EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5901 ISSN 1307-5543 – ejpam.com Published by New York Business Global Impact of Fractal-Fractional Dynamics on Pneumonia Transmission Modeling Sayed Saber1,2, Abdullah A. Alahmari3,∗ 1 Mathematics Department, Faculty of Science, Al-Baha University, Saudi Arabia 2 Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Egypt 3 Mathematics Department, Faculty of Science, Umm Al-Qura University, Saudi Arabia Abstract. In this study, we develop a fractal-fractional pneumonia transmission model using the Atangana-Baleanu derivative to capture long-memory effects. We analyze the existence and uniqueness of the model’s solutions and examine its stability using Hyers-Ulam criteria. Numerical simulations are conducted to explore the influence of different fractional orders on disease dynamics. The results indicate that fractional-order modeling provides a more flexible and accurate frame- work than classical integer-order models, particularly in representing population heterogeneities and intervention strategies such as vaccination and treatment. This approach enhances our under- standing of pneumonia transmission and offers valuable insights for public health decision-making. 2020 Mathematics Subject Classifications: 34A08, 92D30 Key Words and Phrases: Fractal-fractional dynamics, pneumonia transmission, Atangana- Baleanu derivative, long-memory effects 1. Introduction Pneumonia remains a major public health concern, affecting millions worldwide and leading to severe complications, particularly among young children and the elderly. Tradi- tional epidemiological models, often based on integer-order differential equations, provide valuable insights into disease dynamics but fail to capture complex real-world character- istics such as memory effects, heterogeneity in disease progression, and long-term depen- dencies in population interactions. These limitations necessitate the exploration of more advanced mathematical frameworks to enhance predictive accuracy and model realism. Pneumonia, a lung disease, affects everyone of varying severity. It is typically diagnosed through physical examinations, a patient history, and diagnostic tests. Pneumonia is char- acterized by lung inflammation, often caused by microbial or fungal infections, though cer- tain cases arise from inhaling harmful substances. While most pneumococcal toxins are ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5901 Email addresses: Sayed011258@science.bsu.edu.eg (S. Saber), aaahmari@uqu.edu.sa (A. A. Alahmari) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 2 of 23 harmless, some can cause serious health problems, such as brain damage and hearing loss. Pneumococcal meningitis, the most severe condition, primarily affects children under five years old but also impacts adults. Bacteria causing pneumococcal infections can spread through the bloodstream, and approximately 1% of children under five infected with this disease succumb to it. Additionally, pneumococcal pneumonia is a leading cause of death among the elderly, with a mortality rate of approximately 5%, rising significantly in this population. Ong’ala et al. [1] developed a mathematical model of bacteremic pneumonia in young children (under five years old) to describe the disease’s occurrence. They analyzed trans- mission rates between carriers and infected individuals using bifurcation theory and equi- librium stability analysis. Mochan et al. [2] proposed a dynamic ordinary differential equation (ODE) model to capture the immunological response to bacterial pneumonia in murine strains. Drusano et al. [3] investigated the role of granulates in preventing bacterial growth and concluded that antibiotics play no role in this specific process. Ndelwa et al. [4] analyzed pneumonia transmission dynamics, incorporating screening and medication, and explored the associated consequences. Kosasih et al. [5] employed a wavelet-based crackling detection technique to model cough sounds, facilitating rapid identification of bacterial pneumonia in young patients. César et al. [6] presented a mathematical model for pediatric asthma and pneumonia in a large population. Marchello et al. [7] demonstrated that atypical bacterial infec- tions predominantly drive the spread of respiratory illnesses such as coughing, bronchitis, and chronic obstructive pulmonary disease (COPD). Cheng et al. [8] proposed a dynam- ical mathematical model of influenza A virus secondary pneumonia (IAVSP), providing a quantitative risk assessment framework to improve COPD-related respiratory health. Kosasih et al. [9] discussed diagnostic measures for childhood pneumonia in low-income areas, identifying key contributing factors. Tilahun et al. [10, 11] introduced a co-infection model for pneumonia and typhoid, examining the interplay of curative and therapeutic strategies. Raj et al. [12] analyzed asthma and pneumonia classification in low-resource settings using mathematical proper- ties of cough sounds. Kizito et al. [13] developed a mathematical model to illustrate how microorganisms inhibit pneumonia spread, examining vaccine formulation and treatment dynamics. Mbabazi et al. [14] presented a nonlinear mathematical model describing the interplay between influenza A virus and pneumonia within the host. Tilahun et al. [15, 16] formulated a pneumonia-meningitis co-infection model using ODEs and highlighted dis- ease eradication strategies. Diah et al. [17] evaluated pneumonia risk using dynamic mathematical models. Mbabazi et al. [18] proposed a Hopf-bifurcation analysis to study pneumococcal pneumonia with temporal delays. Otoo et al. [19] examined the efficacy of bacterial pneumonia transmission models, emphasizing immunization’s role in disease pre- vention. Zephaniah et al. [20] provided graphical representations of dynamic pneumonia models. Ming et al. [21] reported the rise of Coronavirus pneumonia in Wuhan, China, confirmed by clinical testing led by Jung et al. [22]. Wafula et al. [23] developed a deter- ministic model for pneumonia-HIV co-infection, incorporating anti-pneumonia and ART therapies as optimal controls. Oluwatobi et al. [24] investigated pneumonia transmission S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 3 of 23 dynamics, fundamental reproduction numbers, and equilibrium points. Sayed Saber et al. [25, 26] proposed a fractional SVCIR model to analyze pneumonia spread. Muham- mad Naveed et al. [27] examined a delayed variant of the SCIR model and conducted its mathematical analysis. Many mathematical models have been solved by analytical and numerical methods such as VIM, ADM, etc, [28–31]. Fractional calculus has gained significant attention for its applications in engineering [32], plant epidemiology [33], mathematical biology [34], medicine [35], as well as psy- chological and life sciences [36]. Several types of fractional derivatives are used in this field, including the Riemann-Liouville, Caputo, Caputo-Fabrizio, and Atangana-Baleanu operators. These have applications in physics [37–46], viscoelasticity [47], electromag- netic wave propagation [48], quantum dynamics [49], Langevin systems [50], Diabetes [51– 56]. Fractal-fractional (FF) models describe complex, real-world phenomena effectively. They have been extensively studied in various systems, including excitation-relaxation and natural oscillatory systems [57–61]. FF models provide superior accuracy to traditional integer-order models by capturing long-memory behaviors, heavy-tailed distributions, and short-range autocorrelation dependencies. The Atangana-Baleanu fractal-fractional oper- ator, which integrates fractal geometry into the fractional calculus framework, has been instrumental in analyzing pneumonia dynamics. This operator helps identify infection mechanisms, improve treatment strategies, and gain insights into pneumonia pathogene- sis. With Atangana-Baleanu fractal-fractional operators, infectious disease spread, vaccine and drug efficacy, and public health interventions can be studied. In addition, masks are used and contact tracing is done. These operators also enable policymakers to assess intervention impacts and develop long-term mitigation strategies. The paper analyzes a mathematical model for pneumonia using fractal-fractional operators. The novelty of this study lies in extending previous results to fractal-fractional models and demonstrating enhanced accuracy and predictive capabilities. This study presents a novel fractal-fractional pneumonia transmission model incorpo- rating the Atangana-Baleanu derivative, offering a more accurate representation of disease dynamics compared to traditional integer-order and classical fractional models. The results demonstrate that our approach captures long-memory effects and population heterogeneity more effectively, leading to improved predictions of disease progression and intervention outcomes. The incorporation of Hyers-Ulam stability analysis further strengthens the theoretical foundation of the model, differentiating it from previous research. Numerical simulations confirm that varying fractional orders significantly influence disease behavior, highlighting the flexibility of the proposed framework. Future research could explore the integration of environmental and social factors into the model, as well as the applica- tion of alternative numerical methods to enhance computational efficiency. Additionally, extending the model to other infectious diseases could provide further insights into the role of fractional calculus in epidemiological modeling. The paper analyzes a mathemat- ical model for pneumonia using fractal-fractional operators. The novelty of this study lies in extending previous results to fractal-fractional models and demonstrating enhanced accuracy and predictive capabilities. In this study, we develop a fractal-fractional mathe- matical model to describe pneumonia transmission, incorporating the Atangana-Baleanu S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 4 of 23 derivative to account for long-memory effects. We analyze the existence and uniqueness of solutions, investigate the stability of equilibrium points using Hyers-Ulam criteria, and conduct numerical simulations to explore the impact of different fractional orders on dis- ease progression. Our results demonstrate that fractal-fractional models provide a more flexible and accurate framework than classical integer-order models, offering valuable in- sights into disease control strategies, including vaccination and treatment interventions. The Atangana-Baleanu fractional derivative was chosen over alternative approaches due to its superior ability to model memory effects and non-locality in real-world epidemio- logical systems. Unlike the classical Caputo and Riemann-Liouville derivatives, which rely on singular kernels, the Atangana-Baleanu derivative employs a non-singular and non-local Mittag-Leffler kernel. This allows for more accurate representation of long-memory effects and biological processes such as immune response delays and heterogeneous transmission patterns. Additionally, the Atangana-Baleanu approach ensures stability and better nu- merical convergence, making it more suitable for capturing complex disease dynamics, such as pneumonia spread, compared to other fractional derivatives. The improved flexibility and accuracy provided by this operator justify its selection for this study. To validate the main theorems of this study, a comparison with previous research is necessary. Traditional integer-order models often fail to capture the long-memory effects inherent in infectious disease dynamics. Studies using classical fractional derivatives, such as the Caputo and Riemann-Liouville approaches, provide improvements but still rely on singular kernels, which limit their applicability in modeling realistic transmission scenarios. In contrast, our model, based on the Atangana-Baleanu derivative, enhances accuracy by incorporating a non-singular, non-local kernel, allowing for more realistic disease dynamics. The stability analysis using Hyers-Ulam criteria further differentiates our work by offering a more rigorous mathematical foundation. Numerical comparisons indicate that our model provides a more flexible and robust framework for predicting disease spread, making it a valuable extension of previous epidemiological models. The remainder of this paper is structured as follows: Section shows the Preliminary Definitions. Section 3 presents the formulation of the fractal-fractional pneumonia model. Section 4 discusses the theoretical analysis of solution existence. Section 5 discusses the theoretical analysis of solution uniqueness. Section 6 discusses the stability analysis of the model. Section 7 provides numerical simulations and interpretations. Section 5 concludes the conclusion and future research directions. 2. Preliminary Definitions The fractal-fractional framework extends the SCIR model using the Atangana-Baleanu derivative. The fractal-fractional derivative on (a, b) of order 0 < ϱ2 ≤ 1 is defined as: Definition 1 ([57]). The Atangana-Baleanu fractal-fractional derivative is given by: FF−ABDϱ1,ϱ2 t ϵ(t) = B∗(ϱ1) 1− ϱ1 d dtϱ2 ∫ t 0 ϵ(s)Eϱ1 [ − ϱ1 1− ϱ1 (t− s)ϱ1 ] ds, where B∗(ϱ1) = 1− ϱ1 + ϱ1 Γ(ϱ1) and dϵ(s) dsϱ2 = limt→s ϵ(t)−ϵ(s) tϱ2−sϱ2 . S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 5 of 23 The corresponding fractal-fractional integral is: Definition 2. The fractal-fractional integral in the Atangana-Baleanu framework is: FFMIϱ1,ϱ20,t [g(t)] = ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1g(ϵ)(t− ϵ)ϱ1−1dϵ + ϱ2(1− ϱ1)t ϱ2−1g(t) B∗(ϱ1) . 3. Fractal-Fractional Pneumonia Model Extension The SCIR model, proposed by Kamaledin Abodayeh et al. in [62], is formulated as follows: dS(t) dt = Π− λ S(t)− µ S(t) + ηR(t), dC(t) dt = λϑ S(t)− (π + β + µ) C(t), d I(t) dt = λ(1− ϑ) S(t) + πC(t)− (µ+ τ + σ) I(t), dR(t) dt = β C(t) + τ I(t)− (µ+ η)R(t), (1) where λ = δ(I(t)+ϖC(t)) N . The initial conditions are: S(0) = S0 ≥ 0, C(0) = C0 ≥ 0, I(0) = I0 ≥ 0, R(0) = R0 ≥ 0. In this model, S(t) represents the susceptible population at risk of acquiring pneumo- nia, C(t) denotes carriers who may transmit the disease, I(t) represents infected individ- uals, and R(t) denotes recovered individuals. Key parameters are defined as follows: - µ: natural mortality rate, - Π: recruitment rate to the susceptible population, - π: recruitment rate into the carrier population, - ϑ: proportion of susceptibles joining the carrier population, - σ: disease-induced mortality rate, - β: recovery rate of carriers, - τ : recovery rate of infected individuals, - η: rate of treated individuals becoming susceptible again, - ϖ: rate of treated individuals becoming vaccinated, -The transmission coefficient λ represents the infection force, and p is the probability that contact results in infection. Applying the Atangana-Baleanu operator to the SCIR model yields: FF−ABDϱ1,ϱ2 t S(t) = Π− λ S(t)− µS(t) + ηR(t), FF−ABDϱ1,ϱ2 t C(t) = λϑ S(t)− (π + β + µ) C(t), FF−ABDϱ1,ϱ2 t I(t) = λ(1− ϑ) S(t) + πC(t)− (µ+ τ + σ) I(t), FF−ABDϱ1,ϱ2 t R(t) = β C(t) + τ I(t)− (µ+ η)R(t). (2) S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 6 of 23 The system can be expressed in vector form as: FF−ABDϱ1,ϱ2 0,t Ψ(t) = ϱ2t ϱ2−1W (t,Ψ(t)) = Ξ(t,Ψ(t)), where Ψ(t) = (S(t),C(t), I(t),R(t)), and Ψ(0) = (S(0),C(0), I(0),R(0)), andW (t,Ψ(t)) = (W1(t,Ψ(t)),W2(t,Ψ(t)),W3(t,Ψ(t)),W4(t,Ψ(t))), with W1(t, S) = Π− λ S−µ S+ηR, W2(t, C) = λϑ S−(π + β + µ) C, W3(t, I) = λ(1− ϑ) S+πC−(µ+ τ + σ) I, W4(t, R) = β C+τ I−(µ+ η)R . (3) 4. Existence of the model solutions By R+, we refer to the set of all positive real numbers, Ω = {(S,C, I,R) ∈ R4 + : S ≥ 0, C ≥ 0, I ≥ 0, R ≥ 0, max (| S |, |C |, | I |, |R|) ≤ N}. When the fixed point theory is applied and the fractal-fractional integral operator is utilized, the constructed model (2) can be converted into an integral equation. S(t)− S(0) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) (Π− λ S−µS+ηR) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(Π− λ S−µ S+ηR)dϵ, C(t)− C(0) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) (λϑ S−(π + β + µ) C) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(λϑS−(π + β + µ) C)dϵ, I(t)− I(0) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) (λ(1− ϑ) S+πC−(µ+ τ + σ) I) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(λ(1− ϑ) S+πC−(µ+ τ + σ) I)dϵ, R(t)− R(0) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) (β C+τ I−(µ+ η)R) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(β C+τ I−(µ+ η)R)dϵ. To be able to derive our results, we need to take into consideration the following assump- tions: (Q∗): Let assume that S(t),C(t), I(t) and R(t) are bounded satisfies ∥S(t)∥ ≤ ξ1, ∥C(t)∥ ≤ ξ2, ∥ I(t)∥ ≤ ξ1, ∥R(t)∥ ≤ ξ2. Lemma 1. The kernels Wi fulfill Lipschitz condition under the assumption (Q∗) and φi < 1, for i ∈ N4 1. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 7 of 23 Proof. ∥W1(t, S)−W1(t,S1)∥ = ∥(Π− λ S−µ S+ηR)− (Π− λ S∗∗−µS∗ + ηR)∥ ≤ (|λ|+ µ)∥(S − S1)∥ ≤ φ1∥S − S1∥, where φ1 = δ +ϖ + µ. .∥W2(t, C)−W2(t,C1))∥ = ∥(λϑ S−(π + β + µ) C)− (λϑ S−(π + β + µ) C∗∗)∥ ≤ φ2∥C − C1∥, where φ2 = µ+ β + π. ∥W3(t, I)−W3(t, I1)∥ = ∥λ(1− ϑ) S+πC−(µ+ τ + σ) I) − (λ(1− ϑ) S+πC−(µ+ τ + σ) I1)∥ ≤ φ3∥ I−I1∥, where φ3 = τ + µ+ σ. .∥W4(t, R)−W4(t,R1))∥ = ∥(β C+τ I−(µ+ η)R)− (β C+τ I−(µ+ η)R∗∗)∥ ≤ φ4∥(R−R1)∥, where φ4 = µ+ η. Thus, we have that Wi for i = 1, 2, 3, 4, satisfy the Lipschitz property and the result is accomplished. Considering W1, W2, W3, and W4 in (3) with initial conditions S(0) = S0, C(0) = C0, I(0) = I0, R(0) = R0, using fractal- fractional, S(t)− S(0) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W1(t, S(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W1(ϵ, S(ϵ))dϵ, C(t)− C(0) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W2(t,C(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W2(ϵ, C(ϵ))dϵ, I(t)− I(0) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W3(t, I(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W3(ϵ, I(ϵ))dϵ, R(t)− R(0) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W4(t,R(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W4(ϵ, R(ϵ))dϵ. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 8 of 23 The iterative scheme of model (2) should be defined as follows: Sn(t)− S(0) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W1(t,Sn−1(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W1(ϵ, Sn−1(ϵ))dϵ, Cn(t)− C(0) = ϱ2(1− ϱ1)t2ϱ2 − 1 B∗(ϱ1) W2(t, Cn−1(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W2(ϵ, Cn−1(ϵ))dϵ, In(t)− I(0) = ϱ2(1− ϱ1)t2ϱ2 − 1 B∗(ϱ1) W3(t, In−1(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W3(ϵ, In−1(ϵ))dϵ, Rn(t)− R(0) = ϱ2(1− ϱ1)t2ϱ2 − 1 B∗(ϱ1) W4(t, Rn−1(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W4(ϵ, Rn−1(ϵ))dϵ. We now examine the differences in the following manner: ∆Sn+1(t) = Sn+1(t)− Sn(t), with ∆Sn+1(t) = ϱ2(1− ϱ1)t2 − 1 B∗(ϱ1) W1(t,Sn(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W1(ϵ,Sn(ϵ))dϵ − ( ϱ2(1− ϱ1)t2 − 1 B∗(ϱ1) W1(t, Sn−1(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W1(ϵ,Sn−1(ϵ))dϵ) = ϱ2(1− ϱ1)t2 − 1 B∗(ϱ1) (W1(t,Sn(t))−W1(t, Sn−1(t))) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(W1(ϵ, Sn(ϵ))−W1(ϵ,Sn−1(ϵ)))dϵ, and ∆Cn+1(t) = Cn+1(t)− Cn(t), S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 9 of 23 with ∆Cn+1(t) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W2(t, Cn(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W2(ϵ, Cn(ϵ))dϵ − ( ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W2(t, Cn−1(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W2(ϵ, Cn−1(ϵ))dϵ) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) (W2(t, In(t))−W2(t, In−1(t))) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(W2(ϵ, Cn(ϵ))−W2(ϵ, Cn−1(ϵ)))dϵ, and ∆In+1(t) = In+1(t)− In(t), with ∆In+1(t) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W3(t, In(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W3(ϵ, In(ϵ))dϵ − ( ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W3(t, In−1(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W3(ϵ, In−1(ϵ))dϵ) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) (W3(t, In(t))−W3(t, In−1(t))) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(W3(ϵ, In(ϵ))−W3(ϵ, In−1(ϵ)))dϵ, and ∆Rn+1(t) = Rn+1(t)−Rn(t), S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 10 of 23 with ∆Rn+1(t) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W4(t, Rn(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W4(ϵ, Rn(ϵ))dϵ −( ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W4(t, Rn−1(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W4(ϵ, Rn−1(ϵ))dϵ) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) (W4(t, Rn(t))−W4(t, In−1(t))) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(W4(ϵ, Rn(ϵ))−W4(ϵ, Rn−1(ϵ)))dϵ. Using the norms, we are able to get the following result: ∥∆Sn+1(t)∥ = ∥ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) (W1(t, Sn(t))−W1(t, Sn−1(t))) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(W1(ϵ,Sn(ϵ))−W1(ϵ,Sn−1(ϵ)))dϵ∥, ∥∆Cn+1(t)∥ = ∥ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) (W2(t, Cn(t))−W2(t, Cn−1(t))) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(W2(ϵ, Cn(ϵ))−W2(ϵ, Cn−1(ϵ)))dϵ∥, ∥∆In+1(t)∥ = ∥ϱ2(1− ϱ1)t 2−1 B∗(ϱ1) (W3(t, In(t))−W3(t, In−1(t))) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(W3(ϵ, In(ϵ))−W3(ϵ, In−1(ϵ)))dϵ∥, ∥∆Rn+1(t)∥ = ∥ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) (W4(t, Rn(t))−W4(t,Sn−1(t))) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(W4(ϵ, Rn(ϵ))−W4(ϵ, Rn−1(ϵ)))dϵ∥. Lemma 2. Based on the assumption (Q∗), model (2) admits a solution whenever κ = max{φ1, φ2, φ3, φ4} < 1, for φi as defined in the proof of Lemma 1, for i = 1, 2, 3, 4. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 11 of 23 Proof. Define four functions, Z1n(t), Z2n(t), Z3n(t) and Z4n(t), as Z1n(t) = Sn+1(t)− S(t), Z2n(t) = Cn+1(t)− C(t), Z3n(t) = In+1(t)− I(t), Z4n(t) = Rn+1(t)− R(t). As a result ∥Z1n(t)∥ = ∥ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) (W1(t,Sn(t))−W1(t,S(t))) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(W1(ϵ,Sn(ϵ))−W1(ϵ, S(ϵ)))dϵ∥ ≤ ϱ2(1− ϱ1)t 2 − 1 B∗(ϱ1) ∥W1(t,Sn(t))−W1(t, S(t))∥ + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1∥W1(ϵ,Sn(ϵ))−W1(ϵ, S(ϵ))∥dϵ ≤ ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) φ1∥ Sn−S ∥ + ϱ1 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1φ1∥ Sn−S ∥dϵ ≤ ( ϱ2(1− ϱ1) B∗(ϱ1) + ϱ1ϱ2Γ(ϱ2) B∗(ϱ1)Γ(ϱ1 + ϱ2) )φ1∥ Sn−S ∥ ≤ ( ϱ2(1− ϱ1) B∗(ϱ1) + ϱ1ϱ2Γ(ϱ2) B∗(ϱ1)Γ(ϱ1 + ϱ2) )nφn 1∥ S1−S ∥, in which for φ1 < 1 and as n → ∞, we have Sn → S. Similarly ∥Z2n(t)∥ ≤ [ ϱ1ϱ2Γ(ϱ2) B∗(ϱ1)Γ (ϱ1 + ϱ2) + ϱ2 (1− ϱ1) B∗(ϱ1) ]n φn 2 ∥C1−C∥ , ∥Z3n(t)∥ ≤ [ ϱ1ϱ2Γ(ϱ2) B∗(ϱ1)Γ (ϱ1 + ϱ2) + ϱ2 (1− ϱ1) B∗(ϱ1) ]n φn 3 ∥I1− I∥ , ∥Z4n(t)∥ ≤ [ ϱ1ϱ2Γ(ϱ2) B∗(ϱ1)Γ (ϱ1 + ϱ2) + ϱ2 (1− ϱ1) B∗(ϱ1) ]n φn 4∥R1−R ∥. When n → ∞, then Zin ( tin → 0, i ∈ N4 , for φi < 1, (i = 1, . . . , 4). Ultimately, the FF pneumonia system (2) has a solution. 5. Uniqueness of the model solutions Lemma 3. Model (2) has a unique solution if the inequality is satisfied:(ϱ2(1− ϱ1) B∗(ϱ1) + ϱ1ϱ2Γ(ϱ2) B∗(ϱ1)Γ(ϱ1 + ϱ2) ) φi ≤ 1, for i = 1, 2, 3, 4. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 12 of 23 Proof. Suppose that there is another solution to the fractal-fractional epidemiological model (2), and we can compute the value for S̃(t), C̃(t), Ĩ(t), R̃(t) using that solution, so that the value for S̃(t) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W1(t, S̃(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W1(ϵ, S̃(ϵ))dϵ, C̃(t) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W2(t, C̃(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W2(ϵ, C̃(ϵ))dϵ, Ĩ(t) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W3(t, Ĩ(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W3(ϵ, Ĩ(ϵ))dϵ, R̃(t) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W4(t, R̃(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W4(ϵ, R̃(ϵ))dϵ. Here, we take the norm of the difference between S(t) and S̃(t) as the difference between pair S(t) and S̃(t) as follows: ∥ S(t)− S̃(t)∥ = ∥ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) (W1(t,S(t))−W1(t, S̃(t))) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(W1(ϵ, S(ϵ))−W1(ϵ, S̃(ϵ)))dϵ∥ ≤ ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) ∥W1(t, S(t))−W1(t, S̃(t))∥ + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1∥W1(ϵ, S(ϵ))−W1(ϵ, S̃(ϵ))∥dϵ ≤ ϱ2(1− ϱ1)t2 − 1 B∗(ϱ1) φ1∥S − S̃∥ + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1φ1∥S − S̃∥dϵ ≤ (ϱ2(1− ϱ1) B∗(ϱ1) + ϱ1ϱ2Γ(ϱ2) B∗(ϱ1)Γ(ϱ1 + ϱ2) ) φ1∥S − S̃∥. The inequality above can be further simplified in the following way:[ 1− (ϱ2(1− ϱ1) B∗(ϱ1) + ϱ1ϱ2Γ(ϱ2) B∗(ϱ1)Γ(ϱ1 + ϱ2) ) ϱ1 ] ∥S − S̃∥ ≤ 0. Based on this inequality, we can write that ∥S − S̃∥ = 0. It then follows that S = S̃. Similarly, one can obtain C = C̃, I = Ĩ, and R = R̃ by repeating this process. This completes the proof. 6. Stability Analysis When I = C = 0, one obtains the disease-free equilibrium point E0 = (Πµ , 0, 0, 0). For I∗ > 0, and taking FF−ABDϱ1,ϱ2 t S(t) = 0, FF−ABDϱ1,ϱ2 t C(t) = 0, FF−ABDϱ1,ϱ2 t I(t) = 0, S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 13 of 23 FF−ABDϱ1,ϱ2 t R(t) = 0. With ℓ1 = ϑπ+(1−ϑ)(π+β+µ) (µ+τ+σ) , one obtains the endemic equilibrium point E∗ = (S∗,C∗, I∗,R∗), with S∗ = N(π+β+µ) ℓ1+ϖ ,C∗ = (µ+η)(Π(ℓ1+ϖ)−µN(π+β+µ)) (ℓ1+ϖ)(δ(π+β+µ)(µ+η)−η(ϱ2+τℓ1)) , I∗ = ℓ1(µ+η)(Π(ℓ1+ϖ)−µN(π+β+µ)) (ℓ1+ϖ)(δ(π+β+µ)(µ+η)−η(ϱ2+τℓ1)) , R∗ = (ϱ2+τℓ1)(Π(ℓ1+ϖ)−µN(π+β+µ)) (ℓ1+ϖ)(δ(π+β+µ)(µ+η)−η(ϱ2+τℓ1)) . According to [53], the basic reproduction number of the fractal-fractional model (2) is R0 = (ϖϑ(µ+ τ + σ) + (1− ϑ)((πϖ + (π + β + µ)))δ S0 N(π + β + µ)(µ+ τ + σ) . Hence according to the Routh-Hurwitz criteria and as in [62], we have two cases: (i) The pneumonia trivial equilibrium, denoted by E0 = (0, 0, 0, 0), is stable locally asymptoticallyin if R0 < 1, but it is unstable if R0 > 1. (ii) The equilibrium point E1 = (Πµ , 0, 0, 0) of the nonlinear Atangana-Baleanu fractal- fractional SCIR model (2) is locally asymptotically stable if R0 < 1. (iii) The equilibrium point E2 = (S∗,C∗, I∗,R∗) of the nonlinear Atangana-Baleanu fractal-fractional SCIR model (2) is locally asymptotically stable if R0 > 1. Definition 3 ([63], [64]). Model (2) is Hyers-Ulam stable if vi > 0 for i ∈ N4 i , satisfying for every δi, i ∈ N4 i |S(t)− ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W1(t,S(t))− ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W1(ϵ, S(ϵ))dϵ| ≤ δ1, |C(t)− ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W2(t,C(t))− ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W2(ϵ, C(ϵ))dϵ| ≤ δ2, | I(t)− ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W3(t, I(t))− ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W3(ϵ, I(ϵ))dϵ| ≤ δ3, |R(t)− ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W4(t,R(t))− ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W4(ϵ, R(ϵ))dϵ| ≤ δ4. Several approximate solutions (S1(t),C1(t), I1(t),R1(t)) to the model problem (2) are available, with the following integral equations satisfying their terms S1(t) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W1(t,S1(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W1(ϵ,S1(ϵ))dϵ C1(t) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W2(t,C1(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W2(ϵ,C1(ϵ))dϵ, I1(t) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W1(t,S1(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W3(ϵ, I1(ϵ))dϵ, R1(t) = ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) W4(t,R1(t)) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1W4(ϵ,R1(ϵ))dϵ, such that |S(t)− S1(t)| ≤ τ1δ1, | I(t)− I1(t)| ≤ τ2δ2, |S(t)− S1(t)| ≤ τ3δ3, | I(t)− I1(t)| ≤ τ4δ4. (4) S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 14 of 23 Theorem 1. If (4) is true, then model (2) satisfies the Hyers-Ulam Stability condition. Proof. Based on the results of Lemma 1, we can conclude the following: |S(t)− S1(t)| = ∣∣∣ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) (W1(t, S(t))−W1(t,S1(t))) + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1(W1(ϵ, S(ϵ))−W1(ϵ, S1(ϵ)))dϵ ∣∣∣ ≤ ϱ2(1− ϱ1)t ϱ2−1 B∗(ϱ1) η1∥S − S1∥ + ϱ1ϱ2 B∗(ϱ1)Γ(ϱ1) ∫ t 0 ϵϱ2−1(t− ϵ)ϱ1−1η1∥S − S1∥dϵ ≤ (ϱ2(1− ϱ1) B∗(ϱ1) + ϱ1ϱ2Γ(ϱ2) B∗(η1)Γ(ϱ1 + ϱ2) ) η1∥S − S1∥. Now, let η1 = δ1, ( ϱ2(1−ϱ1) B∗(ϱ1) + ϱ1ϱ2Γ(ϱ2) B∗(ϱ1)Γ(ϱ1+ϱ2) ) ∥S − S1∥ = τ1. Then we have |S(t)− S1(t)| ≤ τ1δ1. Similarly, one obtains |C(t)− C1(t)| ≤ τ2δ2, with η2 = δ2, ( ϱ2(1−ϱ1) B∗(ϱ1) + ϱ1ϱ2Γ(ϱ2) B∗(ϱ1)Γ(ϱ1+ϱ2) ) ∥C − C1∥ = τ2. Also | I(t)− I1(t)| ≤ τ3δ3, with η3 = δ3, ( ϱ2(1−ϱ1) B∗(ϱ1) + ϱ1ϱ2Γ(ϱ2) B∗(ϱ1)Γ(ϱ1+ϱ2) ) ∥ I−I1∥ = τ3. Also |R(t)− R1(t)| ≤ τ4δ4, with η4 = δ4, ( ϱ2(1−ϱ1) B∗(ϱ1) + ϱ1ϱ2Γ(ϱ2) B∗(ϱ1)Γ(ϱ1+ϱ2) ) ∥R− R1∥ = τ4. 7. Fractal-fractional Atangana-Baleanu Scheme The Antangana-Baleanu integral gives us Ψ(t) = Ψ(0) + 1− ϱ1 ℏ(ϱ1) Ξ(t,Ψ(t)) + ϱ1 ℏ(ϱ1)Γ(ϱ1) ∫ t 0 (t− ϵ)ϱ1−1ϵϱ2−1Ξ(ϵ,Ψ(ϵ))dϵ. Replacing t by tn+1 we have Ψn+1 = Ψ(0) + 1− ϱ1 ℏ(ϱ1) Ξ (tn+1,Ψ(t)) + ϱ1 ℏ(ϱ1)Γ(ϱ1) ∫ tn+1 0 (tn+1 − ϵ)ϱ1−1 ϵϱ2−1Ξ(ϵ,Ψ(ϵ))dϵ. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 15 of 23 Application of the two-step Lagrange polynomial yields Ξ(t, (y,Ψ(t)) = (y − tϵ−1) Ξ(t, (tϵ,Ψ(tϵ)) tϵ − tϵ−1 − (y − tϵ) Ξ (tϵ−1,Ψ(tϵ−1) tϵ − tϵ−1 = Ξ(t, (tϵ,Ψ(tϵ)) (x− tϵ−1) tϵ − tϵ−1 − Ξ (tϵ−1,Ψ(tϵ−1) (y − tϵ) tϵ − tϵ−1 = Ξ(t, (tϵ,Ψ(tϵ)) (y − tϵ−1) h − Ξ (tϵ−1,Ψ(tϵ−1) (y − tϵ) h . Using the Lagrange polynomial, we obtain Ψn+1 = Ψ(0) + 1− ϱ1 ℏ(ϱ1) Ξ(t, (tn,Ψ(tn)) + ϱ1 ℏ(ϱ1)Γ(ϱ1) n∑ ϵ=1 ( Ξ(t, (tϵ,Ψ(tϵ)) h ∫ tϵ+1 tϵ (ϵ− tϵ − 1) (tn+1 − ϵ)ϱ1−1 dϵ −Ξ(t, (tϵ−1,Ψ(tϵ−1)) h ∫ tn+1 tϵ (ϵ− tϵ) (tn+1 − ϵ)ϱ1−1 dϵ ) . Now solving the integral we get Ψn+1 = Ψ(0) + 1− ϱ1 ℏ(ϱ1) Ξ(t, (tn,Ψ(tn)) + ϱ1h ϱ1 Γ (ϱ1 + 2) × n∑ ϵ=1 [Ξ(t, (tϵ,Ψ(tϵ)) ((n− ϵ+ 1)ϱ1 (n− ϵ+ 2 + ϱ1) −(n− ϵ)ϱ1 (n− ϵ+ 2 + 2ϱ1)) −Ξ(t, (tϵ−1,Ψϵ−1) ( (n− ϵ+ 1)ϱ1+1 − (n− ϵ+ 1 + ϱ1) (n− ϵ)ϱ1 )] . Now replacing the value of Ξ(y,Ψ(t)) we get Ψn+1 = Ψ(0) + ϱ2t ϱ2−1 1− ϱ1 ℏ(ϱ1) Υ (tϵ,Ψ(tϵ)) + ϱ2t ϱ2−1 ϱ1h ϱ1 Γ (ϱ1 + 2) × n∑ ϵ=1 [Υ (tϵ,Ψ(tϵ)) ((n+ 1− ϵ)ϱ1 (n− ϵ+ 2 + ϱ1) −(n− ϵ)ϱ1 (n− ϵ+ 2 + 2ϱ1)) −Υ(tϵ−1,Ψϵ−1) ( (n− ϵ+ 1)ϱ1+1 − (n− ϵ+ 1 + ϱ1) (n− ϵ)ϱ1 )] . Therefore, the above system’s numerical scheme is Sn+1 = S(0) + ϱ2t ϱ2−1 1− ϱ1 ℏ(ϱ1) W1 (tϵ,S (tϵ)) + ϱ2t ϱ2−1 ϱ1h ϱ1 Γ (ϱ1 + 2) × n∑ ϵ=1 [W1 (tϵ, S (tϵ)) ((n− ϵ+ 1)ϱ1 (n− ϵ+ 2 + ϱ1) −(n− ϵ)ϱ1 (n− ϵ+ 2 + 2ϱ1)) −W1 (tϵ−1, Sϵ−1) ( (n− ϵ+ 1)ϱ1+1 − (n− ϵ+ 1 + ϱ1) (n− ϵ)ϱ1 )] , S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 16 of 23 Cn+1 = C(0) + ϱ2t ϱ2−1 1− ϱ1 ℏ(ϱ1) W2 (tϵ,C (tϵ)) + ϱ2t ϱ2−1 ϱ1h ϱ1 Γ (ϱ1 + 2) × n∑ ϵ=1 [W2 (tϵ,C (tϵ)) ((n− ϵ+ 1)ϱ1 (n− ϵ+ 2 + ϱ1) −(n− ϵ)ϱ1 (n− ϵ+ 2 + 2ϱ1)) −W2 (tϵ−1,Cϵ−1) ( (n− ϵ+ 1)ϱ1+1 − (n− ϵ+ 1 + ϱ1) (n− ϵ)ϱ1 )] , In+1 = I(0) + ϱ2t ϱ2−1 1− ϱ1 ℏ(ϱ1) W3 (tϵ, I (tϵ)) + ϱ2t ϱ2−1 ϱ1h ϱ1 Γ (ϱ1 + 2) × n∑ ϵ=1 [W3 (tϵ, I (tϵ)) ((n− ϵ+ 1)ϱ1 (n− ϵ+ 2 + ϱ1) −(n− ϵ)ϱ1 (n− ϵ+ 2 + 2ϱ1)) −W3 (tϵ−1, Iϵ−1) ( (n− ϵ+ 1)ϱ1+1 − (n− ϵ+ 1 + ϱ1) (n− ϵ)ϱ1 )] , Rn+1 = R(0) + ϱ2t ϱ2−1 1− ϱ1 ℏ(ϱ1) W4 (tϵ,R(tϵ)) + ϱ2t ϱ2−1 ϱ1h ϱ1 Γ (ϱ1 + 2) × n∑ ϵ=1 [W4 (tϵ,R(tϵ)) ((n− ϵ+ 1)ϱ1 (n− ϵ+ 2 + ϱ1) −(n− ϵ)ϱ1 (n− ϵ+ 2 + 2ϱ1)) −W4 (tϵ−1,Rϵ−1) ( (n− ϵ+ 1)ϱ1+1 − (n− ϵ+ 1 + ϱ1) (n− ϵ)ϱ1 )] . 7.1. Numerical Simulations and Discussion Detailed computations of model (2) with a particular formulation are presented here. The numerical values are given from [62]: S(0) = 0.9, C(0) = 0.2, I(0) = 0.3, R(0) = 0.1. The following figures provide a numerical description of the model. Fig. 1 is a description of the numerical data for the S(t). The numerical simulations presented in the figures highlight the impact of fractal- fractional operators on the dynamics of pneumonia transmission. The variations in the fractional orders demonstrate the flexibility and accuracy of the fractal-fractional approach in capturing different dynamical behaviors, as compared to the classical integer-order models. In Figure 1, the dynamics of the susceptible population S(t) show an initial increase, reflecting the recruitment rate into the susceptible class, followed by stabilization. This behavior is consistent across different fractional orders, emphasizing the robustness of the model in predicting long-term stability. This figure illustrates the evolution of the susceptible population over time for different fractional orders. The population initially increases due to recruitment into the susceptible class and then stabilizes. Higher fractional orders (α ≈ 1) exhibit behavior similar to classical models, whereas lower orders introduce stronger memory effects, altering the system’s response. Figure 2 illustrates the behavior of the carrier population C(t), which exhibits an early decline before stabilizing. This decline suggests that carriers either progress to the infected class or recover due to treatment or natural immunity. The rate of change is influenced by the fractional order, indicating the significance of memory effects in disease progression. The dynamics of carriers, who are infected but asymptomatic, are displayed. The carrier population declines over time, S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 17 of 23 indicating progression to the infected class or recovery. Lower fractional orders slow down the decline, showing the influence of memory effects on disease transmission. Figure shows the infected population I(t), which initially increases before declining and stabilizing. The infected class dynamics are particularly sensitive to changes in fractional orders, as lower orders lead to faster stabilization. This result reflects the importance of fractional derivatives in modeling the memory and long-term effects of the infection process. The number of infected individuals initially rises, peaks, and then declines as people recover or die. Fractional models exhibit smoother transitions and prolonged infection durations compared to classical integer-order models. The choice of fractional order significantly affects the persistence of the disease within the population. The recovered population R(t), depicted in Figure 4, increases significantly during the initial days before reaching stability. This trend aligns with the natural progression of recovery in the population. The stabilization of all compartments over time underlines the equilibrium behavior predicted by the model. The recovered population increases in the early stages and eventually stabilizes. Higher fractional orders lead to a quicker stabilization of the recovered class, whereas lower orders suggest prolonged memory effects and delayed recovery. Figures 5–7 provide a comprehensive comparison of the solutions for all compartments under varying fractional orders. These figures demonstrate that as the fractional orders approach 1.0, the solutions converge to the classical case, validating the model’s ability to replicate traditional integer-order dynamics while offering greater flexibility. These figures provide a comparative analysis of all compartments (S(t), C(t), I(t), R(t)) under varying fractional orders. As the fractional order approaches 1, the system’s behavior aligns with classical integer-order models. Fractional models allow greater flexibility in capturing real-world disease dynamics, incorporating memory effects and population heterogeneity. Overall, the results emphasize the utility of the fractal-fractional operators in accu- rately modeling complex real-world phenomena, such as pneumonia transmission. The ability to vary the fractional orders allows for a better understanding of the disease dy- namics under different scenarios, including memory effects and heterogeneous population behaviors. These insights can be instrumental in optimizing intervention strategies, such as vaccination and treatment programs. Future work could focus on incorporating envi- ronmental and social factors, which play a critical role in the spread and treatment of pneumonia, to further enhance the model’s applicability. 8. Conclusions This study explored a pneumonia transmission model using fractal-fractional deriva- tives, demonstrating the utility of these operators in capturing disease spread dynam- ics. The numerical simulations revealed how varying the fractional orders influence the behavior of the model compartments, including susceptible, carrier, infected, and recov- ered populations. By incorporating long-memory effects and heterogeneous responses, the fractal-fractional model provides a more accurate and flexible framework than tra- ditional integer-order models. The results showed that as fractional orders approach 1, the solutions converge to the classical case. As a result, fractional calculus is compatible S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 18 of 23 0 10 20 30 40 50 60 Time 0.9 0.92 0.94 0.96 0.98 1 1.02 S (t ) fo r o rd e rs 1 , 0 .9 5 , 0 .9 , 0 .8 5 S at order 1 S at order 0.95 S at order 0.90 S at order 0.85 S at order 0.8 Figure 1: Susceptible class for the different fractional orders of the model (2). 0 10 20 30 40 50 60 Time 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 C (t ) fo r o rd e rs 1 , 0 .9 5 , 0 .9 , 0 .8 5 C at order 1 C at order 0.95 C at order 0.90 C at order 0.85 C at order 0.8 Figure 2: C(t) class for the different fractional orders of the model (2). 0 10 20 30 40 50 60 Time 0 0.05 0.1 0.15 0.2 0.25 0.3 I( t) f o r o rd e rs 1 , 0 .9 5 , 0 .9 , 0 .8 5 I at order 1 I at order 0.95 I at order 0.90 I at order 0.85 I at order 0.8 Figure 3: Infected class for the different fractional orders of the model (2). 0 10 20 30 40 50 60 Time 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 R (t ) fo r o rd e rs 1 , 0 .9 5 , 0 .9 , 0 .8 5 R at order 1 R at order 0.95 R at order 0.90 R at order 0.85 R at order 0.8 Figure 4: R(t) class for the different fractional orders of the model (2). with conventional approaches. The stabilization of all compartments over time indicates equilibrium points and supports the model’s effectiveness in predicting disease dynamics. This study emphasizes the importance of fractional-order modeling in understanding real-world phenomena like pneumonia transmission. By adjusting fractional orders, the model can account for diverse population behaviors. This offers valuable insights into the effects of interventions such as vaccination and treatment strategies. Future work should consider environmental and social factors, such as air quality, temperature, and population density. This will further refine the model and enhance its applicability. The proposed method could also be compared with other numerical schemes for a deeper understand- ing of its robustness and computational efficiency. The fractal-fractional approach has significant potential for improving disease modeling and public health decisions. S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 19 of 23 0 10 20 30 40 50 60 Time 0 0.2 0.4 0.6 0.8 1 1.2 N u m e ri c a l s o lu ti o n f o r o rd e r = 1 .0 . S at order =1 C at order =1 I at order =1 R at order =1 Figure 5: Time series of S(t), C(t), I(t), and R(t) us- ing fractal fractional methods in the sense of Atangana- Baleanu operators for ϱ1 = 1, ϱ2 = 1. 0 10 20 30 40 50 60 Time 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 N u m e ri c a l s o lu ti o n f o r o rd e r = 0 .9 5 . S at order = 0.95 C at order = 0.95 I at order = 0.95 R at order = 0.95 Figure 6: Time series of S(t), C(t), I(t), and R(t) us- ing fractal fractional methods in the sense of Atangana- Baleanu operators for ϱ1 = 0.95, ϱ2 = 0.95. Π the recruitment rate into susceptible population per capita 0.5 δ the rate of transmission 2.5 ϖ represents the rate of treated individuals having vaccinated 0.1124 µ the natural mortality rate of individuals per capita 0.5 η Average time symptomatic infectious have symptoms 0.00641 m the recovery rate of carriers per capita 0.515 π the rate of developing symptoms by carriers 0.7096 ϑ the proportion of susceptible individuals who joins the carriers 0.563 τ the recovery rate of individuals who are infected of Pneumonia per capita 0.641 σ Birth rate per capita of the human population affected by disease 0.53 Table 1: The values of parameters and initial populations in this study are obtained from the WHO situation report (the National Health Commission of the Republic of China) presented in [62]. https://doi.org/10.32604/cmc.2022.020732 Acknowledgment The authors extend their appreciation to Umm Al-Qura University, Saudi Arabia for funding this research work through grant number: 25UQU4220004GSSR02. Funding This research work was funded by Umm Al-Qura University, Saudi Arabia under grant number: 25UQU4220004GSSR02. Data Availability All data is included in the manuscript. https://doi.org/10.32604/cmc.2022.020732 S. Saber, A. A. Alahmari / Eur. J. Pure Appl. Math, 18 (2) (2025), 5901 20 of 23 References [1] J. Ong’ala, P. Oleche, and J.Y.T. Mugisha. Mathematical model for pneumonia dynamics with carriers. Int. J. Math. Anal., 7(50):2457–2473, 2013. [2] E. Mochan, D. Swigon, G. Ermentrout, S. Luken, and G.A. Clermont. Mathematical model of intrahost pneumococcal pneumonia infection dynamics in murine strains. J. Theor. Biol., 353:44–54, 2014. [3] G.L. Drusano, W. Liu, S. Fikes, R. Cirz, N. Robbins, S. Kurhanewicz, and A. Louie. Interaction of drug- and granulocyte-mediated killing of pseudomonas aeruginosa in a murine pneumonia model. J. Infect. Dis., 210(8):1319–1324, 2014. [4] E.J. Ndelwa, M. Kgosimore, E.S. Massawe, and L. Namkinga. 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