EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6981 ISSN 1307-5543 – ejpam.com Published by New York Business Global Malaria and Malnutrition Dynamics in Children Using the Caputo–Fabrizio Fractional Derivative Amjad Shaikh1, Sunil Howal2, Kottakkaran Sooppy Nisar3,∗ 1 Department of Mathematics, AKI’s Poona College of Arts, Science and Commerce, Camp, Pune, India 2 Department of Mathematics, Sir Parshurambhau College, Savitribai Phule Pune University, Pune, India 3 Department of Mathematics, College of Science and Humanities in Al Kharj, Prince Sattam bin Abdulaziz University, Al Kharj, 11942, Saudi Arabia Abstract. Malaria poses a significant global public health challenge as an infectious disease trans- mitted by vectors, particularly affecting young children, with substantial morbidity and mortality rates. This study formulates criteria ensuring the stability, uniqueness, and existence of a fractional- order malaria-malnutrition framework utilizing the Caputo-Fabrizio differential operator, employing the fixed-point methodology. The adoption of this fractional differentiation technique is an inno- vative approach within such biological contexts. Furthermore, we obtain the earliest approximate solutions for the formulated model through the iterative Laplace transform procedure. At last, numerical simulations are performed using the selected model parameter values. Our findings indi- cate that ensuring a well-rounded diet is crucial for mitigating the spread of malaria among young children, thereby reducing both morbidity and mortality. 2020 Mathematics Subject Classifications: 26A33, 33E12, 34A08, 65R10 Key Words and Phrases: Mathematical models, malaria fever, Caputo-Fabrizio derivative, exis- tence, uniqueness and stability, numerical simulations 1. Introduction Malaria is a disease spread by infected female Anopheles mosquitoes transmitting the Plasmodium parasite. It is a significant health issue in numerous tropical and subtropical areas. The parasite first grows in the liver and then attacks Erythrocytes, resulting in symptoms like tiredness, muscle pain, headaches, sweating, chills and fever. Malaria does not spread directly from person to person. It is only transmitted through mosquito bites, when an infected person is bitten by a mosquito, it picks up the infectious agent and can pass it on to others. Within the mosquito, these parasites undergo a complex life cycle. While stopping malaria could potentially make children grow heavier, this improvement might only be temporary [1–4]. Historically, public health initiatives prioritized communicable diseases and malnutrition due to their significant health risks. Yet, understanding the interplay ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6981 Email addresses: amjad.shaikh@poonacollege.edu.in (A. Shaikh), sunil.howal@fergusson.edu (S. Howal), n.sooppy@psau.edu.sa (K. S. Nisar) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 2 between infection and nutrition remains complex, making it challenging to determine the order of addressing these issues. This study aimed to find out how many children in the area suffer from malnutrition. It also examined whether treating diseases, especially malaria, helps children grow and develop properly. Researchers examined the relationship between malaria and malnutrition in children under five in a region heavily affected by malaria[5]. Animal studies suggested that a poor diet might help reduce malaria. This led to the idea that malnourished children could be less likely to get malaria, experience severe symptoms, or die from the disease [6–9] Malaria and malnutrition are key factors in child mortality in sub-Saharan Africa. Repeated malaria infections weaken children’s nutrition and overall well-being [10, 11]. In this article, we summarize modelling in malaria research to help more researchers in epidemiology, transmission, and other areas understand it better[12]. A mathematical model was developed and analyzed to better understand how malaria spreads and identify effective ways to curb and oversee[13]. Recent papers have started considering environ- mental factors and the development of drug resistance in malaria [14–18]. Ngwa and Shu [19], as well as Ngwa [20], introduced an ODE compartmental model for malaria spread. Addo [21] and Tuwiine Mugisha, and Luboobi [22] have developed compartmental models for malaria transmission, using susceptible-infected-recovered-susceptible (SIRS) patterns for humans and susceptible-infected (SI) patterns for mosquitoes respectively. Yang, Wei, and Li [23]proposed a compartment model with SIR for humans and SI for vectors. These studies establish the reproduction number and investigate the conditions for the existence and stability of both disease-free and endemic states. Mathematics, especially fractional calculus, is vital for modeling epidemics and biological processes. This study uses fractional operators and numerical methods to analyze disease dynamics, compute key parameters, and support decision-making, with potential for future hybrid approaches[24]. This study models poliomyelitis transmission using fractional-order ABC models with vaccination and post-paralytic compartments, analyzes stability and con- vergence, and integrates deep neural networks for accurate simulation and prediction of disease dynamics[25]. In this paper, we developed criteria for the uniqueness, stability and existence of a fractional-order typhoid fever model using the Caputo–Fabrizio operator and fixed-point theory[26]. Infectious diseases can be accurately modelled using the non-local Caputo–Fabrizio fractional derivative operator. Additionally, we have verified the stability conditions for steady-state solutions and established their existence using Banach fixed- point theory[27]. In this paper, we created a model to study how the new coronavirus spreads. This model helps us understand how the disease spreads over time, both in the short term and long term[28]. A fractional-order hepatitis B model with vaccine effects is analyzed using Caputo derivatives, simulations, and ANN to understand disease dynam- ics and support public health strategies[29]. This paper presents a fractional SIQR model with Caputo–Fabrizio derivative, proving key properties, simulating memory effects, and applying neural networks for analysis[30]. 2. Model Information We determine a Susceptible-Exposed-Infectious-Susceptible (SEIS) model of ailment to place the institutions under concern and teenagers under age five into two groups, similarly by their digestive rank: the first group is those the individuals are well augmented, named for individual follow-up 1, and the second group is composed of those the individuals are A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 3 thin, named for each follow-up 2. The total babies organization is conveyed by N. Influx into the youthful category progresses at the rate η following a disposal p entering the well augment childlike U , and 1 − p into the thin childlike V . Note that we only trust drafts into the unprotected division. A well-augmented childlike U because a feeble diet permits an action to reinforce a thin, unprotected V at a rate ψ, and a thin, trusting following position/correct diet acknowledges feasibility and embellishes a well-augmented trusting at a measure ϕ. Commonly, a trusting individual, for that reason, can take a wound caused by extending scourge with nausea; therefore, it can improve at the rate, λ, or fix and embellish unprotected recurring at a measure α. Concerning the well-augmented epidemic, a percentage b that restores holds improves thin, unprotected belongings; thus, b = 1 − b improves well-augmented, trusting belongings. Individuals exit the infected compartment by recovery or malaria-induced death, common obliteration n, or by erasure from the ailment e1 for the class C1 and e2 for the class C2. The nausea broadcast probabilities per pest bite on U and V are likely individually by γ1 and γ2. both trusting groups of youth are defenseless to sickness and move separately to the K1 or K2 classes. The strength of contamination, or the measure at which naive belongings get ailment, is ρ1, ρ2 and ρ3 for the contagion-heading folk. The compartments include susceptible (U) individuals who are at risk of contracting malaria, exposed (E) individuals who have been infected but are not yet infectious, infected (I) individuals capable of transmitting the disease, vaccinated (V) individuals with partial or full immunity, fully infected or febrile (F) individuals displaying severe symptoms, and quarantined (Q) individuals who are isolated for treatment or control. By presenting these compartments clearly and linking them directly to the biological processes they represent, readers can better understand the dynamics of malaria transmission. A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 4 U E I V F Q λ χ bξ θ ϵ bξ µ ψϕ ϱ ϱ+ ∂ ϱ+ κϱ ϱ qπ ϱ (1− q)π From the model flowcharts of the affliction broadcast described in the above Figure, we evolve the following equations. dU dt = qπ + (1− b)ξI + ϕV − (σ + ψ + ϱ)U dE dt = σU − (χ+ ϱ)E dI dt = χE − (ξ + ϱ+ ∂)I dV dt = (1− q)π + µQ+ ψU + bξI − (θ + ϕ+ ϱ)V dF dt = θV − (ε+ ϱ)F dQ dt = ϵF − (µ+ ϱ+ κ)Q Motivated by the previously listed works of literature, the Malaria fever model pro- posed in [11] is analyzed using the Caputo-Fabrizio operator of order η where η ∈ (0, 1). A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 5  CFDη t U(t) = qπ + (1− b)ξI + ϕV − (σ + ψ + ϱ)U CFDη tE(t) = σU − (χ+ ϱ)E CFDη t I(t) = χE − (ξ + ϱ+ ∂)I CFDη t V (t) = (1− q)π + µQ+ ψU + bξI − (θ + ϕ+ ϱ)V CFDη t F (t) = θV − (ε+ ϱ)F CFDη tQ(t) = ϵF − (µ+ ϱ+ κ)Q (2.1) with initial conditions U(0) ≥ 0, E(0) ≥ 0, I(0) ≥ 0, V (0) ≥ 0, F (0) ≥ 0, Q(0) ≥ 0. (2.2) The classic model given in [11] is accomplished for η = 1. Given that the Caputo- Fabrizio fractional derivative accurately illustrates the previously discussed occurrences. The quantitative findings are obtained repeatedly using the Laplace transform approach. To validate our results, we assign random values to the parameters and initial conditions. The remainder of the paper is organized as follows: Section 3 provides some definitions and preliminary information on fractional calculus. The existence and uniqueness of the model’s solution with the method and operator under consideration are covered in Section 4. Section 5 presents the stability of the approximated solution. The solution’s graphical representation is related to Section 6. In section 7, numerical results are discussed. Finally, section 8 provides a summary of the study’s main findings and observations. 3. Preliminaries Definition 1. ([27]) Let u ∈ H1(0, d), d > 0, 0 < η < 1, then time-fractional Caputo- Fabrizio differential operator the time fractional Caputo–Fabrizio fractional differential op- erator (C-FFDO) is expressed as CFDη t u(t) = N(η) (1− η) ∫ t 0 exp [ − η(t− s) 1− η ] u′(s)ds, t ≥ 0, 0 < η < 1, (3.1) where N(η) is a normalization function dependent on η, ensuring that N(0) = N(1) = 1. Definition 2. ([27]) The Caputo-Fabrizio integral operator of fractional order 0 < η < 1 is given by CFJη t u(t) = 2(1− η) (2− η)N(η) u(t) + 2η (2− η)N(η) ∫ t 0 u(ℵ)dℵ, (3.2) Like the traditional Caputo derivative, this new operator offers CFDη t u(t) = 0, pro- vided that u does not change. The key advantage of the Caputo-Fabrizio operator applied to the traditional Caputo op- erator is that the updated kernel avoids singularity at t = s. A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 6 Definition 3. ([27]) The Laplace transformation of the Caputo-Fabrizio fractional operator of order 0 < η ≤ 1 and m ∈ N is given by L (CF Dm+η t u(t) ) (s) = 1 1− η L(u(m+1)(t))L ( exp ( − η 1− η t )) = sm+1L(u(t))− smu(0)− sm−1u′(0)− · · · − u(m)(0) s+ η(1− s) . (3.3) Specifically, we have L (CF Dη t u(t) ) (s) = sL(u(t))− u(0) s+ η(1− s) , m = 0. L (CF Dη+1 t u(t) ) (s) = s2L(u(t))− su(0)− u′(0) s+ η(1− s) , m = 1. 4. Existence and Uniqueness In this section, we examine the existence and uniqueness of the solution for the system (2.1) using the fixed-point theory. Considering equation (3.2), we get U(t) = U(0) + 2(1− η) 2ηN(η) (qπ + bξI(t) + ϕV (t)− (σ + ψ + ϱ)U(t)) + 2(1− η) (2− η)N(η) ∫ t 0 (qπ + bξI(ℵ) + ϕV (ℵ)− (σ + ψ + ϱ)U(ℵ))dℵ E(t) = E(0) + 2(1− η) (2η)N(η) (σU(t)− (χ+ ϱ)E(t)) + 2(1− η) (2− η)N(η) ∫ t 0 (σU(ℵ)− (χ+ ϱ)E(ℵ))dℵ I(t) = I(0) + 2(1− η) (2η)N(η) (χE(t)− (ξ + ϱ+ ∂)I(t)) + 2(1− η) (2− η)N(η) ∫ t 0 (χE(ℵ)− (ξ + ϱ+ ∂)I(ℵ))dℵ V (t) = V (0) + 2(1− η) 2ηN(η) ((1− q)π + µQ(t) + ψU(t) + bξI(t)− (θ + ϕ+ ϱ)V (t)) + 2(1− η) (2− η)N(η) ∫ t 0 ((1− q)π + µQ(ℵ) + ψU(ℵ) + bξI(ℵ)− (θ + ϕ+ ϱ)V (ℵ))dℵ F (t) = F (0) + 2(1− η) (2η)N(η) (θV (t)− (µ+ n)F (t)) + 2(1− η) (2− η)N(η) ∫ t 0 (θV (ℵ)− (µ+ ϱ)F (ℵ))dℵ Q(t) = Q(0) + 2(1− η) (2η)N(η) (ϵF (t)− (µ+ ϱ+ κ)Q(t)) + 2(1− η) (2− η)N(η) ∫ t 0 (ϵF (ℵ)− (µ+ ϱ+ κ)Q(ℵ))dℵ A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 7 Let’s now examine the subsequent kernels: ψ1(t, U(t)) = qπ + bξI + ϕV (t)− (σ + ψ + ϱ)U(t) ψ2(t, E(t)) = σU(t)− (χ+ ϱ)E(t) ψ3(t, I(t)) = χE(t)− (ξ + ϱ+ ∂)I(t) (4.1) ψ4(t, V (t)) = (1− q)π + µQ(t) + ψU(t) + bξI(t)− (θ + ϕ+ ϱ)V (t) ψ5(t, F (t)) = θV (t)− (µ+ ϱ)F (t) ψ6(t, Q(t)) = ϵF (t)− (µ+ ϱ+ κ)Q(t) Lemma 1. The kernels ψ1, ψ2, ψ3, ψ4, ψ5, and ψ6 in (4.1) satisfy the Lipschitz condition, i.e., for any two state vectors X1 = (U1, E1, I1, V1, F1, Q1) and X2 = (U2, E2, I2, V2, F2, Q2), there exist constants 0 < Λi < 1 such that ∥ψi(t,X1) − ψi(t,X2)∥ ≤ Λi∥X1 − X2∥, i = 1, 2, . . . , 6. Proof: Let U1 and U2, for the kernel ψ1, E1 and E2, for the kernel ψ2, I1 and I2, for the kernel ψ3, V1 and V2, for the kernel ψ4, F1 and F2, for the kernel ψ5 and Q1 and Q2, the respective functions associated with the kernel ψ6 correspond to the following: ∥ψ1(t, U1(t))− ψ1(t, U2(t))∥ = ∥(−(σ + ψ + ϱ)U1(t)) + (σ + ψ + ϱ)U2(t))∥, ∥ψ2(t, E1(t))− ψ2(t, E2(t))∥ = ∥(−((χ+ ϱ))E1(t)) + ((χ+ ϱ))E2(t))∥, ∥ψ3(t, I1(t))− ψ3(t, I2(t))∥ = ∥(−(ξ + ϱ+ ∂)I1(t)) + (ξ + ϱ+ ∂)I2(t))∥, ∥ψ4(t, V1(t))− ψ4(t, V2(t))∥ = ∥(−(θ + ϕ+ ϱ)V1(t)) + (θ + ϕ+ ϱ)V2(t))∥, ∥ψ5(t, F1(t))− ψ5(t, F2(t))∥ = ∥(−(µ+ ϱ)F1(t)) + (µ+ ϱ)F2(t))∥, ∥ψ6(t, Q1(t))− ψ6(t, Q2(t))∥ = ∥(−(µ+ ϱ+ κ)Q1(t)) + (µ+ ϱ+ κ)Q2(t))∥, (4.2) Consider ∥ψ1(t, U1(t))− ψ1(t, U2(t))∥ = ∥(−(σ + ψ + ϱ)U1(t)) + (σ + ψ + ϱ)U2(t))∥, ≤ ∥((σ + ψ + ϱ)(U1(t))− U2(t)))∥, ≤ Λ1∥(U1(t))− U2(t)∥, (4.3) where Λ1 = σ + ψ + ϱ. Similarly, we can obtain ∥ψ2(t, E1(t))− ψ2(t, E2(t))∥ ≤ Λ2∥(E1(t))− E2(t)∥, ∥ψ3(t, I1(t))− ψ3(t, I2(t))∥ ≤ Λ3∥(I1(t))− I2(t)∥, ∥ψ4(t, V1(t))− ψ4(t, V2(t))∥ ≤ Λ4∥(V1(t))− V2(t)∥, ∥ψ5(t, F1(t))− ψ5(t, F2(t))∥ ≤ Λ5∥(F1(t))− F2(t)∥, ∥ψ6(t, Q1(t))− ψ6(t, Q2(t))∥ ≤ Λ6∥(Q1(t))−Q2(t)∥, (4.4) where Λ2 = χ+ ϱ,Λ3 = ξ + ϱ+ ℵ, Λ4 = θ + ϕ+ ϱ, Λ5 = µ+ ϱ, Λ6 = µ+ ϱ+ κ. Utilising the subsequent recursive formula, we obtain Un(t) = 2(1− η) (2− η)N(η) ψ1(t, Un−1(t)) + 2η (2− η)N(η) ∫ t 0 ψ1(ℵ, Un−1(ℵ))dℵ, A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 8 En(t) = 2(1− η) (2− η)N(η) ψ2(t, En−1(t)) + 2η (2− η)N(η) ∫ t 0 ψ2(ℵ, En−1(ℵ))dℵ, In(t) = 2(1− η) (2− η)N(η) ψ3(t, In−1(t)) + 2η (2− η)N(η) ∫ t 0 ψ3(ℵ, In−1(ℵ))dℵ, Vn(t) = 2(1− η) (2− η)N(η) ψ4(t, Vn−1(t)) + 2η (2− η)N(η) ∫ t 0 ψ4(ℵ, Vn−1(ℵ))dℵ, Fn(t) = 2(1− η) (2− η)N(η) ψ5(t, Fn−1(t)) + 2η (2− η)N(η) ∫ t 0 ψ5(ℵ, Fn−1(ℵ))dℵ, Qn(t) = 2(1− η) (2− η)N(η) ψ6(t, Qn−1(t)) + 2η (2− η)N(η) ∫ t 0 ψ6(ℵ, Qn−1(ℵ))dℵ. Now, by triangle inequality, we get ∥∆1n(t)∥ = ∥Un(t)− Un−1(t)∥ ≤ 2(1− η) (2− η)N(η) ∥ψ1 ( t, Un−1(t) ) − ψ1 ( t, Un−2(t) ) ∥ + 2η (2− η)N(η) ∥∥∥∥∫ t 0 [ ψ1 ( ℵ, Un−1(ℵ) ) − ψ1 ( ℵ, Un−2(ℵ) )] dℵ ∥∥∥∥, ∥∆2n(t)∥ = ∥En(t)− En−1(t)∥ ≤ 2(1− η) (2− η)N(η) ∥ψ2 ( t, En−1(t) ) − ψ2 ( t, En−2(t) ) ∥ + 2η (2− η)N(η) ∥∥∥∥∫ t 0 [ ψ2 ( ℵ, En−1(ℵ) ) − ψ2 ( ℵ, En−2(ℵ) )] dℵ ∥∥∥∥, ∥∆3n(t)∥ = ∥In(t)− In−1(t)∥ ≤ 2(1− η) (2− η)N(η) ∥ψ3 ( t, In−1(t) ) − ψ3 ( t, In−2(t) ) ∥ + 2η (2− η)N(η) ∥∥∥∥∫ t 0 [ ψ3 ( ℵ, In−1(ℵ) ) − ψ3 ( ℵ, In−2(ℵ) )] dℵ ∥∥∥∥, ∥∆4n(t)∥ = ∥Vn(t)− Vn−1(t)∥ ≤ 2(1− η) (2− η)N(η) ∥ψ4 ( t, Vn−1(t) ) − ψ4 ( t, Vn−2(t) ) ∥ + 2η (2− η)N(η) ∥∥∥∥∫ t 0 [ ψ4 ( ℵ, Vn−1(ℵ) ) − ψ4 ( ℵ, Vn−2(ℵ) )] dℵ ∥∥∥∥, ∥∆5n(t)∥ = ∥Fn(t)− Fn−1(t)∥ ≤ 2(1− η) (2− η)N(η) ∥ψ5 ( t, Fn−1(t) ) − ψ5 ( t, Fn−2(t) ) ∥ + 2η (2− η)N(η) ∥∥∥∥∫ t 0 [ ψ5 ( ℵ, Fn−1(ℵ) ) − ψ5 ( ℵ, Fn−2(ℵ) )] dℵ ∥∥∥∥, ∥∆6n(t)∥ = ∥Qn(t)−Qn−1(t)∥ ≤ 2(1− η) (2− η)N(η) ∥ψ6 ( t, Qn−1(t) ) − ψ6 ( t, Qn−2(t) ) ∥ + 2η (2− η)N(η) ∥∥∥∥∫ t 0 [ ψ6 ( ℵ, Qn−1(ℵ) ) − ψ6 ( ℵ, Qn−2(ℵ) )] dℵ ∥∥∥∥, (4.5) Un(t) = ∞∑ j=0 ∆1j(t), En(t) = ∞∑ j=0 ∆2j(t), In(t) = ∞∑ j=0 ∆3j(t), Vn(t) = ∞∑ j=0 ∆4j(t), Fn(t) = ∞∑ j=0 ∆5j(t), Qn(t) = ∞∑ j=0 ∆6j(t). (4.6) A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 9 Since the kernels ψ1, ψ2, ψ3, ψ4, ψ5 and ψ6 fulfils the Lipschitz condition, we get ∥∆1n(t)∥ = ∥Un(t)− Un−1(t)∥ ≤ 2(1− η) (2− η)N(η) Λ1∥Un−1(t)− Un−2(t)∥ + 2η (2− η)N(η) Λ1 ∫ t 0 ∥Un−1(ℵ)− Un−2(ℵ)∥dℵ, ∥∆2n(t)∥ = ∥En(t)− En−1(t)∥ ≤ 2(1− η) (2− η)N(η) Λ2∥En−1(t)− En−2(t)∥ + 2η (2− η)N(η) Λ2 ∫ t 0 ∥En−1(ℵ)− En−2(ℵ)∥dℵ, ∥∆3n(t)∥ = ∥In(t)− In−1(t)∥ ≤ 2(1− η) (2− η)N(η) Λ3∥In−1(t)− In−2(t)∥ + 2η (2− η)N(η) Λ3 ∫ t 0 ∥In−1(ℵ)− In−2(ℵ)∥dℵ, ∥∆4n(t)∥ = ∥Vn(t)− Vn−1(t)∥ ≤ 2(1− η) (2− η)N(η) Λ4∥Vn−1(t)− Vn−2(t)∥ + 2η (2− η)N(η) Λ4 ∫ t 0 ∥Vn−1(ℵ)− Vn−2(ℵ)∥dℵ, ∥∆5n(t)∥ = ∥Fn(t)− Fn−1(t)∥ ≤ 2(1− η) (2− η)N(η) Λ5∥Fn−1(t)− Fn−2(t)∥ + 2η (2− η)N(η) Λ5 ∫ t 0 ∥Fn−1(ℵ)− Fn−2(ℵ)∥dℵ, ∥∆6n(t)∥ = ∥Qn(t)−Qn−1(t)∥ ≤ 2(1− η) (2− η)N(η) Λ6∥Qn−1(t)−Qn−2(t)∥ + 2η (2− η)N(η) Λ6 ∫ t 0 ∥Qn−1(ℵ)−Qn−2(ℵ)∥dℵ, (4.7) it validates the outcome. Theorem 1. Prove that the system (2.1) admits solution. Proof. We have arrived at the following using the equation (4.7) and in accordance with the recursive formula: ∥∆n 1 (t)∥ ≤ ∥U(0)∥+ [( 2(1− η) (2− η)N(η) Λ1 ) + ( 2η (2− η)N(η) Λ1t )]n , ∥∆n 2 (t)∥ ≤ ∥E(0)∥+ [( 2(1− η) (2− η)N(η) Λ2 ) + ( 2η (2− η)N(η) Λ2t )]n , ∥∆n 3 (t)∥ ≤ ∥I(0)∥+ [( 2(1− η) (2− η)N(η) Λ3 ) + ( 2η (2− η)N(η) Λ3t )]n , ∥∆n 4 (t)∥ ≤ ∥V (0)∥+ [( 2(1− η) (2− η)N(η) Λ4 ) + ( 2η (2− η)N(η) Λ4t )]n , ∥∆n 5 (t)∥ ≤ ∥F (0)∥+ [( 2(1− η) (2− η)N(η) Λ5 ) + ( 2η (2− η)N(η) Λ5t )]n , A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 10 ∥∆n 6 (t)∥ ≤ ∥Q(0)∥+ [( 2(1− η) (2− η)N(η) Λ6 ) + ( 2η (2− η)N(η) Λ6t )]n . (4.8) The recursive inequalities (4.8) can be written an ∥∆n i (t)∥ ≤ ∥Xi(0)∥+ [ (2− η)N(η) 2(1− η) Λi + (2− η)N(η) 2η tΛi ]n , i = 1, . . . , 6. The recursive bound can be written as ∥∆n i (t)∥ ≤ ∥Xi(0)∥+ Cn i , i = 1, 2, . . . , 6, where Ci is a constant independent of the iteration, given by Ci = (2− η)N(η) 2(1− η) Λi + (2− η)N(η) 2η tΛi, i = 1, 2, . . . , 6, where X1 = U, X2 = E, X3 = I, X4 = V, X5 = F, X6 = Q. Therefore, (4.8) exists. Additionally, we demonstrate that the functions in (4.8) constitute a solution system for (2.1) under the assumption that U(t) = Un(t)−Υ1n(t), E(t) = En(t)−Υ2n(t), I(t) = In(t)−Υ3n(t), V (t) = Vn(t)−Υ4n(t), F (t) = Fn(t)−Υ5n(t), Q(t) = Qn(t)−Υ6n(t), (4.9) where Υ1n(t),Υ2n(t),Υ3n(t),Υ4n(t),Υ5n(t) and Υ6n(t) are the leftover terms of the solution. Therefore, we derive U(t)− Un(t) = 2(1− η) (2− η)N(η) ψ1 ( t, U(t)− Un(t) ) + 2η (2− η)N(η) ∫ t 0 ψ1 ( ℵ, U(ℵ)− Un(ℵ) ) dℵ = 2(1− η) (2− η)N(η) ψ1 ( t,−Υ1n(t) ) + 2η (2− η)N(η) ∫ t 0 ψ1 ( ℵ,−Υ1n(ℵ) ) dℵ E(t)− En(t) = 2(1− η) (2− η)N(η) ψ2 ( t, E(t)− En(t) ) + 2η (2− η)N(η) ∫ t 0 ψ2 ( ℵ, E(ℵ)− En(ℵ) ) dℵ = 2(1− η) (2− η)N(η) ψ2 ( t,−Υ2n(t) ) + 2η (2− η)N(η) ∫ t 0 ψ2 ( ℵ,−Υ2n(ℵ) ) dℵ I(t)− In(t) = 2(1− η) (2− η)N(η) ψ3 ( t, I(t)− In(t) ) + 2η (2− η)N(η) ∫ t 0 ψ3 ( ℵ, I(ℵ)− In(ℵ) ) dℵ = 2(1− η) (2− η)N(η) ψ3 ( t,−Υ3n(t) ) + 2η (2− η)N(η) ∫ t 0 ψ3 ( ℵ,−Υ3n(ℵ) ) dℵ V (t)− Vn(t) = 2(1− η) (2− η)N(η) ψ4 ( t, V (t)− Vn(t) ) + 2η (2− η)N(η) ∫ t 0 ψ4 ( ℵ, V (ℵ)− Vn(ℵ) ) dℵ A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 11 = 2(1− η) (2− η)N(η) ψ4 ( t,−Υ4n(t) ) + 2η (2− η)N(η) ∫ t 0 ψ4 ( ℵ,−Υ4n(ℵ) ) dℵ F (t)− Fn(t) = 2(1− η) (2− η)N(η) ψ5 ( t, F (t)− Fn(t) ) + 2η (2− η)N(η) ∫ t 0 ψ5 ( ℵ, F (ℵ)− Fn(ℵ) ) dℵ = 2(1− η) (2− η)N(η) ψ5 ( t,−Υ5n(t) ) + 2η (2− η)N(η) ∫ t 0 ψ5 ( ℵ,−Υ5n(ℵ) ) dℵ Q(t)−Qn(t) = 2(1− η) (2− η)N(η) ψ6 ( t, Q(t)−Qn(t) ) + 2η (2− η)N(η) ∫ t 0 ψ6 ( ℵ, Q(ℵ)−Qn(ℵ) ) dℵ = 2(1− η) (2− η)N(η) ψ6 ( t,−Υ6n(t) ) + 2η (2− η)N(η) ∫ t 0 ψ6 ( ℵ,−Υ6n(ℵ) ) dℵ (4.10) By taking the norm on both sides, we get ∥∥U(t)−Un(0)− 2(1− η) (2− η)N(η) ψ1 ( t, U(t)−Un(t) ) − 2η (2− η)N(η) ∫ t 0 ψ1 ( ℵ, U(ℵ)−Un(ℵ) ) dℵ ∥∥ ≤ ∥Υ1n(t)∥ { 1 + ( 2(1− η) (2− η)N(η) Λ1 ) + ( 2η (2− η)N(η) Λ1t )} (4.11) Now take limit n→ ∞ in an equation (4.11), we get ∥Υ1n(t)∥ → 0. Hence, we get U(t) = U(0)− 2(1− η) (2− η)N(η) ψ1 ( t, U(t) ) − 2η (2− η)N(η) ∫ t 0 ψ1 ( ℵ, U(ℵ) ) dℵ (4.12) Similarly, as limit n→ ∞, we get ∥Υ2n(t)∥ → 0, ∥Υ3n(t)∥ → 0, ∥Υ4n(t)∥ → 0, ∥Υ5n(t)∥ → 0, ∥Υ6n(t)∥ → 0. Consequently, like (4.12) there exists solutions of (2.1) Theorem 2. Show that the system (2.1) has a only one solution. Proof. Let there is another solution of the system (2.1), say U∗(t), E∗(t), I∗(t), V ∗(t), F ∗(t), and Q∗(t), then we get U(t)− U∗(t) = 2(1− η) (2− η)N(η) [ ψ1 ( t, U(t) ) − ψ1 ( t, U∗(t) )] + 2η (2− η)N(η) ∫ t 0 [ ψ1 ( ℵ, U(ℵ) ) − ψ1 ( ℵ, U∗(ℵ) )] dℵ, E(t)− E∗(t) = 2(1− η) (2− η)N(η) [ ψ2 ( t, E(t) ) − ψ2 ( t, E∗(t) )] + 2η (2− η)N(η) ∫ t 0 [ ψ2 ( ℵ, E(ℵ) ) − ψ2 ( ℵ, E∗(ℵ) )] dℵ, I(t)− I∗(t) = 2(1− η) (2− η)N(η) [ ψ3 ( t, I(t) ) − ψ3 ( t, I∗(t) )] + 2η (2− η)N(η) ∫ t 0 [ ψ3 ( ℵ, I(ℵ) ) − ψ3 ( ℵ, I∗(ℵ) )] dℵ, V (t)− V ∗(t) = 2(1− η) (2− η)N(η) [ ψ4 ( t, V (t) ) − ψ4 ( t, V ∗(t) )] A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 12 + 2η (2− η)N(η) ∫ t 0 [ ψ4 ( ℵ, V (ℵ) ) − ψ4 ( ℵ, V ∗(ℵ) )] dℵ, F (t)− F ∗(t) = 2(1− η) (2− η)N(η) [ ψ5 ( t, F (t) ) − ψ5 ( t, F ∗(t) )] + 2η (2− η)N(η) ∫ t 0 [ ψ5 ( ℵ, F (ℵ) ) − ψ5 ( ℵ, F ∗(ℵ) )] dℵ, Q(t)−Q∗(t) = 2(1− η) (2− η)N(η) [ ψ6 ( t, Q(t) ) − ψ6 ( t, Q∗(t) )] + 2η (2− η)N(η) ∫ t 0 [ ψ6 ( ℵ, Q(ℵ) ) − ψ6 ( ℵ, Q∗(ℵ) )] dℵ, (4.13) Again by using norm on (4.13), we get ∥U(t)− U∗(t)∥ = 2(1− η) (2− η)N(η) ∥∥∥ψ1 ( t, U(t) ) − ψ1 ( t, U∗(t) )∥∥∥ + 2η (2− η)N(η) ∫ t 0 ∥∥∥ψ1 ( ℵ, U(ℵ) ) − ψ1 ( ℵ, U∗(ℵ) )∥∥∥dℵ, ∥E(t)− E∗(t)∥ = 2(1− η) (2− η)N(η) ∥∥∥ψ2 ( t, E(t) ) − ψ2 ( t, E∗(t) )∥∥∥ + 2η (2− η)N(η) ∫ t 0 ∥∥∥ψ2 ( ℵ, E(ℵ) ) − ψ2 ( ℵ, E∗(ℵ) )∥∥∥dℵ, ∥I(t)− I∗(t)∥ = 2(1− η) (2− η)N(η) ∥∥∥ψ3 ( t, I(t) ) − ψ3 ( t, I∗(t) )∥∥∥ + 2η (2− η)N(η) ∫ t 0 ∥∥∥ψ3 ( ℵ, I(ℵ) ) − ψ3 ( ℵ, I∗(ℵ) )∥∥∥dℵ, ∥V (t)− V ∗(t)∥ = 2(1− η) (2− η)N(η) ∥∥∥ψ4 ( t, V (t) ) − ψ4 ( t, V ∗(t) )∥∥∥ + 2η (2− η)N(η) ∫ t 0 ∥∥∥(ψ4(ℵ, V (ℵ)) ) − ψ4 ( ℵ, V ∗(ℵ) )] dℵ, ∥F (t)− F ∗(t)∥ = 2(1− η) (2− η)N(η) ∥∥∥ψ5 ( t, F (t) ) − ψ5 ( t, F ∗(t) )∥∥∥ + 2η (2− η)N(η) ∫ t 0 ∥∥∥ψ5 ( ℵ, F (ℵ) ) − ψ5 ( ℵ, F ∗(ℵ) )∥∥∥dℵ, ∥Q(t)−Q∗(t)∥ = 2(1− η) (2− η)N(η) ∥∥∥ψ6 ( t, Q(t) ) − ψ6 ( t, Q∗(t) )∥∥∥ + 2η (2− η)N(η) ∫ t 0 ∥∥∥ψ6 ( ℵ, Q(ℵ) ) − ψ6 ( ℵ, Q∗(ℵ) )∥∥∥dℵ. (4.14) Based on Theorems 4.1 and 4.2, the results are ∥U(t)− U∗(t)∥ = Λ1 2(1− η) (2− η)N(η) ∥U(t)− U∗(t)∥ + Λ1 2η (2− η)N(η) t ∥∥∥U(t)− U∗(t) ∥∥∥, A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 13 ∥E(t)− E∗(t)∥ = Λ2 2(1− η) (2− η)N(η) ∥E(t)− E∗(t)∥ + Λ2 2η (2− η)N(η) t ∥∥∥E(t)− E∗(t) ∥∥∥, ∥I(t)− I∗(t)∥ = Λ3 2(1− η) (2− η)N(η) ∥I(t)− I∗(t)∥ + Λ3 2η (2− η)N(η) t ∥∥∥I(t)− I∗(t) ∥∥∥, ∥V (t)− V ∗(t)∥ = Λ4 2(1− η) (2− η)N(η) ∥V (t)− V ∗(t)∥ + Λ4 2η (2− η)N(η) t ∥∥∥V (t)− V ∗(t) ∥∥∥, ∥F (t)− F ∗(t)∥ = Λ5 2(1− η) (2− η)N(η) ∥F (t)− F ∗(t)∥ + Λ5 2η (2− η)N(η) t ∥∥∥F (t)− F ∗(t) ∥∥∥, ∥Q(t)−Q∗(t)∥ = Λ6 2(1− η) (2− η)N(η) ∥Q(t)−Q∗(t)∥ + Λ6 2η (2− η)N(η) t ∥∥∥Q(t)−Q∗(t) ∥∥∥. (4.15) The following is how the computed functions in (4.11) fulfill the non-equalities: ∥U(t)− U∗(t)∥ { 1− 2Λ1 (2− η)N(η) ( 1− η − ηt )} ≤ 0, ∥E(t)− E∗(t)∥ { 1− 2Λ2 (2− η)N(η) ( 1− η − ηt )} ≤ 0, ∥I(t)− I∗(t)∥ { 1− 2Λ3 (2− η)N(η) ( 1− η − ηt )} ≤ 0, ∥V (t)− V ∗(t)∥ { 1− 2Λ4 (2− η)N(η) ( 1− η − ηt )} ≤ 0, ∥F (t)− F ∗(t)∥ { 1− 2Λ5 (2− η)N(η) ( 1− η − ηt )} ≤ 0, ∥Q(t)−Q∗(t)∥ { 1− 2Λ6 (2− η)N(η) ( 1− η − ηt )} ≤ 0. (4.16) From the final equation, we establish that U(t) = U∗(t), E(t) = E∗(t), I(t) = I∗(t), V (t) = V ∗(t), F (t) = F ∗(t), Q(t) = Q∗(t). (4.17) 5. Stability This section examines the stability conditions of approximate solutions and explores the use of the iterative Laplace transform method in the fractional malaria fever model. A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 14 5.1. Iterative Laplace Transform Method Examine the Malaria infection framework (2.1) with starting values (2.2). Utilizing the Laplace method on both ends of equation (2.1), we get sL(U(t))− U(0) s+ η(1− s) = L ( qπ + (1− b)ξI + ϕV − (σ + ψ + ϱ)U ) sL(E(t))− E(0) p+ η(1− s) = L ( σU − (χ+ ϱ)E ) sL(I(t))− I(0) s+ η(1− s) = L ( χE − (ξ + ϱ+ ∂)I ) sL(V (t))− V (0) s+ η(1− s) = L ( (1− q)π + µQ+ ψU + bξI − (θ + ϕ+ ϱ)V ) sL(F (t))− F (0) s+ η(1− s) = L ( θV − (ε+ ϱ)F ) sL(Q(t))−Q(0) s+ η(1− s) = L ( ϵF − (µ+ ϱ+ κ)Q ) (5.1) After reordering, we get L(U(t)) = U(0)) s + ( s+ η(1− s) s ) L ( qπ + (1− b)ξI + ϕV − (σ + ψ + ϱ)U ) , L(E(t)) = E(0)) s + ( s+ η(1− s) s ) L ( σU − (χ+ ϱ)E ) , L(I(t)) = I(0)) s + ( s+ η(1− s) s ) L ( χE − (ξ + ϱ+ ∂)I ) , L(V (t)) = V (0)) s + ( s+ η(1− s) s ) L ( (1− q)π + µQ+ ψU + bξI − (θ + ϕ+ ϱ)V ) , L(F (t)) = F (0)) s + ( s+ η(1− s) s ) L ( θV − (ε+ ϱ)F ) , L(Q(t)) = Q(0)) s + ( s+ η(1− s) s ) L ( ϵF − (µ+ ϱ+ κ)Q ) . (5.2) In addition, the inverse Laplace transform applied to equation (5.2) produces U(t) = U(0) + L−1 [( s+ η(1− s) s ) L ( qπ + (1− b)ξI + ϕV − (σ + ψ + ϱ)U )] , E(t) = E(0) + L−1 [( s+ η(1− s) s ) L ( σU − (χ+ ϱ)E )] , I(t) = I(0) + L−1 [( s+ η(1− s) s ) L ( χE − (ξ + ϱ+ ∂)I )] , V (t) = V (0) + L−1 [( s+ η(1− s) s ) L ( (1− q)π + µQ+ ψU + bξI − (θ + ϕ+ ϱ)V )] , A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 15 F (t) = F (0) + L−1 [( s+ η(1− s) s ) L ( θV − (ε+ ϱ)F )] , Q(t) = Q(0) + L−1 [( s+ η(1− s) s ) L ( ϵF − (µ+ ϱ+ κ)Q )] . (5.3) The method’s infinite series solutions are provided as, U = ∞∑ n=0 Un, E = ∞∑ n=0 En, I = ∞∑ n=0 In, V = ∞∑ n=0 Vn, F = ∞∑ n=0 Fn, Q = ∞∑ n=0 Qn (5.4) The recursive formula that follows is then obtained by applying beginning conditions. Un+1(t) = U(0) + L−1 [( s+ η(1− s) s ) L ( qπ + (1− b)ξIn + ϕVn − (σ + ψ + ϱ)Un )] , En+1(t) = E(0) + L−1 [( s+ η(1− s) s ) L ( σUn − (χ+ ϱ)En )] , In+1(t) = I(0) + L−1 [( s+ η(1− s) s ) L ( χEn − (ξ + ϱ+ ∂)In )] , Vn+1(t) = V (0) + L−1 [( s+ η(1− s) s ) L ( (1− q)π + µQn + ψUn + bξIn − (θ + ϕ+ ϱ)Vn )] , Fn+1(t) = F (0) + L−1 [( s+ η(1− s) s ) L ( θVn − (ε+ ϱ)Fn )] , Qn+1(t) = Q(0) + L−1 [( s+ η(1− s) s ) L ( ϵFn − (µ+ ϱ+ κ)Qn )] . (5.5) 5.2. Analysis of iteration method As a Banach space, consider (B, ∥ · ∥). Define F as the self-map of B1, whose the repeating process is indicated by zn+1 = g(J, zn). Let J(F ) represent the fixed-point set on F . Additionally, F must have At least one item for zn to approach a point i ∈ J(F ). After defining kn = ∥yn+1 − g(F, yn)∥, let {yn ∈ B1}. The iteration technique zn+1 = g(F, zn) is referred to as F -stable if lim n→∞ kn = 0 implies that lim n→∞ yn = i. In contrast, we declare that there is an upper constraint on the sequence {yn}. If all these requirements are met for zn+1 = Fzn, then this iteration-also referred to as Picard’s iteration-is F− stable. Theorem 3. Suppose (B, ∥ · ∥) as a Banach space and let F be a self-map on B fulfilling ∥Fw −Fz∥ ≤M∥w − z∥+m∥w − z∥. For each w, z ∈ B with 0 ≤ M, 0 ≤ m < 1. Assume F is Picard stable. Suppose the equations (5.5), which are connected to (2.1). Un+1(t) = U(0) + L−1 [( s+ η(1− s) s ) L ( qπ + (1− b)ξIn + ϕVn − (σ + ψ + ϱ)Un )] A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 16 En+1(t) = E(0) + L−1 [( s+ η(1− s) s ) L ( σUn − (χ+ ϱ)En )] In+1(t) = I(0) + L−1 [( s+ η(1− s) s ) L ( χEn − (ξ + ϱ+ ∂)In )] Vn+1(t) = V (0) + L−1 [( s+ η(1− s) s ) L ( (1− q)π + µQn + ψUn + bξIn − (θ + ϕ+ ϱ)Vn )] Fn+1(t) = F (0) + L−1 [( s+ η(1− s) s ) L ( θVn − (ε+ ϱ)Fn )] Qn+1(t) = Q(0) + L−1 [( s+ η(1− s) s ) L ( ϵFn − (µ+ ϱ+ κ)Qn )] where s+ η(1− s) s is a fractional Lagrange multiplier Theorem 4. Let F be a mapping on itself defined as F(Un(t)) = Un+1(t) = U(0) + L−1 [( s+ η(1− s) s ) L ( qπ + (1− b)ξIn + ϕVn − (σ + ψ + ϱ)Un )] F(En(t)) = En+1(t) = E(0) + L−1 [( s+ η(1− s) s ) L ( σUn − (χ+ ϱ)En )] F(In(t)) = In+1(t) = I(0) + L−1 [( s+ η(1− s) s ) L ( χEn − (ξ + ϱ+ ∂)In )] F(Vn(t)) = Vn+1(t) = V (0) + L−1 [( s+ η(1− s) s ) L ( (1− q)π + µQn + ψUn + bξIn − (θ + ϕ+ ϱ)Vn )] F(Fn(t)) = Fn+1(t) = F (0) + L−1 [( s+ η(1− s) s ) L ( θVn − (ε+ ϱ)Fn )] F(Qn(t)) = Qn+1(t) = Q(0) + L−1 [( s+ η(1− s) s ) L ( ϵFn − (µ+ ϱ+ κ)Qn )] is F-stable in L1(a, b) if ( (1− b)ξh1(s) + ϕh2(s) + (σ + ψ + ϱ)h3(s) ) < 1( σf1(s)− (χ+ ϱ)f2(s) ) < 1( χg1(s)− (ξ + ϱ+ ∂)g2(s) ) < 1( qπ + (1− b)ξz1(s) + ϕz1(s)− (σ + ψ + ϱ)z1(s) ) < 1( θx1(s)− (ε+ ϱ)x2(s) ) < 1( ϵy1(s)− (µ+ ϱ+ κ)y2(s) ) < 1 Proof. Here, we shall demonstrate that F has a fixed point. So for any (m,n) in R A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 17 where R = N× N, we evaluate the followings. F(Un(t))−F(Um(t)) = L−1 [( s+ η(1− s) s ) L ( qπ + (1− b)ξIn + ϕVn − (σ + ψ + ϱ)Un )] − L−1 [( s+ η(1− s) s ) L ( qπ + (1− b)ξIm + ϕVm − (σ + ψ + ϱ)Um )] Taking norm ∥F(Un(t))−F(Um(t))∥ = ∥∥∥∥L−1 [( s+ η(1− s) s ) L ( qπ + (1− b)ξIn + ϕVn − (σ + ψ + ϱ)Un )] − L−1 [( s+ η(1− s) s ) L ( qπ + (1− b)ξIm + ϕVm − (σ + ψ + ϱ)Um )]∥∥∥∥ ∥F(Un(t))−F(Um(t))∥ = ∥∥∥∥L−1 [( s+ η(1− s) s ) L ( qπ + (1− b)ξIn + ϕVn − (σ + ψ + ϱ)Un )] − L−1 [( s+ η(1− s) s ) L ( qπ + (1− b)ξIm + ϕVm − (σ + ψ + ϱ)Um )]∥∥∥∥ ∥F(Un(t))−F(Um(t))∥ = ∥∥∥∥L−1 [( s+ η(1− s) s )( L ( (1− b)ξIn + ϕVn − (σ + ψ + ϱ)Un )] − L ( (1− b)ξIm + ϕVm − (σ + ψ + ϱ)Um ))]∥∥ ∥F(Un(t))−F(Um(t))∥ ≤ L−1 [( s+ η(1− s) s ) L [∥∥((1− b)ξ(In − Im)) ∥∥ (5.6) + ∥∥(ϕ(Vn − Vm)) ∥∥+ ∥∥− ((σ + ψ + ϱ)(Un − Um)) ∥∥]] (5.7) As the obtained solutions perform a comparable role, we suppose that ∥Un(t)− Um(t)∥ = ∥Vn(t)− Vm(t)∥, ∥Un(t)− Um(t)∥ = ∥In(t)− Im(t)∥ Also, Un, En, In, Vn, and Fn are convergent sequences, thus they are bounded. Consider the equation in (5.6). We have. ∥F(Un(t))−F(Um(t))∥ ≤ L−1 [( s+ η(1− s) s ) L [( (1− b)ξ + ϕ+ (σ + ψ + ϱ) ) ∥Un − Um∥ ]] ≤ ( (1− b)ξh1(s) + ϕh2(s) + (σ + ψ + ϱ)h3(s) ) ∥Un − Um∥ (5.8) where h1, h2 and h3 are functions from L−1 [ L ( s+η(1−s) s )] In the same manner, we can get ∥F(Un(t))−F(Um(t))∥ ≤ ( (1− b)ξh1(s) + ϕh2(s) + (σ + ψ + ϱ)h3(s) ) ∥Un − Um∥ A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 18 ∥F(En(t))−F(Em(t))∥ ≤ ( σf1(s)− (χ+ ϱ)f2(s) ) ∥En − Em∥ ∥F(In(t))−F(Im(t))∥ ≤ ( χg1(s)− (ξ + ϱ+ ∂)g2(s) ) ∥In − Im∥ ∥F(Vn(t))−F(Vm(t))∥ ≤ ( qπ + (1− b)ξz1(s) + ϕz1(s)− (σ + ψ + ϱ)z1(s) ) ∥Vn − Vm∥ ∥F(Fn(t))−F(Fm(t))∥ ≤ ( θx1(s)− (ε+ ϱ)x2(s) ) ∥Fn − Fm∥ ∥F(Qn(t))−F(Qm(t))∥ ≤ ( ϵy1(s)− (µ+ ϱ+ κ)y2(s) ) ∥Qn −Qm∥ (5.9) Hence, the function F− has a fixed point. Now show that F meets every the conditions listed above. Theorem 4.1. Here (5.8) and (5.9) is valid, likewise by using Φ = (0, 0, 0, 0, 0, 0), Φ =  ( (1− b)ξh1(s) + ϕh2(s) + (σ + ψ + ϱ)h3(s) ) < 1( σf1(s)− (χ+ ϱ)f2(s) ) < 1( χg1(s)− (ξ + ϱ+ ∂)g2(s) ) < 1( qπ + (1− b)ξz1(s) + ϕz1(s)− (σ + ψ + ϱ)z1(s) ) < 1( θx1(s)− (ε+ ϱ)x2(s) ) < 1( ϵy1(s)− (µ+ ϱ+ κ)y2(s) ) < 1 All of the requirements in Theorem 4.2 are met by F . Thus, F is Picard F− stable. 6. Illustration At this stage, we perform computational experiments of the Caputo-Fabrizio operator model for Malaria and Malnutrition, represented by equation (2.1), considering initial con- ditions U0 = 0, E0 = 0, I0 = 0, V0 = 0, F0 = 0, Q0 = 0, across various fractional order values η ∈ (0, 1). The corresponding physical factors together have values as provided. A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 19 Table 1: 1 Model parameters Variable Explanation Value Reference π The rate at which the host population is being re- cruited 1520 [31] χ, ϵ The rate of transition from infected to infectious 0.05, 0.001333 [11] ξ, µ The rate of translation from infectious to suscepti- ble 0.10333, 0.08333 [11] b Proportion of people who successfully overcome malaria 0.027 Assumed ϱ Rate of mortality due to natural causes 1 365× 5 Assumed ∂ Malaria mortality rate 0.00183542 [32] κ Mortality rate caused by malaria and malnutrition 9.4 [32] ϕ Transition rate from malnourished susceptible indi- viduals to properly nourished susceptible individu- als 0.00000986 [33] q Fraction of new entries as properly nourished sus- ceptible 2 3 Figure 1: Near estimate of nourished susceptible individuals A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 20 Figure 2: Near estimate of carrier exposed mosquitoes individuals Figure 3: Near estimate of infectious mosquitoes individuals A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 21 Figure 4: Near estimate of malnourished susceptible individuals Figure 5: Near estimate of malnourished exposed individuals A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 22 Figure 6: Near estimate of malnourished Infected individuals 7. Numerical Results and Discussion Diagrams 1 through 6 depict the results of computational experiments illustrating nourished susceptible individuals U(t), nourished exposed E(t), nourished contagious per- sons I(t), malnourished susceptible persons V (t), malnourished exposed F (t),and malnour- ished infectious individuals Q(t) obtained for various values of (η = 1, 0.9, 0.8, 0.7) utilizing the Iterative Laplace Transform Method(ILTM) applied to a fractional Malaria and Malnu- trition model. These simulations demonstrate that ILTM accurately predicts the nature of these parameters within the specified area. Moreover, the simulations reveal that variations in the parameter values significantly impact the model dynamics. Specifically, fractional orders exhibit minimal influence on the spread patterns of the Malaria and Malnutrition model. Given that mathematical models serve as symbolic representations of biological sys- tems, they inherently inherit the loss of information during construction, potentially leading to imprecise predictions of model outcomes. Therefore, we opt to conduct a graphical sen- sitivity analysis of model parameters to delve into this issue further. Figure 1 illustrates a sharp increase in nourished susceptible individuals U(t) within the initial days across various values of the order η. In Figure 2, the graph depicting nour- ished exposed individuals E(t) demonstrates a rapid increase in the early days followed by a decrease. Rapid growth of nourished infectious individuals I(t) is observed in Figure 3, particularly with non-integer values of η. Conversely, Figure 4 shows a decline in the popu- lation of malnourished susceptible individuals V (t) after a few days. Malnourished exposed individuals F (t) in Figure 5 exhibit a rapid increase in the initial days, while malnourished infectious individuals Q(t) in Figure 6 experience a slower increase with varying values of η. Notably, it is observed that computational outcomes consistently rely on the fractional-time derivative η, indicating the significant influence of specific fractional operators such as the A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 23 of 25 Caputo–Fabrizio operator in providing precise predictions with reduced noise. Furthermore, the hybrid nature of the Caputo–Fabrizio operator proves robust in capturing the complex- ity of the model and offering meaningful predictions. Malaria exacerbates the incidence of malnutrition. As anticipated, implementing protective strategies, including the application of mosquito repellent (periodic residual spraying) and insecticide-coated bed nets to de- crease the daily count of bites from female mosquitoes will significantly reduce the number of children contracting malaria. Additionally, since malaria-infected children are vulnerable to malnutrition due to factors like imbalanced eating habits or reduced appetite, improving children’s diets could be beneficial to alleviate the transmission of critical malaria episodes. 8. Conclusions In this study, we investigated the Caputo-Fabrizio fractional-order system of malaria and malnutrition, examining the spread patterns through both direct and indirect transmis- sion pathways of the disease through the iterative Laplace transform method. Additionally, by employing the Banach theorem, we established results concerning the presence, unique- ness, and stability of equilibrium solutions. The sequence solutions derived from this robust approach demonstrate a promising ability to mitigate the devastating impact of malaria and malnutrition over different time intervals and to combat a significant mortality factor. It is evident that the efficacy of this method can be significantly improved by streamlining processes and incorporating additional components. We utilized a randomized set of pa- rameters in the behavior of the previously mentioned epidemic model. Malaria and malnutrition stand out as the primary contributors to childhood mor- tality, particularly as repeated malaria exposure notably affects the nutritional well-being of children. The current analysis, while informative, is not comprehensive and offers po- tential avenues for extension. While malaria predominantly impacts children under five years old, malnutrition impacts the whole family. Looking ahead, research endeavors might explore expanded, intricate models that incorporate the dynamics of individuals aged five and above. malaria control, noting that innovative tools and climate adaptation strategies should be integrated into national programs with adequate funding, community engage- ment, and equitable implementation, alongside interventions such as nutrition programs and insecticide-treated nets. Acknowledgements “The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2025/01/5180)” Authors’ contributions : Conceptualization: AS, SH; Formal analysis: KSN; Inves- tigation: AS, SH, KSN; Methodology: AS; Software: SH, KSN; Validation: KSN; Writing - original draft: MAS, SH, KSN References [1] J. F. Friedman, M. Kurtis, J. D. Ramadhan, M. Opollo, D. E. Lanar, and P. E. Duffy. Malaria is related to decreased nutritional status among male adolescents and adults A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 24 of 25 in the setting of intense perennial transmission. The Journal of Infectious Diseases, 188(3):449–457, 2003. [2] C. Shiff, W. Checkley, P. Winch, Z. Premji, J. Minjas, and P. Lubega. Changes in weight gain and anaemia attributable to malaria in tanzanian children living under holoendemic conditions. Transactions of the Royal Society of Tropical Medicine and Hygiene, 90(3):262–265, 1996. [3] F. O. Ter Kuile, D. J. Terlouw, S. K. Kariuki, P. A. Phillips-Howard, L. B. Mirel, W. A. Hawley, and B. L. Nahlen. Impact of permethrin-treated bed nets on malaria, anemia and growth in infants in an area of intense perennial malaria transmission in western kenya. The American Journal of Tropical Medicine and Hygiene, 68:68–77, 2003. [4] L. Q. Hung, P. J. De Vries, P. T. Giao, T. Q. Binh, N. V. Nam, M. T. Chong, and P. A. Kager. Nutritional status following malaria control in a vietnamese ethnic minority commune. European Journal of Clinical Nutrition, 59:891–899, 2005. [5] T. Gone, F. Lemango, E. Eliso, S. Yohannes, and T. Yohannes. The association be- tween malaria and malnutrition among under-five children in shashogo district, south- ern ethiopia: a case-control study. Infectious Diseases of Poverty, 6(9), 2017. [6] I. A. McGregor. Malaria: nutritional implications. Reviews of Infectious Diseases, 4:798–804, 1982. [7] W. H. Wernsdorfer and I. A. McGregor. Malaria: Principles and practice of malariology. London: Churchill Livingstone, pages 753–767, 1988. [8] J. S. Edirisinghe. Infections in the malnourished: with special reference to malaria and malnutrition in the tropics. Annals of Tropical Paediatrics, 6:233–237, 1986. [9] M. C. Latham. Needed research on the interactions of certain parasitic diseases and nutrition in humans. Reviews of Infectious Diseases, 4:896–900, 1982. [10] S. Y. Tchoumi, E. Z. Dongmo, J. C. Kamgang, and J. M. Tchuenche. Dynamics of a two-group structured malaria transmission model. Informatics in Medicine Unlocked, 29:100897, 2022. [11] S. Y. Tchoumi, N. Y. Njintang, J. C. Kamgang, and J. M. Tchuenche. Malaria and malnutrition in children: A mathematical model. Franklin Open, 3:100013, 2023. [12] S. Mandal, R. R. Sarkar, and S. Sinha. Mathematical models of malaria - a review. Malaria Journal, 10:202, 2011. [13] A. A. Gebremeskel and H. E. Krogstad. Mathematical modelling of endemic malaria transmission. American Journal of Applied Mathematics, 3(2):36–46, 2015. [14] J. Welch, R. M. Li, U. S. Nair, T. L. Sever, D. E. Irwin, C. Cordon-Rosales, and N. Padilla. Dynamic malaria models with environmental changes. In Proceedings of the Thirty-fourth Southeastern Symposium on System Theory, pages 396–400, Huntsville, 2002. [15] H. M. Yang. Malaria transmission model for different levels of acquired immunity and temperature-dependent parameters(vector). Revista de Saúde Pública, 34:223– 231, 2000. [16] M. U. Ferreira and H. M. Yang. Assessing the effects of global warming and local social and economic conditions on the malaria transmission. Revista de Saúde Pública, 34:214–222, 2000. [17] J. C. Koella and R. Antia. Epidemiological models for the spread of anti-malarial resistance. Malaria Journal, 2, 2003. [18] N. Bacaer and C. Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences and Engineering, 2:227–238, 2005. A. Shaikh, S. Howal, K. S. Nisar / Eur. J. Pure Appl. Math, 18 (4) (2025), 6981 25 [19] G. A. Ngwa and W. S. Shu. A mathematical model for endemic malaria with variable human and mosquito populations. Mathematical and Computer Modeling Journal, 32:747–763, 2000. [20] G. A. Ngwa. Modelling the dynamics of endemic malaria in growing populations. Discrete and Continuous Dynamical Systems - Series B, 4:1173–1202, 2004. [21] Danso Addo. Mathematical model for the control of Malaria. PhD thesis, University of Cape Coast, 2009. [22] J. Tumwiine, J. Y. T. Mugisha, and L. S. Luboobi. A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity. Journal of Applied Mathematics and Computation, 189:1953–1965, 2005. [23] H. Yang, H. Wei, and X. Li. Global stability of an epidemic model for vector borne disease. Journal of Systems Science and Complexity, 23:279–292, 2010. [24] K. S. Nisar, M. Farman, M. Abdel-Aty, and C. Ravichandran. A review of fractional order epidemic models for life sciences problems: Past, present and future. Alexandria Engineering Journal, 95:283–305, 2024. [25] A. Alsaadi, R. Shafqat, A. Al-Quran, and A. M. Djaouti. Poliomyelitis dynamics with fractional order derivatives and deep neural networks. Scientific Reports, 15(1):32023, 2025. [26] A. S. Shaikh and K. S. Nisar. Transmission dynamics of fractional order typhoid fever model using caputo–fabrizio operator. Chaos, Solitons and Fractals, 128:355–365, 2019. [27] A. S. Shaikh, I. N. Shaikh, and K. S. Nisar. A mathematical model of covid-19 using fractional derivative: outbreak in india with dynamics of transmission and control. Advances in Difference Equations, page 373, 2020. [28] S. Khajanchi, K. Sarkar, J. Mondal, K. S. Nisar, and S. F. Abdelwahab. Mathematical modeling of the covid-19 pandemic with intervention strategies. Results in Physics, 25:2211–3797, 2021. [29] A. Turab, R. Shafqat, and S. Muhammad. Predictive modeling of hepatitis b viral dy- namics: a caputo derivative-based approach using artificial neural networks. Scientific Reports, 14:21853, 2024. [30] R. Shafqat and A. Alsaadi. Mathematical and numerical analysis of a fractional siqr epidemic model with normalized caputo-fabrizio operator and machine learning ap- proaches. AIMS Mathematics, 10(9):20235–20261, 2025. [31] F. Forouzannia and A. Gumel. Dynamics of an age-structured two-strain model for malaria transmission. Applied Mathematics and Computation, 250:860–886, 2015. [32] Unicef. https://data.unicef.org/topic/child-health/malaria/. Accessed: [current date]. [33] S. D. Hove-Musekwa, F. Nyabadza, C. Chiyaka, P. Das, A. Tripathi, and Z. Mukan- davire. Modelling and analysis of the effects of malnutrition in the spread of cholera. Mathematical and Computer Modelling, 53(9-10):1583–1595, 2011. Introduction Model Information Preliminaries Existence and Uniqueness Stability Iterative Laplace Transform Method Analysis of iteration method Illustration Numerical Results and Discussion Conclusions Acknowledgements