A Fractional Framework for Modeling Malicious Code Spread in Wireless Sensor Networks

Abstract

This paper develops a fractional six-compartment model to describe malware spread in wireless sensor networks. To represent actual network activity, the model is constructed using generalized proportional-Caputo operators that incorporate memory and tempering effects. The existence and uniqueness of solutions are proved by applying fixed-point theorems. The stability of the system is then studied using the Ulam–Hyers approach and its extended form. A fractional Adams predictor–corrector method is employed to illustrate the dynamics. The results suggest that memory and tempering play an important role in shaping infection patterns, and they indicate that fractional calculus can provide a useful framework for studying and managing malware in distributed sensor networks.

1. Introduction

In centuries past, societies faced deadly epidemics such as influenza, malaria, and cholera at times when limited medical knowledge caused diseases to spread quickly. Even then, people understood that immunity gained from infection could fade, leaving them vulnerable again [1,2]. Misunderstandings often followed: those who had recovered from illness were thought to be no longer contagious, so they mixed freely with others, unknowingly making things worse. Over time, advances in biomedical science made this picture better: diagnostics, vaccines, treatments, and public health strategies transformed how outbreaks were managed. The eighteenth century marked a turning point for mathematics: real populations began to be described using mathematical laws, starting with basic growth models, then evolving into nonlinear equations like the logistic model, and eventually more complex predator–prey dynamics. By the late 20th century, these ideas solidified into the first mechanism that grouped people by infection status, known as compartmental models. Since then, researchers have continued to explore how diseases spread and how people’s behavior and awareness affect that spread. They have carefully analyzed how these models behave and stay stable over time [3,4,5] while tailoring them to specific diseases and settings from pine-wilt and leishmaniasis to HIV, tuberculosis, viral infections, and, most recently, COVID-19 [6,7,8]. This extensive work has helped define the conditions under which diseases persist or fade out, improving our understanding of thresholds, long-term dynamics, and effective control strategies [9,10]. When the discipline evolved, researchers ordered structures of modeling by age-structure characteristics, predictability, qualitative dynamics by virtue of bifurcations, and by prey and predator interactions [9,11,12]. The tools soon found use in investigations not confined to communicable illnesses. Closely related constructions in turn found application in addressing social processes, like smoking and alcoholic beverage consumption behaviors among different population groups. The above cross-disciplinary generalization is representative of mathematical models being able to shed and guide understanding of complex behaviors in a wide range of areas.

According to Akyildiz et al. [13], wireless sensor nodes can monitor a wide range of environmental and physical conditions, such as temperature, humidity, pressure, sound, and pollution. When deployed in the field, these nodes interconnect wirelessly to create sensing networks that may be as small as a few devices or as large as several thousand spread over the area of interest. The measured data are transferred to base stations either through direct communication with nearby nodes or by relaying information across multiple intermediate nodes. Standard radio protocols, adjusted to the operating environment, usually support these connections. Deployment strategies may be random, with minimal later intervention, or carefully planned to allow for easier monitoring and reduced maintenance. Tentative deployments usually reach good coverage at small numbers of devices, while random deployment may leave swaths uncovered [14]. Zhao et al. [15] note that sensor networks can be designed in two main ways. In one approach, all nodes perform the same functions without hierarchy. In another, tasks such as sensing, data collection, and coordination are divided across several layers. As the field develops, two choices remain critical: whether nodes should be placed randomly or in a planned layout, and what overall structure best supports reliable monitoring. Meanwhile, investigations of artificial neural networks are extensive and problematic to translate, with work currently going on in portions of this field as in disease modeling, models of random processes, deterministic chaos, and miscellaneous complex systems [16,17,18].

The calculus of derivatives and integrals of non-integer order, traditionally called fractional calculus, expands the classical calculus by enabling the order of differentiation and integration to take arbitrary real values. Concern in this area has grown because it retains memory and hereditary impacts that classical integer-order models cannot express with the same accuracy. In recent years, a variety of new definitions of fractional operators have emerged. A purpose has been to expand the class of acceptable kernels exceeding simple power functions so that a greater variety of physical and biological conducts can be modeled. With these more elastic operators, researchers have formulated new classes of nonlinear differential and integral equations [19,20,21,22,23]. Exact solutions are infrequently available for such models, so effectiveness is typically evaluated through numerical simulation. Several approximation strategies have therefore been refined, providing computational validation for chaotic limit cycles and other complex behaviors. Based on combinations of power law, modified exponential decay, and Mittag–Leffler laws, Atangana presented differential operators with fractal derivatives [24]. In related instructions, Akgul et al. analyzed a computer virus system, forming conditions for solution existence, steady states, equilibrium points, and stability within the structure of the Atangana–Baleanu operator [25]. Abdel-Gawad et al. analyzed an antivirus model on a computer network, gained an equivalent system of nonautonomous ordinary differential equations from a fractional formulation, and estimated numerical solutions for both Caputo and Caputo–Fabrizio contexts [26]. Singh et al. investigated a fractional computer virus epidemiology model, proved existence results, and supplied numerical experiments [27]. Ozdemir et al. presented a qualitative analysis for fractional computer virus spread and investigated existence, uniqueness, and numerical results [28]. Mishra and Keshri validated the reproduction threshold, equilibria, and stability properties for a model explaining wireless sensor networks [29]. Kumar and Singh investigated existence and uniqueness for a fractional epidemiological model for computer viruses using fixed-point techniques with a Mittag–Leffler kernel [30].

The aim of this paper is to build a fractional model that explains how malware can spread through wireless sensor networks. The model uses a six-part structure with generalized proportional-Caputo operators, which make it possible to reflect memory, delay, and the gradual loss of protection in a realistic way. Fixed-point theory is applied to show the existence and uniqueness of solutions for the investigated model. The stability of the model is then investigated via Ulam–Hyers techniques. A fractional Adams predictor–corrector method is used to explore the system in practice, and the results highlight the influence of infection rate, vaccination, and quarantine on the spread of malware.

Now, we outline the paper’s flow. In Section 2, we gather the fractional tools needed later, including Riemann–Liouville and Caputo operators alongside the generalized proportional-Caputo operators. Section 3 formulates the six-compartment model for wireless sensor networks under the generalized proportional-Caputo framework. Section 4 develops the analytical approach: we recast the system into an equivalent integral form, prove existence and uniqueness under the fixed-point principle, and establish Ulam–Hyers-type stability bounds. In Section 5, we turn to computation, designing a fractional Adams predictor–corrector scheme that respects the nonlocal memory kernel and reporting simulations. Finally, Section 6 provides the concluding remarks.

2. Fundamentals of Fractional Operators

This section introduces the primary fractional operators utilized in our study. These operators play a crucial role in the examination and analysis of fractional differential equations.

Consider $\varrho \in \mathbb{C}$ with $\mathrm{Re}\left(\varrho \right)>0$, and let $\chi :[\kappa ,\sigma ]\to \mathbb{R}$, where $\kappa <\sigma$. Under these conditions, we present the following definitions.

Definition 1 

([31]). The left -sided Riemann–Liouville fractional integral of χ of order ϱ is given by

$$\left({}_{\kappa }{\rm Y}^{\varrho }\chi \right)\left(\eta \right)={\displaystyle \frac{1}{\Gamma \left(\varrho \right)}}{\int }_{\kappa }^{\eta }{(\eta -\nu )}^{\varrho -1}\chi \left(\nu \right)d\nu ,\eta >\kappa .$$

(1)

The right-sided Riemann–Liouville fractional integral of χ with order ϱ is expressed as

$$\left({{\rm Y}}_{\sigma }^{\varrho }\chi \right)\left(\eta \right)={\displaystyle \frac{1}{\Gamma \left(\varrho \right)}}{\int }_{\eta }^{\sigma }{(\nu -\eta )}^{\varrho -1}\chi \left(\nu \right)d\nu ,\eta <\sigma .$$

(2)

Definition 2 

([31]). Given that $\mathrm{Re}\left(\varrho \right)\ge 0$ and defining $k=\left[\varrho \right]+1$, the left-sided Riemann–Liouville fractional derivative of χ of order ϱ is formulated as

$$\left({}_{\kappa }\Delta ^{\varrho }\chi \right)\left(\eta \right)={\left({\displaystyle \frac{d}{d\eta }}\right)}^k\left({}_{\kappa }{\rm Y}^{k-\varrho }\chi \right)\left(\eta \right).$$

(3)

The right-sided Riemann–Liouville fractional derivative of χ of order ϱ is given by

$$\left({\Delta }_{\sigma }^{\varrho }\chi \right)\left(\eta \right)={\left(-{\displaystyle \frac{d}{d\eta }}\right)}^k\left({{\rm Y}}_{\sigma }^{k-\varrho }\chi \right)\left(\eta \right).$$

(4)

Definition 3. 

If $k=\left[\varrho \right]+1$, the left-sided Caputo fractional derivative of order ϱ for the function χ is expressed as

$$\left({}_{\kappa }{}^C\Delta ^{\varrho }\chi \right)\left(\eta \right)=\left({}_{\kappa }{\rm Y}^{k-\varrho }{\chi }^{\left(k\right)}\right)\left(\eta \right).$$

(5)

The right-sided Caputo fractional derivative of order ϱ for the function χ is expressed as

$$\left({}^C\Delta _{\sigma }^{\varrho }\chi \right)\left(\eta \right)=\left({{\rm Y}}_{\sigma }^{k-\varrho }{(-1)}^k{\chi }^{\left(k\right)}\right)\left(\eta \right).$$

(6)

In [32], a novel class of generalized fractional operators was proposed. These operators exhibit three key characteristics: their corresponding kernels involve an exponential component, the resulting fractional integrals maintain a semigroup property, and they serve as a natural extension of both the Riemann–Liouville and Caputo frameworks. Below, we define these operators.

Definition 4 

([32]). Let $\varrho \in (0,1]$ and $\upsilon \in (0,1]$. The fractional integral of the generalized proportional type of order ϱ applied to a function δ is given by

$${}^{\mathit{GP}}{\rm Y}^{\varrho ,\upsilon }\delta \left(\eta \right)={\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }exp\left({\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )\right){(\eta -\mu )}^{\varrho -1}\delta \left(\mu \right)d\mu .$$

Definition 5 

([32]). Under the same conditions on ϱ and υ, define $k=\lfloor \varrho \rfloor +1$. Then, the one-sided generalized proportional-Caputo fractional derivative of order ϱ is given by

$${}^{\mathit{GPC}}\Delta ^{\varrho ,\upsilon }\delta \left(\eta \right)={\displaystyle \frac{1}{{\upsilon }^{k-\varrho }\Gamma (k-\varrho )}}{\int }_0^{\eta }exp\left({\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )\right){(\eta -\mu )}^{k-\varrho -1}\left({\displaystyle \frac{d^k\delta \left(\mu \right)}{d{\mu }^k}}\right)d\mu .$$

A fundamental result supporting our analysis is presented in [32]:

Lemma 1 

([32]). Let ${\mathcal{AC}}^{\alpha }[0,U]$ represent the collection of absolutely continuous functions $\mathfrak{h}$ whose $(\alpha -1)$-th derivative remains absolutely continuous over $[0,U]$. Suppose $\varrho \in \mathbb{C}$ satisfies $1\ge \mathrm{Re}\left(\varrho \right)>0$, let $\upsilon \in (0,1]$, and set $\alpha =\lfloor \mathrm{Re}\left(\varrho \right)\rfloor +1$. Assume $\mathfrak{h}\in L_1[0,U]$. If $\left({}^{\mathit{GP}}{\rm Y}^{\varrho ,\upsilon }\mathfrak{h}\right)\left(\eta \right)\in {\mathcal{AC}}^{\alpha }[0,U],$ then

$${}^{\mathit{GP}}{\rm Y}^{\varrho ,\upsilon } {}^{\mathit{GPC}}\Delta ^{\varrho ,\upsilon }\mathfrak{h}\left(\eta \right)=\mathfrak{h}\left(\eta \right)-e^{{\displaystyle \frac{\upsilon -1}{\upsilon }}\eta }\sum _{j=1}^{\alpha }{\displaystyle \frac{{}^{\mathit{GPC}}\Delta ^{\varrho -j,\upsilon }\mathfrak{h}\left(0\right)}{\Gamma (\varrho -j+1){\upsilon }^{\varrho -j}}}{\eta }^{\varrho -j}.$$

(7)

3. Fractional-Order Model Description

The spread of malware in wireless sensor networks has traditionally been described by integer-order differential equations [29,33,34,35,36]. These models assume that the change rate in each compartment depends only on the state at the current moment, ruling out the existence of memory effects possibly caused by network latency, slow response rate, or varying behavior by the sensors as time progresses. The classical model presented in [37] divides the network into six classes: susceptible $\mathfrak{S}\left(\eta \right)$, exposed $\mathfrak{E}\left(\eta \right)$, infectious $\mathfrak{I}\left(\eta \right)$, quarantined $\mathfrak{Q}\left(\eta \right)$, recovered $\mathfrak{R}\left(\eta \right)$, and vaccinated $\mathfrak{V}\left(\eta \right)$. The integer-order system governing these dynamics is given by

$$\left\{\begin{array}{c}{\displaystyle \frac{d\mathfrak{S}}{d\eta }}=\Theta -{\displaystyle \frac{\mathfrak{i}\mathfrak{d}\pi {\mathfrak{c}}^2}{\mathfrak{a}}}\mathfrak{S}\mathfrak{I}-({\mathfrak{m}}_1+{\mathfrak{t}}_1)\mathfrak{S}+{\mathfrak{t}}_2\mathfrak{R}+{\mathfrak{t}}_3\mathfrak{V},\hfill \\ {\displaystyle \frac{d\mathfrak{E}}{d\eta }}={\displaystyle \frac{\mathfrak{i}\mathfrak{d}\pi {\mathfrak{c}}^2}{\mathfrak{a}}}\mathfrak{S}\mathfrak{I}-({\mathfrak{m}}_1+{\mathfrak{t}}_4)\mathfrak{E},\hfill \\ {\displaystyle \frac{d\mathfrak{I}}{d\eta }}={\mathfrak{t}}_4\mathfrak{E}-({\mathfrak{m}}_1+{\mathfrak{m}}_2+{\mathfrak{t}}_5+\alpha )\mathfrak{I},\hfill \\ {\displaystyle \frac{d\mathfrak{Q}}{d\eta }}=\alpha \mathfrak{I}-({\mathfrak{m}}_1+{\mathfrak{m}}_2+{\mathfrak{t}}_5)\mathfrak{Q},\hfill \\ {\displaystyle \frac{d\mathfrak{R}}{d\eta }}={\mathfrak{t}}_5\mathfrak{I}+{\mathfrak{t}}_6\mathfrak{Q}-({\mathfrak{m}}_1+{\mathfrak{t}}_2)\mathfrak{R},\hfill \\ {\displaystyle \frac{d\mathfrak{V}}{d\eta }}={\mathfrak{t}}_1\mathfrak{S}-({\mathfrak{m}}_1+{\mathfrak{t}}_3)\mathfrak{V}.\hfill \end{array}\right.$$

(8)

For clarity, the model parameters appearing in (8) are summarized in

Table 1. Parameter definitions for the fractional-order model.

Parameter	Meaning	Parameter	Meaning
$\Theta$	Node inclusion rate in the population of the sensor network.	$\mathfrak{i}$	Rate of infectivity contact.
$\mathfrak{d}$	Distribution density of the nodes.	$\mathfrak{c}$	Nodes’ communication radius.
$\mathfrak{a}$	Area over which the nodes are deployed.	$\alpha$	Transmission rate between the confined and infected compartments.
${\mathfrak{m}}_1$	Node death rate due to software/hardware malfunction.	${\mathfrak{m}}_2$	Crash rate caused by a malicious attack.
${\mathfrak{t}}_1$	Vaccination rate of susceptible nodes.	${\mathfrak{t}}_2$	Reversion rate from recovered to susceptible.
${\mathfrak{t}}_3$	Reversion rate from vaccinated to susceptible.	${\mathfrak{t}}_4$	Transition rate from exposed to infectious.
${\mathfrak{t}}_5$	Transition rate from infectious to recovered.	${\mathfrak{t}}_6$	Transition rate from quarantined to recovered.

.

Though this model describes the transmission of malicious code, it does not account for memory effects, which dominate in realistic wireless networks where past interactions influence future conduct. To account for this shortcoming, Khan et al. [38] introduced fractional-order derivatives to model the hereditary nature and nonlocal interactions of the model. Indeed, they replaced the classical integer-order derivatives with the Caputo fractional derivative of order $\varrho$ and suggested the following model:

$$\left\{\begin{array}{c}{\displaystyle \frac{1}{{\Lambda }^{1-\varrho }}}\left({}^C\Delta _0^{\varrho }\mathfrak{S}\right)\left(\eta \right)=\Theta -{\displaystyle \frac{\mathfrak{i}\mathfrak{d}\pi {\mathfrak{c}}^2}{\mathfrak{a}}}\mathfrak{S}\mathfrak{I}-({\mathfrak{m}}_1+{\mathfrak{t}}_1)\mathfrak{S}+{\mathfrak{t}}_2\mathfrak{R}+{\mathfrak{t}}_3\mathfrak{V},\hfill \\ {\displaystyle \frac{1}{{\Lambda }^{1-\varrho }}}\left({}^C\Delta _0^{\varrho }\mathfrak{E}\right)\left(\eta \right)={\displaystyle \frac{\mathfrak{i}\mathfrak{d}\pi {\mathfrak{c}}^2}{\mathfrak{a}}}\mathfrak{S}\mathfrak{I}-({\mathfrak{m}}_1+{\mathfrak{t}}_4)\mathfrak{E},\hfill \\ {\displaystyle \frac{1}{{\Lambda }^{1-\varrho }}}\left({}^C\Delta _0^{\varrho }\mathfrak{I}\right)\left(\eta \right)={\mathfrak{t}}_4\mathfrak{E}-({\mathfrak{m}}_1+{\mathfrak{m}}_2+{\mathfrak{t}}_5+\alpha )\mathfrak{I},\hfill \\ {\displaystyle \frac{1}{{\Lambda }^{1-\varrho }}}\left({}^C\Delta _0^{\varrho }\mathfrak{Q}\right)\left(\eta \right)=\alpha \mathfrak{I}-({\mathfrak{m}}_1+{\mathfrak{m}}_2+{\mathfrak{t}}_5)\mathfrak{Q},\hfill \\ {\displaystyle \frac{1}{{\Lambda }^{1-\varrho }}}\left({}^C\Delta _0^{\varrho }\mathfrak{R}\right)\left(\eta \right)={\mathfrak{t}}_5\mathfrak{I}+{\mathfrak{t}}_6\mathfrak{Q}-({\mathfrak{m}}_1+{\mathfrak{t}}_2)\mathfrak{R},\hfill \\ {\displaystyle \frac{1}{{\Lambda }^{1-\varrho }}}\left({}^C\Delta _0^{\varrho }\mathfrak{V}\right)\left(\eta \right)={\mathfrak{t}}_1\mathfrak{S}-({\mathfrak{m}}_1+{\mathfrak{t}}_3)\mathfrak{V}.\hfill \end{array}\right.$$

(9)

Now, we will give the generalized version of the Caputo model (9) with respect to the proportional-Caputo derivative to provide dual control of memory: the order $\varrho$ governs how strongly the past persists, and the tempering $\upsilon$ limits that influence over time; this yields a tighter match to WSN realities (latency, expiring patches, and shifting contacts) without giving up standard initial conditions as follows:

Let $\eta \in [0,U]$, fractional order $\varrho \in (0,1]$, tempering $\upsilon \in (0,1]$, and a time-scale factor $\Lambda >0$. The proportional-Caputo (GPC) SEIQRV system reads as follows:

$$\left\{\begin{array}{c}{\Lambda }^{\varrho -1}\left({}^{\mathrm{GPC}}\Delta ^{\varrho ,\upsilon }\mathfrak{S}\right)\left(\eta \right)=\Theta -{\displaystyle \frac{\mathfrak{i}\mathfrak{d}\pi {\mathfrak{c}}^2}{\mathfrak{a}}}\mathfrak{S}\left(\eta \right)\mathfrak{I}\left(\eta \right)-({\mathfrak{m}}_1+{\mathfrak{t}}_1)\mathfrak{S}\left(\eta \right)+{\mathfrak{t}}_2\mathfrak{R}\left(\eta \right)+{\mathfrak{t}}_3\mathfrak{V}\left(\eta \right),\hfill \\ {\Lambda }^{\varrho -1}\left({}^{\mathrm{GPC}}\Delta ^{\varrho ,\upsilon }\mathfrak{E}\right)\left(\eta \right)={\displaystyle \frac{\mathfrak{i}\mathfrak{d}\pi {\mathfrak{c}}^2}{\mathfrak{a}}}\mathfrak{S}\left(\eta \right)\mathfrak{I}\left(\eta \right)-({\mathfrak{m}}_1+{\mathfrak{t}}_4)\mathfrak{E}\left(\eta \right),\hfill \\ {\Lambda }^{\varrho -1}\left({}^{\mathrm{GPC}}\Delta ^{\varrho ,\upsilon }\mathfrak{I}\right)\left(\eta \right)={\mathfrak{t}}_4\mathfrak{E}\left(\eta \right)-({\mathfrak{m}}_1+{\mathfrak{m}}_2+{\mathfrak{t}}_5+\alpha )\mathfrak{I}\left(\eta \right),\hfill \\ {\Lambda }^{\varrho -1}\left({}^{\mathrm{GPC}}\Delta ^{\varrho ,\upsilon }\mathfrak{Q}\right)\left(\eta \right)=\alpha \mathfrak{I}\left(\eta \right)-({\mathfrak{m}}_1+{\mathfrak{m}}_2+{\mathfrak{t}}_5)\mathfrak{Q}\left(\eta \right),\hfill \\ {\Lambda }^{\varrho -1}\left({}^{\mathrm{GPC}}\Delta ^{\varrho ,\upsilon }\mathfrak{R}\right)\left(\eta \right)={\mathfrak{t}}_5\mathfrak{I}\left(\eta \right)+{\mathfrak{t}}_6\mathfrak{Q}\left(\eta \right)-({\mathfrak{m}}_1+{\mathfrak{t}}_2)\mathfrak{R}\left(\eta \right),\hfill \\ {\Lambda }^{\varrho -1}\left({}^{\mathrm{GPC}}\Delta ^{\varrho ,\upsilon }\mathfrak{V}\right)\left(\eta \right)={\mathfrak{t}}_1\mathfrak{S}\left(\eta \right)-({\mathfrak{m}}_1+{\mathfrak{t}}_3)\mathfrak{V}\left(\eta \right).\hfill \end{array}\right.$$

(10)

For $X\in \{\mathfrak{S},\mathfrak{E},\mathfrak{I},\mathfrak{Q},\mathfrak{R},\mathfrak{V}\}$, impose $X\left(0\right)=X_0\ge 0.$ A clear view of the model parameters is provided by Table 1 to facilitate the recognition of the suggested fractional-order system. Together with its exact explanation, the symbol’s significance in describing the dynamics of virus dissemination in wireless sensor networks is explained.

To clarify the movement of nodes among the six states and the associated transition rates prior to the formulation of (10), we summarize the model structure in Figure 1.

4. Main Results

This section presents the analytical context for the wireless sensor network malware propagation model at the fractional order. This section begins by specifying the function space and the norms that are to be used for the space. Let the space $\mathfrak{X}=C[0,U]$ of continuous functions defined on the interval $[0,U]$ be furnished by the norm:

$$\parallel y\parallel =\underset{\eta \in [0,U]}{sup}\left|\{\mathfrak{S}\left(\eta \right)+\mathfrak{E}\left(\eta \right)+\mathfrak{I}\left(\eta \right)+\mathfrak{Q}\left(\eta \right)+\mathfrak{R}\left(\eta \right)+\mathfrak{V}\left(\eta \right)\}\right|.$$

where

$$y\left(\eta \right)=\left[\begin{array}{c}\mathfrak{S}\left(\eta \right)\\ \mathfrak{E}\left(\eta \right)\\ \mathfrak{I}\left(\eta \right)\\ \mathfrak{Q}\left(\eta \right)\\ \mathfrak{R}\left(\eta \right)\\ \mathfrak{V}\left(\eta \right)\end{array}\right].$$

We specify $\mathcal{A}$ as the nonlinear operator:

$$\mathcal{A}(\eta ,y\left(\eta \right))={\Lambda }^{1-\varrho }\left[\begin{array}{c}{\mathcal{A}}_1\left(\eta ,\mathfrak{S},\mathfrak{E},\mathfrak{I},\mathfrak{Q},\mathfrak{R},\mathfrak{V}\right)\\ ⋮\\ {\mathcal{A}}_6\left(\eta ,\mathfrak{S},\mathfrak{E},\mathfrak{I},\mathfrak{Q},\mathfrak{R},\mathfrak{V}\right)\end{array}\right].$$

For each $\eta \in [0,U]$, the nonlinear mappings ${\mathcal{A}}_k(k=1,\dots ,6)$ are defined as follows:

$$\left\{\begin{array}{c}{\mathcal{A}}_1\left(\eta ,\mathfrak{S},\mathfrak{E},\mathfrak{I},\mathfrak{Q},\mathfrak{R},\mathfrak{V}\right)=\Theta -{\displaystyle \frac{\mathfrak{i}\mathfrak{d}\pi {\mathfrak{c}}^2}{\mathfrak{a}}}\mathfrak{S}\left(\eta \right)\mathfrak{I}\left(\eta \right)-({\mathfrak{m}}_1+{\mathfrak{t}}_1)\mathfrak{S}\left(\eta \right)+{\mathfrak{t}}_2\mathfrak{R}\left(\eta \right)+{\mathfrak{t}}_3\mathfrak{V}\left(\eta \right),\hfill \\ {\mathcal{A}}_2\left(\eta ,\mathfrak{S},\mathfrak{E},\mathfrak{I},\mathfrak{Q},\mathfrak{R},\mathfrak{V}\right)={\displaystyle \frac{\mathfrak{i}\mathfrak{d}\pi {\mathfrak{c}}^2}{\mathfrak{a}}}\mathfrak{S}\left(\eta \right)\mathfrak{I}\left(\eta \right)-({\mathfrak{m}}_1+{\mathfrak{t}}_4)\mathfrak{E}\left(\eta \right),\hfill \\ {\mathcal{A}}_3\left(\eta ,\mathfrak{S},\mathfrak{E},\mathfrak{I},\mathfrak{Q},\mathfrak{R},\mathfrak{V}\right)={\mathfrak{t}}_4\mathfrak{E}\left(\eta \right)-({\mathfrak{m}}_1+{\mathfrak{m}}_2+{\mathfrak{t}}_5+\alpha )\mathfrak{I}\left(\eta \right),\hfill \\ {\mathcal{A}}_4\left(\eta ,\mathfrak{S},\mathfrak{E},\mathfrak{I},\mathfrak{Q},\mathfrak{R},\mathfrak{V}\right)=\alpha \mathfrak{I}\left(\eta \right)-({\mathfrak{m}}_1+{\mathfrak{m}}_2+{\mathfrak{t}}_5)\mathfrak{Q}\left(\eta \right),\hfill \\ {\mathcal{A}}_5\left(\eta ,\mathfrak{S},\mathfrak{E},\mathfrak{I},\mathfrak{Q},\mathfrak{R},\mathfrak{V}\right)={\mathfrak{t}}_5\mathfrak{I}\left(\eta \right)+{\mathfrak{t}}_6\mathfrak{Q}\left(\eta \right)-({\mathfrak{m}}_1+{\mathfrak{t}}_2)\mathfrak{R}\left(\eta \right),\hfill \\ {\mathcal{A}}_6\left(\eta ,\mathfrak{S},\mathfrak{E},\mathfrak{I},\mathfrak{Q},\mathfrak{R},\mathfrak{V}\right)={\mathfrak{t}}_1\mathfrak{S}\left(\eta \right)-({\mathfrak{m}}_1+{\mathfrak{t}}_3)\mathfrak{V}\left(\eta \right).\hfill \end{array}\right.$$

(11)

subject to the initial data:

$$\left[\begin{array}{c}\mathfrak{S}\left(0\right)\\ \mathfrak{E}\left(0\right)\\ \mathfrak{I}\left(0\right)\\ \mathfrak{Q}\left(0\right)\\ \mathfrak{R}\left(0\right)\\ \mathfrak{V}\left(0\right)\end{array}\right]=\left[\begin{array}{c}{\mathfrak{S}}_0\\ {\mathfrak{E}}_0\\ {\mathfrak{I}}_0\\ {\mathfrak{Q}}_0\\ {\mathfrak{R}}_0\\ {\mathfrak{V}}_0\end{array}\right]\mathrm{and}\mathcal{A}(0,y\left(0\right))={\Lambda }^{1-\varrho }\left[\begin{array}{c}{\mathcal{A}}_1\left(0,\mathfrak{S}\left(0\right),\mathfrak{E}\left(0\right),\mathfrak{I}\left(0\right),\mathfrak{Q}\left(0\right),\mathfrak{R}\left(0\right),\mathfrak{V}\left(0\right)\right)\\ ⋮\\ {\mathcal{A}}_6\left(0,\mathfrak{S}\left(0\right),\mathfrak{E}\left(0\right),\mathfrak{I}\left(0\right),\mathfrak{Q}\left(0\right),\mathfrak{R}\left(0\right),\mathfrak{V}\left(0\right)\right)\end{array}\right].$$

Under the GPC fractional derivative, the dynamics expressed in (10) can be rewritten as

$$\left\{\begin{array}{cc}{}^{\mathrm{GPC}}\Delta ^{\varrho ,\upsilon }y\left(\eta \right)=\mathcal{A}(\eta ,y\left(\eta \right)),\hfill & \eta \in [0,U],\hfill \\ y\left(0\right)=y_0.\hfill \end{array}\right.$$

(12)

Employing Lemma 1, the system (12) can be converted into its equivalent integral form:

$$\begin{array}{ccc}\hfill y\left(\eta \right)& =& y_0{+}^{GP}{{\rm Y}}^{\varrho ,\upsilon }\mathcal{A}(\eta ,y\left(\eta \right))\hfill \\ & =& y_0+{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }exp\left({\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )\right){(\eta -\mu )}^{\varrho -1}\mathcal{A}(\mu ,y\left(\mu \right))d\mu ,\eta \in [0,U].\hfill \end{array}$$

(13)

To demonstrate the existence of a solution for the problem (12), we use the following assumptions:

Assumption A1 

(AS1). There exists a fixed ${\lambda }_{\mathcal{A}}>0$ such that

$$\parallel \mathcal{A}(\eta ,y_1)-\mathcal{A}(\eta ,y_2)\parallel \le {\lambda }_{\mathcal{A}}\parallel y_1-y_2\parallel ,$$

For all $y_1,y_2\in \mathfrak{X}$ and $\eta \in [0,U]$. This assumption guarantees the Lipschitz continuity of $\mathcal{A}$.

Assumption A2 

(AS2). Suppose there exists a continuous and nondecreasing function $\Omega :[0,\infty )\to [0,\infty )$ with $\Omega \left(\varsigma z\right)\le \varsigma \Omega \left(z\right)$ for all $\varsigma \ge 1,z\in [0,\infty )$, and a function $\varphi \in {\mathcal{AC}}^{\alpha }[0,U]$ such that

$$\parallel \mathcal{A}(\eta ,y(\eta \left)\right)\parallel \le \varphi \left(\eta \right)\Omega \left(\parallel y\left(\eta \right)\parallel \right),$$

for all $y\in \mathfrak{X}$ and $\eta \in [0,T]$. This assumption governs the growth of the operator $\mathcal{A}$ through the auxiliary functions Ω and ϕ.

Assumption A3 

(AS3). The appropriate positive constants ϖ and $\mathcal{D}$ are selected so that

$${\displaystyle \frac{\mathcal{D}}{\parallel y_0\parallel +\frac{{sup}_{\eta \in [0,T]}\varphi \left(\eta \right)\Omega \left(\varpi \right)U^{\varrho }}{{\upsilon }^{\varrho }\Gamma (\varrho +1)}}}>1.$$

This guarantees the solution norm is a priori bounded and meets requirements demanded in fixed-point theorems.

4.1. Existence and Uniqueness Analysis

We now formulate the mapping $\mathcal{E}:\mathfrak{X}\to \mathfrak{X}$, which corresponds to the system (12), and is given by

$$\begin{array}{c}\hfill \mathcal{E}y\left(\eta \right)=y_0+{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{(\eta -\mu )}^{\varrho -1}exp\left({\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )\right)\mathcal{A}(\mu ,y\left(\mu \right))d\mu ,\eta \in [0,T].\end{array}$$

(14)

Immediately after introducing the operator $\mathcal{E}$ in (14), we note that the subsequent estimates are carried out while retaining the exponentially tempered kernel. This yields bounds, and consequently existence–uniqueness and stability criteria, that depend explicitly on the tempering parameter $\upsilon$. In addition, rather than invoking a purely abstract hypothesis, we verify that the nonlinear vector field associated with the proposed system is Lipschitz on a suitable bounded admissible set, ensuring that the fixed-point framework is consistent with the intended memory-tempering interpretation.

Theorem 1. 

Assuming that conditions (AS2) and (AS3) hold true, the fractional differential system described by (12) ensures the existence of at least one solution over the interval $[0,T]$.

Proof. 

Let $\epsilon >0$ and define the closed ball

$${\mathbb{B}}_{\epsilon }=\{y\in \mathfrak{X}:\parallel y\parallel \le \epsilon \}.$$

For each $\eta \in [0,U]$, consider the operator $\mathcal{E}$ introduced in (14). Using assumption (AS2), we estimate

$$\parallel \mathcal{E}y\left(\eta \right)\parallel \le \parallel y_0\parallel +{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{(\eta -\mu )}^{\varrho -1}{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )}\parallel \mathcal{A}(\mu ,y\left(\mu \right))\parallel d\mu .$$

By the growth condition (AS2), we have

$$\parallel \mathcal{A}(\mu ,y(\mu \left)\right)\parallel \le \varphi \left(\mu \right)\Omega \left(\parallel y\left(\mu \right)\parallel \right).$$

Substituting this into the integral yields

$$\parallel \mathcal{E}y\left(\eta \right)\parallel \le \parallel y_0\parallel +{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{(\eta -\mu )}^{\varrho -1}{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )}\varphi \left(\mu \right)\Omega \left(\parallel y\left(\mu \right)\parallel \right)d\mu .$$

If $\parallel y\left(\mu \right)\parallel \le \epsilon$ for all $\mu \in [0,U]$, then

$$\parallel \mathcal{E}y\left(\eta \right)\parallel \le \parallel y_0\parallel +{\displaystyle \frac{\Omega \left(\epsilon \right)}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{(\eta -\mu )}^{\varrho -1}{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )}\varphi \left(\mu \right)d\mu .$$

Let $C_{\varphi }={sup}_{\eta \in [0,U]}\varphi \left(\eta \right)$. Then,

$$\parallel \mathcal{E}y\left(\eta \right)\parallel \le \parallel y_0\parallel +{\displaystyle \frac{\Omega \left(\epsilon \right)C_{\varphi }}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{(\eta -\mu )}^{\varrho -1}{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )}d\mu .$$

Since

$${\int }_0^{\eta }{(\eta -\mu )}^{\varrho -1}{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )}d\mu \le {\displaystyle \frac{T^{\varrho }}{\varrho }},$$

it follows that

$$\parallel \mathcal{E}y\left(\eta \right)\parallel \le \parallel y_0\parallel +{\displaystyle \frac{\Omega \left(\epsilon \right)C_{\varphi }{T}^{\varrho }}{{\upsilon }^{\varrho }\Gamma (\varrho +1)}}.$$

Thus, $\mathcal{E}$ maps bounded sets into bounded sets.

Next, we prove equicontinuity. Let ${\eta }_1,{\eta }_2\in [0,U]$ with ${\eta }_1<{\eta }_2$. Then,

$$\parallel \mathcal{E}y\left({\eta }_2\right)-\mathcal{E}y\left({\eta }_1\right)\parallel \le {\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_{{\eta }_1}^{{\eta }_2}{(\eta -\mu )}^{\varrho -1}{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )}\parallel \mathcal{A}(\mu ,y\left(\mu \right))\parallel d\mu .$$

Applying (AS2) again gives

$$\parallel \mathcal{E}y\left({\eta }_2\right)-\mathcal{E}y\left({\eta }_1\right)\parallel \le {\displaystyle \frac{C_{\varphi }\Omega \left(\epsilon \right)}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_{{\eta }_1}^{{\eta }_2}{(\eta -\mu )}^{\varrho -1}{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )}d\mu .$$

The kernel is continuous, and the integral tends to zero as ${\eta }_2\to {\eta }_1$. Hence, $\mathcal{E}$ is equicontinuous.

By the Arzelà–Ascoli theorem, $\mathcal{E}$ maps bounded sets into relatively compact sets, and thus it is completely continuous.

Finally, let $y\in \mathfrak{X}$ be a fixed point of $\mathcal{E}$, i.e., $y=\mathcal{E}y$. Then,

$$\parallel y\left(\eta \right)\parallel \le \parallel y_0\parallel +{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{(\eta -\mu )}^{\varrho -1}{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )}\varphi \left(\mu \right)\Omega (\parallel z\left(\mu \right)\parallel )d\mu .$$

Let $C_{\varphi }={sup}_{\mu \in [0,U]}\varphi \left(\mu \right)$. Then,

$$\parallel y\left(\eta \right)\parallel \le \parallel y_0\parallel +{\displaystyle \frac{C_{\varphi }\Omega \left(\parallel z\parallel \right)T^{\varrho }}{{\upsilon }^{\varrho }\Gamma (\varrho +1)}}.$$

By assumption (AS3), suitable constants exist such that the right-hand side is bounded. Hence, the set of solutions $\{z:z=\mathcal{E}z\}$ is bounded.

Applying the Leray–Schauder alternative, we conclude that $\mathcal{E}$ admits at least one fixed point in $\mathfrak{X}$. Therefore, system (12) has at least one solution on $[0,U]$. □

To assert uniqueness after checking for existence, we employ the Lipschitz condition (AS1) for the nonlinear operator $\mathcal{A}$. The condition asserts controlled variations in the output for the slightest variations in the input in such a way that the uniqueness in the solution can be ascertained.

Theorem 2. 

Assume that the nonlinear operator $\mathcal{A}:[0,T]\times \mathfrak{X}\to {\mathbb{R}}^6$ satisfies the Lipschitz condition (AS1) for all $y_1,y_2\in \mathfrak{X}$. Then, the generalized proportional-Caputo fractional system (12) possesses a unique solution on $[0,T]$ provided that

$$\begin{array}{c}\hfill {\lambda }_{\mathcal{A}}{T}^{\varrho }<{\upsilon }^{\varrho }\Gamma (\varrho +1),\end{array}$$

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where ${\lambda }_{\mathcal{A}}>0$ is the Lipschitz constant of $\mathcal{A}$, $\varrho >0$ is the fractional order, and $\Gamma (·)$ denotes the Gamma function.

Proof. 

Let $y\in \mathfrak{X}$ and consider the closed ball

$${\left[B\right]}_k=\left\{y\in \mathfrak{X}:\parallel y\parallel \le k\right\},$$

where

$$k\ge {\displaystyle \frac{{\upsilon }^{\varrho }\Gamma (\varrho +1)\parallel y_0\parallel +C_{\varphi }\Omega \left(k\right)T^{\varrho }}{{\upsilon }^{\varrho }\Gamma (\varrho +1)-{\lambda }_{\mathcal{A}}{T}^{\varrho }}}.$$

For $\eta \in [0,U]$, using the definition of the operator $\mathcal{E}$ in (14) and applying (AS1), we estimate

$$\begin{array}{ccc}\hfill \parallel \mathcal{E}y\left(\eta \right)\parallel & \le & \parallel y_0\parallel +{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{(\eta -\mu )}^{\varrho -1}{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )}\parallel \mathcal{A}(\mu ,y\left(\mu \right))\parallel d\mu \hfill \\ & \le & \parallel y_0\parallel +{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{(\eta -\mu )}^{\varrho -1}{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )}\left[\parallel \mathcal{A}(\mu ,y(\mu \left)\right)-\mathcal{A}(\mu ,0)\parallel +\parallel \mathcal{A}(\mu ,0)\parallel \right]d\mu \hfill \\ & \le & \parallel y_0\parallel +{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{(\eta -\mu )}^{\varrho -1}{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )}\left[{\lambda }_{\mathcal{A}}k+C_{\varphi }\Omega \left(k\right)\right]d\mu .\hfill \end{array}$$

The kernel integral can be bounded as

$${\int }_0^{\eta }{(\eta -\mu )}^{\varrho -1}{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )}d\mu \le {\displaystyle \frac{T^{\varrho }}{\varrho }}.$$

Substituting this bound yields

$$\parallel \mathcal{E}y\left(\eta \right)\parallel \le \parallel y_0\parallel +{\displaystyle \frac{[{\lambda }_{\mathcal{A}}k+C_{\varphi }\Omega \left(k\right)]T^{\varrho }}{{\upsilon }^{\varrho }\Gamma (\varrho +1)}}\le k.$$

Thus, $\mathcal{E}$ maps the ball ${\left[B\right]}_k$ into itself.

Next, for $y_1,y_2\in {\left[B\right]}_k$ and $\eta \in [0,U]$, we calculate

$$\parallel \mathcal{E}{y}_1\left(\eta \right)-\mathcal{E}{y}_2\left(\eta \right)\parallel \le {\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{(\eta -\mu )}^{\varrho -1}{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )}\parallel \mathcal{A}(\mu ,y_1\left(\mu \right))-\mathcal{A}(\mu ,y_2\left(\mu \right))\parallel d\mu .$$

By the Lipschitz condition (AS1), we have

$$\parallel \mathcal{A}(\mu ,y_1(\mu \left)\right)-\mathcal{A}(\mu ,y_2(\mu \left)\right)\parallel \le {\lambda }_{\mathcal{A}}\parallel y_1\left(\mu \right)-y_2\left(\mu \right)\parallel .$$

Therefore,

$$\parallel \mathcal{E}{y}_1\left(\eta \right)-\mathcal{E}{y}_2\left(\eta \right)\parallel \le {\displaystyle \frac{{\lambda }_{\mathcal{A}}}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{(\eta -\mu )}^{\varrho -1}{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -\mu )}\parallel y_1\left(\mu \right)-y_2\left(\mu \right)\parallel d\mu .$$

Evaluating the kernel integral gives

$$\parallel \mathcal{E}{y}_1-\mathcal{E}{y}_2\parallel \le {\displaystyle \frac{{\lambda }_{\mathcal{A}}{T}^{\varrho }}{{\upsilon }^{\varrho }\Gamma (\varrho +1)}}\parallel y_1-y_2\parallel <\parallel y_1-y_2\parallel .$$

Hence, $\mathcal{E}$ is a contraction mapping on ${\left[B\right]}_k$. By the Banach fixed-point theorem, $\mathcal{E}$ admits a unique fixed point $y\in {\left[B\right]}_k$. Consequently, the fractional system (12) has a unique solution on $[0,U]$. □

4.2. Stability Properties of the Proposed Model

In this section, we establish the conditions under which the generalized proportional-Caputo system (12) remains stable. The analysis centers on Ulam–Hyers stability (UHS) and its extended variant. Before presenting the main results, we introduce a few preliminary definitions.

Let $\wp >0$ be a fixed constant, $\varpi \in \mathfrak{X}$, and $𝘍:[0,U]\to {\mathbb{R}}^{+}$ be a continuous function. The stability conditions can be described using the following inequalities:

$$\parallel {}^{\mathrm{GPC}}\Delta ^{\varrho ,\upsilon }\varpi \left(\tau \right)-\mathcal{A}(\tau ,\varpi (\tau \left)\right)\parallel \le \wp ,\forall \tau \in [0,U],$$

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$$\parallel {}^{\mathrm{GPC}}\Delta ^{\varrho ,\upsilon }\varpi \left(\tau \right)-\mathcal{A}(\tau ,\varpi (\tau \left)\right)\parallel \le \wp 𝘍\left(\tau \right),\forall \tau \in [0,U],$$

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Definition 6. 

The system (12) is said to exhibit Ulam–Hyers stability under condition (16) if there exists a solution $\varpi \in \mathfrak{X}$ such that, for every $\wp >0$ and $\beth \in \mathfrak{X}$, there exists a constant ${\mathbb{Q}}_{\wp }>0$ satisfying

$$\parallel \beth \left(\tau \right)-\varpi \left(\tau \right)\parallel \le \wp {\mathbb{Q}}_{\wp },\forall \tau \in [0,U],$$

where ${\mathbb{Q}}_{\wp }=max({\mathbb{Q}}_{{\wp }_1},{\mathbb{Q}}_{{\wp }_2},{\mathbb{Q}}_{{\wp }_3},{\mathbb{Q}}_{{\wp }_4},{\mathbb{Q}}_{{\wp }_5},{\mathbb{Q}}_{{\wp }_6})$, and each ${\mathbb{Q}}_{{\wp }_i}$ is a constant specific to the problem.

Definition 7. 

The system (12) is said to possess extended Ulam–Hyers stability under condition (17) if for each $\beth \in \mathfrak{X}$, there exists a solution $\varpi \in \mathfrak{X}$, and a function ${𝘍}_{\wp }:[0,U]\to {\mathbb{R}}^{+}$ with ${𝘍}_{\wp }\left(0\right)=0$ such that, the inequality

$$\parallel \beth \left(\tau \right)-\varpi \left(\tau \right)\parallel \le \wp {𝘍}_{\wp }\left(\tau \right),\forall \tau \in [0,U],$$

holds, where ${𝘍}_{\wp }\left(\tau \right)=max\{{𝘍}_{{\wp }_1}\left(\tau \right),{𝘍}_{{\wp }_2}\left(\tau \right),{𝘍}_{{\wp }_3}\left(\tau \right),{𝘍}_{{\wp }_4}\left(\tau \right)\}$, and each ${𝘍}_{{\wp }_i}$ is an auxiliary function associated with the model parameters.

Lemma 2. 

Let $\varrho >0$ and $\upsilon \in (0,1]$. Then, the following estimate is valid:

$$\begin{array}{c}\hfill ∥\varpi \left(\tau \right)-{\varpi }_0-{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\tau }{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\tau -s)}{(\tau -s)}^{\varrho -1}\mathcal{A}(s,\varpi \left(s\right))ds∥\le {\displaystyle \frac{\wp U^k}{{\upsilon }^{\varrho }\Gamma (\varrho +k)}},\end{array}$$

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whenever $\varpi \in \mathfrak{X}$ satisfies condition (16).

Proof. 

Since $\varpi$ fulfills inequality (16), there exists an auxiliary function $\beth \in \mathfrak{X}$, depending on $\varpi$, such that

$$\parallel \beth \left(\eta \right)\parallel \le \wp ,\beth =max({\beth }_1,{\beth }_2,{\beth }_3,{\beth }_4,{\beth }_5,{\beth }_6),\forall \eta \in [0,U],$$

and

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}^{\mathrm{GPC}}\Delta ^{\varrho ,\upsilon }\varpi \left(\eta \right)=\mathcal{A}(\eta ,\varpi \left(\eta \right))+\beth \left(\eta \right),\hfill & \eta \in [0,U],\hfill \\ \varpi \left(0\right)={\varpi }_0\ge 0.\hfill \end{array}\right.\end{array}$$

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By applying the integral representation associated with the GPC operator, the solution of (19) can be expressed as

$$\varpi \left(\eta \right)={\varpi }_0+{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -s)}{(\eta -s)}^{\varrho -1}\mathcal{A}(s,\varpi \left(s\right))ds$$

$$+{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -s)}{(\eta -s)}^{\varrho -1}\beth \left(s\right)ds.$$

Evaluating the norm of the difference from the integral representation gives

$$\parallel \varpi \left(\eta \right)-{\varpi }_0-{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -s)}{(\eta -s)}^{\varrho -1}\mathcal{A}(s,\varpi \left(s\right))ds\parallel$$

$$\le {\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -s)}{(\eta -s)}^{\varrho -1}\parallel \beth \left(s\right)\parallel ds.$$

Since $\parallel \beth \left(s\right)\parallel \le \wp$, it follows that

$$\parallel \varpi \left(\eta \right)-{\varpi }_0-{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -s)}{(\eta -s)}^{\varrho -1}\mathcal{A}(s,\varpi \left(s\right))ds\parallel \le {\displaystyle \frac{\wp }{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -s)}{(\eta -s)}^{\varrho -1}ds.$$

Finally, the kernel integral is bounded by

$${\int }_0^{\eta }{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -s)}{(\eta -s)}^{\varrho -1}ds\le {\displaystyle \frac{U^{\varrho }}{\varrho }},$$

which yields

$$\parallel \varpi \left(\eta \right)-{\varpi }_0-{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\eta }{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\eta -s)}{(\eta -s)}^{\varrho -1}\mathcal{A}(s,\varpi \left(s\right))ds\parallel \le {\displaystyle \frac{\wp U^{\varrho }}{{\upsilon }^{\varrho }\Gamma (\varrho +1)}}.$$

This completes the proof. □

Theorem 3. 

Assume that the nonlinear operator $\mathcal{A}:[0,U]\times \mathfrak{X}\to {\mathbb{R}}^6$ satisfies the Lipschitz condition (AS1). If the inequality (15) holds, then the unique solution of system (12) is both Ulam–Hyers stable and extended Ulam–Hyers stable.

Proof. 

Let $\varpi \in \mathfrak{X}$ denote the exact solution of (12), and let $\beth \in \mathfrak{X}$ be an approximate solution such that

$${\parallel }^{\mathrm{GPC}}{\Delta }^{\varrho ,\upsilon }\beth \left(\tau \right)-\mathcal{A}(\tau ,\beth \left(\tau \right))\parallel \le \wp ,\forall \tau \in [0,U].$$

Thus, we may write

$${}^{\mathrm{GPC}}\Delta ^{\varrho ,\upsilon }\beth \left(\tau \right)=\mathcal{A}(\tau ,\beth \left(\tau \right))+\mathcal{E}\left(\tau \right),\beth \left(0\right)={\beth }_0,$$

where $\parallel \mathcal{E}\left(\tau \right)\parallel \le \wp$.

The mild form of ℶ is then

$$\beth \left(\tau \right)={\beth }_0+{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\tau }{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\tau -s)}{(\tau -s)}^{\varrho -1}\mathcal{A}(s,\beth \left(s\right))ds+{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\tau }{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\tau -s)}{(\tau -s)}^{\varrho -1}\mathcal{E}\left(s\right)ds.$$

Hence,

$$\begin{array}{ccc}\hfill \parallel \beth \left(\tau \right)-\varpi \left(\tau \right)\parallel & \le & {\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\tau }{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\tau -s)}{(\tau -s)}^{\varrho -1}\parallel \mathcal{A}(s,\beth \left(s\right))-\mathcal{A}(s,\varpi \left(s\right))\parallel ds\hfill \\ & & +{\displaystyle \frac{1}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\tau }{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\tau -s)}{(\tau -s)}^{\varrho -1}\parallel \mathcal{E}\left(s\right)\parallel ds.\hfill \end{array}$$

By (AS1),

$$\parallel \mathcal{A}(s,\beth \left(s\right))-\mathcal{A}(s,\varpi \left(s\right))\parallel \le {\lambda }_{\mathcal{A}}\parallel \beth \left(s\right)-\varpi \left(s\right)\parallel .$$

Thus,

$$\parallel \beth \left(\tau \right)-\varpi \left(\tau \right)\parallel \le {\displaystyle \frac{{\lambda }_{\mathcal{A}}}{{\upsilon }^{\varrho }\Gamma \left(\varrho \right)}}{\int }_0^{\tau }{e}^{{\displaystyle \frac{\upsilon -1}{\upsilon }}(\tau -s)}{(\tau -s)}^{\varrho -1}\parallel \beth \left(s\right)-\varpi \left(s\right)\parallel ds+{\displaystyle \frac{\wp {\tau }^{\varrho }}{{\upsilon }^{\varrho }\Gamma (\varrho +1)}}.$$

Applying the fractional Grönwall inequality, one obtains

$$\parallel \beth \left(\tau \right)-\varpi \left(\tau \right)\parallel \le {\displaystyle \frac{\wp T^{\varrho }}{{\upsilon }^{\varrho }\Gamma (\varrho +1)-{\lambda }_{\mathcal{A}}{T}^{\varrho }}}=\wp {\mathbb{Q}}_{\wp },$$

where ${\mathbb{Q}}_{\wp }>0$ is a constant depending on $U,\varrho ,\upsilon ,{\lambda }_{\mathcal{A}}$.

Since ${\mathbb{Q}}_{\wp }$ is finite and positive, and the function ${𝘍}_{\wp }(\wp )=\wp {𝘍}_{\wp }$ satisfies ${𝘍}_{\wp }\left(0\right)=0$, the solution $\varpi$ is Ulam–Hyers stable and extended Ulam–Hyers stable. □

5. Numerical Simulations

This section develops a predictor–corrector Adams scheme for the model (10), explicitly accounting for fractional memory and nonlocal effects. To enhance accuracy, we employ iterative correction within each time step.

Throughout the simulations we take a uniform grid on $[0,U]$ with N time steps and $N+1$ grid points, so $h=U/N$, and ${\eta }_n=nh$ for $n=0,1,2,\dots ,N$ denotes the transformed time variable in the proportional-Caputo setting. For a generic right-hand side $\mathcal{F}(\eta ,y)$, the fractional Adams predictor–corrector method [39,40] reads

$$\begin{array}{cc}\hfill y_n^p& =y\left(0\right)+h^{\varrho }\sum _{j=1}^{n-1}{a}_{n-j-1}^{\varrho }exp\left(\frac{\upsilon -1}{\upsilon }({\eta }_n-{\eta }_j)\right)\mathcal{F}({\eta }_j,y_j),\hfill \\ \hfill y_n& =y\left(0\right)+h^{\varrho }\left({\tilde{b}}_n^{\varrho }\mathcal{F}({\eta }_0,y_0)+\sum _{j=1}^{n-1}{b}_{n-j}^{\varrho }exp\left(\frac{\upsilon -1}{\upsilon }({\eta }_n-{\eta }_j)\right)\mathcal{F}({\eta }_j,y_j)+b_0^{\varrho }exp\left(\frac{\upsilon -1}{\upsilon }h\right)\mathcal{F}({\eta }_n,y_n^p)\right).\hfill \end{array}$$

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To improve the approximation, we apply $\eta$ corrector iterations at time level n:

$$\begin{array}{cc}\hfill y_n^{\left[0\right]}& =y\left(0\right)+h^{\varrho }\sum _{j=1}^{n-1}{a}_{n-j-1}^{\varrho }exp\left(\frac{\upsilon -1}{\upsilon }({\eta }_n-{\eta }_j)\right)\mathcal{F}({\eta }_j,y_j),\hfill \\ \hfill y_n^{\left[\eta \right]}& =y\left(0\right)+h^{\varrho }\left({\tilde{b}}_n^{\varrho }\mathcal{F}({\eta }_0,y_0)+\sum _{j=1}^{n-1}{b}_{n-j}^{\varrho }exp\left(\frac{\upsilon -1}{\upsilon }({\eta }_n-{\eta }_j)\right)\mathcal{F}({\eta }_j,y_j)+b_0^{\varrho }exp\left(\frac{\upsilon -1}{\upsilon }h\right)\mathcal{F}({\eta }_n,y_n^{[\eta -1]})\right).\hfill \end{array}$$

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The convolution weights are given by

$$\begin{array}{cc}\hfill a_n^{\varrho }& ={\displaystyle \frac{{(n+1)}^{\varrho }-n^{\varrho }}{\Gamma (\varrho +1)}},\hfill \\ \hfill {\tilde{b}}_n^{\varrho }& ={\displaystyle \frac{{(n-1)}^{\varrho +1}-n^{\varrho }(n-\varrho -1)}{\Gamma (\varrho +2)}},\hfill \\ \hfill b_n^{\varrho }& =\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (\varrho +2)}},\hfill & n=0,\hfill \\ {\displaystyle \frac{{(n-1)}^{\varrho +1}-2n^{\varrho +1}+{(n+1)}^{\varrho +1}}{\Gamma (\varrho +2)}},\hfill & n=1,2,\dots \hfill \end{array}\right.\hfill \end{array}$$

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The parameter values and initial conditions used in the simulations are listed in

Table 2. Parameter values and initial conditions.

Parameter	Value	Parameter	Value
$\Theta$	100	$\mathfrak{i}$	0.007
$\mathfrak{d}$	0.5	$\mathfrak{c}$	1
$\mathfrak{a}$	10	$\alpha$	0.1
${\mathfrak{m}}_1$	0.05	${\mathfrak{m}}_2$	0.035
${\mathfrak{t}}_1$	0.45	${\mathfrak{t}}_2$	0.05
${\mathfrak{t}}_3$	0.55	${\mathfrak{t}}_4$	0.65
${\mathfrak{t}}_5$	0.35	${\mathfrak{t}}_6$	0.07
$\mathfrak{S}\left(0\right)$	2400	$\mathfrak{E}\left(0\right)$	1700
$\mathfrak{I}\left(0\right)$	1400	$\mathfrak{Q}\left(0\right)$	900
$\mathfrak{R}\left(0\right)$	5200	$\mathfrak{V}\left(0\right)$	5600

.

The numerical values in Table 2 are adopted as a scaled configuration to illustrate the model response under variations of the fractional order and tempering parameter. The initial conditions represent scaled compartment sizes in a reference network, while $\mathfrak{a},\mathfrak{d},\mathfrak{c}$ act as effective network-contact factors through $\beta ={\displaystyle \frac{\mathfrak{i}\mathfrak{d}\pi {\mathfrak{c}}^2}{\mathfrak{a}}}$.

This section reports graphical results for the fractal–fractional-order model (10), computed with the numerical methodology described in Section 5. Figure 2 and Figure 3 display the time evolution of the susceptible $\mathfrak{S}$, exposed $\mathfrak{E}$, infected $\mathfrak{I}$, quarantined $\mathfrak{Q}$, recovered $\mathfrak{R}$, and vaccinated $\mathfrak{V}$ node populations.

In Figure 2, we vary the fractional order $\varrho$ and fractal time parameter $\upsilon$ over $[0.60,0.95]$. As $\varrho ,\upsilon \to 1$, memory effects weaken: $\mathfrak{S}\left(\eta \right)$ increases, $\mathfrak{E}\left(\eta \right)$ and $\mathfrak{Q}\left(\eta \right)$ decrease toward negligible levels (by $t\approx 100$), $\mathfrak{I}\left(\eta \right)$ also declines, $\mathfrak{R}\left(\eta \right)$ falls (partly due to returns to susceptibility), and $\mathfrak{V}\left(\eta \right)$ grows more rapidly consistent with the classical-limit behavior.

Figure 3 shows sensitivity to the infection contact rate $\mathfrak{i}$. Larger $\mathfrak{i}$ accelerates the drop in $\mathfrak{S}\left(\eta \right)$ and yields the typical rise–peak–decay pattern for $\mathfrak{I}\left(\eta \right)$ with a faster increase and higher peak, followed by a sharper decline as quarantine and recovery take effect.

6. Conclusions

This study introduced a fractional model with six compartments to explore how malware spreads in wireless sensor networks. Two fractional parameters, the order $\varrho$ and the tempering $\upsilon$, were shown to play a key role in shaping the system. The order $\varrho$ controls how strongly past events influence the present: values below one keep earlier infections active for longer, while $\varrho =1$ leads back to the standard, memory-free case. The tempering $\upsilon$ then adjusts how far this memory extends; smaller values strengthen the effect of history, while $\upsilon =1$ corresponds to the well-known Caputo operator. Together, these parameters create a flexible way to represent both short-term responses and long-term effects in the network. The analysis confirmed that the model is mathematically consistent and stable, while numerical experiments showed how different parameter choices influence infection levels and the impact of control strategies. Overall, the results point to fractional modeling as a promising approach for better understanding malware in sensor networks and for supporting more effective protection methods.