EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6884 ISSN 1307-5543 – ejpam.com Published by New York Business Global Fuzzy Fractal-Fractional Derivative-Based Modelling and Analysis of Monkeypox Transmission Vediyappan Govindan1, Busayamas Pimpunchat2,∗, Kamran3,∗, Mohammad Javad Ebadi4, Ioan-Lucian Popa5,6 1 Department of Mathematics, Hindustan Institute of Technology and Science, Chennai, India 2 Department of Mathematics, School of Science, King Mongkut’s Institute of Technology Ladkrabang (KMITL), Bangkok, 10520, Thailand 3 Department of Mathematics, Islamia College Peshawar, Peshawar 25120, Khyber Pakhtunkhwa, Pakistan 4 Department of Mathematics, Chabahar Maritime University, Chabahar, Iran 5 Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania 6 Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania Abstract. This study investigates the transmission dynamics of monkeypox using a fuzzy frac- tal fractional-order mathematical model. The model incorporates fuzzy triangular numbers into both system parameters and initial conditions to effectively represent uncertainty in real-world epidemiological data. By employing fractal-fractional derivatives, the model captures memory and hereditary effects, providing a more accurate representation of disease progression. The mathe- matical analysis, supported by Ulam-Hyers stability and fixed-point theory, confirms the model’s reliability. Numerical simulation reveals the effects of changing the fractional order and uncertain inputs on the susceptible, exposed, infected, and recovered populations. The results underscore the enhanced comprehension of monkeypox spread and recovery provided by the combined effects of fractional dynamics and fuzziness and provide important insights for public health policy. 2020 Mathematics Subject Classifications: 26A33, 34A08, 92D30, 03E72 Key Words and Phrases: Mathematical model, fractal fractional operator, fuzzy values, trian- gular fuzzy number, Lipschitz function, Ulam-Hyers stability ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6884 Email addresses: busayamas.pi@kmitl.ac.th (B. Pimpunchat), kamran.maths@icp.edu.pk (Kamran) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 2 of 29 1. Introduction Mpox, previously referred to as monkeypox, is an emerging viral infection with growing worldwide importance. The CDC (2024) reports its prevention, management, and spread [1]. A WHO-sponsored study by Hoxha et al. (2023) reported Mpox infections in children during the outbreak in 2022–2023, citing distinct clinical findings [2]. Lu et al. (2023) offered a review of Mpox pathogenesis and therapy, focusing on vaccine and antiviral im- provements [3]. Karagoz et al. (2023) presented in detail the origin, genome, transmission, and diagnostic methods of the virus [4]. The WHO (2024) is still publishing global health advice and prevention measures [5]. Mpox re-emerged in 2022 with a worldwide epidemic that had predominantly beset the MSM population in the presentation of genital ulcers and lymphadenopathy [6]. Transmission is caused by direct contact with an infected le- sion, body fluids, respiratory droplets, or contaminated material. Although previously rare, human-to-human transmission is now recognized as a significant mode of spread. Preventive strategies include isolation, symptomatic treatment, and post-exposure vacci- nation with the modified vaccinia Ankara (MVA) vaccine, which has shown a favourable safety profile even in immunocompromised individuals [7]. Enhanced surveillance, vacci- nation strategies, and global awareness are essential in mitigating the ongoing outbreak and addressing the challenges posed by this re-emerging disease. Researchers in infectious diseases have highlighted the significant role of mathemat- ics and mathematical modelling in providing a clear framework for understanding the transmission dynamics of infectious diseases among individuals, communities, and animal populations. They have developed simplified models that incorporate essential clinical and biological parameters to describe disease development and spread [8–11]. In this con- text, numerous mathematical studies have examined the dynamics of various epidemic phenomena, such as alcoholism, smoking, COVID-19, and obesity [12–19]. To study the dynamics of monkeypox specifically, several compartmental models com- monly used in epidemiology have been proposed [20–22]. Although monkeypox received limited attention in the past, leading to gaps in understanding its transmission mecha- nisms, more recent studies have started to address this issue through mathematical mod- elling techniques. A deterministic mathematical model was developed in [23] to examine the monkeypox outbreak, revealing that isolating infected individuals significantly reduces disease incidence. A system of nonlinear differential equations was proposed in [24] to analyse multiple transmission pathways of monkeypox. Several foundational studies have contributed to understanding monkeypox transmis- sion dynamics. For instance, the transmission dynamics of pox-like diseases using monkey- pox as a case study were first analysed in [25]. The potential for eradication of monkeypox from both human and non-human primate populations through structured treatment in- terventions was demonstrated in [26]. A detailed stability analysis involving both human hosts and rodents was carried out in [27]. Additional contributions using compartmental modelling to enhance the understanding of monkeypox dynamics can be found in [28, 29]. In [30] considered both human-to-human and rodent-to-human transmission, offering an in-depth analysis without relying on specific outbreak data. V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 3 of 29 Recent studies have introduced innovative approaches to modelling monkeypox dy- namics. A study [31] analyses backward bifurcation using real observed data, highlighting how key parameters influence disease persistence. Another work [32] examines the trans- mission dynamics of the monkeypox virus using compartmental models to identify critical factors in its spread. Additionally, optimal control strategies have been applied in [33] to evaluate the effectiveness of interventions such as vaccination and isolation in reducing infection rates. Fractional-order differential equations have gained prominence in epi- demiology due to their ability to capture memory and hereditary effects, which classical models often overlook [34–38]. This approach has been used to model both infectious and non-infectious diseases, including COVID-19 [39–41]. In the context of monkeypox, a stochastic model addressing cross-infection was proposed in [42], and zoonotic transmis- sion using fractal-fractional operators was explored in [43]. Fractional-order operators like the Mittag–Leffler kernel and Caputo–Fabrizio derivative have improved model accuracy in various epidemiological studies [44–46]. Specifically, monkeypox spread in Nigeria was modeled using real data in [47], highlighting the role of public health measures, while [48] emphasized the importance of isolating infected individuals to reduce transmission. Fuzzy logic has also been applied in epidemic modelling to manage uncertainty in pa- rameters. The impact of fuzzy transmission and treatment rates was studied in [49], while [50] conducted sensitivity analysis under fuzzy imprecision. Combining fuzzy logic with fractional calculus enhances model realism, as demonstrated in fuzzy fractional COVID-19 models [51, 52] and malaria modelling [53]. Theoretical advancements in fuzzy fractional operators have been presented in [54], along with efficient numerical methods for solving such systems [55]. This research introduces a new fuzzy fractal fractional-order model to study the spread of monkeypox. It is the first approach that combines fuzzy logic with fractal-fractional derivatives to analyse disease transmission. Unlike existing models, this framework uniquely integrates fuzzy triangular numbers into model parameters and initial conditions, effec- tively capturing the imprecision and uncertainty present in real-world epidemiological data. The use of fractal-fractional operators allows the model to capture memory effects and genetic factors, making the analysis more realistic and detailed. The study also makes additional contributions by developing the theoretical validity of the intended model using fixed-point theory and Ulam-Hyers stability to guarantee the consistency of the system. Numerical computations illustrate the impact of varying fractional orders and fuzzy val- ues on main population compartments (susceptible, exposed, infected, recovered), thereby highlighting the model’s ability to reflect complex disease behaviour more accurately than classical or purely fractional models. This integrated modelling framework provides valu- able insights for designing effective public health strategies against monkeypox, making it a significant advancement in epidemic modelling literature. 2. Preliminaries This section presents the essential definitions and foundational concepts related to fractal-fractional differential equations and fuzzy systems, which serve as the theoretical V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 4 of 29 basis for this study. The treatment of fractional differential equations under uncertainty is examined in the work of [56], while the formulation of fuzzy differential equations and fractional differential equations with fuzzy initial conditions is detailed in [57, 58]. Collec- tively, these studies provide the theoretical framework underpinning the present research. Definition 1. The fuzzy set A consists of arranged collections A = {(x, µA(x)) : x ∈ X}, and the value or grade µA(x) of a membership function is an element from X → [0, 1]. Definition 2. The Triangular Fuzzy Number represented by three numbers ǎ, b̌, č has its membership function defined as µA(x) =  x−ǎ b̌−ǎ if ǎ ≤ x ≤ b̌ č−x č−b̌ if b̌ ≤ x ≤ č 0 otherwise, where ǎ ≤ b̌ ≤ č. Therefore A = (ǎ, b̌, č) can be defined as illustrated in [59]. An α-cut of fuzzy number à can be expressed by [Ãl(α), Ãr(α)] = [(1− α)ǎ+ αb̌, (1− α)č+ αb̌], ∀α ∈ [0, 1]. Definition 3. A fuzzy set à has an associated α-cut at level α ∈ [0, 1], which is the crisp subset of X containing all elements whose membership degree exceeds or is equivalent to α, i.e., Ãα = {x ∈ X | Ã(x) ≥ α}, α ∈ [0, 1]. Definition 4. (Graded Mean Integral Value Theorem [60]). Let m ∈ [0, 1] be the optimism degree for a fuzzy number. Now, for a fuzzy number Ã, the graded mean integral value (GMIV) is defined in turn. Gf (Ã) = ∫ 1 0 tf { (1−m)I−1 L (tf ) +mI−1 R (tf ) } dt∫ 1 0 tf dt = 2 ∫ 1 0 tf { (1−m)I−1 L (tf ) +mI−1 R (tf ) } dt, where IL and IR are the left and right integral values of a fuzzy number Ã. For triangular membership function, IL(tf ) = tf − ǎ b̌− ǎ , and IR(tf ) = č− tf č− b̌ , I−1 L (tf ) = ǎ+ (b̌− ǎ)tf , I−1 R (tf ) = č− (č− b̌)tf . Therefore, the GMIV is given as à = 2 ∫ 1 0 tf { (1−m)I−1 L (tf ) +mI−1 R (tf ) } dtf = 1 3 [(1−m)ǎ+ 2b̌+mč]. V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 5 of 29 Definition 5. Consider a function f : X → Y be such that X and Y are crisp sets. Let à and B̃ be fuzzy sets on X and Y respectively. If the membership values satisfy the condition µÃ(x) ≤ µB̃(x), then the function is termed a fuzzy function with constraints defined over the fuzzy domain à and fuzzy range B̃. Definition 6. Given a fuzzy number Ṽ , its β-level set is defined as [Ṽ ]β = {x ∈ R | Ṽ (x) ≥ β}, where β ∈ (0, 1] and x ∈ R. Definition 7. A fuzzy number can be described using a parametric form as a pair [V (ω), V (ω)] where 0 ≤ ω ≤ 1, satisfying the following conditions: • V (ω) is a bounded and non-decreasing function, which is right-continuous at 0, and left-continuous for (0, 1). • V (ω) is a non-increasing and bounded function on [0, 1], right-continuous at 0. • It always holds that V (ω) ≤ V (ω). • When V (ω) = V (ω) = 0, the fuzzy number becomes a crisp value. Definition 8. Consider a function α : E × E → R, let two fuzzy numbers be p = (p(ω), p(ω)) and q = (q(ω), q(ω)). The Hausdorff-type metric α(p, q) is defined by: α(p, q) = sup ω∈[0,1] max { |p(ω)− q(ω)|, |p(ω)− q(ω)| } . This function satisfies the properties below: • α(p+ q, q + p) = α(p, q) for all p, q ∈ E. • α(pϱ, qϱ) = |ϱ|α(p, q) for all p, q ∈ E, with ϱ ∈ R. • α(p+ r, q + s) ≤ α(p, q) + α(r, s) for all p, q, r, s ∈ E. The space (E,α) form a complete metric space. Definition 9. If two fuzzy numbers i1, i2 ∈ E satisfy the relation i1 = i2 + i3 for some fuzzy number i3 ∈ E, then i3 is called the H-difference of i1 and i2, and is denoted by i1 ⊖ i2. Definition 10. Consider a fuzzy-valued function Θ : R → E. The function Θ is said to be continuous at a point Λ0 ∈ [ζ1, ζ2], if for any ε > 0, there exists δ > 0 such that: α(Θ(Λ),Θ(Λ0)) < ε whenever |Λ− Λ0| < δ. V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 6 of 29 Definition 11. Let ω(x) be a fuzzy function belongs to CF[0, b]∩LF[0, b], and represented as ω = [ωp(x), ωp(x)] for 0 ≤ p ≤ 1. The Caputo type fuzzy fractional derivative is: [Dβω(x0)]p = [Dβωp(x0), D βωp(x0)], 0 ≤ β ≤ 1, where Dβωp(x0) = 1 Γ(n− β) [∫ x 0 (x− ζ)n−β−1 d n dζn ωp(ζ) dζ ] x=x0 , Dβωp(x0) = 1 Γ(n− β) [∫ x 0 (x− ζ)n−β−1 d n dζn ωp(ζ) dζ ] x=x0 . With n = ⌈β⌉ provided the integral exists. Definition 12. Suppose a fuzzy function Ũ is defined as Ũ = [Up(x), Up(x)] for 0 ≤ p ≤ 1. The Riemann-Liouville type Fuzzy fractal fractional derivative involving the generalized Mittag-Leffler Kernel function is given by (FFM 0 )D(η,ϑ) τ (Ũ(τ)) = AB(ϑ) 1− ϑ d dtη ∫ τ 0 Eϑ [ − ϑ 1− ϑ (τ − ψ)ϑŨ(ψ) dψ ] . (1) Where 0 < ϑ, η ≤ 1 and AB(ϑ) = 1− ϑ+ ϑ ⌈ϑ⌉ . The associated integral operator for this derivative is defined as: (FFM 0 )I(η,ϑ)τ (Ũ(τ)) = η(1− ϑ)τη−1Ũ(τ) AB(ϑ) + ηϑ AB(ϑ)⌈ϑ⌉ × ∫ τ 0 ψη−1(τ − ψ)ϑ−1Ũ(ψ) dψ. (2) 3. Model Formulation This section formulates a fuzzy fractal-fractional mathematical framework to study monkeypox disease dynamics considering interactions between human-animal populations. The operator of the fractional derivative Atangana-Baleanu-Caputo (ABC), involving memory effects as well as local behaviours in existence within biological environments, is adopted in the proposed model. To further enhance the model’s realism, fuzzy set theory is employed to account for uncertainty and imprecision in parameter estimation, such as contact rates, recovery rates, and transition probabilities. Moreover, the model integrates fractal dimension through a fractal-fractional framework, allowing a more generalized and flexible representation of the complex dynamics of disease spread across heterogeneous populations. By merging these higher-level mathematical tools, the model offers a strong and inclusive framework for analysing the dynamics and control of monkeypox in an un- certain and complex biological memory environment. 3.1. Evolution of Human Population There are various compartments in the human population: • SH(τ): Susceptible humans V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 7 of 29 • IH(τ): Infected humans • TH(τ): Treated (or isolated) humans • RH(τ): Recovered humans • PH(τ): Protected humans Let NH = SH(τ) + IH(τ) + TH(τ) +RH(τ) + PH(τ) denote the total human population. The Atangana-Baleanu logic of a Fractal-Fractional derivative regulates the propagation dynamics, and the power-law kernel is defined as (FFM 0 )D (η,ϑ) τ . The model of the human population is provided by the given system of fractional differential equations: (FFM 0 )D(η,ϑ) τ SH(τ) = ΠH − λHSH − (µH + ξH)SH (FFM 0 )D(η,ϑ) τ IH(τ) = λHSH − (ΥH + E1 + δ1 + ωH + µH)IH (FFM 0 )D(η,ϑ) τ TH(τ) = ΥHIH − (E2 + δ2 + µH)TH (FFM 0 )D(η,ϑ) τ RH(τ) = E1IH + E2TH − (µH + θH)RH (FFM 0 )D(η,ϑ) τ PH(τ) = ξHSH + ωHIH + θHRH − µHPH . Where • ΠH : Recruitment rate into the human population • λH : Force of infection combining contact with infected animals and humans • µH : Natural human death rate • ξH : Susceptible individuals’ rate entering the protected class to avoid infection • ΥH : Infection cases’ rate entering treatment or isolation • E1, E2: Recovery rates from infection and treatment, respectively • δ1, δ2: Disease-induced death rates from infected and treated individuals • ωH : Rate at which infection cases are transferred into protection in order to prevent further spread • θH : Recovery rate at which cured persons shift to the protected class This structure captures the dynamic transitions in the human population, integrating memory and non-local effects via the fractal-fractional operator. V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 8 of 29 3.2. Animal Population Dynamics The animal population is divided into: • SA(τ): Susceptible animals • IA(τ): Infected animals • RA(τ): Recovered animals. Let NA = SA + IA +RA be the total animal population. The animal model follows a similar nonlinear system of differential equations based on classical or fractional derivatives (depending on your specification): (FFM 0 )D(η,ϑ) τ SA(τ) = ΠA − λASA − µASA (FFM 0 )D(η,ϑ) τ IA(τ) = λASA − (ΥA + µA + dA)IA (FFM 0 )D(η,ϑ) τ RA(τ) = ΥAIA − µARA. Where • ΠA: Recruitment rate into the animal population • λA: Contact rate between infected and susceptible animals that combines into force of infection • µA: Animal natural death rate • dA: Mortality rate due to disease in animals • ΥA: Rate of recovery of infected animals. This model framework facilitates a comprehensive exploration of monkeypox transmis- sion dynamics and control strategies within a fractional-order setting, effectively capturing the influence of memory and hereditary behaviour in population of humans and animals. The corresponding equations are formulated using a fractal-fractional framework charac- terized by the order η (0 < η < 1) and fractal dimension ϑ. The model also uses fuzzy initial conditions to reflect the uncertainty and imprecision associated with the starting values of each compartment, ensuring a more realistic and flexible simulation framework. V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 9 of 29 (FFM 0 )D(η,ϑ) τ SH(τ) = (Π̃H)− (λ̃H)SH − ((µ̃H) + (ξ̃H))SH (FFM 0 )D(η,ϑ) τ IH(τ) = (λ̃H)SH − ((Υ̃H) + (Ẽ1) + (δ̃1) + (ω̃H) + (µ̃H))IH (FFM 0 )D(η,ϑ) τ TH(τ) = (Υ̃H)IH − ((Ẽ2) + (δ̃2) + (µ̃H))TH (FFM 0 )D(η,ϑ) τ RH(τ) = (Ẽ1)IH + (Ẽ2)TH − ((µ̃H) + (θ̃H))RH (FFM 0 )D(η,ϑ) τ PH(τ) = (ξ̃H)SH + (ω̃H)IH + (θ̃H)RH − (µ̃H)PH (FFM 0 )D(η,ϑ) τ SA(τ) = (Π̃A)− (λ̃A)SA − (µ̃A)SA (FFM 0 )D(η,ϑ) τ IA(τ) = (λ̃A)SA − ((Υ̃A) + (µ̃A) + (d̃A))IA (FFM 0 )D(η,ϑ) τ RA(τ) = (Υ̃A)IA − (µ̃A)RA. (3) The transmission terms are given by (λ̃H) = (( ˜βH1 )IA+( ˜βH2 )IH) NH for humans and (λ̃A) = (β̃A)IA NA for animals, representing the respective force of infection. All related parameters such as Π̃H , λ̃H , µ̃H , ξ̃H , Υ̃H , Ẽ1, δ̃1, Ẽ2, δ̃2, ω̃H , θ̃H , Π̃A, λ̃A, Υ̃A, µ̃A and d̃A are modeled as triangular fuzzy numbers (TFNs) with narrow supports ranges to represent the uncertainty present in real-world scenarios. These parameters define the dynamic behavior of the state variables SH ≥ 0, IH ≥ 0, TH ≥ 0, RH ≥ 0, PH ≥ 0 for humans and SA ≥ 0, IA ≥ 0, RA ≥ 0 for animals within the fuzzy fractal-fractional monkeypox model. 4. Qualitative analysis Theorem 1. The solution paths of the proposed model (3) are positive at any time instant in R8 +. Proof. It follows from the model (3) that (FFM 0 )D(η,ϑ) τ SH(τ) ∣∣∣ SH=0 = (Π̃H) ≥ 0, (FFM 0 )D(η,ϑ) τ IH(τ) ∣∣∣ IH=0 = (λ̃H)SH ≥ 0, (FFM 0 )D(η,ϑ) τ TH(t) ∣∣∣ TH=0 = (Υ̃H)IH ≥ 0, (FFM 0 )D(η,ϑ) τ RH(t) ∣∣∣ RH=0 = (Ẽ1)IH + (Ẽ2)TH ≥ 0, (FFM 0 )D(η,ϑ) τ PH(t) ∣∣∣ PH=0 = (ξ̃H)SH + (ω̃H)IH + (θ̃H)RH ≥ 0, (FFM 0 )D(η,ϑ) τ SA(τ) ∣∣∣ SA=0 = (Π̃A) ≥ 0, (FFM 0 )D(η,ϑ) τ IA(τ) ∣∣∣ IA=0 = (λ̃A)SA ≥ 0, (FFM 0 )D(η,ϑ) τ RA(τ) ∣∣∣ RA=0 = (Υ̃A)IA ≥ 0. V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 10 of 29 All rates are above-described are nonnegative, as are the solution paths in the non-negative of R8 +. 5. Existence and Uniqueness of Proposed scheme This part outlines the existence and uniqueness of the solution to the following fuzzy fractional model. The analysis focuses on a system of fuzzy fractional-order differential equations formulated using the Atangana–Baleanu Caputo operator. To reformulate the system in a more analysable form, the Atangana–Baleanu fractal–fractional operator is applied to transform the set of equations (3). With fixed-point theorems, it is shown that the model admits at least one solution uniquely. In below, the suggested model is discussed, considering fractional differentiation and integration techniques. (AB 0 )D(η,ϑ) τ SH(τ) = ϑtϑ−1g1(τ, SH , IH , TH , RH , PH , SA, IA, RA) (AB 0 )D(η,ϑ) τ IH(τ) = ϑtϑ−1g2(τ, SH , IH , TH , RH , PH , SA, IA, RA) (AB 0 )D(η,ϑ) τ TH(τ) = ϑtϑ−1g3(τ, SH , IH , TH , RH , PH , SA, IA, RA) (AB 0 )D(η,ϑ) τ RH(τ) = ϑtϑ−1g4(τ, SH , IH , TH , RH , PH , SA, IA, RA) (AB 0 )D(η,ϑ) τ PH(τ) = ϑtϑ−1g5(τ, SH , IH , TH , RH , PH , SA, IA, RA) (AB 0 )D(η,ϑ) τ SA(τ) = ϑtϑ−1g6(τ, SH , IH , TH , RH , PH , SA, IA, RA) (AB 0 )D(η,ϑ) τ IA(τ) = ϑtϑ−1g7(τ, SH , IH , TH , RH , PH , SA, IA, RA) (AB 0 )D(η,ϑ) τ RA(τ) = ϑtϑ−1g8(τ, SH , IH , TH , RH , PH , SA, IA, RA). Where the fuzzy functions are g1, g2, g3, g4, g5, g6, g7, g8 then, for γ ∈ [0, 1], model (3) gets the form (FFM 0 )D (η,ϑ) τ SH(τ) = g1(τ, SH , IH , TH , RH , PH , SA, IA, RA) (FFM 0 )D (η,ϑ) τ IH(τ) = g2(τ, SH , IH , TH , RH , PH , SA, IA, RA) (FFM 0 )D (η,ϑ) τ TH(τ) = g3(τ, SH , IH , TH , RH , PH , SA, IA, RA) (FFM 0 )D (η,ϑ) τ RH(τ) = g4(τ, SH , IH , TH , RH , PH , SA, IA, RA) (FFM 0 )D (η,ϑ) τ PH(τ) = g5(τ, SH , IH , TH , RH , PH , SA, IA, RA) (FFM 0 )D (η,ϑ) τ SA(τ) = g6(τ, SH , IH , TH , RH , PH , SA, IA, RA) (FFM 0 )D (η,ϑ) τ IA(τ) = g7(τ, SH , IH , TH , RH , PH , SA, IA, RA) (FFM 0 )D (η,ϑ) τ RA(τ) = g8(τ, SH , IH , TH , RH , PH , SA, IA, RA). (4) The analysis begins by considering uncertain initial conditions. S̃H(0, γ) = [SH(0, γ), SH(0, γ)] ĨH(0, γ) = [IH(0, γ), IH(0, γ)] T̃H(0, γ) = [TH(0, γ), TH(0, γ)] V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 11 of 29 R̃H(0, γ) = [RH(0, γ), RH(0, γ)] P̃H(0, γ) = [PH(0, γ), PH(0, γ)] S̃A(0, γ) = [SA(0, γ), SA(0, γ)] ĨA(0, γ) = [IA(0, γ), IA(0, γ)] R̃A(0, γ) = [RA(0, γ), RA(0, γ)]. The system can be further written as (ABR 0 )D(η,ϑ) τ Φ̃(τ) = ϑτϑ−1Ω̃(τ, Φ̃(τ)) with Φ̃(0) = Φ̃0. Replacing (ABR 0 )D (η,ϑ) τ with (ABC 0 )D (η,ϑ) τ , applying the fuzzy fractional integral operator Iϑ together with the provided initial conditions, the system is reformulated. Φ̃(τ) = Φ̃(0, γ)+ ϑτϑ−1(1− η) AB(η) Ω(τ, Φ̃(τ))+ ηϑ ⌈η⌉AB(η) × ∫ τ 0 mϑ−1(τ−m)η−1Ω(τ, Φ̃(τ)) dm. (5) Where Φ̃(t) =  SH(τ) IH(t) TH(t) RH(t) PH(t) SA(t) IA(t) RA(τ), Φ̃(0, γ) =  S̃H(0, γ) ĨH(0, γ) T̃H(0, γ) R̃H(0, γ) P̃H(0, γ) S̃A(0, γ) ĨA(0, γ) R̃A(0, γ), Ω(t, Φ̃(t)) =  ψ1(t, SH(t)) ψ2(t, IH(t)) ψ3(t, TH(t)) ψ4(t, RH(t)) ψ5(t, PH(t)) ψ6(t, SA(t)) ψ7(t, IA(t)) ψ7(t, RA(t)). A Banach Space I = C × C × C × C × C × C × C is constructed, where C[0,K] denotes the space of continuous functions over the interval [0,K], equipped with the norm: ∥Φ̃∥ = max τ∈[0,K] |SH(τ), IH(τ), TH(τ), RH(τ), PH(τ), SA(τ), IA(τ), RA(τ)|. To examine the existence of a solution, the operator ψ : I → I is defined as follows: ψ(Φ̃)(τ) = Φ̃(0, γ)+ ϑτϑ−1(1− η) AB(η) Ω(τ, Φ̃(τ))+ ηϑ ⌈η⌉AB(η) × ∫ τ 0 mϑ−1(τ−m)η−1Ω(τ, Φ̃(τ)) dm. Assume the nonlinear function Ω(τ, Φ̃(τ)) satisfies the Lipschitz conditions and a bounded growth condition. That is, for every Φ̃ ∈ I, there exists a constant CΦ > 0 and GΦ such that: |Ω(τ, Φ̃(τ))| ≤ CΦ|Φ̃(τ)|+GΦ. V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 12 of 29 And for all Φ̃1, Φ̃2 ∈ I, there exists a constant HΦ > 0 such that: |Ω(τ, Φ̃1(τ))− Ω(τ, Φ̃2(τ))| ≤ HΦ|Φ̃1(τ)− Φ̃2(τ)|. (6) These conditions are used to establish the uniqueness and stability of the solution for the proposed fuzzy fractal-fractional model under the Atangana–Baleanu framework. Theorem 2. On the premise that the conditions (4) hold. A continuous mapping Ω : [0,K]×I → L is defined, in order to assure that the postulated model is uniquely solvable. Proof. The continuity of the operator ψ is established by first verifying its overall continuity. Given that Ω is constant, the continuity of ψ is consequently ensured. Let Φ̃ = {Φ̃ ∈ I : ∥Φ̃∥ ≤ L,L > 0}. Whenever Φ̃ ∈ I, we obtain |ψ(Φ̃)| = max τ∈[0,K] ∣∣∣∣Φ̃(0) + ϑτϑ−1(1− η) AB(η) Ω(τ, Φ̃(τ)) + ηϑ ⌈η⌉AB(η) × ∫ τ 0 mϑ−1(τ −m)η−1Ω(τ, Φ̃(τ))dm ∣∣∣∣ ≤ Φ̃(0) + ϑKϑ−1(1− η) AB(η) (CΦ∥Φ̃∥+GΦ) + max τ∈[0,K] ηϑ ⌈η⌉AB(η) × ∫ τ 0 mϑ−1(τ −m)η−1|Ω(m, Φ̃(m))|dm |ψ(Φ̃)| ≤ Φ̃(0) + ϑKϑ−1(1− η) AB(η) (CΦ∥Φ̃∥+GΦ) + ηϑ ⌈η⌉AB(η) (CΦ∥Φ̃∥+GΦ)×Kη+ϑ−1Φ̃(η, ϑ) ≤ L. (7) Thus, if Φ̃(η, ϑ) represents a function, the operator ψ is bound by homogeneity. Since ψ is Equi-continuous, when τ1, τ2 ≤ K denote a function. Then consider |ψ(Φ̃)(τ2)− ψ(Φ̃)(τ1)| = ∣∣∣∣∣ϑτϑ−1 2 (1− η) AB(η) Ω(τ2, Φ̃(τ2)) + ηϑ ⌈η⌉AB(η) × ∫ τ2 0 mϑ−1(τ2 −m)η−1Ω(m, Φ̃(m)) dm −ϑτ ϑ−1 1 (1− η) AB(η) Ω(t1, Φ̃(t1)) + ηϑ ⌈η⌉AB(η) × ∫ τ1 0 mϑ−1(τ1 −m)η−1Ω(m, Φ̃(m)) dm ∣∣∣∣∣ ≤ ϑτϑ−1 2 (1− η) AB(η) (CΦ∥Φ̃∥+GΦ) + ηϑ ⌈η⌉AB(η) (CΦ∥Φ̃∥+GΦ)× τη+ϑ−1 2 Φ̃(η, ϑ) − ϑτϑ−1 1 (1− η) AB(η) (CΦ∥Φ̃∥+GΦ) + ηϑ ⌈η⌉AB(η) (CΦ∥Φ̃∥+GΦ)× τη+ϑ−1 1 Φ̃(η, ϑ). V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 13 of 29 When τ1, τ2 then |ψ(Φ̃)(τ2)− ψ(Φ̃)(τ1)| → 0. As a result, we have ∥ψ(Φ̃)(τ2)− ψ(Φ̃)(τ1)∥ → 0 when τ1 → τ2. Due to the fact that ψ is also continuous, Arzela-Ascoli theory is also entirely completely continuous. As such, Schauder’s fixed point theorem proves that the suggested model has at least one solution. Theorem 3. If condition (7) holds such that ϖ < 1, where ϖ = ( ϑKϑ−1(1− η) AB(η) + ηϑ ⌈η⌉AB(η) Kη+ϑ−1Φ̃(η, ϑ) ) HΦ, then the system described by equation (3) admits a unique solution. Proof. Let Φ̃1, Φ̃2 ∈ I, we have |ψ(Φ̃1)− ψ(Φ̃2)| = max τ∈[0,K] ∣∣∣∣ϑτϑ−1(1− η) AB(η) [ Ω(τ, Φ̃1(τ))− Ω(τ, Φ̃2(τ)) ] + ηϑ ⌈η⌉AB(η) × ∫ τ 0 mϑ−1(τ −m)η−1 dm [ Ω(m, Φ̃1(m))− Ω(m, Φ̃2(m)) ]∣∣∣∣ ≤ [ ϑKϑ−1(1− η) AB(η) + ηϑ ⌈η⌉AB(η) Kη+ϑ−1Φ̃(η, ϑ) ] ∥Φ̃1 − Φ̃2∥ ≤ ϖ∥Φ̃1 − Φ̃2∥. As a result, the operator ψ qualifies as a contraction, and the model guarantees a unique solution in accordance with the Banach fixed-point theorem. 5.1. Ulam- Hyers Stability Definition 13. The model is referred to as Ulam-Hyers stable if, for every small pertur- bation Q(τ) such that |Q(τ)| ≤ ε for ε > 0, there is a unique solution P (τ) such that the approximate solution Φ̃(τ) satisfies the inequality |Φ̃(τ)− P (τ)| ≤ ψη,ϑε, for every τ ∈ [0,K]. Where ψη,ϑ is a constant that dependent on the parameters of the proposed model. Now, consider the fractional-order dynamical system governed by (FFM 0 )D(η,ϑ) τ Φ̃(τ) = Ω(τ, Φ̃(τ)) +Q(τ) with initial condition Φ̃(0) = Φ̃0. Lemma 1. Suppose P (τ) is the precise solution to the unperturbed problem, and be the solution of the perturbed problem. Provided |Q(τ)| ≤ ε, then |Φ̃(τ)− P (τ)| ≤ αη,ϑε+ϖ|Φ̃(τ)− P (τ)|. V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 14 of 29 Proof. This is a very simple illustration. It can be explained by considering (6). (FFM 0 )D(η,ϑ) τ (Φ̃(τ)− P (τ)) = Ω(τ, Φ̃(τ)) +Q(τ)− P (τ). Applying the Lipschitz condition to Ω: |(FFM 0 )D(η,ϑ) τ (Φ̃(τ)− P (τ))| ≤ L|Φ̃(τ)− P (τ)|+ |Q(τ)|. Where L stands for Lipschitz the terms. Expanding the fractional derivative yields |Φ̃(τ)− P (τ)| = ∣∣∣∣Φ̃(τ)− ( P (0) + ϑτϑ−1(1− η) AB(η) Ω(τ, P (τ)) + ηϑ ⌈η⌉AB(η) × ∫ τ 0 mϑ−1(τ −m)η−1Ω(m,P (m)) dm )∣∣∣∣ ≤ ϑKϑ−1(1− η) AB(η) + ηϑ ⌈η⌉AB(η) Kη+ϑ−1Φ̃(η, ϑ) +ϖ|Φ̃(τ)− P (τ)|. Satisfies the required criterion |Φ̃(τ)− P (τ)| ≤ αη,ϑε+ϖ|Φ̃(τ)− P (τ)|, where αη,ϑε = ϑKϑ−1(1−η) AB(η) + ηϑ ⌈η⌉AB(η)K η+ϑ−1Φ̃(η, ϑ). Consequently, the proof has been validated. Lemma 2. If ϖ < 1, the suggested model’s Ulam-Hyers stable result engages with Lemma 1. The Ulam-Hyers stability for the fractional-order system holds if ϖ < 1. Under this condition, the perturbed solution Υ(τ) holds the following: |Φ̃(τ)− P (τ)| ≤ αη,ϑ 1−ϖ . Proof. From Lemma 1, |Φ̃(τ)−P (τ)| ≤ αη,ϑε+ϖ|Φ̃(τ)−P (τ)|. Another representation of the above relation is in the form |Φ̃(τ)− P (τ)| ≤ ψη,ϑε, where ψη,ϑ = αη,ϑ 1−ϖ . The system is shown to be Ulam-Hyers stable using the lemmas hereinabove. The outcome guarantees that any perturbation Q(τ) produces bounded errors in the solution without affecting system stability. V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 15 of 29 6. Numerical structure by fractal – fractional representation The numerical scheme is constructed at the point τ = τd+1, permitting an approximate solution of system (3) in terms of Atangana–Baleanu fractal–fractional techniques in the Caputo sense [59]. Sd+1 H = S̃H(0, γ) + ϑτϑ−1(1− η) AB(η) κ1(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) ∫ τ 0 sϑ−1(s− τ)η−1 × κ1(τ, SH , IH , TH , RH , PH , SA, IA, RA)ds Id+1 H = ĨH(0, γ) + ϑτϑ−1(1− η) AB(η) κ2(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) ∫ τ 0 sϑ−1(s− τ)η−1 × κ2(τ, SH , IH , TH , RH , PH , SA, IA, RA)ds T d+1 H = T̃H(0, γ) + ϑτϑ−1(1− η) AB(η) κ3(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) ∫ τ 0 sϑ−1(s− τ)η−1 × κ3(τ, SH , IH , TH , RH , PH , SA, IA, RA)ds Rd+1 H = R̃H(0, γ) + ϑτϑ−1(1− η) AB(η) κ4(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) ∫ τ 0 sϑ−1(s− τ)η−1 × κ4(τ, SH , IH , TH , RH , PH , SA, IA, RA)ds P d+1 H = P̃H(0, γ) + ϑτϑ−1(1− η) AB(η) κ5(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) ∫ τ 0 sϑ−1(s− τ)η−1 × κ5(τ, SH , IH , TH , RH , PH , SA, IA, RA)ds Sd+1 A = S̃A(0, γ) + ϑτϑ−1(1− η) AB(η) κ6(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) ∫ τ 0 sϑ−1(s− τ)η−1 × κ6(τ, SH , IH , TH , RH , PH , SA, IA, RA)ds Id+1 A = ĨA(0, γ) + ϑτϑ−1(1− η) AB(η) κ7(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) ∫ τ 0 sϑ−1(s− τ)η−1 × κ7(τ, SH , IH , TH , RH , PH , SA, IA, RA)ds Rd+1 A = R̃A(0, γ) + ϑτϑ−1(1− η) AB(η) κ8(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) ∫ τ 0 sϑ−1(s− τ)η−1 × κ8(τ, SH , IH , TH , RH , PH , SA, IA, RA)ds. (8) From approximating the integrals on the right side of that Equation (8) yields the V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 16 of 29 following expression. Sd+1 H = S̃H(0, γ) + ϑτϑ−1 d (1− η) AB(η) κ1(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) d∑ q=0 ∫ τq+1 τq sϑ−1(τd+1 − s)η−1 × κ1(τ, SH , IH , TH , RH , PH , SA, IA, RA) ds Id+1 H = ĨH(0, γ) + ϑτϑ−1 d (1− η) AB(η) κ2(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) d∑ q=0 ∫ τq+1 τq sϑ−1(τd+1 − s)η−1 × κ2(τ, SH , IH , TH , RH , PH , SA, IA, RA) ds T d+1 H = T̃H(0, γ) + ϑτϑ−1 d (1− η) AB(η) κ3(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) d∑ q=0 ∫ τq+1 τq sϑ−1(τd+1 − s)η−1 × κ3(τ, SH , IH , TH , RH , PH , SA, IA, RA) ds Rd+1 H = R̃H(0, γ) + ϑτϑ−1 d (1− η) AB(η) κ4(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) d∑ q=0 ∫ τq+1 τq sϑ−1(τd+1 − s)η−1 × κ4(τ, SH , IH , TH , RH , PH , SA, IA, RA) ds P d+1 H = P̃H(0, γ) + ϑτϑ−1 d (1− η) AB(η) κ5(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) d∑ q=0 ∫ τq+1 τq sϑ−1(τd+1 − s)η−1 × κ5(τ, SH , IH , TH , RH , PH , SA, IA, RA) ds Sd+1 A = S̃A(0, γ) + ϑτϑ−1 d (1− η) AB(η) κ6(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) d∑ q=0 ∫ τq+1 τq sϑ−1(τd+1 − s)η−1 × κ6(τ, SH , IH , TH , RH , PH , SA, IA, RA) ds Id+1 A = ĨA(0, γ) + ϑτϑ−1 d (1− η) AB(η) κ7(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ηϑ ⌈η⌉AB(η) d∑ q=0 ∫ τq+1 τq sϑ−1(τd+1 − s)η−1 × κ7(τ, SH , IH , TH , RH , PH , SA, IA, RA) ds Rd+1 A = R̃A(0, γ) + ϑτϑ−1 d (1− η) AB(η) κ8(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 17 of 29 + ηϑ ⌈η⌉AB(η) d∑ q=0 ∫ τq+1 τq sϑ−1(τd+1 − s)η−1 × κ8(τ, SH , IH , TH , RH , PH , SA, IA, RA) ds Afterward, we possess Pq(s) = (s− τq−1) (τq − τq−1) τϑ−1 q κ1(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) − (s− τq) (τq − τq−1) τϑ−1 q−1 κ1(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) Qq(s) = (s− τq−1) (τq − τq−1) τϑ−1 q κ2(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) − (s− τq) (τq − τq−1) τϑ−1 q−1 κ2(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) Uq(s) = (s− τq−1) (τq − τq−1) τϑ−1 q κ3(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) − (s− τq) (τq − τq−1) τϑ−1 q−1 κ3(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) Vq(s) = (s− τq−1) (τq − τq−1) τϑ−1 q κ4(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) − (s− τq) (τq − τq−1) τϑ−1 q−1 κ4(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) Wq(s) = (s− τq−1) (τq − τq−1) τϑ−1 q κ5(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) − (s− τq) (τq − τq−1) τϑ−1 q−1 κ5(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) Xq(s) = (s− τq−1) (τq − τq−1) τϑ−1 q κ6(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) − (s− τq) (τq − τq−1) τϑ−1 q−1 κ6(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) Yq(s) = (s− τq−1) (τq − τq−1) τϑ−1 q κ7(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) − (s− τq) (τq − τq−1) τϑ−1 q−1 κ7(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) Zq(s) = (s− τq−1) (τq − τq−1) τϑ−1 q κ8(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) − (s− τq) (τq − τq−1) τϑ−1 q−1 κ8(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) (9) Evaluating the partitioned Lagrange polynomial from Equation (9) provides results, V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 18 of 29 which are used as the basis for the subsequent numerical analysis. Sd+1 H = S̃H(0, γ) + ϑτϑ−1 d (1− η) AB(η) × κ1(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ϑ(δτ)η ⌈η + 2⌉AB(η) × d∑ q=0 [ τϑ−1 q κ1(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) × ((d+ 1− q)η(d− q + 2 + η)− (d− q)(2 + 2η + d− q)) − τϑ−1 q−1 × κ1(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) × ( (d+ 1− q)η+1 − (d− q)η × (1 + η + d− q) )] . Id+1 H = ĨH(0, γ) + ϑτϑ−1 d (1− η) AB(η) × κ2(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ϑ(δτ)η ⌈η + 2⌉AB(η) × d∑ q=0 [ τϑ−1 q κ2(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) × ((d+ 1− q)η(d− q + 2 + η)− (d− q)(2 + 2η + d− q)) − τϑ−1 q−1 × κ2(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) × ( (d+ 1− q)η+1 − (d− q)η × (1 + η + d− q) )] . T d+1 H = T̃H(0, γ) + ϑτϑ−1 d (1− η) AB(η) × κ3(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ϑ(δτ)η ⌈η + 2⌉AB(η) × d∑ q=0 [ τϑ−1 q κ3(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) × ((d+ 1− q)η(d− q + 2 + η)− (d− q)(2 + 2η + d− q)) − τϑ−1 q−1 × κ3(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) × ( (d+ 1− q)η+1 − (d− q)η × (1 + η + d− q) )] . Rd+1 H = R̃H(0, γ) + ϑτϑ−1 d (1− η) AB(η) × κ4(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ϑ(δτ)η ⌈η + 2⌉AB(η) × d∑ q=0 [ τϑ−1 q κ4(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) × ((d+ 1− q)η(d− q + 2 + η)− (d− q)(2 + 2η + d− q)) − τϑ−1 q−1 × κ4(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) × ( (d+ 1− q)η+1 − (d− q)η × (1 + η + d− q) )] . P d+1 H = P̃H(0, γ) + ϑτϑ−1 d (1− η) AB(η) × κ5(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ϑ(δτ)η ⌈η + 2⌉AB(η) × d∑ q=0 [ τϑ−1 q κ5(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) × ((d+ 1− q)η(d− q + 2 + η)− (d− q)(2 + 2η + d− q)) − τϑ−1 q−1 × κ5(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) × ( (d+ 1− q)η+1 − (d− q)η × (1 + η + d− q) )] . (10) V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 19 of 29 Sd+1 A = S̃A(0, γ) + ϑτϑ−1 d (1− η) AB(η) × κ6(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ϑ(δτ)η ⌈η + 2⌉AB(η) × d∑ q=0 [ τϑ−1 q κ6(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) × ((d+ 1− q)η(d− q + 2 + η)− (d− q)(2 + 2η + d− q)) − τϑ−1 q−1 × κ6(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) × ( (d+ 1− q)η+1 − (d− q)η × (1 + η + d− q) )] . Id+1 A = ĨA(0, γ) + ϑτϑ−1 d (1− η) AB(η) × κ7(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ϑ(δτ)η ⌈η + 2⌉AB(η) × d∑ q=0 [ τϑ−1 q κ7(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) × ((d+ 1− q)η(d− q + 2 + η)− (d− q)(2 + 2η + d− q)) − τϑ−1 q−1 × κ7(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) × ( (d+ 1− q)η+1 − (d− q)η × (1 + η + d− q) )] . Rd+1 A = R̃A(0, γ) + ϑτϑ−1 d (1− η) AB(η) × κ8(τd, S d H , I d H , T d H , R d H , P d H , S d A, I d A, R d A) + ϑ(δτ)η ⌈η + 2⌉AB(η) × d∑ q=0 [ τϑ−1 q κ8(τq, S q H , I q H , T q H , R q H , P q H , S q A, I q A, R q A) × ((d+ 1− q)η(d− q + 2 + η)− (d− q)(2 + 2η + d− q)) − τϑ−1 q−1 × κ8(τq−1, S q−1 H , Iq−1 H , T q−1 H , Rq−1 H , P q−1 H , Sq−1 A , Iq−1 A , Rq−1 A ) × ( (d+ 1− q)η+1 − (d− q)η × (1 + η + d− q) )] . 7. Results and Discussion The numerical solutions given in this research provide valuable insights into the trans- mission dynamics of monkeypox under fuzzy uncertainty and fractional-order modelling. Use of the fractal–fractional method in solving the derived system, that is, Equation (10), offers an efficient paradigm for the solution of the fractional differential equations in- volved in disease transmission among human and animal populations. Utilizing MATLAB, the simulations incorporate the fuzzified parameters expressed through Triangular Fuzzy Numbers (TFNs) as defined in Definition 2, providing a robust mechanism to capture the uncertainty inherent in real-world epidemiological data. Each parameter’s nominal value is obtained through the GMIV (Generalized Mean Integral Value) method outlined in Def- inition 4, which facilitates the analysis from three key perspectives pessimistic (m = 0), moderate (m = 0.5), and optimistic (m = 1). This methodology makes it possible to gain V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 20 of 29 deeper insights into how the model outputs are sensitive to different levels of optimism, representing different levels of information available within a vague environment. The pa- rameters utilized in the suggested fuzzy fractional-order monkeypox disease transmission model are presented in Table 1, and the compartmental variations in human and animal populations for different optimism levels m = 0, m = 0.5, m = 1 using triangular fuzzy numbers are shown in Figure 1. Table 1: Parameter values utilized in the suggested fuzzy fractional-order monkeypox disease transmission model. Parameter Values ΠH 0.0056 ΠA 0.002 λH 0.004 ξH 0.094 ΥH 0.025 E1 0.02 E2 0.01 δ1 0.005 δ2 0.001 ωH 0.04 θH 0.02 λA 0.015 ΥA 0.08 µH 0.5 µA 0.5 dA 0.2 V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 21 of 29 Figure 1: Compartmental variations in human and animal populations for different opti- mism levels m = 0,m = 0.5,m = 1using triangular fuzzy numbers. Figure 2 explores the influence of varying fractal dimensions, which represent the environmental complexity and irregular spatial structures impacting disease spread. In the human compartments, the susceptible population shows a varied depletion pattern depending on the fractal geometry, with more irregular environments slowing down ex- posure. The infected group displays delayed and more dispersed peaks in higher fractal dimensions, highlighting how environmental complexity affects contact and transmission rates. The treatment and recovered compartments are also influenced by fractal structure, with spatial irregularities potentially reducing accessibility to medical care or delaying the healing process. The protected compartment shows mixed behaviour, as its effectiveness depends on how fractal patterns impact the spread and reach of protective measures. Within the animal population, complicated fractal environments are likely to decrease the rate of exposure of susceptible animals to the disease. The infected animal compartment reflects altered peaks based on habitat structure, and the recovery in animals is also sensi- V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 22 of 29 tive to fractal properties more irregular settings may hinder movement or isolate infected hosts, thus influencing recovery timing. Figure 2: Effect of varying fractal dimensions on the dynamics of human and animal monkeypox compartments. Figure 3 illustrates the influence of varying fractional orders specifically 0.5, 0.7, and 0.9 on the dynamics of monkeypox transmission in the proposed model. The analysis reveals that incorporating fractional-order derivatives enhances the model’s capacity to capture memory and hereditary effects across human and animal compartments. In the human population, increasing the fractional order from 0.5 to 0.9 accelerates the decline of susceptible individuals, indicating that stronger memory effects at higher orders lead to faster responses to infection or preventive measures. Similarly, the infected human compartment decreases more rapidly at higher orders, such as 0.9, reflecting improved adaptability and responsiveness of the system. Treatment and recovery compartments also have more rapid and efficient transitions with increasing order, reflecting diminished V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 23 of 29 system inertia and better health outcomes. The shielded human group increases more significantly at higher orders, showing more effective immunity accumulation. In the animal compartments, lower fractional order (e.g., 0.5) will result in slower susceptibility reduction and increased infection duration because of enhanced memory retention. A sharper decline in infection and quicker recovery rate is obtained by higher orders (0.7 and 0.9), suggesting improved control and robustness in the animal population. (a) (b) Figure 3: (a) Effect of various fractional orders on disease development in human and animal populations. (a) Susceptible human in various fractional order, (b) infected human in various fractional order. (a) (b) Figure 4: (a) Treated human in various fractional order, (b) Recovered human in various fractional order. V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 24 of 29 (a) (b) Figure 5: (a) Protected human in various fractional order. (b) Susceptible animal in various fractional order. (a) (b) Figure 6: (a) Infected animal in various fractional order. (b) Recovered animal in various fractional order. In summary, the combination of fuzzy logic, fractal dimensions, and fractional calculus results in a more complete modelling framework that captures actual-world uncertainty and complexity more reliably than conventional models. This multiple-paradigm strategy increases the sensitivity and responsiveness of epidemiological models, ultimately leading to better-informed and more flexible public health decision-making under uncertainty. V. Govindan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6884 25 of 29 8. Conclusion In this research, a fuzzy fractal fractional-order model of monkeypox transmission was established that captures parameter uncertainty and memory effects. The findings identify how uncertainty in important parameters like transmission and recovery rates can signif- icantly modify outbreak size and timing and how the incorporation of memory effects indicates that history of past infection continues to influence future epidemic patterns. In capturing imprecise or incomplete data, the model more accurately represents heterogene- ity present in real populations. Epidemiologically, this suggests that neglecting uncertainty and memory could underestimate the size and duration of outbreaks. The simulations also give insights into the conditions under which monkeypox propagates rapidly, stabilizes, or disappears, in support of designing effective interventions. Practically, this means that approaches like targeted vaccination, early isolation, and adaptive public health policy should incorporate parameter uncertainty and long-term impacts explicitly in order to enhance outbreak prediction and control. Future research could involve augmenting the framework with optimal control methods and time delays and validating the model using real-world epidemiological data for enhanced predictive ability. Acknowledgements Dr .Busayamas Pimpunchat, School of Science, Department of Mathematics,King Mongkut’s Institute of Technology Ladkrabang, Bangkok, 10520, Thailand,helped in the completion of this manuscript. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Author’s Contribution All authors contributed equally to the manuscript and typed, read, and approved the final manuscript. References [1] P.D.A Qizi. Do you know about monkeypox or mpox? Multidisciplinary Journal of Science and Technology, 5(4):908–910, 2025. [2] Ana Hoxha, Steven M Kerr, Henry Laurenson-Schafer, Nikola Sklenovská, Bernadette Basuta Mirembe, et al. Mpox in children and adolescents during mul- ticountry outbreak, 2022–2023. Emerging Infectious Diseases, 29(10):2125, 2023. 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