EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6682 ISSN 1307-5543 – ejpam.com Published by New York Business Global Modeling Dengue Dynamics Using Classical, Caputo Fractional, and Fractal Derivatives Rania Saadeh1,∗, Naseam Al-kuleab2, Maha S. Al Soudi3, Khadeeja A. A. Helal4, Aymen Imam4, Ibrahim Elshamy4,5, Suliman Jamiel M. Abdalla6 1 Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun, Jordan 2 Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia 3 Department of Basic Scientific Sciences, Applied Science Private University, Amman 11931, Jordan 4 Department of Mathematics, Faculty of Science, Al-Baha University, AlBaha, Saudi Arabia 5 Higher Institute of Engineering Technology, Almanzala, Egypt 6 Department of Mathematics and Applied Mathematics, University of Johannesburg, PO Box 524, Auckland Park 2006, Johannesburg, South Africa Abstract. Dengue is a major problem in Sudan. Dengue spreads in Gedaref, Sudan. In this work, we compare classical, Caputo fractional derivative, and fractal SIAR-SI pandemic models. We investigate the disease-free and endemic equilibrium stability, two important stable states of the proposed dynamical model. This study identifies the factors guiding a disease’s persistence or extinction in the population. The model is fitted using real data from Gedaref, Sudan, and the Markov Chain Monte Carlo method is used for estimating parameters. The computed R0 = 3.27, indicating that the disease is endemic. The classical model fits the real data better than the other methods. The fractional model, involving Caputo derivatives, explains memory effects. Thus, they provide a more realistic formulation of disease transmission. The fractal model with Hausdorff derivatives offers a new direction for complicated transmission dynamics and is worthy of further research. The fractional and fractal models outperform the classical model by treating long-term dependency with fractional derivatives and disease spread accurately with fractal operators. 2020 Mathematics Subject Classifications: 26A33, 34A08, 92D30 Key Words and Phrases: Dengue fever, mathematical modeling, fractional derivatives, fractals, epidemiology, forecasting ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6682 Email addresses: r.saadeh@bau.edu.jo (R. Saadeh), nkuleab@ksu.edu.sa (N. Al-kuleab), m alsoudi@asu.edu.jo (M. S. Al Soudi), khilal@bu.edu.sa (K. A. A. Helal), aimam@bu.edu.sa (A. Imam), elshamyii@hotmail.com (I. Elshamy), jamiel@aims.ac.za (S. J. M. Abdalla) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 2 of 21 1. Introduction Dengue fever is a viral infection spread by mosquitoes. The infection usually occurs in warm and subtropical regions with a high population density. The virus has four serotypes: DENV-1, DENV-2, DENV-3, and DENV-4. Dengue fever can appear in various ways, ranging from minor signs such as fever, headache, and joint pain to severe complications such as dengue hemorrhagic fever and dengue shock syndrome [1–5]. Dengue fever is a worldwide health issue. It causes millions of reported cases annually in Asia, Africa, and South America [6]. Weather factors like temperature, humidity, and rainfall influence viral spread due to their direct effect on the breeding of mosquitoes and virus reproduction. Dengue fever burst into epidemic proportions in eastern Sudan in 2023, specifically in Gedaref state, with morbidity and mortality [7]. The extremely high dengue fever epidemics in Gedaref are due to the interaction between social, environmental, and economic determinants. Climate change and the intensification of seasonal rains create favorable breeding conditions for mosquitoes. However, water courses and wetland areas represent breeding sites that contribute to the spread of the epidemic. Inadequate health- care impacts the timely detection and care of new cases, increasing outbreaks’ severity. These linked determinants explain the region’s chronic and endemic dengue fever epidemic. In 2023, Gedaref State recorded one of Sudan’s highest dengue fever incidence rates. Thus, it is critical to investigate epidemiological characteristics of the disease spread in the region [8–11]. Mathematical models are of central importance to comprehending and predicting infectious disease transmission to enable public health authorities to design effective disease surveillance systems, respond to outbreaks, and formulate policy. The models utilize accurate data and computer methods to understand infection transmission, intervention effectiveness, and environmental factors that affect disease transmission [12– 14]. Scientists have devised various modelling methods, each with advantages and disadvan- tages. Classical Susceptible-Infected-Recovered (SIR) and Susceptible-Exposed-Infected- Recovered (SEIR) are among the most commonly used integer-order models. While pro- viding a straightforward framework for understanding disease spread, these models fail to capture the complexity of actual epidemics [15]. Fractional and fractal models emerged and are used to address these limitations. These models incorporate memory effects and improve prediction accuracy [16, 17]. Fractional-order models enhance conventional frameworks by incorporating non-integer derivatives. They are of special use in long-term epidemic forecasting, particularly for dis- eases with latency in immune response or with long incubation periods [16]. Fractional- order models were found to increase accuracy in prediction, particularly for vector-borne disease, which is influenced by changing mosquito populations and environmental condi- tions [18, 19]. Studies on dengue showed that fractional order predicted epidemic patterns more precisely than traditional models [20]. Fractional models, however, cannot account for time variations [21–23]. Fractal models account for time variations, assuming that dis- ease transmission occurs in irregular patterns influenced by population density and vector mobility. Expanding epidemiological modeling into non-Euclidean space allows fractal R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 3 of 21 models to represent disease transmission in complex ecosystems accurately [24]. Fractal models improved epidemic predictions, particularly in densely populated urban areas where human mobility and clustering are critical elements in disease spread [23]. Fractal surpassed traditional models in forecasting infection, as indicated by the study of the influenza epidemic that showed more congruence with empirical data [25]. The advantages of each modelling technique, integer, fractional, and fractal, are dis- tinct. The integer models give a simple and practical foundation, whereas fractional models increase accuracy by incorporating memory effects, and fractal models introduce consciousness to improve predictions [26]. Due to these complementary qualities, there is an increasing interest in developing better and more flexible epidemiological models by incorporating integer, fractional, and fractal components in hybrid modelling strategies. In this study, we aim to compare the performance of three modelling approaches in fore- casting dengue fever outbreaks in Gedaref, Sudan, by integrating actual epidemiological data with computational approaches to determine the most accurate model. Specifically, we explore how the role of memory effects in fractional models and spatial heterogeneity in fractal models influence disease transmission dynamics and outbreak generation. Fur- thermore, the study assesses the predictability and applicability of integer, fractional, and fractal models in enhancing disease control interventions. This research has six chapters, starting with the Introduction, which discusses dengue fever and its public health impact in Gedaref, Sudan, which was chosen for the investi- gation. In the Classical describe chapter, integer-order differential equations have been utilized to describe disease transmission. Theoretical analysis analyzes the model’s be- haviour and stability to understand the disease’s dynamics. The chapter on fractional and fractal models is called Non-Integer Calculus. In the Discussion, classical, fractional, and fractal models are compared to actual data. Finally, the Conclusion summarises key findings and emphasizes the importance of sophisticated modeling in predicting illnesses. 2. Classical model Let N1 and N2 denote the total populations of humans and mosquitoes, respectively. Dengue, transmitted by Aedes aegypti, has four distinct serotypes (DENV-1 to DENV-4). Infection grants serotype-specific immunity, but reinfection with a different serotype in- creases the risk of severe complications due to Antibody-Dependent Enhancement (ADE). The human population is divided into: S1 (susceptible), I1 (infected and infectious), A1 (partially immune), and R1 (fully immune). For mosquitoes: S2 (susceptible) and I2 (infected and infectious). We assume constant total populations for humans and mosquitoes, i.e., N1 = S1 + I1 +A1 +R1, N2 = S2 + I2, With natural births balancing natural deaths. The parameters of the model in Table (1) and Figure((1)). The model equations are described below, where β1 represents the transmission rate at which an infected mosquito infects susceptible humans. R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 4 of 21 dS1 dt = µ1N1 − β1 S1I2 N1 − µ1S1, dI1 dt = β1I2 N1 S1 + δ1A1 − (µ1 + γ)I1, dA1 dt = pγI1 − (µ1 + δ1)A1, dR1 dt = (1− p)γI1 − µ1R1. dS2 dt = µ2N2 − β2 S2I1 N1 − µ2S2, dI2 dt = β2 S2I1 N1 − µ2I2. (1) System (1) is naturally appended with initial conditions: S1(0) = N1−I1(0)−A1(0)−R1(0), I1(0) = 8, A1(0) = 0, R1(0) = 0, S2(0) = N2−I2(0), I2(0) = 10. Figure 1: Enhanced schematic of Dengue transmission dynamics. R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 5 of 21 Table 1: Model parameters, values, and sources. Parameter Interpretation Value Source µ1 Birth/death rate (humans). 0.0000391 Fixed µ2 Birth/death rate (mosquitoes). 0.1 Fixed β1 Mosquito to human transmission rate. 0.1003 Fitted β2 Human to mosquito transmission rate. 0.1001 Fitted γ Human recovery rate. 0.2101 Fitted p Proportion of partial immunity. 0.8452 Fitted δ1 Reinfection rate. 0.0250 Fitted N1 Total humans. 2,208,385 Fixed N2 Total mosquitoes. 3.5×N1 Assumed Precise estimation of model parameters is crucial for adequately delineating dengue transmission patterns and forecasting capacities. This study applied the Markov Chain Monte Carlo (MCMC) technique [27, 28] to estimate parameter values derived from daily data about the dengue outbreak in Gedaref State, eastern Sudan, reported in 2023. The data were obtained from the reports of the Ministry of Health in Gedaref State, Sudan. The use of MCMC provides excellent accuracy in obtaining parameter values, enabling a more reliable assessment of dengue management efforts. Table 1 presents the estimated values. 3. Analysis of model In this part, we verify that all state variables are nonnegative and bounded for all time. We also obtain the model equilibrium points and compute the basic reproduction number. Further, we explore bifurcation and stability properties. 3.1. Model non-negativity and boundedness Proposition 1. Let S1,0, I1,0, A1,0, R1,0, S2,0, I2,0 be non-negative initial conditions. Then the solution of (1) is non-negative for all t ≥ 0. Proof. For y ∈ {S1, I1, A1, R1, S2, I2}, dy dt ∣∣∣ y=0 ≥ 0 ensures non-negativity. Proposition 2. The solution of (1) is bounded for all t. R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 6 of 21 Proof. If N1 and N2 are bounded, all state variables remain bounded. Thus, system (1) remains in the compact biologically feasible set: Ω = { (S1(t), I1(t), A1(t), R1(t), S2(t), I2(t)) ⊂ R6 + : N1(t) +N2(t) = N1,0 +N2,0 } . 3.2. Model equilibria and basic reproduction number As the total population N1(t) = S1(t)+ I1(t)+A1(t)+R1(t) and N2(t) = S2(t)+ I2(t) are constants, System (1) simplifies to: dS1 dt = µ1N1 − β1 N1 S1I2 − µ1S1, dI1 dt = β1 N1 S1I2 − (µ1 + γ)I1 + δ1A1, dA1 dt = pγI1 − (µ1 + δ1)A1, dI2 dt = β2 N1 (N2 − I2)I1 − µ2I2. (2) This system has two equilibrium points: the disease-free equilibrium (DFE) P0 = (N1, 0, 0, 0) and the endemic equilibrium (EE) P ∗ = (S∗ 1 , I ∗ 1 , A ∗ 1, S ∗ 2 , I ∗ 2 ) with: S∗ 1 = N1(N1N −1 2 µ1R̂0 + β1) R̂0(N1N −1 2 µ1 + β1) , I∗1 = ( µ1N1(µ1 + δ1)β1 (µ1 + γ)(µ1 + δ1)− δ1pγ )( R̂0 − 1 R̂0(N1N −1 2 µ1 + β1) ) , A∗ 1 = ( µ1N1β1pγ (µ1 + γ)(µ1 + δ1)− δ1pγ )( R̂0 − 1 R̂0(N1N −1 2 µ1 + β1) ) , I∗2 = µ1N1(R̂0 − 1) N1N −1 2 µ1R̂0 + β1 . (3) where R̂0 is: R̂0 = β1β2N2(µ1 + δ1) N1µ2[(µ1 + γ)(µ1 + δ1)− δ1pγ] . (4) We calculate the basic reproductive number, R0, using the next-generation matrix method. The Jacobian matrices are: F =  0 0 β1 0 0 0 β2N2 N1 0 0  , V = µ1 + γ −δ1 0 −pγ µ1 + δ1 0 0 0 µ2  . The basic reproduction number is the dominant eigenvalue of FV −1: R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 7 of 21 R0 = √ R̂0. (5) 3.3. Stability Analysis Theorem 1. (i) If R0 < 1, the disease-free equilibrium (DFE) of System (2) is locally asymptotically stable. (ii) If R0 > 1, the DFE is unstable. Proof. The Jacobian matrix of System (2), denoted by J , is expressed as: J =  − β1 N1 I2 − µ1 0 0 − β1 N1 S1 β1 N1 I2 −(µ1 + γ) δ1 β1 N1 S1 0 pγ −(µ1 + δ1) 0 0 β2 N1 (N2 − I2) 0 − β2 N1 I1 − µ2  . At the disease-free equilibrium P0, where I1 = I2 = 0, S1 = N1, and S2 = N2, the Jacobian matrix simplifies to: JP0 =  −µ1 0 0 −β1 0 −(µ1 + γ) δ1 β1 0 pγ −(µ1 + δ1) 0 0 β2N2 N1 0 −µ2  . The eigenvalues of JP0 are {λ1, λ2, λ3, λ4}, where: λ1 = −µ1, λ2 = −(µ1 + γ). The remaining eigenvalues, λ3 and λ4, are determined by solving the following char- acteristic equations: 1. For λ3, we have: λ2 3 + (γ + 2µ1 + δ1)λ3 + γδ1(1− p) + γµ1 + µ1(δ1 + µ1) = 0. R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 8 of 21 2. For λ4, we solve: λ3 4 + (δ1 + γ + 2µ1 + µ2)λ 2 4 + [ µ2(δ1 + µ1) + (δ1 + µ1)(γ + µ1)− pγδ1 + µ2(µ1 + γ)− N2β1β2 N1 ] λ4 + 1 µ2 ((δ1 + µ1)(γ + µ1)− pγδ1) [ 1− R̂0 ] = 0. To analyze the stability of the DFE, we examine the condition.R0 < 1. From the definition of R0, R0 = √ R̂0, where: R̂0 = β1β2N2(µ1 + δ1) N1µ2[(µ1 + γ)(µ1 + δ1)− δ1pγ] . If R0 < 1, it follows that β1β2N2 N1 < µ2(µ1 + γ). This ensures that all coefficients of the characteristic equations are positive, and using the Routh-Hurwitz criterion, all eigenvalues have negative real parts. Thus, the DFE is locally asymptotically stable when R0 < 1. In contrast, if R0 > 1, according to Descartes’ sign rule, the characteristic equation for λ4 has at least one positive root. It causes the DFE to be unstable. Theorem 2. The disease-free equilibrium (DFE) is globally asymptotically stable when R0 < 1. Proof. To establish the global stability of the DFE, we analyze the subsystem corre- sponding to the disease-free state variables (S1, A1, R1, S2) and rewrite the full system (1) in the following form: dX dt = P (X,Y ), dY dt = G(X,Y ), with G(X, 0) = 0, (6) where X = (S1, A1, R1, S2) ∈ R4 + and Y = (I1, I2) ∈ R2 +. We use Castillo-Chavez’s method to determine the DFE’s global stability. (P0). Specif- ically, we define: Ĝ(X,Y ) = A3|P0Y −G(X,Y ), (7) R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 9 of 21 where G(X,Y ) is given by G(X,Y ) = β1S1I2 N1 − (µ1 + γ)I1 β2S2I1 N1 − µ2I2  . Here A3|P0 is the Jacobian matrix G(X,Y ) evaluated at the DFE P0. The Jacobian A3|P0 is expressed as: A3|P0 = −(µ1 + γ) β1 β2N2 N1 −µ2  . Substituting G(X,Y ) and A3|P0 into equation (7), we obtain: Ĝ(X,Y ) =  β1 N1 I2(N1 − S1) β2 N1 I1(N2 − S2)  . Consequently, Ĝ(X,Y ) ≥ 0, (X,Y ) ∈ Ω. Therefore, P0 is globally asymptotically stable when R0 < 1. Theorem 3. The endemic equilibrium point P ∗ of system (1) is globally asymptotically stable. Proof. First, we normalize state variables: s1 = S1 N1 , i1 = I1 N1 , a1 = A1 N1 , r1 = R1 N1 , s2 = S2 N2 , i2 = I2 N2 . Under this normalization, the total human and mosquito populations satisfy: s1 + i1 + a1 + r1 = 1, s2 + i2 = 1. Furthermore, the endemic equilibrium P ∗ corresponds to the equilibrium point (s∗1, i ∗ 1, a ∗ 1, r ∗ 1, s ∗ 2, i ∗ 2). To prove global stability, we construct the following Lyapunov function: V (s1, i1, a1, r1, s2, i2) = 1 2 [ (s1 − s∗1) 2 + (i1 − i∗1) 2 + (a1 − a∗1) 2 + (r1 − r∗1) 2 + (s2 − s∗2) 2 + (i2 − i∗2) 2 ] . The derivative of V in (1) is: dV dt = [ (s1 − s∗1) ds1 dt + (i1 − i∗1) di1 dt + (a1 − a∗1) da1 dt + (r1 − r∗1) dr1 dt + (s2 − s∗2) ds2 dt + (i2 − i∗2) di2 dt ] . R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 10 of 21 Since all normalized states are in [0, 1], we obtain dV dt = [ (s1 − s∗1) ds1 dt + (i1 − i∗1) di1 dt + (a1 − a∗1) da1 dt + (r1 − r∗1) dr1 dt + (s2 − s∗2) ds2 dt + (i2 − i∗2) di2 dt ] ≤ d dt (s1 + i1 + a1 + r1 + s2 + i2) = 0. Thus dV dt ≤ 0. Furthermore, it dV dt = 0 occurs only at P ∗. By LaSalle’s Invariance Principle, all trajectories asymptotically approach. P ∗. Thus, the endemic equilibrium is globally asymptotically stable. 4. Non-integer calculus Research of differential operators of fractional orders dates back to 1695 when L’Hospital asked Leibniz the meaning of duy dxu for u = 1/2 [29]. This led to the investigation of frac- tional calculus, and mathematicians like Euler, Lagrange, and Fourier took an interest in it [30]. The fractional calculus was later used in various fields such as electrical spectroscopy, wave propagation, quantum mechanics, fluid mechanics, and epidemiology [31–34]. Fractal derivatives and hybrid fractal-fractional models were also studied [35–37]. In this paper, the effects of noninteger-order derivatives on the dengue fever epidemic model are considered. 4.1. Fractional operators Here, we introduce fundamental concepts of fractional calculus [38–41], which will be used to develop our analysis. Due to the non-linearity of the epidemiological equations, obtaining analytical solutions is usually very difficult or, in most cases, impossible. To overcome this challenge,we present the system 1 under the Caputo operator: CDα t S1 = µ1N1 − β1 S1I2 N1 − µ1S1, CDα t I1 = β1I2 N1 S1 + δ1A1 − (µ1 + γ)I1, CDα t A1 = pγI1 − (µ1 + δ1)A1, CDα t R1 = (1− p)γI1 − µ1R1, CDα t S2 = µ2N2 − β2 S2I1 N1 − µ2S2, CDα t I2 = β2 S2I1 N1 − µ2I2. (8) Now, we present a numerical method for solving fractional differential equations, re- ferred to as the fractional Euler method. The method is based on the following formula: R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 11 of 21 yk+1 = y0 + hα Γ(α+ 1) k∑ j=0 [(k + 1− j)α − (k − j)α] f(tj , yj), (9) When α = 1, Equation 9 reduces to the Euler method for ODEs. This scheme is used for solving fractional differential equations with h = 0.01 in simulations. Convergence is ensured if f(t, y) satisfies the Lipschitz condition, guaranteeing uniqueness and stability of the solution. (a) S1 (b) I1 (c) A1 (d) R1 (e) S2 (f) I2 Figure 2: Fractional-order dynamics of the compartmental variables S1(t), I1(t), A1(t), R1(t), S2(t), and I2(t) under varying (α). The figure illustrates the fractional-order dynamics of model () compartments under varying fractional-order parameters (α = 0.90, 0.91, 0.92, 0.94). Fractional-order models effectively capture memory effects and the long-term dependence of disease progression, making them superior to classical integer-order models in representing dengue transmission dynamics. The (S1) progressively diminishes over time as a result of infection transmission. Lower values of α lead to an accelerated decline, indicating that fractional derivatives affect the rate of disease transmission by integrating memory effects. The (I1) demonstrates an exponential-like growth, where lower α values result in a greater peak and a more gradual decay. This behaviour illustrates the non-local effects and varied characteristics of dengue transmission, as fractional models consider changes in infection duration. The (A1) exhibits a comparable growth trajectory to I1, albeit with diminished inten- sity. The varying α values affect the trajectory, suggesting that fractional-order models more effectively represent the delay in disease progression and concealed carriers. R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 12 of 21 4.2. Fractal operator In this section, we present the concept of fractal derivatives and their significance in dengue fever spread modeling. Traditional differential operators fail to capture the com- plexities in disease dynamics in heterogeneous environments. Unlike fractional derivatives, which are typically defined in terms of convolution integrals, the fractal derivative is local, thus being computationally efficient and structurally distinct [42–44]. It is formulated in a non-Euclidean fractal metric space [45, 46], making it possible to describe the irregularity and self-similarity of real phenomena. The novelty of this mathematical strategy provides an improved description of dengue fever transmission, particularly in situations where spa- tial heterogeneity is of the utmost importance. The application of fractal derivatives to epidemiological modeling provides an improved understanding of the transmission of the disease and increases predictive capacity. We present system 1 with a fractal operator: dS1 dt = αtα−1µ1N1 − β1 S1I2 N1 − µ1S1, dI1 dt = αtα−1β1I2 N1 S1 + δ1A1 − (µ1 + γ)I1, dA1 dt = αtα−1pγI1 − (µ1 + δ1)A1, dR1 dt = αtα−1(1− p)γI1 − µ1R1. dS2 dt = αtα−1µ2N2 − β2 S2I1 N1 − µ2S2, dI2 dt = αtα−1β2 S2I1 N1 − µ2I2. (10) Now, we apply the 4th-order Runge-Kutta method to solve the fractal system and perform simulations using RK4. R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 13 of 21 (a) S1 (b) I1 (c) A1 (d) R1 (e) S2 (f) I2 Figure 3: Fractal dynamics of the compartmental variables S1(t), I1(t), A1(t), R1(t), S2(t), and I2(t) under varying fractal dimensions (τ). Figure 2 shows how the model compartments (system 1) changed over time when the fractal values changed (α = 0.90, 0.91, 0.92, 0.94). The human susceptible population (S1) slowly decreases over time as more people become infected. The rate of decline changes with lower values of alpha, where lower values of fractal order accelerate the rate of decline. The asymmetric pattern shows the importance of fractal derivatives in modeling complex disease spread patterns in the real world. The number of infected people (I1) grows at an exponential rate over time, reaching a peak before starting to fall. The quantity I1 increases exponentially over time, attaining a maximum before declining thereafter. Lower α values result in a higher peak and a slower decay, suggesting that fractal dynamics contribute to disease persistence by capturing delayed recovery and spatially heterogeneous transmission. Over time, the asymptomatic human population (A1) follows a growth trend similar to I1 but at a lower intensity. With time, the recovered population (R1) gradually increases. The recovery trajectory depends on α, where lower values correspond to faster accumulation of immunity, likely due to shorter infectious periods and delayed waning immunity, as captured by fractal operators. Based on this behavior, it seems that fractal models are a better way to show immune responses, especially when there is partial immunity and infections keep happening. The susceptible vector population (S2) declines over time, similar to S1. However, its trajectory changes, suggesting that reduced sensitivity or resistance to reinfection influ- ences secondary exposure risks. The infected vector population (I2) exhibits a multiwave pattern, suggesting delayed R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 14 of 21 secondary infections or reinfections. The fractal-order operator modifies the transmission dynamics, incorporating erratic disease spread and persistent outbreaks. This means that fractal models are a better way to show the chances of getting dengue again and secondary waves of the disease, which are very important for understanding how long-term epidemics work. 5. Discussion In this section, we conduct a comparative analysis of three different dengue models: classical, fractal, and fractional models. The predictions obtained from these models are compared with accurate data for the 2023 dengue epidemic in Gedaref State, Eastern Sudan. The classical model is used as a baseline, whereas the fractal and fractional models offer more complex and general frameworks that take into account memory effects in the spread of epidemics. Compared to the classical model, the fractal and fractional models generally exhibit superior predictive abilities, with the fractional model offering the best fit to actual data. Figures 4, 5, and 6 provide more detailed insights into the evolution of the susceptible (S1, S2) and infected (I1, I2) populations (humans and mosquitoes) over a period of 200 days. Figure 4, The left panel (a) illustrates the decline in the susceptible human pop- ulation (S1), while the right panel (b) depicts the corresponding increase in the infected population (I1). The results indicate that individuals become infected more rapidly in the fractional and fractal models compared to the classical model. After day 100, we observe a notable reduction in susceptibility, followed by a slow decline in infection due to recovery. Figure 5,This figure illustrates the dynamics of the susceptible (S2) and infected (I2) mosquito populations. The fractional and fractal models predict a more significant reduction in the susceptible mosquito population and a rapid increase in the infected population. This finding confirms the ability of these models to capture realistic disease progression patterns. Figure 6,This figure compares A1 and R1, which represent asymp- tomatic infections and recovered individuals. Both quantities increase over time, with the fractional model predicting a slightly faster growth rate. The results show that the fractal and fractional models better show how complicated and nonlinear dengue transmission is. These models are better than the classical model because they give a more accurate picture of how the disease spreads between susceptible and infected groups and make predicting long-term epidemics easier. R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 15 of 21 (a) S1 (b) I1 Figure 4: Comparison of fractal and fractional dynamics for S1(t) and I1(t) over time. (a) S2 (b) I2 Figure 5: Comparison of fractal and fractional dynamics for S2(t) and I2(t) over time.. (a) A1 (b) R1 Figure 6: Comparison of fractal and fractional dynamics for A1(t) and R1(t) over time Figure 7 illustrates the temporal evolution of the infected human population as pre- dicted by the three models, using precise dengue data. The curves reveal marked discrep- ancies in each model’s ability to accurately depict the epidemic’s trajectory. The fractional model (blue line) captures non-linear and varied patterns of disease R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 16 of 21 transmission. It closely matches the data in later epidemic stages, indicating that frac- tional derivatives effectively capture long-memory effects and fluctuations in disease trans- mission. The fractal model (red line) captures disease dynamics moderately and lies be- tween the classical and fractional models. By incorporating memory-dependent transmis- sion effects, this model leads to smoother and more gradual disease progression compared to the classical model. The classical model (black line) significantly underestimates the actual cases and exhibits slower infection progression. The results indicate that this model fails to capture the long-term dynamics of the epidemic, especially when secondary waves and delayed transmission effects are considered. The comparative study emphasizes the inadequacy of the classical model in the cor- rect forecasting of dengue fever outbreaks since it does not take into account the memory effects, spatial heterogeneity, and time-delay transmission dynamics. However, the fractal and fractional models accurately outline the disease transmission process, the best among them being the fractional model, to actual epidemiological data. With the inclusion of long-memory effects and nonlinear transmission patterns, predictive power is introduced by the fractional model, and thus it can be a useful tool for epidemic prediction as well as public health intervention planning. The fractal model also introduces robustness to disease modeling by the inclusion of spatial heterogeneity, which is applicable in describing heterogeneous transmission patterns. Such sophisticated models provide more flexibility and reliability for improved epidemic control interventions and policy planning effective- ness in provinces such as Gedaref. Figure 7: Comparison between fractal, fractional, and classical models against actual data for I1(t) over time. R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 17 of 21 6. Conclusion In this article, we have modeled the spread of dengue by three different SIAR-SI epi- demic models based on classical, fractional (Caputo), and fractal (Hausdorff) derivatives. To solve these models numerically, we employed the classical Euler method to solve the standard model, the fractional Euler method to solve the fractional model, and the fourth- order Runge-Kutta method to solve the fractal model. We also examined the stability of both the disease-free and endemic equilibrium states, which reveals the conditions under which the disease will become extinct or persistent. The model was calibrated using real data from Gedaref, Sudan, and parameters were estimated using the Markov Chain Monte Carlo method. The basic reproduction number for 2023 was estimated as mathcalR0 = 3.27, confirming the endemicity of the disease. Among the three models, the classical model best fits real data. The fractional model, in the form of Caputo deriva- tives, could include memory effects and hence give a more realistic representation of disease spread. The fractal model, in the form of Hausdorff derivatives, offered a new insight into complex transmission dynamics. For further research, some directions of extension are available. The extension of the fractal model into time-evolving parameters will make it even more realistic in its ability to cope with real situations. The integration of stochas- tic influence into fractional and fractal models will provide a better representation of the diffusion of uncertainty-causing diseases. In future work, we aim to solve new fractional models [47–49] and compare with other numerical methods [50–56]. Conflicts of Interest: The authors declare that they have no conflict of interest. References [1] Victor Henrique Ferreira-de Lima and Tamara Nunes Lima-Camara. Natural vertical transmission of dengue virus in Aedes aegypti and Aedes albopictus: a systematic review. Parasites & Vectors, 11:1–8, 2018. [2] Sudipta Kumar Roy and Soumen Bhattacharjee. Dengue virus: epidemiology, biology, and disease aetiology. Canadian Journal of Microbiology, 67(10):687–702, 2021. [3] Sweety Trivedi and Ambar Chakravarty. Neurological complications of dengue fever. Current Neurology and Neuroscience Reports, 22(8):515–529, 2022. [4] Viroj Wiwanitkit. Dengue fever: diagnosis and treatment. Expert Review of Anti- infective Therapy, 8(7):841–845, 2010. [5] Fridous Jahan. Dengue fever (df) in pakistan. Asia Pacific Family Medicine, 10:1–4, 2011. [6] Carlos Letacio Silveira Lessa, Katharine Valéria Saraiva Hodel, Marilda de Souza Gonçalves, and Bruna Aparecida Souza Machado. Dengue as a disease threatening global health: a narrative review focusing on latin america and brazil. Tropical Medicine and Infectious Disease, 8(5):241, 2023. [7] Ayman Ahmed, Adel Elduma, Babiker Magboul, Tarig Higazi, and Yousif Ali. The R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 18 of 21 first outbreak of dengue fever in greater darfur, western sudan. Tropical Medicine and Infectious Disease, 4(1):43, 2019. [8] Duane J. Gubler. Epidemic dengue and dengue hemorrhagic fever: a global public health problem in the 21st century. Emerging Infections 1, pages 1–14, 1997. [9] Xiaorong Yang, Mikkel B. M. Quam, Tongchao Zhang, and Shaowei Sang. Global burden for dengue and the evolving pattern in the past 30 years. Journal of Travel Medicine, 28(8):taab146, 2021. [10] Fathelrhman E. L. Guma, Ossama M. Badawy, Mohammed Berir, and Mohamed A. Abdoon. Numerical analysis of fractional-order dynamic dengue disease epidemic in sudan. Journal of the Nigerian Society of Physical Sciences, page 1464, 2023. [11] Arwa Elaagip, Khider Alsedig, Omnia Altahir, Tellal Ageep, Ayman Ahmed, Hanaa Adli Siam, Abdallah M. Samy, Waleed Mohamed, Fatima Khalid, Suhaib Gumaa, et al. Seroprevalence and associated risk factors of dengue fever in kassala state, eastern sudan. PLoS Neglected Tropical Diseases, 14(12):e0008918, 2020. [12] Máıra Aguiar, Vizda Anam, Konstantin B. Blyuss, Carlo Delfin S. Estadilla, Bruno V. Guerrero, Damián Knopoff, Bob W. Kooi, Akhil Kumar Srivastav, Vanessa Steindorf, and Nico Stollenwerk. Mathematical models for dengue fever epidemiology: A 10-year systematic review. Physics of Life Reviews, 40:65–92, 2022. [13] Priyanka Harjule et al. Mathematical modelling and analysis of dengue transmission dynamics. Procedia Computer Science, 235:539–548, 2024. [14] M. A. Abdoon. Fractional derivative approach for modeling chaotic dynamics: Ap- plications in communication and engineering systems. In Proceedings of the Inter- national Conference on Mathematical Modelling, Applied Analysis and Computation, pages 82–95. Springer, 2024. [15] Md Rafiul Islam, Angela Peace, Daniel Medina, and Tamer Oraby. Integer versus fractional order seir deterministic and stochastic models of measles. International Journal of Environmental Research and Public Health, 17(6):2014, 2020. [16] Nekmat Ullah, Zahir Shah, Rashid Jan, Narcisa Vrinceanu, Muhammad Farhan, and Elisabeta Antonescu. Modeling the non-integer dynamics of a vector-borne infection with nonlocal and nonsingular kernel. Scientific Reports, 15(1):6262, 2025. [17] Tao-Qian Tang, Rashid Jan, Adil Khurshaid, Zahir Shah, Narcisa Vrinceanu, and Mihaela Racheriu. Analysis of the dynamics of a vector-borne infection with the effect of imperfect vaccination from a fractional perspective. Scientific Reports, 13(1):14398, 2023. [18] Parvaiz Ahmad Naik, Mehmet Yavuz, Sania Qureshi, Kolade M. Owolabi, Amanul- lah Soomro, Abdul Hamid Ganie, et al. Memory impacts in hepatitis c: A global analysis of a fractional-order model with an effective treatment. Computer Methods and Programs in Biomedicine, 254:108306, 2024. [19] N. E. Alsubaie, Fathelrhman EL Guma, Kaouther Boulehmi, Naseam Al-kuleab, and Mohamed A. Abdoon. Improving influenza epidemiological models under caputo fractional-order calculus. Symmetry, 16(7):929, 2024. [20] Mutum Zico Meetei, Shahbaz Zafar, Abdullah A. Zaagan, Ali M. Mahnashi, and Muhammad Idrees. Dengue transmission dynamics: A fractional-order approach with R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 19 of 21 compartmental modeling. Fractal and Fractional, 8(4):207, 2024. [21] Nausheen Razi, Ambreen Bano, Umar Ishtiaq, Tayyab Kamran, Mubariz Garayev, and Ioan-Lucian Popa. Probing malware propagation model with variable infection rates under integer, fractional, and fractal–fractional orders. Fractal and Fractional, 9(2):90, 2025. [22] Enrique C. Gabrick, Ervin K. Lenzi, and Antonio M. Batista. Fractional and frac- tal extensions of epidemiological models. In Mathematical Methods in Medical and Biological Sciences, pages 39–62. Elsevier, 2025. [23] Enrique C. Gabrick, Elaheh Sayari, Diogo L. M. Souza, Fernando S. Borges, José Trobia, Ervin K. Lenzi, and Antonio M. Batista. Fractal and fractional sis model for syphilis data. Chaos: An Interdisciplinary Journal of Nonlinear Science, 33(9), 2023. [24] Rami Ahmad El-Nabulsi. Qualitative financial modelling in fractal dimensions. Fi- nancial Innovation, 11(1):42, 2025. [25] Zilong Hu, Yanzhao Wu, Puxi Li, Ruofu Xiao, and Ran Tao. Comparative study on the fractal and fractal dimension of the vortex structure of hydrofoil’s tip leakage flow. Fractal and Fractional, 7(2):123, 2023. [26] Enrique C. Gabrick, Ervin K. Lenzi, and Antonio M. Batista. Fractional and frac- tal extensions of epidemiological models. In Mathematical Methods in Medical and Biological Sciences, pages 39–62. Elsevier, 2025. [27] Chien-Hung Lee, Ko Chang, Yao-Mei Chen, Jinn-Tsong Tsai, Yenming J. Chen, and Wen-Hsien Ho. Epidemic prediction of dengue fever based on vector compartment model and markov chain monte carlo method. BMC Bioinformatics, 22:1–11, 2021. [28] Muhammad Fahmi, Norhayati Rosli, and Noryanti Muhammad. Estimation of epi- demiological parameter of covid-19 using the markov chain monte carlo method. In AIP Conference Proceedings, volume 3150. AIP Publishing, 2024. [29] Luiz Roberto Evangelista and Ervin Kaminski Lenzi. Fractional Diffusion Equations and Anomalous Diffusion. Cambridge University Press, 2018. [30] Richard Herrmann. Fractional Calculus: An Introduction for Physicists. World Sci- entific, 2011. [31] Wei Cai, Wen Chen, Jun Fang, and Sverre Holm. A survey on fractional deriva- tive modeling of power-law frequency-dependent viscous dissipative and scattering attenuation in acoustic wave propagation. Applied Mechanics Reviews, 70(3):030802, 2018. [32] HongGuang Sun, Mark M. Meerschaert, Yong Zhang, Jianting Zhu, and Wen Chen. A fractal richards’ equation to capture the non-boltzmann scaling of water transport in unsaturated media. Advances in Water Resources, 52:292–295, 2013. [33] Yuli Chen, Fawang Liu, Qiang Yu, and Tianzeng Li. Review of fractional epidemic models. Applied Mathematical Modelling, 97:281–307, 2021. [34] Vladimir V. Kulish and José L. Lage. Application of fractional calculus to fluid mechanics. Journal of Fluids Engineering, 124(3):803–806, 2002. [35] Abdon Atangana. Fractal-fractional differentiation and integration: connecting frac- tal calculus and fractional calculus to predict complex system. Chaos, Solitons & Fractals, 102:396–406, 2017. R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 20 of 21 [36] Zeeshan Ali, Faranak Rabiei, Kamal Shah, and Touraj Khodadadi. Modeling and analysis of novel covid-19 under fractal-fractional derivative with case study of malaysia. Fractals, 29(01):2150020, 2021. [37] Ji-Huan He. A tutorial review on fractal spacetime and fractional calculus. Interna- tional Journal of Theoretical Physics, 53:3698–3718, 2014. [38] Jatin Bansal, Amit Kumar, Anoop Kumar, Aziz Khan, and Thabet Abdeljawad. Investigation of monkeypox disease transmission with vaccination effects using frac- tional order mathematical model under atangana-baleanu caputo derivative. Modeling Earth Systems and Environment, 11(1):40, 2025. [39] Kang-Jia Wang. An effective computational approach to the local fractional low-pass electrical transmission lines model. Alexandria Engineering Journal, 110:629–635, 2025. [40] Hasib Khan, Jehad Alzabut, and DK Almutairi. Applications of artificial intelligence for clusters analysis of uranium decay via a fractional order discrete model. Partial Differential Equations in Applied Mathematics, 13:101056, 2025. [41] Mohamed A. Abdoon and Abdulrahman BM Alzahrani. Comparative analysis of influenza modeling using novel fractional operators with real data. Symmetry, 16(9):1126, 2024. [42] HongGuang Sun, Mark M. Meerschaert, Yong Zhang, Jianting Zhu, and Wen Chen. A fractal richards’ equation to capture the non-boltzmann scaling of water transport in unsaturated media. Advances in Water Resources, 52:292–295, 2013. [43] Yong Zhang and Charalambos Papelis. Particle-tracking simulation of fractional diffusion-reaction processes. Physical Review E—Statistical, Nonlinear, and Soft Mat- ter Physics, 84(6):066704, 2011. [44] Yingjie Liang, Wen Chen, Wei Xu, and HongGuang Sun. Distributed order haus- dorff derivative diffusion model to characterize non-fickian diffusion in porous media. Communications in Nonlinear Science and Numerical Simulation, 70:384–393, 2019. [45] Wen Chen. Time–space fabric underlying anomalous diffusion. Chaos, Solitons & Fractals, 28(4):923–929, 2006. [46] Wen Chen, Fajie Wang, Bin Zheng, and Wei Cai. Non-euclidean distance fundamental solution of hausdorff derivative partial differential equations. Engineering Analysis with Boundary Elements, 84:213–219, 2017. [47] F. Hasan, M. A. Abdoon, R. Saadeh, M. Berir, and A. Qazza. A new perspective on the stochastic fractional order materialized by the exact solutions of allen-cahn equa- tion. International Journal of Mathematical, Engineering and Management Sciences, 8(5):912, 2023. [48] M. Ali, S. M. Alzahrani, R. Saadeh, M. A. Abdoon, A. Qazza, N. Al-kuleab, and F. E. Guma. Modeling covid-19 spread and non-pharmaceutical interventions in south africa: A stochastic approach. Scientific African, 24:e02155, 2024. [49] R. Saadeh, M. A. Abdoon, A. Qazza, M. Berir, F. E. Guma, N. Al-Kuleab, and A. M. Degoot. Mathematical modeling and stability analysis of the novel fractional model in the caputo derivative operator: A case study. Heliyon, 10(5), 2024. [50] A. B. M. Alzahrani, R. Saadeh, M. A. Abdoon, M. Elbadri, M. Berir, and A. Qazza. R. Saadeh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6682 21 of 21 Effective methods for numerical analysis of the simplest chaotic circuit model with atangana–baleanu caputo fractional derivative. Journal of Engineering Mathematics, 144(1):9, 2024. [51] E. Salah, A. Qazza, R. Saadeh, and A. El-Ajou. A hybrid analytical technique for solving multi-dimensional time-fractional navier-stokes system. AIMS Mathematics, 8(1):1713–1736, 2023. [52] M. Abu-Ghuwaleh, R. Saadeh, and A. Qazza. General master theorems of integrals with applications. Mathematics, 10(19):3547, 2022. [53] R. Allogmany, A. Sarrah, M. A. Abdoon, F. J. Alanazi, M. Berir, and S. A. Alharbi. A comprehensive analysis of complex dynamics in the fractional-order rössler system. Mathematics, 13(19):3089, 2025. [54] I. Kadri, R. Saadeh, D. M. AlMutairi, M. E. Dafaalla, M. Berir, and M. A. Abdoon. Analytical and numerical investigation of a fractional order 4d chaotic system via caputo fractional derivative. European Journal of Pure and Applied Mathematics, 18(3):6381–6381, 2025. [55] M. Berir. The impact of white noise on chaotic behavior in a financial fractional system with constant and variable order: A comparative study. European Journal of Pure and Applied Mathematics, 17(4):3915–3931, 2024. [56] M. Berir. A fractional study for solving the smoking model and the chaotic engineer- ing model. In Proceedings of the 2023 2nd International Engineering Conference on Electrical, Energy, and Artificial Intelligence (EICEEAI), pages 1–6. IEEE, 2023.