructed Wetland in a Humid Environment’ International Journal of 
Innovative Science and Research Technology ISSN No:-2456-2165, Volume 8, Issue 9, September – 
2023,www.ijisrt.com 

19. Olayiwola, M. O, Alaje, A.I. (2024) Mathematical Modelling of Diphtheria transmission and Vaccine 
efficacy using Nigeria, ModelEarth Syst.Environ.10, 3941-3967 https://doi.org/10.1007/s40808-024-
01976-7 

20. Ulam SM, A collection of Mathematical Problems; New York; 1960, p.29  
21. Ulam SM, Problems in Modern  Mathematics; Courier Co; 2004  

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