Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks

Hassan Tahir; Anwarud Din; Kamal Shah; Maggie Aphane; Thabet Abdeljawad

doi:10.1515/phys-2023-0190

Article Open Access

Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks

Hassan Tahir , Anwarud Din , Kamal Shah , Maggie Aphane and Thabet Abdeljawad

Published/Copyright: February 26, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

Due to inherent operating constraints, wireless sensor networks (WSNs) need help assuring network security. This problem is caused by worms entering the networks, which can spread uncontrollably to nearby nodes from a single node infected with computer viruses, worms, trojans, and other malicious software, which can compromise the network’s integrity and functionality. This article discusses a fractional S E 1 E 2 I R model to explain worm propagation in WSNs. For capturing the dynamics of the virus, we use the Mittag–Leffler kernel and the Atangana–Baleanu (AB) Caputo operator. Besides other characteristics of the problem, the properties of superposition and Lipschitzness of the AB Caputo derivatives are studied. Standard numerical methods were employed to approximate the Atangana–Baleanu–Caputto fractional derivative, and a detailed analysis is presented. To illustrate our analytical conclusions, we ran numerical simulations.

Keywords: epidemic model; wireless sensor networks; fractional derivative; differential operator; computational method

1 Introduction

The advancement of information technology has resulted in an increase in worrying scenarios involving wireless networks. These advancements, particularly in the industry, not only pose a security threat to the entire world but also put at risk the human lives. For the time being, having a wireless network that is dependable, effective, and secure will be of numerous benefits to people. In a wireless network, a sensor node is a small, smart, and inexpensive device. In many deployments, including those involving mission-critical situations, wireless sensor networks (WSNs) are used to gather recurring data. These networks have various important applications, such as object monitoring in agriculture, military target tracking, pollution and environmental monitoring, disaster management, flood detection, exploring the environmental dangers, traffic monitoring, gas monitoring, tracking of vehicles, seismic sensing, healthcare applications, and monitoring the quality of water (Akyildiz, Su, Sankarasubramaniam, [1–3]). Contrarily, sensor nodes are not only inexpensive gadgets but also intelligent ones that function as a system. They do, however, have resource limitations, including short battery lives, low memory capacities, and processing speed limitations [4,5]. As WSNs have very limited resources and an architecture of decentralized type, it is too difficult to develop wireless communication and ensure sufficient security measures among these networks. Wireless networks are more susceptible to threats compared to other networks, as they exhibit greater vulnerability [6]. Software bugs and vulnerabilities are the typical difficulties that hackers may exploit, despite the network having several security measures in place in order to protect it from external threats.

Comprehending the intricate dynamics of multimalware attacks necessitates knowledge of viral modeling in WSNs. This modeling illustrates how many virus strains and malicious software may influence network sensor nodes simultaneously. Furthermore, analyzing the vulnerabilities and hazards related to these assaults requires modeling their occurrence. Not to mention, WSNs are subject to a number of restrictions. Malware attacks can force sensor nodes to perform additional computations or communication activities, leading to increased energy consumption. These energy-intensive activities from different types of malware can quickly deplete the node’s battery, potentially causing node failure or network degradation [7,8]. They also target various resources within the sensor nodes, such as CPU, memory, or communication bandwidth. When multiple types of malware are active, they can compete for these resources, causing resource depletion and a slowdown in normal network operations. Vulnerabilities in a software could occur due to multiple reasons, such as poor coding, poor system design, or insufficient security controls. With WSNs, these issues take on even greater importance. WSN sensor nodes rely on multi-hop data delivery because of its limited range of communication [7]. These constraints limit network nodes’ ability to defend themselves against virus attacks, including those created by viruses, worms, and other malicious software [8]. The long-term viability of the network depends on controlling and managing worm reproduction. Consequently, understanding and reducing these threats within a WSN heavily depend on the study of malicious signal propagation, and mathematical modeling is the best tool to analyze and control its spread [9–12].

In ordinary calculus, Riemann–Liouville, Fourier, and Euler made a lot of contributions to discovering beneficial results in the eighteenth century. At that time, in the sequel, a number of researchers had significantly improved the field of fractional calculus (FC) [13–16]. This rapid growth in FC is due to the fact that traditional calculus is unable to adequately explain how modern calculus is used in a variety of mathematical modeling contexts, including the study of the memory process and different hereditary properties. FC, which is a general case of the integer-order derivative, has a significant degree of freedom that could not be noticed in ordinary calculus as these ordinary derivatives have local behavior in nature. A wide range of applications of FC could be seen in the literature [17–20] and references cited therein. Due to its wide range of applications, the noninteger-order differential and integral calculus has drawn more attention from academics and researchers in recent years. In this emerging field, a new concept was presented in different articles, where the kernels of singular and local have been replaced by nonlocal and nonsingular ones, respectively. The best feature of this newly developed kernel is its memory capabilities for the system under investigation. A novel fractional differential operator based on the generalized nonlocal and nonsingular kernel of Mittag–Leffler (ML) mapping was developed by Atangana, Baleanu, and Caputo (ABC) in 2016 [21]. As one can note in [22–26], numerous real-world phenomena have been modeled using ABC differential operator and were rigorously investigated. In this operator, the generalized ML function deals with real-world phenomena considerably more accurately and precisely.

In this article, we propose a fractional model for WSNs, the model suggests limiting malware’s ability to attack the sensor network. Each of the nodes has a sensing range of r and a spreading area with a dimension of π r 2 . The quantity of susceptible nodes S ( t ) per unit area is denoted by the expression ρ ( t ) = S ( t ) Ł × Ł . The size of network nodes in the sensing area of the node is captured mathematically by S ′ ( t ) = S ( t ) π r 2 Ł 2 . The proposed model extends network lifetime and enhances data efficiency in WSNs. These discoveries have direct applications for the software industry since they may be used to improve antivirus programs that can successfully prevent malware assaults on WSNs. This research will help users recover the compromised nodes, and they can carefully apply an antivirus software on these sensor nodes, improving the whole security architecture to lessen the threats.

In the present work, we considered a fractional-order WSN model using the Atangana–Baleanu (AB) derivative’s novel fractional differential operator. The structure of the remaining article is as follows: a brief explanation of the model’s formulation can be found in Section 2. The fundamental idea of the AB operator is discussed in Section 3. The qualitative investigation and the study of the fractional stability of the suggested model were reported in Section 4. In Section 5, the Hyers–Ulam stability of the model is presented. In Section 6, a scheme is developed by using the Newton polynomial techniques for approximating an iterative solution to the model. The theory is validated with the help of simulations in Section 7, where the authors discussed the effect of each parameter of the model. Section 8 concludes the research and puts forward future research interests to the readers.

2 Proposed fractional S E 1 E 2 I R multimalware model

A model based on WSNs was recently developed by Awasthi et al. [27] using the deterministic approach of modeling. The five different states that make up the suggested model are described below:

Denote the sensor nodes that are not yet virus effected; however, they can catch malwares at any time within the network.
These malware-infected sensor nodes are not contagious despite their infection. But over time, they spread contagiousness. Due to the assumption that two different types of malware would attack the network at the same time, therefore, two different types of exposed nodes E 1 and E 2 are taken into consideration.
The network’s sensor nodes that are effected by the malware and have the capacity to spread that infection to the nearby sensor nodes.
This class contains those sensor nodes of the network that have been recovered after passing from an infectious state.

The model suggests limiting malware’s ability to attack the sensor network. The N number of sensor nodes is dispersed over the Ł 2 region. These nodes are used to collect the necessary data from the surroundings. The total nodes of the network are divided into distinct groups. These include the states of susceptible ( S ) , exposed category 1 ( E 1 ) , exposed category 2 ( E 2 ) , infectious ( I ) , and the recovered ( R ) . The transition of the nodes within the network of the underlying S E 1 E 2 I R model is diagrammatically demonstrated in Figure 1.

$Figure 1 Schematic representation of S E 1 E 2 I R {\mathsf{S}}{{\mathsf{E}}}_{1}{{\mathsf{E}}}_{2}{\mathsf{I}}{\mathsf{R}} model [27].$

Figure 1

Schematic representation of S E 1 E 2 I R model [27].

At any instant time t ≥ 0 , we can represent the total nodes by N ( t ) , and we have

(1) N ( t ) = S ( t ) + E 1 ( t ) + E 2 ( t ) + I ( t ) + R ( t ) .

Each of the nodes has a sensing range of r and a spreading area with a dimension of π r 2 . The quantity of susceptible nodes per unit area is denoted by the expression ρ ( t ) = S ( t ) Ł × Ł . The size of network nodes within the sensing area of the node is captured mathematically by S ′ ( t ) = S ( t ) π r 2 Ł 2 . To proceed further, for the sake of brevity, we defined another two parameters ζ = π r 2 Ł 2 β and φ = π r 2 Ł 2 β . According to the flow diagram in Figure 1, the various states of the sensing nodes display dynamics that are governed by the following equations:

(2) d S d t = b − φ S ( t ) I ( t ) − σ S ( t ) − ω S ( t ) , d E 1 d t = p φ S ( t ) I ( t ) − ( λ 1 + σ ) E 1 ( t ) , d E 2 d t = q φ S ( t ) I ( t ) − ( λ 2 + σ ) E 2 ( t ) d I d t = λ 1 E 1 ( t ) + λ 2 E 2 ( t ) − ( σ + γ ) I ( t ) , d R d t = γ I ( t ) − σ R ( t ) + ω S ( t ) .

A detailed interpretation of the model’s parameters is presented in Table 1.

Table 1

Parameters of the model and their interpretation

Parameters	Description of parameters	Source
b	Rate of new nodes in WSN	[27]
σ	Rate crashing of sensor nodes due to depletion of battery	[27]
β	Coefficient of malware transmission in WSN	[27]
p	Rate of exposed and entered into state E 1	[27]
q	Rate of exposure from a different malware type, changing the state from S to E 2	[27]
λ 1 , λ 2 ( λ 1 > λ 2 )	The exposed state of nodes become infectious	[27]
γ	The rate at which infected nodes recover.	[27]
ω	Rate of susceptible nodes that immure and enter a state of recovery is R	[27]

The threshold parameter R 0 is given by the expression

(3) R 0 = b β π r 2 L 2 ( σ + γ ) ( σ + ω ) p λ 1 ( σ + λ 1 ) + q λ 2 ( σ + λ 2 ) .

There has recently been a claim made that the theory of FC has a wide range of uses. In addition, it has been demonstrated that utilizing fractional systems to represent identical situations yields more accurate results than the traditional method of using ordinary derivatives [28–34]. Motivated by these and many other facts, we will extend system (2) to a fractional model while using the generalized ML kernel as follows:

(4) D 0 , t Ψ 1 A B C [ S ] = b − φ S ( t ) I ( t ) − σ S ( t ) − ω S ( t ) D 0 , t Ψ 1 A B C [ E 1 ] = p φ S ( t ) I ( t ) − ( λ 1 + σ ) E 1 ( t ) D 0 , t Ψ 1 A B C [ E 2 ] = q φ S ( t ) I ( t ) − ( λ 2 + σ ) E 2 ( t ) D 0 , t Ψ 1 A B C [ I ] = λ 1 E 1 ( t ) + λ 2 E 2 ( t ) − ( σ + γ ) I ( t ) D 0 , t Ψ 1 A B C [ R ] = γ I ( t ) − σ R ( t ) + ω S ( t ) .

Under the starting approximation,

S ( 0 ) = S 0 , E 1 ( 0 ) = E 1 0 , E 2 ( 0 ) = E 2 0 , I ( 0 ) = I 0 , R ( 0 ) = R 0 ≥ 0 .

3 Preliminaries

In this part of the study, we intend to present the basics of integral and differential operators in FC. To proceed further, we define the Caputo derivative as [35] follows:

(5) D t Ψ 1 0 C ℱ ( t ) = 1 Γ ( 1 − Ψ 1 ) ∫ 0 t d d T ℱ ( T ) ( t − T ) − Ψ 1 d T .

Based on the above definition, the Caputo–Fabrizio fractional derivative [35] can be defined as follows:

(6) D t Ψ 1 0 C F ℱ ( t ) = M ( Ψ 1 ) 1 − Ψ 1 ∫ 0 t d d T ℱ ( T ) exp − Ψ 1 1 − Ψ 1 ( t − T ) d T ,

where M ( t ) is known as the normalization function that satisfies M ( 0 ) = M ( 1 ) = 1 . Another differential operator due to Atangana and Baleanu [21] is defined as follows:

(7) D t Ψ 1 A B C ℱ ( t ) = A B ( Ψ 1 ) 1 − Ψ 1 ∫ 0 t d d T ℱ ( T ) E Ψ 1 − Ψ 1 1 − Ψ 1 ( t − T ) Ψ 1 d T .

If one utilize the kernel of power-law and of the ML-type, the fractal-fractional derivative of exponential decay is defined as [35] follows:

(8) D t Ψ 1 , Φ 1 0 F F P ℱ ( t ) = 1 Γ ( 1 − Ψ 1 ) d d t Φ 1 ∫ 0 t ℱ ( T ) ( t − T ) − Ψ 1 d T , D t Ψ 1 , Φ 1 0 F F E ℱ ( t ) = M ( Ψ 1 ) 1 − Ψ 1 d d t Φ 1 ∫ 0 t ℱ ( T ) exp − Ψ 1 1 − Ψ 1 ( t − T ) d T , D t Ψ 1 , Φ 1 0 F F M ℱ ( t ) = A B ( Ψ 1 ) 1 − Ψ 1 d d t Φ 1 ∫ 0 t ℱ ( T ) E Ψ 1 − Ψ 1 1 − Ψ 1 ( t − T ) Ψ 1 d T ,

where [36]

(9) d ℱ ( t ) d t Φ 1 = lim t → t 1 ℱ ( t ) − ℱ ( t 1 ) t 2 − Φ 1 − t 1 2 − Φ 1 ( 2 − Φ 1 ) .

In the similar way, by utilizing the kernel of power-law and of the ML-type, the fractal-fractional integral of exponential decay is defined as follows [35]:

(10) J t Ψ 1 , Φ 1 0 F F P ℱ ( t ) = 1 Γ ( Ψ 1 ) ∫ 0 t ( t − T ) Ψ 1 − 1 T 1 − Φ 1 ℱ ( T ) d T , J t Ψ 1 , Φ 1 0 F F E ℱ ( t ) = 1 − Ψ 1 M ( Ψ 1 ) t 1 − Φ 1 ℱ ( t ) + Ψ 1 M ( Ψ 1 ) ∫ 0 t Φ 1 1 − Φ 1 ℱ ( T ) d T , J t Ψ 1 , Φ 1 0 F F M ℱ ( t ) = 1 − Ψ 1 A B ( Ψ 1 ) t 1 − Φ 1 ℱ ( t ) + Ψ 1 A B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − Φ 1 ) Ψ 1 − 1 T 1 − Φ 1 ℱ ( T ) d T .

Theorem 1

Schauder’s fixed point theorem [37,38]: “Let X be a Banach space and F : X → X is compact and continuous, and if the set

S = { δ ∈ X : δ = m F δ , m ∈ ( 0 , 1 ) }

is bounded, then F has a unique fixed point.”

4 Existence results

In the part of the manuscript, we will use the Schauder’s fixed point theorem 1 to prove that the solution to model (4) exists. Let us define a real-valued and continuous function B ( J ) with the property of supremum norm space, and J = [ 0 , b ] is a Banach Space with P = ℬ ( J ) × ℬ ( J ) × ℬ ( J ) × ℬ ( J ) × ℬ ( J ) , having norm ∥ ( S , E 1 , E 2 , I , R ) ∥ = ‖ S ‖ + ‖ E 1 ‖ + ∥ E 2 ∥ + ‖ I ‖ + ‖ R ‖ , where ‖ S ‖ = sup t ∈ J ∣ S ( t ) ∣ , ‖ E 1 ‖ = sup t ∈ j ∣ E 1 ( t ) ∣ , ∥ E 2 ∥ = sup t ∈ j ∣ E 2 ( t ) ∣ , ∥ I ∥ = sup t ∈ j ∣ I ( t ) ∣ , ∥ R ∥ = sup t ∈ j ∣ R ( t ) ∣ . By applying the AℬC fractional integral operator to both sides of Eq. (4), we obtain the following system of equations:

(11) S ( t ) − S ( 0 ) = D 0 , t Ψ 1 A B C [ S ] ( b − φ S ( t ) I ( t ) − σ S ( t ) − ω S ( t ) ) , E 1 ( t ) − E 1 ( 0 ) = D 0 , t Ψ 1 A B C [ E 1 ] ( p φ S ( t ) I ( t ) − ( λ 1 + σ ) E 1 ( t ) ) , E 2 ( t ) − E 2 ( 0 ) = D 0 , t Ψ 1 A B C [ E 2 ] ( q φ S ( t ) I ( t ) − ( λ 2 + σ ) E 2 ( t ) ) , I ( t ) − I ( 0 ) = D 0 , t Ψ 1 A B C [ I ] ( λ 1 E 1 ( t ) + λ 2 E 2 ( t ) − ( σ + γ ) I ( t ) ) , R ( t ) − R ( 0 ) = D 0 , t Ψ 1 A B C [ R ] ( γ I ( t ) − σ R ( t ) + ω S ( t ) ) .

By using Definition 7, we have

(12) S ( t ) − S ( 0 ) = 1 − Ψ 1 B ( Ψ 1 ) M 1 ( Ψ 1 , t , S ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 1 ( Ψ 1 , ξ , S ( ξ ) ) d ξ , E 1 ( t ) − E 1 ( 0 ) = 1 − Ψ 1 B ( Ψ 1 ) M 2 ( Ψ 1 , t , E 1 ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 2 ( Ψ 1 , ξ , E 1 ( ξ ) ) d ξ , E 2 ( t ) − E 2 ( 0 ) = 1 − Ψ 1 B ( Ψ 1 ) M 3 ( Ψ 1 , t , E 2 ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 3 ( Ψ 1 , ξ , E 2 ( ξ ) ) d ξ , I ( t ) − I ( 0 ) = 1 − Ψ 1 B ( Ψ 1 ) M 4 ( Ψ 1 , t , I ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 4 ( Ψ 1 , ξ , I ( ξ ) ) d ξ , R ( t ) − R ( 0 ) = 1 − Ψ 1 B ( Ψ 1 ) M 5 ( Ψ 1 , t , R ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 5 ( Ψ 1 , ξ , R ( ξ ) ) d ξ ,

where

(13) M 1 ( Ψ 1 , t , S ( t ) ) = b − φ S ( t ) I ( t ) − σ S ( t ) − ω S ( t ) M 2 ( Ψ 1 , t , E 1 ( t ) ) = p φ S ( t ) I ( t ) − ( λ 1 + σ ) E 1 ( t ) , M 3 ( Ψ 1 , t , E 2 ( t ) ) = q φ S ( t ) I ( t ) − ( λ 2 + σ ) E 2 ( t ) , M 4 ( Ψ 1 , t , I ( t ) ) = λ 1 E 1 ( t ) + λ 2 E 2 ( t ) − ( σ + γ ) I ( t ) , M 5 ( Ψ 1 , t , R ( t ) ) = γ I ( t ) − σ R ( t ) + ω S ( t ) .

Under the condition of the maximum upper bounds of the functions S , E 1 , E 2 , I , and R , then M 1 , M 2 , M 3 , M 4 , and M 5 should satisfy the Lipschitzian condition. For different values of S * and S , we have the following relation:

(14) ∥ M 1 ( Φ 1 , t , S ) − M 1 ( Φ 1 , t , S * ) ∥ = ∥ − ( φ I ( t ) + σ + ω ) ( S − S * ) ∥ ≤ ∥ − ( φ I ( t ) + σ + ω ) ∥ ∥ ( S − S * ) ∥ ≤ ∥ φ I ( t ) + σ + ω ∥ ∥ ( S − S * ) ∥ .

Taking into account

η 1 ≔ ( φ M 1 + σ + ω ) , M 1 = max t ∈ J ‖ I ( t ) ‖ ,

one reaches

(15) ∥ M 1 ( Ψ 1 , t , S ) − M 1 ( Ψ 1 , t , S * ) ∥ ≤ η 1 ∥ S − S * ∥ .

Adapting the similar procedure, we have

(16) ∥ M 2 ( Ψ 1 , t , E 1 ) − M 2 ( Ψ 1 , t , E 1 * ) ∥ ≤ η 2 ∥ E 1 − E 1 * ∥ , ∥ M 3 ( Ψ 1 , t , E 2 ) − M 3 ( Ψ 1 , t , E 2 * ) ∥ ≤ η 4 ∥ E 2 − E 2 * ∥ , ∥ M 4 ( Ψ 1 , t , I ) − M 4 ( Ψ 1 , t , I * ) ∥ ≤ η 5 ∥ I − I * ∥ , ∥ M 5 ( Ψ 1 , t , R ) − M 5 ( Ψ 1 , t , R * ) ∥ ≤ η 6 ∥ R − R * ∥ .

The Lipschitz property is described by the above relations, and the condition holds for all of the underlying functions. With the help of these relations, one can write Eq. (12) as follows:

(17) S n ( t ) − S ( 0 ) = 1 − Ψ 1 B ( Ψ 1 ) M 1 ( Ψ 1 , t , S n − 1 ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 1 ( Ψ 1 , ξ , S n − 1 ( ξ ) ) d ξ , E 1 n ( t ) − E 1 ( 0 ) = 1 − Ψ 1 B ( Ψ 1 ) M 2 ( Ψ 1 , t , E 1 n − 1 ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 2 ( Ψ 1 , ξ , E 1 n − 1 ( ξ ) ) d ξ , E 2 n ( t ) − E 2 ( 0 ) = 1 − Ψ 1 B ( Ψ 1 ) M 3 ( Ψ 1 , t , E 2 n − 1 ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 3 ( Ψ 1 , ξ , E 2 n − 1 ( ξ ) ) d ξ , I n ( t ) − I ( 0 ) = 1 − Ψ 1 B ( Ψ 1 ) M 4 ( Ψ 1 , t , I n − 1 ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 4 ( Ψ 1 , ξ , I n − 1 ( ξ ) ) d ξ , R n ( t ) − R ( 0 ) = 1 − Ψ 1 B ( Ψ 1 ) M 5 ( Ψ 1 , t , R n − 1 ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 5 ( Ψ 1 , ξ , R n − 1 ( ξ ) ) d ξ ,

together with S ( 0 ) = S 0 , E 1 ( 0 ) = E 1 0 , E 2 ( 0 ) = E 2 0 , I ( 0 ) = I 0 , and R ( 0 ) = R 0 . If we took the repeated terms in discernment, we have the following expressions:

(18) Π S , n = S n − S n − 1 = 1 − Ψ 1 B ( Ψ 1 ) ( M 1 ( Ψ 1 , t , S n − 1 ) − M 1 ( Ψ 1 , t , S n − 2 ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) ∫ 0 t ( t − ξ ) Ψ 1 − 1 ( M 1 ( Ψ 1 , ξ , S n − 1 ( ξ ) ) − M 1 ( Ψ 1 , ξ , S n − 2 ( ξ ) ) ) d ξ Π E 1 , n = E 1 n − E 1 n − 1 = 1 − Ψ 1 B ( Ψ 1 ) ( M 2 ( Ψ 1 , t , E 1 n − 1 ) − M 2 ( Ψ 1 , t , E 1 n − 2 ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) ∫ 0 l ( t − ξ ) Ψ 1 − 1 ( M 2 ( Ψ 1 , ξ , E 1 n − 1 ( ξ ) ) − M 2 ( Ψ 1 , ξ , E 1 n − 2 ( ξ ) ) ) d ξ Π E 2 , n = E 22 n − E 2 n − 1 = 1 − Ψ 1 B ( Ψ 1 ) ( M 3 ( Ψ 1 , t , E 2 n − 1 ) − M 3 ( Ψ 1 , t , E 2 n − 2 ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) ∫ 0 t ( t − ξ ) Ψ 1 − 1 ( M 3 ( Ψ 1 , ξ , E 2 n − 1 ( ξ ) ) − M 3 ( Ψ 1 , ξ , E 2 n − 2 ( ξ ) ) ) d ξ Π I , n = I n − I n − 1 = 1 − Ψ 1 B ( Ψ 1 ) ( M 4 ( Ψ 1 , t , I n − 1 ) − M 4 ( Ψ 1 , t , I n − 2 ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) ∫ 0 t ( t − ξ ) Ψ 1 − 1 ( M 4 ( Ψ 1 , ξ , I n − 1 ( ξ ) ) − M 4 ( Ψ 1 , ξ , I n − 2 ( ξ ) ) ) d ξ Π R , n = R n − R n − 1 = 1 − Ψ 1 B ( Ψ 1 ) ( M 5 ( Ψ 1 , t , R n − 1 ) − M 5 ( Ψ 1 , t , R n − 2 ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) ∫ 0 t ( t − ξ ) Ψ 1 − 1 × ( M 5 ( Ψ 1 , ξ , R n − 1 ( ξ ) ) − M 5 ( Ψ 1 , ξ , R n − 2 ( ξ ) ) ) d ξ .

It is worthy to note that

S n = ∑ i = 0 n Π ( S , i ) , E 1 n = ∑ i = 0 n Π ( E 1 , i ) , E 2 n = ∑ i = 0 n Π ( E 2 , i ) I n = ∑ i = 0 n Π ( I , i ) , R n = ∑ i = 0 n Π ( R , i ) .

Furthermore, considering Eqs (15)–(16) and letting

Π S , n − 1 = S n − 1 − S n − 2 , Π E 1 , n − 1 = E 1 n − 1 − E 1 n − 2 , Π E 2 , n − 1 = E 2 n − 1 − E 2 n − 2 , Π I , n − 1 = I n − 1 − I n − 2 , Π R , n − 1 = R n − 1 − R n − 2 ,

we reach

(19) ∥ Π S , n ( t ) ∥ ≤ 1 − Ψ 1 B ( Ψ 1 ) η 1 ∥ Π S , n − 1 ( t ) ∥ Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) η 1 × ∫ 0 t ( t − ξ ) Ψ 1 − 1 ∥ Π S , n − 1 ( ξ ) ∥ d ξ ∥ Π E 1 , n ( t ) ∥ ≤ 1 − Ψ 1 B ( Ψ 1 ) η 2 ∥ Π E 1 , n − 1 ( t ) ∥ Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) η 2 × ∫ 0 t ( t − ξ ) Ψ 1 − 1 ∥ Π E 1 , n − 1 ( ξ ) ∥ d ξ ∥ Π E 2 , n ( t ) ∥ ≤ 1 − Ψ 1 B ( Ψ 1 ) η 4 ∥ Π E 2 , n − 1 ( t ) ∥ Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) η 3 × ∫ 0 t ( t − ξ ) Ψ 1 − 1 ∥ Π E 2 , n − 1 ( ξ ) ∥ d ξ ∥ Π I , n ( t ) ∥ ≤ 1 − Ψ 1 B ( Ψ 1 ) η 5 ∥ Π I , n − 1 ( t ) ∥ Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) η 4 × ∫ 0 t ( t − ξ ) Ψ 1 − 1 ∥ Π I , n − 1 ( ξ ) ∥ d ξ ∥ Π R , n ( t ) ∥ ≤ 1 − Ψ 1 B ( Ψ 1 ) η 6 ∥ Π R , n − 1 ( t ) ∥ Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) η 5 × ∫ 0 t ( t − ξ ) Ψ 1 − 1 ∥ Π R , n − 1 ( ξ ) ∥ d ξ .

Theorem 2

Note that if the following conditions hold:

(20) 1 − Ψ 1 B ( Ψ 1 ) η i + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) b Ψ 1 η i < 1 , i = 1 , 2 , … , 5 ,

model (4) possesses a unique solution for t ∈ [ 0 , b ] .

Proof

As we have noted earlier that under the condition of the maximum upper bounds of the functions S , E 1 , E 2 , I , and R , then M 1 , M 2 , M 3 , M 4 , and M 5 should satisfy the Lipschitzian condition. Therefore, by using Eq. (19), we obtain the following relation:

(21) ∥ Π S , n ( t ) ∥ ≤ ∥ S 0 ( t ) ∥ 1 − Ψ 1 B ( Ψ 1 ) η 1 + Ψ 1 b Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) η 1 n ∥ Π E 1 , n ( t ) ∥ ≤ ∥ E 10 ( t ) ∥ 1 − Ψ 1 B ( Ψ 1 ) η 2 + Ψ 1 b Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) η 2 n ∥ Π E 2 , n ( t ) ∥ ≤ ∥ E 20 ( t ) ∥ 1 − Ψ 1 B ( Ψ 1 ) η 3 + Ψ 1 b Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) η 3 n ∥ Π I , n ( t ) ∥ ≤ ∥ I 0 ( t ) ∥ 1 − Ψ 1 B ( Ψ 1 ) η 4 + Ψ 1 b Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) η 4 n ∥ Π R , n ( t ) ∥ ≤ ∥ R 0 ( t ) ∥ 1 − Ψ 1 B ( Ψ 1 ) η 5 + Ψ 1 b Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) η 5 n .

Thus, by letting n → ∞ , the entire functions will have the limits:

∥ Π S , n ∥ → 0 , ∥ Π E 1 , n ∥ → 0 , ∥ Π E 2 , n ∥ → 0 , ∥ Π I , n ∥ → 0 , ∥ Π R , n ∥ → 0 .

Furthermore, by using the triangle inequality for each value of k , Eq. (17) will take the following form:

(22) ∥ S n + k − S n ∥ ≤ ∑ j = n + 1 n + k Z 1 j = Z 1 n + 1 − Z 1 n + k + 1 1 − Z 1 , ∥ E 1 n + k − E 1 n ∥ ≤ ∑ j = n + 1 n + k Z 2 j = Z 2 n + 1 − Z 2 n + k + 1 1 − Z 2 , ∥ E 2 n + k − E 2 n ∥ ≤ ∑ j = n + 1 n + k Z 3 j = Z 3 n + 1 − Z 3 n + k + 1 1 − Z 3 , ∥ I n + k − I n ∥ ≤ ∑ i = n + 1 n + k Z 4 j = Z 4 n + 1 − Z 4 n + k + 1 1 − Z 4 , ∥ R n + k − R n ∥ ≤ ∑ i = n + 1 n + k Z 5 j = Z 5 n + 1 − Z 5 n + k + 1 1 − Z 5 ,

with Z i = 1 − Ψ 1 B ( Ψ 1 ) η i + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) b Ψ 1 η i < 1 by supposition. It may be observed that in space B ( J ) , the functions S n , E 1 n , E 2 n , I n and R n are Cauchy sequences, and hence implies its uniform convergence. By letting n → ∞ , relation (18) guarantees that Eq. (4) has a unique solution. Therefore, one can conclude that under condition (20), there does exist a unique solution of Eq. (4) and hence the result.□

5 Hyers–Ulam stability

In the field of functional analysis, the concept of Hyers–Ulam stability, also known as Hyers–Ulam–Rassias stability, refers to the steadiness of functional equations. It was named after David Hyers, Stanislaw Ulam, and Themistocles Rassias, who independently showed that the Cauchy equation for functional equations preserves stability in certain situations. In other words, it states that a function meets the criteria for being considered a solution to a functional equation if it strongly resembles a solution to the problem. The scope of this idea has been extended to include a wide range of functional equations, including those involving nonlinear processes. Overall, Hyers–Ulam stability plays a key role in the investigation of functional equations and has been used in a wide range of mathematical and scientific fields.

Definition 1

A model having integration of AB type as in system (12) is said to be H–U stable [38] if there exists some positive constants Δ i , for i ∈ N 5 satisfying the following:

(23) S ( t ) − 1 − Ψ 1 B ( Ψ 1 ) M 1 ( Ψ 1 , t , S ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 1 ( Ψ 1 , ξ , S ( ξ ) ) d ξ ≤ γ 1 , E 1 ( t ) − 1 − Ψ 1 B ( Ψ 1 ) M 2 ( Ψ 1 , t , E 1 ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 2 ( Ψ 1 , ξ , E 1 ( ξ ) ) d ξ ≤ γ 2 , E 2 ( t ) − 1 − Ψ 1 B ( Ψ 1 ) M 3 ( Ψ 1 , t , E 2 ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 3 ( Ψ 1 , ξ , E 2 ( ξ ) ) d ξ ≤ γ 3 , I ( t ) − 1 − Ψ 1 B ( Ψ 1 ) M 4 ( Ψ 1 , t , I ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 4 ( Ψ 1 , ξ , I ( ξ ) ) d ξ ≤ γ 4 , R ( t ) − 1 − Ψ 1 B ( Ψ 1 ) M 5 ( Ψ 1 , t , R ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 5 ( Ψ 1 , ξ , R ( ξ ) ) d ξ ≤ γ 5 ,

where γ i > 0 , i ∈ N 5 . Furthermore, it is easy to show that ( S ˙ , E 1 ˙ , E 2 ˙ , I ˙ , R ˙ ) satisfies

(24) S ˙ ( t ) = 1 − Ψ 1 B ( Ψ 1 ) M 1 ( Ψ 1 , t , S ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 1 ( Ψ 1 , ξ , S ˙ ( ξ ) ) d ξ , E 1 ˙ ( t ) = 1 − Ψ 1 B ( Ψ 1 ) M 2 ( Ψ 1 , t , E 1 ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 2 ( Ψ 1 , ξ , E 1 ˙ ( ξ ) ) d ξ , E 2 ˙ ( t ) = 1 − Ψ 1 B ( Ψ 1 ) M 3 ( Ψ 1 , t , E 2 ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 3 ( Ψ 1 , ξ , I 12 ˙ ( ξ ) ) d ξ , I ˙ ( t ) = 1 − Ψ 1 B ( Ψ 1 ) M 4 ( Ψ 1 , t , I ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 4 ( Ψ 1 , ξ , I ˙ ( ξ ) ) d ξ , R ˙ ( t ) = 1 − Ψ 1 B ( Ψ 1 ) M 5 ( Ψ 1 , t , R ( t ) ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) × ∫ 0 t ( t − ξ ) Ψ 1 − 1 M 5 ( Ψ 1 , ξ , R ˙ ( ξ ) ) d ξ .

Such that

∣ S − S ˙ ∣ ≤ ζ 1 γ 1 , ∣ E 1 − E 1 ˙ ∣ ≤ ζ 2 γ 2 , ∣ E 2 − E 2 ˙ ∣ ≤ ζ 3 γ 3 , ∣ I − I ˙ ∣ ≤ ζ 4 γ 4 , ∣ R − R ˙ ∣ ≤ ζ 5 γ 5 .

Theorem 3

If J holds, then the fractional-order model presented in system (4) is H–U stable.

Proof

Referred to Theorem 2, one can say that model (4) has a unique solution ( S , E 1 , E 2 , I , R ) satisfying model (12). Thus, one can write

(25) ‖ S − S ˙ ‖ ≤ 1 − Ψ 1 B ( Ψ 1 ) ∥ M 1 ( Ψ 1 , t , S ) − M 1 ( Ψ 1 , t , S ˙ ) ∥ + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) ∫ 0 t ( t − ξ ) Ψ 1 − 1 ∥ M 1 ( Ψ 1 , t , S ) − M 1 ( Ψ 1 , t , S ˙ ) ∥ d ξ ≤ 1 − Ψ 1 B ( Ψ 1 ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) Ψ 11 ‖ S − S ˙ ‖ ,

(26) ‖ E 1 − E 1 ˙ ‖ ≤ 1 − Ψ 1 B ( Ψ 1 ) ∥ M 2 ( Ψ 1 , t , E 1 ) − M 2 ( Ψ 1 , t , E 1 ˙ ) ∥ + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) ∫ 0 t ( t − ξ ) Ψ 1 − 1 ∥ M 2 ( Ψ 1 , t , E 1 ) − M 2 ( Ψ 1 , t , E 1 ˙ ) ∥ d ξ ≤ 1 − Ψ 1 B ( Ψ 1 ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) Ψ 12 ‖ E 1 − E 1 ˙ ‖ ,

(27) ‖ E 2 − E 2 ˙ ‖ ≤ 1 − Ψ 1 B ( Ψ 1 ) ∥ M 3 ( Ψ 1 , t , I ) − M 3 ( Ψ 1 , t , I ˙ ) ∥ + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) ∫ 0 t ( t − ξ ) Ψ 1 − 1 ∥ M 3 ( Ψ 1 , t , E 2 ) − M 3 ( Ψ 1 , t , E 2 ˙ ) ∥ d ξ ≤ 1 − Ψ 1 B ( Ψ 1 ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) Ψ 14 ‖ E 2 − E 2 ˙ ‖ ,

(28) ‖ I − I ˙ ‖ ≤ 1 − Ψ 1 B ( Ψ 1 ) ∥ M 4 ( Ψ 1 , t , I ) − M 4 ( Ψ 1 , t , I ˙ ) ∥ + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) ∫ 0 t ( t − ξ ) Ψ 1 − 1 ∥ M 4 ( Ψ 1 , t , A ) − M 4 ( Ψ 1 , t , I ˙ ) ∥ d ξ ≤ 1 − Ψ 1 B ( Ψ 1 ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) Ψ 15 ‖ I − I ˙ ‖ ,

(29) ‖ R − R ˙ ‖ ≤ 1 − Ψ 1 B ( Ψ 1 ) ∥ M 5 ( Ψ 1 , t , R ) − M 5 ( Ψ 1 , t , R ˙ ) ∥ + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) ∫ 0 t ( t − ξ ) Ψ 1 − 1 ∥ M 5 ( Ψ 1 , t , R ( t ) ) − M 5 ( Ψ 1 , t , R ˙ ) ∥ d ξ ≤ 1 − Ψ 1 B ( Ψ 1 ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) Ψ 16 ‖ R − R ˙ ‖ .

Taking γ i = Ψ 1 , Δ i = 1 − Ψ 1 B ( Ψ 1 ) + Ψ 1 B ( Ψ 1 ) Γ ( Ψ 1 ) , which implies

(30) ‖ S − S ˙ ‖ ≤ γ 1 Δ 1 .

Similarly,

(31) ‖ E 1 − E 1 ˙ ‖ ≤ γ 2 Δ 2 , ‖ E 2 − E 2 ˙ ‖ ≤ γ 3 Δ 3 , ‖ I − I ˙ ‖ ≤ γ 4 Δ 4 , ‖ R − R ˙ ‖ ≤ γ 5 Δ 5 ,

which completes the proof.□

6 Iterative solution by Newton polynomial

In this part of the manuscript, we will introduce a numerical approach based on Newton polynomials for the underlying model [39]. As seen in [40,41], by using this approach, Atangana and Seda suggested and resolved new deterministic models for COVID-19. Newton’s interpolation, a well-known technique for polynomial interpolation, is used in both numerical analysis and image processing. Interpolation functions used in traditional approaches are frequently adapted to known data. Due to its several benefits over other polynomial interpolation methods, Newton’s polynomial interpolation is highlighted in this article. In addition, this method provides rapid convergence, simple execution, mathematical dependability, and efficiency in a variety of areas, including differentiation and integration. Likewise, it is very simple to compute the derivatives of polynomials of any order. The Newton-type polynomial interpolation function can be modified within the interpolation zone by choosing the appropriate values of the parameter. It is also feasible to modify the interpolated curves or surfaces to conform to precise geometric design specifications.

Starting with the ML kernel,

(32) D 0 , t Ψ 1 A B C [ S ] = b − φ S ( t ) I ( t ) − σ S ( t ) − ω S ( t ) D 0 , t Ψ 1 A B C [ E 1 ] = p φ S ( t ) I ( t ) − ( λ 1 + σ ) E 1 ( t ) D 0 , t Ψ 1 A B C [ E 2 ] = q φ S ( t ) I ( t ) − ( λ 2 + σ ) E 2 ( t ) D 0 , t Ψ 1 A B C [ I ] = λ 1 E 1 ( t ) + λ 2 E 2 ( t ) − ( σ + γ ) I ( t ) D 0 , t Ψ 1 A B C [ R ] = γ I ( t ) − σ R ( t ) + ω S ( t ) .

In a more compact and simpler form, we have the following system of equations:

(33) D 0 , t Ψ 1 AℬC [ S ( t ) ] = S ∗ ( t , S , E 1 , E 2 , I , R ) , D 0 , t Ψ 1 AℬC [ I 1 ( t ) ] = E 1 ∗ ( t , S , E 1 , E 2 , I , R ) , D 0 , t Ψ 1 AℬC [ E 2 ( t ) ] = E 2 ∗ ( t , S , E 1 , E 2 , I , R ) , D 0 , t Ψ 1 AℬC [ I ( t ) ] = I ∗ ( t , S , E 1 , E 2 , I , R ) , D 0 , t Ψ 1 AℬC [ R ( t ) ] = R ∗ ( t , S , E 1 , E 2 , I , R ) .

If one use the fractional order integral subject to the ML kernel and imply the method of Newton polynomial, the following expression can be easily obtained:

S a + 1 = 1 − Ψ 1 A B ( Ψ 1 ) + S ∗ ( t a , S a , E 1 a , E 2 a , I a , R a ) + Ψ 1 ( Δ t ) Ψ 1 A B ( Ψ 1 ) Γ ( Ψ 1 + 1 ) ∑ μ = 2 a S ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Π + Ψ 1 ( Δ t ) Ψ 1 A B ( Ψ 1 ) Γ ( Ψ 1 + 2 ) ∑ μ = 2 a × S ∗ ( t μ − 1 , S μ − 1 , E 1 μ − 1 , E 2 μ − 1 , I μ − 1 , R μ − 1 ) − S ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Σ + Ψ 1 ( Δ t ) Ψ 1 2 A B ( Ψ 1 ) Γ ( Ψ 1 + 3 ) ∑ μ = 2 a × S ∗ ( t μ , S μ , E 1 μ , E 2 μ , I μ , R μ ) − 2 S ∗ ( t μ − 1 , S μ − 1 , E 1 μ − 1 , E 2 μ − 1 , I μ − 1 , R μ − 1 ) + S ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Δ

E 1 a + 1 = 1 − Ψ 1 A B ( Ψ 1 ) + E 1 ∗ ( t a , S a , E 1 a , E 2 a , I a , R a ) + Ψ 1 ( Δ t ) Ψ 1 A B ( Ψ 1 ) Γ ( Ψ 1 + 1 ) ∑ μ = 2 a E 1 ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Π + Ψ 1 ( Δ t ) Ψ 1 A B ( Ψ 1 ) Γ ( Ψ 1 + 2 ) ∑ μ = 2 a × E 1 ∗ ( t μ − 1 , S μ − 1 , E 1 μ − 1 , E 2 μ − 1 , I μ − 1 , R μ − 1 ) − E 1 ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Σ + Ψ 1 ( Δ t ) Ψ 1 2 A B ( Ψ 1 ) Γ ( Ψ 1 + 3 ) ∑ μ = 2 a × E 1 ∗ ( t μ , S μ , E 1 μ , E 2 μ , I μ , R μ ) − 2 E 1 ∗ ( t μ − 1 , S μ − 1 , E 1 μ − 1 , E 2 μ − 1 , I μ − 1 , R μ − 1 ) + E 1 ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Δ

E 2 a + 1 = 1 − Ψ 1 A B ( Ψ 1 ) + E 2 ∗ ( t a , S a , E 1 a , E 2 a , I a , R a ) + Ψ 1 ( Δ t ) Ψ 1 A B ( Ψ 1 ) Γ ( Ψ 1 + 1 ) ∑ μ = 2 a E 2 ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Π + Ψ 1 ( Δ t ) Ψ 1 A B ( Ψ 1 ) Γ ( Ψ 1 + 2 ) ∑ μ = 2 a × E 2 ∗ ( t μ − 1 , S μ − 1 , E 1 μ − 1 , E 2 μ − 1 , I μ − 1 , R μ − 1 ) − E 2 ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Σ + Ψ 1 ( Δ t ) Ψ 1 2 A B ( Ψ 1 ) Γ ( Ψ 1 + 3 ) ∑ μ = 2 a × E 2 ∗ ( t μ , S μ , E 1 μ , E 2 μ , I μ , R μ ) − 2 E 2 ∗ ( t μ − 1 , S μ − 1 , E 1 μ − 1 , E 2 μ − 1 , I μ − 1 , R μ − 1 ) + E 2 ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Δ

I a + 1 = 1 − Ψ 1 A B ( Ψ 1 ) + I ∗ ( t a , S a , E 1 a , E 2 a , I a , R a ) + Ψ 1 ( Δ t ) Ψ 1 A B ( Ψ 1 ) Γ ( Ψ 1 + 1 ) ∑ μ = 2 a I ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Π + Ψ 1 ( Δ t ) Ψ 1 A B ( Ψ 1 ) Γ ( Ψ 1 + 2 ) ∑ μ = 2 a × I ∗ ( t μ − 1 , S μ − 1 , E 1 μ − 1 , E 2 μ − 1 , I μ − 1 , R μ − 1 ) − I ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Σ + Ψ 1 ( Δ t ) Ψ 1 2 A B ( Ψ 1 ) Γ ( Ψ 1 + 3 ) ∑ μ = 2 a × I ∗ ( t μ , S μ , E 1 μ , E 2 μ , I μ , R μ ) − 2 I ∗ ( t μ − 1 , S μ − 1 , E 1 μ − 1 , E 2 μ − 1 , I μ − 1 , R μ − 1 ) + I ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Δ

R a + 1 = 1 − Ψ 1 A B ( Ψ 1 ) + R ∗ ( t a , S a , E 1 a , E 2 a , I a , R a ) + Ψ 1 ( Δ t ) Ψ 1 A B ( Ψ 1 ) Γ ( Ψ 1 + 1 ) ∑ μ = 2 a R ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Π + Ψ 1 ( Δ t ) Ψ 1 A B ( Ψ 1 ) Γ ( Ψ 1 + 2 ) ∑ μ = 2 a × R ∗ ( t μ − 1 , S μ − 1 , E 1 μ − 1 , E 2 μ − 1 , I μ − 1 , R μ − 1 ) − R ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Σ + Ψ 1 ( Δ t ) Ψ 1 2 A B ( Ψ 1 ) Γ ( Ψ 1 + 3 ) ∑ μ = 2 a × R ∗ ( t μ , S μ , E 1 μ , E 2 μ , I μ , R μ ) − 2 R ∗ ( t μ − 1 , S μ − 1 , E 1 μ − 1 , E 2 μ − 1 , I μ − 1 , R μ − 1 ) + R ∗ ( t μ − 2 , S μ − 2 , E 1 μ − 2 , E 2 μ − 2 , I μ − 2 , R μ − 2 ) Δ ,

where

Δ = ( a − μ + 1 ) Ψ 1 2 ( a − μ ) 2 + ( 3 Ψ 1 + 10 ) ( a − μ ) + 2 Ψ 1 2 + 9 Ψ 1 + 12 − ( a − μ ) Ψ 1 2 ( a − μ ) 2 + ( 5 Ψ 1 + 10 ) ( a − μ ) + 6 Ψ 1 2 + 18 Ψ 1 + 12 , Σ = ( a − μ + 1 ) Ψ 1 ( a − μ + 3 + 2 Ψ 1 ) − ( a − μ ) Ψ 1 ( a − μ + 3 + 3 Ψ 1 ) , Π = [ ( 1 − μ + a ) Ψ 1 − ( a − μ ) Ψ 1 ] .

7 Numerical simulations of the communication radius of nodes

Both the models, based on ODEs and AB fractional derivatives, have nonlinearities. Therefore, it is very important to use approximation techniques which is easy to implement and accurate for obtaining the desired simulation results. The approximation techniques to be utilized here have some interesting results about its well-posedness, convergence, error constraints, and outfitted with stability. In addition, Table 2 displays the values of the state variables at the initial time as well as parameter values pertinent to the models under consideration and to be used for simulation. The desired time-period for simulating the model is specified in weeks, and it last from 0 to 140. A MATLAB built in-function named “fminsearch” is used to obtain the optimal value of the fractional-order parameter for Ψ = 1 . The scheme was developed, the least squares optimization technique was utilized, and the results are displayed in Figure 2(a)–(e). At Ψ = 1 , the differences between the ordinary differential equation (ODE) and the model based on AB fractional derivative is specifically shown in Figure 2. As the model is stable in the long-run, the trajectories of all compartments with fractional orders behave like integer-orders. It is to be noted that increasing sensor node connectivity yields better network connectivity.

Table 2

Model parameters and their values

Parameters	Value	Source
b	0.400	[27]
σ	0.001	[27]
L	10.00	[27]
r	1.500	[27]
β	0.002	[27]
p	0.700	[27]
q	0.500	Estimated
λ 1	0.002	[27]
λ 2	0.001	[27]
γ	0.003	[27]
ω	0.004	[27]
S ( 0 )	100.0	Estimated
E 1 ( 0 )	90.00	Estimated
E 2 ( 0 )	80.00	Estimated
I ( 0 )	70.00	Estimated
R ( 0 )	20.00	Estimated

$Figure 2 Comparison of the classical system (2) with the fractional-order system (4) for the WSNs population when Ψ = 1 \Psi =1 . (a) S ( t ) {\mathsf{S}}\left(t) -sensor node, (b) E 1 ( t ) {{\mathsf{E}}}_{1}\left(t) -sensor node, (c) E 2 ( t ) {{\mathsf{E}}}_{2}\left(t) -sensor node, (d) I ( t ) {\mathsf{I}}\left(t) -sensor node, and (e) R ( t ) {\mathsf{R}}\left(t) -sensor node.$

Figure 2

Comparison of the classical system (2) with the fractional-order system (4) for the WSNs population when Ψ = 1 . (a) S ( t ) -sensor node, (b) E 1 ( t ) -sensor node, (c) E 2 ( t ) -sensor node, (d) I ( t ) -sensor node, and (e) R ( t ) -sensor node.

For the proposed system (4), we ran a variety of simulations using various fractional order values to depict the future behavior of the model. The parameter values and initial data for the models under investigation are also shown in Table 3. The numerical simulations is carried out within the time ranges from 0 to 140 weeks. We used the MATLAB code “fminsearch,” which employs the least squares optimization technique, to find optimum values for the fractional-order parameter. Figure 3(a)--(e) shows the results of these optimizations. Specifically, Figure 3 illustrates the comparison of solution trajectories obtained from the classical model and the AB fractional-order model. The AB operator closely approximates the ODE model and exhibits the highest efficiency rate for fractional-order differential operators. All compartments with fractional orders exhibit behavior similar to integer-orders at stable points. To provide more detail, the solution trajectories of the proposed system are presented using the AB-Caputo operator with a fractional order of 0.9, along with classical integer-order derivatives. Moreover, it is worth noting that an increase in the connectivity of sensor nodes leads to improved network connectivity.

Table 3

Model parameters and their values

Parameters	Value	Source
b	0.400	[27]
σ	0.001	[27]
L	10.00	[27]
r	1.500	[27]
β	0.002	[27]
p	0.700	[27]
q	0.500	Estimated
λ 1	0.002	[27]
λ 2	0.001	[27]
γ	0.003	[27]
ω	0.004	[27]
S ( 0 )	100.0	Estimated
E 1 ( 0 )	90.00	Estimated
E 2 ( 0 )	80.00	Estimated
I ( 0 )	70.00	Estimated
R ( 0 )	20.00	Estimated

$Figure 3 Comparison of the classical system (2) with the fractional-order system (4) for the WSNs population. (a) S ( t ) {\mathsf{S}}\left(t) -sensor node, (b) E ( t ) {\mathsf{E}}\left(t) -sensor node, (c) I ( t ) {\mathsf{I}}\left(t) -sensor node, (d) R ( t ) {\mathsf{R}}\left(t) -sensor node, and (e) R ( t ) {\mathsf{R}}\left(t) -sensor node.$

Figure 3

Comparison of the classical system (2) with the fractional-order system (4) for the WSNs population. (a) S ( t ) -sensor node, (b) E ( t ) -sensor node, (c) I ( t ) -sensor node, (d) R ( t ) -sensor node, and (e) R ( t ) -sensor node.

To draw the solution trajectories of system (4), we ran a number of simulations using various fractional order values, especially Ψ = ( 0.95 , 0.85 , 0.75 , 0.65 , 0.55 ) . In addition, Table 4 lists the initial states as well as the parameter values related to the underlying models. The range of the numerical simulations’ time period is from 0 to 140 weeks. Through the use of the MATLAB program “fminsearch,” which utilize the least squares optimization method, the fractional-order parameter was optimized. Figure 4(a)--(e) shows the outcomes of these optimizations algorithm. In particular, Figure 4 demonstrates the comparison of the classical model and AB fractional-order operator when Ψ = ( 0.95 , 0.85 , 0.75 , 0.65 , 0.55 ) . The AB operator are nearest to the ODE model and highest efficiency rate for the fractional-order differential operator. All of the compartments at fractional orders behave in integer-order at the points of stability. To be more specific, the solution pathways of the proposed system are presented through ABC operator with fractional order Ψ = ( 0.95 , 0.85 , 0.75 , 0.65 , 0.55 ) and with classical integer-order derivatives. Furthermore, it is notable that an increase in the connectivity of sensor nodes leads to improved network connectivity.

Table 4

Model parameters and their values

Parameters	Value	Source
b	0.500	Estimated
σ	0.001	[27]
L	10.00	[27]
r	1.000	Estimated
β	0.002	[27]
p	0.500	Estimated
q	0.500	Estimated
λ 1	0.002	[27]
λ 2	0.001	[27]
γ	0.003	[27]
ω	0.004	[27]
S ( 0 )	100.0	Estimated
E 1 ( 0 )	90.00	Estimated
E 2 ( 0 )	80.00	Estimated
I ( 0 )	70.00	Estimated
R ( 0 )	20.00	Estimated

$Figure 4 Comparison of the classical system (2) with the fractional-order system (4) for the WSNs population when Ψ = ( 0.95 , 0.85 , 0.75 , 0.65 , 0.55 ) \Psi =\left(0.95,0.85,0.75,0.65,0.55) . (a) S ( t ) {\mathsf{S}}\left(t) -sensor node, (b) E ( t ) {\mathsf{E}}\left(t) -sensor node, (c) I ( t ) {\mathsf{I}}\left(t) -sensor node, (d) R ( t ) {\mathsf{R}}\left(t) -sensor node, and (e) R ( t ) {\mathsf{R}}\left(t) -sensor node.$

Figure 4

Comparison of the classical system (2) with the fractional-order system (4) for the WSNs population when Ψ = ( 0.95 , 0.85 , 0.75 , 0.65 , 0.55 ) . (a) S ( t ) -sensor node, (b) E ( t ) -sensor node, (c) I ( t ) -sensor node, (d) R ( t ) -sensor node, and (e) R ( t ) -sensor node.

To determine the ideal values of the fractional-order parameter, we performed simulations using the parameter values from Table 5 and the given initial state of the variables. Figure 5(a)--(e) is produced as a consequence of this simulations using the MATLAB code “fminsearch,” which applies the least squares method. The contrast between the ODE model and the model based on AB fractional-order operator for different values of Ψ = ( 0.95 , 0.85 , 0.75 , 0.65 , 0.55 ) is particularly displayed in Figure 3. The solution trajectories of the suggested system are shown for both the classical and fractional model for suitable orders Ψ = ( 0.95 , 0.85 , 0.75 , 0.65 , 0.55 ) to further increase their specificity. Furthermore, it is important to note that an increase in sensor node connectivity results better connectivity of the network.

Table 5

Model parameters and their values

Parameters	Value	Source
b	0.600	Estimated
σ	0.001	[27]
L	10.00	[27]
r	1.700	Estimated
β	0.002	[27]
p	0.500	Estimated
q	0.500	Estimated
λ 1	0.002	[27]
λ 2	0.001	[27]
γ	0.003	[27]
ω	0.004	[27]
S ( 0 )	100.0	Estimated
E 1 ( 0 )	90.00	Estimated
E 2 ( 0 )	80.00	Estimated
I ( 0 )	70.00	Estimated
R ( 0 )	20.00	Estimated

$Figure 5 Comparison of the compartments of fractional-order system (4) for the WSNs population. (a) S ( t ) {\mathsf{S}}\left(t) and E 1 ( t ) {{\mathsf{E}}}_{1}\left(t) sensor nodes, (b) S ( t ) {\mathsf{S}}\left(t) and E 2 ( t ) {{\mathsf{E}}}_{2}\left(t) sensor nodes, (c) E 1 ( t ) {{\mathsf{E}}}_{1}\left(t) and I ( t ) {\mathsf{I}}\left(t) sensor nodes, (d) S ( t ) , E ( t ) {\mathsf{S}}\left(t),{\mathsf{E}}\left(t) , and R ( t ) {\mathsf{R}}\left(t) sensor nodes, and (e) S ( t ) , E 1 ( t ) {\mathsf{S}}\left(t),{{\mathsf{E}}}_{1}\left(t) and I ( t ) {\mathsf{I}}\left(t) sensor nodes.$

Figure 5

Comparison of the compartments of fractional-order system (4) for the WSNs population. (a) S ( t ) and E 1 ( t ) sensor nodes, (b) S ( t ) and E 2 ( t ) sensor nodes, (c) E 1 ( t ) and I ( t ) sensor nodes, (d) S ( t ) , E ( t ) , and R ( t ) sensor nodes, and (e) S ( t ) , E 1 ( t ) and I ( t ) sensor nodes.

8 Conclusion

For the fractional-order models to remain dimensionally consistent, the appropriate fractional-order parameter has been applied to each dimensional quantity. In the present work, fixed point theory was employed to demonstrate that the AB fractional-order model has a solitary solution in the sense of Caputo. By utilizing the AB operator, we presented a fractional-order model to solve the problem of worm control in WSNs. In the context of the Caputo sense, the AB operator contains a ML-type kernel, giving it the capacity to encompass both Riemann-Liouville and Caputo derivatives for later times while successfully handling the Caputo fractional derivative for earlier times. The solution of the suggested fractional-order system was proven to exist and is unique by applying fixed-point theory. In addition, to prove the stability of the model, the Ulam–Hyers stability approach was utilized. We used the Newton polynomial as an iterative technique to produce a numerical solution for the system under analysis. The results of the simulation have consistently shown convergence and stability across all compartments. These simulation results have received a lot of praise for significantly strengthening the study of infectious illness models at different fractional orders.

Multiple malware strains in WSNs can significantly reduce the network’s reliability and robustness. It becomes more challenging to defend against and recover from attacks when multiple attack vectors are simultaneously in play. The extensive results obtained from this study provide strong evidence that the proposed model contributes to increased network lifetime in WSN. These discoveries have direct applications for the software industry since they may be used to improve antivirus programs that can successfully prevent malware assaults on WSNs. In addition, the research will help users recover the compromised nodes, and they can carefully apply an antivirus software on these sensor nodes, improving the whole security architecture to lessen the threats. Researchers and practitioners often employ various security measures, including intrusion detection systems, encryption, secure routing protocols, and regular software updates to patch vulnerabilities. This approach is innovative in which it aims to improve modeling accuracy by including interactions between nodes that are memory-dependent and nonlocal. This is accomplished by combining the power-law kernel with the ML-type functions in the definition of the derivatives.

A future study may also include other elements, including classes that have been passed from the anti-virus software or the isolation of the nodes affected by the malware, as well as incorporate diversified and mobile nodes. These factors may improve the model’s applicability and offer further information on the dynamics of worm spread and on its defense mechanisms in WSNs.

Funding information: Prince Sultan University is appreciated for article processing charges and support through the theoretical and applied science research Lab. This study was also supported in the part by the Science and Technology Commission of Shanghai Municipality (No. 22DZ2229014).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Because no new data were produced or processed in this study, data sharing is not applicable to this publication.

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Received: 2023-09-11

Revised: 2023-12-17

Accepted: 2024-01-14

Published Online: 2024-02-26

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2023-0190

Keywords for this article

epidemic model; wireless sensor networks; fractional derivative; differential operator; computational method

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