EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5687 ISSN 1307-5543 – ejpam.com Published by New York Business Global Existence and Sensitivity Analysis of a Caputo-Fabrizio Fractional Order Vector-Borne Disease Model Nekmat Ullah1, Zahir Shah1,∗, Rashid Jan2,3, S. Islam4, Narcisa Vrinceanu5,∗, Muhammad Farhan6, Elisabeta Antonescu7 1 Department of Mathematical Sciences, University of Lakki Marwat, KPK, Pakistan 2 Department of Mathematics, Saveetha School of Engineering (SIMATS), Thandalam 600124, Chennai, Tamil Nadu, India 3 Institute of Energy Infrastructure (IEI), Department of Civil Engineering, College of Engineering, Universiti Tenaga Nasional (UNITEN), Putrajaya Campus, Jalan IKRAM-UNITEN, 43000 Kajang, Selangor, Malaysia 4 Department of Mechanical Engineering, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia 5Faculty of Engineering, “Lucian Blaga” University of Sibiu, Romania 6 School of Mathematical Science, Yangzhou University, Yangzhou 225002, China 7Preclinical Department, Faculty of Medicine, “Lucian Blaga” University, Romania Abstract. In this study, we develop a mathematical model for vector-borne infections within a fractional-order framework, employing the Caputo-Fabrizio fractional derivative to enhance the analysis. The steady states of the system are examined, and the basic reproduction number R0 is derived using the next-generation matrix method. The existence and uniqueness of solutions are established through the application of the Banach fixed-point theorem. A detailed sensitivity analysis of R0 identifies the most critical parameters influencing the disease dynamics. Numerical simulations are performed to validate the theoretical findings and highlight key factors impacting the control and prevention of the infection. This work provides insights into the dynamics of vector- borne infections and offers a robust mathematical approach for optimizing intervention strategies. 2020 Mathematics Subject Classifications: 92D25, 92D30 Key Words and Phrases: Vector-Borne disease, CF fractional derivative, Mathematical mod- eling, Existence and uniqueness, Sensitivity analysis, Numerical results ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5687 Email addresses: zahir@ulm.edu.pk (Z. Shah), rashid ash2000@yahoo.com (R. Jan), sislam@pmu.edu.sa (S. Islam), vrinceanu.narcisai@ulbsibiu.ro (N. Vrinceanu) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 2 of 25 1. Introduction Vector-borne diseases are carried through mosquitoes or different kinds of bugs such as houseflies. Vector-borne ailments are mainly studied in vertebrates like mice, rats, nonhu- man primates (NHPs), and birds. Investigators want those animals to assist them explore how pathogens are moved from one host to another in addition to determine the feeding behaviors of immune and non-immune individuals as well as selectivity in pathogenicity at different times when hosts feed[1]. The [2] talks about the difficulties encountered when trying to control vector-borne diseases, mainly focusing on mosquitoes as vectors. While there is progress in vaccine technology, challenges such as; resistance of vectors to insec- ticides, altering ecosystems and the resurgence of some diseases in new regions are still present. The paper states that in order to better handle them, novel surveillance mech- anisms, more efficient diagnosis procedures and new ways of controlling such organisms must be put up[3]. The vector-borne diseases explained in [4] indicate how climate change is re-resurfacing or emerging their re-emergence. Moreover, it pointed out that weather elements affect vectors such as mosquitoes hence increasing their numbers resulting into higher chances transmission. Mathematical modeling is a cornerstone of modern biology, enabling researchers to explore complex systems, generate new hypotheses, and design efficient solutions for real-world problems [5–7]. In [8], the researchers focussed on how mathematical models can be used to estimate how much vector-borne viral infections will spread out as well as the speed at which they might go ahead thereby creating an im- pact. The idea here is that through interventions aimed at lowering R0 using different approaches one can use such models to control epidemics and pandemic caused by these viruses. The mathematical model for the dynamics of vector-borne diseases transmitted by mosquitoes is presented in this paper [9], integrating aquatic stages of mosquitoes as well as the gonotrophic cycle. It is illustrated that these elements have an effect on the spread of the disease. Consequently, mosquito-borne illnesses can only be treated effi- ciently if they aim at these parts. In [10] employs fractional calculus in the representation of malaria transmission with particular reference to impacts of treatment as well as insec- ticides indicating that disease containment can be optimally achieved through fusion of these techniques using nonlinear ODEs. The fractional-order model with Caputo-Fabrizio derivative is utilized in order to analyze vector-host disease dynamics, such that memory effects (which are important for obtaining better predictions) are emphasized. The pa- per [11] further provides analytical as well as numerical analyses for exploring on disease behavior including control strategies.In this particular case, a nonlinear fractional-order model was applied in order to take a closer look at vector-borne diseases instead of just using traditional integer-order solid-state models; thus making an improvement through incorporating memory effects. As far as stability goes along with following pathogene- sis or epidemiology routes are concerned which leads us to know more about how they can affect each other through time or space[12]. The Caputo-Fabrizio Fractional Order Model which captures the dynamics of COVID-19 pandemic exhibiting greater accuracy than integer-order systems prevalent in past literature by showing improved accuracy when N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 3 of 25 compared against ordinary models for disease spread within community settings; moreover it illustrates better stability and fits actual/empirical facts as reported by [13]. In [14], the authors utilized a fractional-order Caputo-Fabrizio operator for the mod- eling of COVID-19 spread; it is shown to provide better descriptions of disease dynamics than those obtained through using integer-order models. It improves comprehension re- garding how COVID-19 evolves while facilitating for better policies aimed at curbing its spread. In [15] created a broad similitude model of a fractional-order Boost converter applying the Caputo-Fabrizio principle in governing its system of operation; this has also led to improvement thereof by analysing its effectiveness and possible ways of operation with inductors contained therein.The identification of key parameters influencing cholera’s basic reproduction number using sensitivity analysis could assist in the development of intervention strategies that are more effective [16]. The spread of monkeypox in Nigeria in the time of COVID-19 was examined by the researcher in [17], where they estimated the effective reproduction number R0 during the pandemic.Using mathematical modeling, the basic and effective reproduction numbers of EBOV during the 2014 outbreak in West Africa were estimated as in [18]. In [19] the transmission dynamics of Ebola virus using a Caputo fractional derivative model are examined, thus emphasizing its important pa- rameters like sensitivity for effective control strategies.The article [20] examines issues of the existence of solutions in fractional differential equations, providing a number of basic theorems that underlie these types of problems. The authors employ fixed point theorems to achieve these ends, thereby making a major contribution to the discipline of fractional calculus. The work in [21] examines how diphtheria spreads in populations by utilizing Caputo fractional derivatives, as it establishes that fixed-point theorems result in solutions which are unique and existent, and in addition carries sensitivity analysis to comprehend the parameters effects towards this model.A fractional-order Caputo operator is used in this study to model the COVID-19 transmission dynamics such that both asymptomatic and symptomatic cases can be effectively captured leading to enhanced accuracy[22]. The article [23] investigates the role of booster vaccination and public enlightenment in tackling infectious diseases using a fractional-order epidemic model (FOEM). The necessity of booster doses and continuous awareness programs in order to successfully curb outbreaks is underscored by the model that shows that even those who have been vaccinated can have the disease. This research delves into the study of that pine wilt disease is transmitted with help from a fractional derivative by Caputo. The main goal of this model is to give insight that the disease can spread across different communities while helping predict what is likely going to happen next as well as coming up with appropriate measures for its containment by using fractional calculus methods that are more efficient in representing memory and hereditary tendencies during its transmission [24]. Caputo derivatives in fractional-order epidemiological models help remember that COVID-19 spreads remotely[25]. The Caputo HIV and Malaria co-infection model applies fractional-order derivatives to appropriately mimic these diseases dynamics and relationships[26]. Investigating rumor spread model dynamics with fractional piecewise derivative[27]. Epidemic models in fractions are a way of using half calculus to explain better how certain diseases, which are contagious spread, N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 4 of 25 accounting for those times-past effect and difference in population with different hosts [28]. While not all patients receiving antibiotic therapy develop diarrhea, HA CDI is an important and potentially devastating complication in hospitalized patients who require post-treatment follow up and therapy[29]. Since each patient has their own differences, it is a good idea to analyze the individual condition study and find the proper method for each patient. The treatment plan can be tailored best based on the factors [size, type] and recovery time patient is expected to have before suggesting the treatment plan. The effective outcome and fast recovery can be improved with personalized plan of care[30]. Advancing robotics have made surgery more accurate, but because of their prices, which are prohibitively expensive compared to laparoscopy, the future of surgery is these ad- vancements. However, robotic surgery is now being used more often because of its advan- tages and we know that there are benefits to the practice, such as using ultrasound with contrast agents during inoperative. These have been followed by many novel developed surgical approach allowed surgical outcome to be improved[31]. The factors to consider for selection of the best treatment approach include the physician’s experience, patient’s characteristics and preference[32]. The importance of fractional epidemic models lies in their ability to generalize tradi- tional approaches, offering greater flexibility in modeling various disease scenarios [33, 34]. In this paper, we present a Caputo-Fabirizo fractional-order VBD model and conduct supplementary qualitative analysis. The effects of parameters associated with R0 were investigated through sensitivity analyses. This helps identify key elements of the model that should be targeted for effective control strategies. The theoretical results of the vector borne disease model were shown using numerical simulations, which provided information regarding the dynamics. Moreover, we prefer Caputo-Fabrizio derivatives to integer or- der derivatives as described by CF derivatives because they address real world problems. Specifically, the equilibrium points, R0, sensitivity as well as its existence and uniqueness are the models under consideration. Section 2 deals with basics preliminaries concept of fractional calculus, In section 3 we formulated CF fractional-order model for VBD and proved the basics results from our derived model. Whereas in section 4 we present the sensitivity analysis. In section 5 we obtain numerical simulation from numerical scheme to look into the impact of various parameters governing the in the dynamics of VBD transmission, and finally, there is the conclusion of our research work. 2. Preliminaries This section covered some of the most important theoretical ideas related to fractional derivatives, which are necessary for demonstrating the theory analysis of the model. Definition 1. [21].The order of the CF fractional derivative α ∈ (0, 1) is equivalent to: for the function g(s) CFDγg(s) = 1 1− γ ∫ s 0 g′(ξ)e − γ 1−γ (s−ξ) dξ. (1) N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 5 of 25 The definition includes a non-singular exponential kernel, which sets it apart from tradi- tional fractional derivatives, and singular kernels are typically found in these definitions. Definition 2. A generalized form of the Caputo-Fabrizio Fractional fraction is given as [21]: cDp 0,t ψ(t) = 1 Γ(1− p) ∫ t 0 1 (t− y)p d dt ψ(y)dy, t > 0, (2) is known as the CF derivative of order p of the ψ function. For this, if p → 1, then EDp 0,tψ(t) = d dtψ(t). Lemma 1. [21]. Assuming that ψ : [0, b] → R is a continuous and x ∈ C1[0, b]. Consider{ cDp 0,t x(t) = ψ(t), t ∈ [0, b], 0 < p ≤ 1, x(0) = x0, x0 ∈ R. (3) If x(t) satisfies, then x(t) is a solution of the problem (3). x(t) = x0 + 1 Γ(q) ∫ t 0 ψ(y)(t− y)p−1dy. 3. Caputo-Fabirizo Fractional VBD Model The Caputo-Fabrizio fractional-order vector-borne disease (VBD) model is formu- lated to achieve improved representation of dynamics of vector-host interactions using the Caputo-Fabrizio derivative, which imparts a non-singular kernel and memory effects into the model. This methodology enhances fidelity in representing the disease trans- mission processes encompassing host and vector populations as opposed to conventional integer-order models[35]. A vector-borne disease transmission model has humans and vec- tors in its population, where people are classified as susceptible, exposed, infected, and recovered (Sh, Eh, Ih, Rh), while vectors are classified as susceptible, exposed, and infected (Sv, Ev, Iv). The population is susceptible to natural mortality rates, and the model im- plies that humans who have recovered are immune. The paper analyzes the dynamics of disease propagation with fractional calculus using Caputo-fabirizo derivatives, showing the improved accuracy of fractional-order models. The Caputo-Fabirizo fractional model for VBD is as follow:  dαSh duα = Λh − β1Sh Iv Nv − µhSh, dαEh duα = β1Sh Iv Nv − (σh + µh)Eh, dαIh duα = σhEh − (γ + µh)Ih, dαRh duα = γIh − µhRh, dαSv duα = Λv − β2Sv Ih Nh − µvSv, dαEv duα = β2Sv Ih Nh − (σv + µv)Ev, dαIv duα = σvEv − µvIv. (4) N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 6 of 25 The definition of its CF derivative operator is as follows: dαf dtα = 1 Γ(1− α) ∫ t 0 (t− s)−αf ′(s) ds, (5) and Γ(1− α) is the gamma function. with I.C: Sh(0) ≥ 0, Eh(0) ≥ 0, Ih(0) ≥ 0, Rh(0) ≥ 0, Sv(0) ≥ 0, Ev(0) ≥ 0, Iv(0) ≥ 0. Table 1 explains each state variable and parameter associated with the model (4). Table 1: Meaning of Variable and Parameter Variable / Parameter Meaning Sh(t) Susceptible Population of human Eh(t) Exposed population of human Ih(t) Infected population of human Rh(t) Recovered population of human Sv(t) Susceptible population of vector Ev(t) Exposed population of vector Iv(t) Infected population of vector α Fractional derivative order, 0 < α ≤ 1 Λh Birth rate for humans Λv Birth rate for vectors β1 Transmission rate from vectors to humans β2 Transmission rate from humans to vectors σh Progression rate from exposed to infected for humans σv Progression rate from exposed to infected for vectors γ Recovery rate for humans µh Mortality rate for humans µv Mortality rate for vectors Nh Total human population Nv Total vector population 3.1. Analysis of the Caputo-Fabrizio Fractional Model In this section, we analyze the and existence uniqueness results of the(4) model by using the Banach fixed point theorem. Note that in Q = [0, b] and sup norm all continuous real value functions and Y(Q) represent Banach spaces N = Y(Q)× Y(Q)× Y(Q)× Y(Q)× Y(Q)× Y(Q)× Y(Q), with norm ∥(Sh, Eh, Ih, Rh, Sv, Ev, Iv)∥ = ∥Sh∥+ ∥Eh∥+ ∥Ih∥+ ∥Rh∥+ ∥Sv∥+ ∥Ev∥+ ∥Iv∥, N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 7 of 25 where ∥Sh∥ = max u∈Q |Sh(u)|, ∥Eh∥ = max u∈Q |Eh(u)|, ∥Ih∥ = max u∈Q |Ih(u)|, ∥Rh∥ = max u∈Q |Rh(u)|, ∥Sv∥ = max u∈Q |Sv(u)|, ∥Ev∥ = max u∈Q |Ev(u)|, ∥Iv∥ = max u∈Q |Iv(u)|. For each equation in the(4) system, apply Hp 0 obtained:  Sh(u) = Sh(0) +Hp 0[Λh − β1Sh Iv Nv − µhSh], Eh(u) = Eh(0) +Hp 0[β1Sh Iv Nv − (σh + µh)Eh], Ih(u) = Ih(0) +Hp 0[σhEh − (γ + µh)Ih], Rh(u) = Rh(0) +Hp 0[γIh − µhRh], Sv(u) = Sv(0) +Hp 0[Λv − β2Sv Ih Nh − µvSv], Ev(u) = Ev(0) +Hp 0[β2Sv Ih Nh − (σv + µv)Ev], Iv(u) = Iv(0) +Hp 0[σvEv − µvIv]. (6) Setting the (6)  ψ1(u, Sh) = Λh − β1Sh Iv Nv − µhSh, ψ2(u,Eh) = β1Sh Iv Nv − (σh + µh)Eh, ψ3(u, Ih) = σhEh − (γ + µh)Ih, ψ4(u,Rh) = γIh − µhRh, ψ5(u, Sv) = Λv − β2Sv Ih Nh − µvSv, ψ6(u,Ev) = β2Sv Ih Nh − (σv + µv)Ev, ψ7(u, Iv) = σvEv − µvIv. (7) The Lipschitz condition is ψi, i = 1, 2, . . . , 7, this implies that there exist an upper bound of Sh(u), Eh(u), Ih(u), Rh(u), Sv(u), Ev(u) and Iv(u). In fact, if we consider two functions, Sh and Sh1, such that ∥ψ1 − ψ1∥ = ∥∥∥∥(β1 ∥Iv∥Nv + µh ) (Sh − Sh1) ∥∥∥∥ ≤ ∥∥∥∥(β1 ∥Iv∥Nv + µh ) (Sh − Sh1) ∥∥∥∥ , (8) η1 = (β1n1 + µh), n1 = maxt∈Q ||Iv || Nv , where ∥ψ1 − ψ1∥ ≤ η1 ∥Sh − Sh1∥ , (9) 0 ≤ η1 < 1 and ψ1 meet Lipschitz standards. By using the same procedure as before in N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 8 of 25 (8) and (9), we can get ∥ψ2 − ψ2∥ ≤ η2 ∥Eh − Eh1∥ , ∥ψ3 − ψ3∥ ≤ η3 ∥Ih − Ih1∥ , ∥ψ4 − ψ4∥ ≤ η4 ∥Rh −Rh1∥ , ∥ψ5 − ψ5∥ ≤ η5 ∥Sv − Sv1∥ , ∥ψ6 − ψ6∥ ≤ η6 ∥Ev − Ev1∥ , ∥ψ7 − ψ7∥ ≤ η7 ∥Iv − Iv1∥ . (10) In the above (10), we have η2 = σh + µh, η3 = γ + µh, η4 = µh, η5 = (β2n5 + µh) , n5 = max u∈Q ∥Ih∥ Nh , η6 = σv + µv, η7 = µv. Thus, system (6) reconstructed the as follows as system (7): Sh(u)− Sh(0) = 1 Γ(p) ∫ u 0 (u− y)p−1ψ1(y, Sh(u))dy, Eh(u)− Eh(0) = 1 Γ(p) ∫ u 0 (u− y)p−1ψ2(y,Eh(u))dy, Ih(u)− Ih(0) = 1 Γ(p) ∫ u 0 (u− y)p−1ψ3(y, Ih(u))dy, Rh(u)−Rh(0) = 1 Γ(p) ∫ u 0 (u− y)p−1ψ4(y,Rh(u))dy, Sv(u)− Sv(0) = 1 Γ(p) ∫ u 0 (u− y)p−1ψ5(y, Sv(u))dy, Ev(u)− Ev(0) = 1 Γ(p) ∫ u 0 (u− y)p−1ψ6(y,Ev(u))dy, Iv(u)− Iv(0) = 1 Γ(p) ∫ u 0 (u− y)p−1ψ7(y, Ih(u))dy. (11) N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 9 of 25 The recursive form of (11) is as follows: Shn(u) = 1 Γ(p) ∫ u 0 (u− y)p−1ψ1 ( y, Sh(n−1)(u) ) dy, Ehn(u) = 1 Γ(p) ∫ u 0 (u− y)p−1ψ2 ( y,Eh(n−1)(u) ) dy, Ihn(u) = 1 Γ(p) ∫ u 0 (u− y)p−1ψ3 ( y, Ih(n−1)(u) ) dy, Rhn(u) = 1 Γ(p) ∫ u 0 (u− y)p−1ψ4 ( y,Rh(n−1)(u) ) dy, Svn(u) = 1 Γ(p) ∫ u 0 (u− y)p−1ψ5 ( y, Sv(n−1)(u) ) dy, Evn(u) = 1 Γ(p) ∫ u 0 (u− y)p−1ψ6 ( y,Ev(n−1)(u) ) dy, Ivn(u) = 1 Γ(p) ∫ u 0 (u− y)p−1ψ7 ( y, Iv(n−1)(u) ) dy, (12) related to Sh0 = Sh(0), Eh0 = Eh(0), Ih0 = Ih(0), Rh0 = Rh(0), Sh0 = Sv(0), Eh0 and Iv0 = Iv(0). As a result, the difference between concepts that follow produces ΞSh,n(u) = Shn(u)− Sh(n−1)(u) = 1 Γ(p) ∫ u 0 (u− y)p−1 ( ψ1(x, Sh(n−1)(y))− ψ1(y, Sh(n−2)(y)) ) dy, ΞEh,n(u) = Ehn(u)− Eh(n−1)(u) = 1 Γ(p) ∫ u 0 (u− y)p−1 ( ψ2(x,Eh(n−1)(y))− ψ2(y,Eh(n−2)(y)) ) dy, ΞIh,n(u) = Ihn(u)− Ih(n−1)(u) = 1 Γ(p) ∫ u 0 (u− y)p−1 ( ψ3(x, Ih(n−1)(y))− ψ3(y, Ih(n−2)(y)) ) dy, ΞRh,n(u) = Rhn(u)−Rh(n−1)(u) = 1 Γ(p) ∫ u 0 (u− y)p−1 ( ψ4(x,Rh(n−1)(y))− ψ4(y,Rh(n−2)(y)) ) dy, ΞSv ,n(u) = Svn(u)− Sv(n−1)(u) = 1 Γ(p) ∫ u 0 (u− y)p−1 ( ψ5(x, Sv(n−1)(y))− ψ5(y, Sv(n−2)(y)) ) dy, ΞEv ,n(u) = Evn(u)− Ev(n−1)(u) = 1 Γ(p) ∫ u 0 (u− y)p−1 ( ψ6(x,Ev(n−1)(y))− ψ6(y,Ev(n−2)(y)) ) dy, ΞIv ,n(u) = Ivn(u)− Iv(n−1)(u) = 1 Γ(p) ∫ u 0 (u− y)p−1 ( ψ7(x, Iv(n−1)(y))− ψ7(y, Iv(n−2)(y)) ) dy. (13) N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 10 of 25 Assuming That Shn(u) = n∑ j=0 ΞShn,j(u), Ehn(u) = n∑ j=0 ΞEhn,j(u), Ihn(u) = n∑ j=0 ΞIhn,j(u), Rhn(u) = n∑ j=0 ΞRhn,j(u), Svn(u) = n∑ j=0 ΞSvn,j(u), Evn(u) = n∑ j=0 ΞEvn,j(u), Ivn(u) = n∑ j=0 ΞIvn,j(u). (14) Thus, based on Equations (9) and (10) as well as the relationships  ΞSh,n−1(u) = Sh(n−1)(u)− Sh(n−2)(u), ΞEh,n−1(u) = Eh(n−1)(u)− Eh(n−2)(u), ΞIh,n−1(u) = Ih(n−1)(u)− Ih(n−2)(u), ΞRh,n−1(u) = Rh(n−1)(u)−Rh(n−2)(u), ΞSv ,n−1(u) = Sv(n−1)(u)− Sv(n−2)(u), ΞEv ,n−1(u) = Ev(n−1)(u)− Ev(n−2)(u), ΞIv ,n−1(u) = Iv(n−1)(u)− Iv(n−2)(u). (15) N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 11 of 25 provides ∥ΞSh,n∥ = η1 Γ(p) ∫ u 0 ∥ΞSh,n−1∥ (u− y)p−1dy, ∥ΞEh,n∥ = η2 Γ(p) ∫ u 0 ∥ΞEh,n−1∥ (u− y)P−1dy, ∥ΞIh,n∥ = η3 Γ(p) ∫ u 0 ∥ΞIh,n−1∥ (u− y)p−1dy, ∥ΞRh,n∥ = η4 Γ(p) ∫ u 0 ∥ΞRh,n−1∥ (u− y)p−1dy, ∥ΞSh,n∥ = η5 Γ(p) ∫ u 0 ∥ΞSv ,n−1∥ (u− y)p−1dy, ∥ΞEv ,n∥ = η6 Γ(p) ∫ u 0 ∥ΞEv ,n−1∥ (u− y)p−1dy, ∥ΞIv ,n∥ = η7 Γ(p) ∫ u 0 ∥ΞIv ,n−1∥ (u− y)p−1dy. (16) We construct and prove the subsequent theorem based on the analysis above: Theorem 1. Suppose that the function ψi : [0, T ] × R7 → R, such that ψi ∈ D([0, U ],R) for any Sh(u), Eh(u), Ih(u), Rh(u), Sv(u), Ev(u), Iv(u) ∈ D([0, T ],R) satisfies the Lipschitz and contraction condition 0 < ηi < 1, i = 1, . . . , 7. Then, the vector-borne Caputo-fabrizio fractional-order (4) possess a unique solution if Up Γ(p+ 1) ηi < 1, i = 1, . . . , 7. (17) is true for t ∈ [0, U ]. Remark 1. Existence and uniqueness results are important in epidemiology, as they need to build mathematical models to predict the spread of diseases and help develop public health policies and vaccination strategies, all of which are necessary to better decision-making. For example, in practical fields such as engineering, physics and biology, the transmission dynamics of COVID-19, HIV/AIDS and disease are just a few examples. Proof. The equation (16) provides, if ψi, i = 1, . . . , 7, meets the Lipchitz condition, N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 12 of 25 while Sh(u), Eh(u), Ih(u), Rh(u), Sv(u), Ev(u) and Iv(u) are confined. ∥ΞSh,n(u)∥ ≤ ∥Sh0(u)∥ ( Up Γ(p+ 1) η1 )n , ∥ΞEh,n(t)∥ ≤ ∥Eh0(u)∥ ( Up Γ(p+ 1) η2 )n , ∥ΞIh,n(tu∥ ≤ ∥Ih0(u)∥ ( Up Γ(p+ 1) η3 )n , ∥ΞRh,n(u)∥ ≤ ∥Rh0(u)∥ ( Up Γ(p+ 1) η4 )n , ∥ΞSv ,n(u)∥ ≤ ∥Sv0(u)∥ ( Up Γ(p+ 1) η5 )n , ∥ΞEv ,n(u)∥ ≤ ∥Ev0(u)∥ ( Up Γ(p+ 1) η6 )n , ∥ΞIv ,n(u)∥ ≤ ∥Ev0(u)∥ ( Up Γ(p+ 1) η7 )n . (18) =⇒ ∥ΞSh,n∥ ≤ 1 Γ(p) ∫ u 0 (u− y)p−1 ( ψ1 (y, Shn(y))− ψ1 ( y, Sh(n−1)(y) )) dy, ∥ΞEh,n∥ ≤ 1 Γ(p) ∫ u 0 (u− y)p−1 ( ψ2 (y,Ehn(y))− ψ2 ( y,Eh(n−1)(y) )) dy, ∥ΞIh,n∥ ≤ 1 Γ(p) ∫ u 0 (u− y)p−1 ( ψ3 (y, Ihn(y))− ψ3 ( y, Ih(n−1)(y) )) dy, ∥ΞRh,n∥ ≤ 1 Γ(p) ∫ u 0 (u− y)p−1 ( ψ4 (y,Rhn(y))− ψ4 ( y,Rh(n−1)(y) )) dy, ∥ΞSv ,n∥ ≤ 1 Γ(p) ∫ u 0 (u− y)p−1 ( ψ5 (y, Svn(y))− ψ5 ( y, Sv(n−1)(y) )) dy, ∥ΞEv ,n∥ ≤ 1 Γ(p) ∫ u 0 (u− y)p−1 ( ψ6 (y,Evn(y))− ψ6 ( y,Ev(n−1)(y) )) dy, ∥ΞIv ,n∥ ≤ 1 Γ(p) ∫ u 0 (u− y)p−1 ( ψ7 (y, Ivn(y))− ψ7 ( y, Iv(n−1)(y) )) dy. (19) N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 13 of 25 recursively, Using the same procedure, we obtain: ∥ΞSh,n(u)∥ ≤ ∥Sh0(u)∥ ( Up Γ(p+ 1) η1 )n+1 , ∥ΞEh,n(u)∥ ≤ ∥Eh0(u)∥ ( Up Γ(p+ 1) η2 )n+1 , ∥ΞIh,n(u)∥ ≤ ∥Ih0(u)∥ ( Up Γ(p+ 1) η3 )n+1 , ∥ΞRh,n(u)∥ ≤ ∥Rh0(u)∥ ( Up Γ(p+ 1) η4 )n+1 , ∥ΞSv ,n(u)∥ ≤ ∥Sv0(u)∥ ( Up Γ(p+ 1) η5 )n+1 , ∥ΞEv ,n(u)∥ ≤ ∥Ev0(u)∥ ( Up Γ(p+ 1) η6 )n+1 , ∥ΞIv ,n(u)∥ ≤ ∥Iv0(u)∥ ( Up Γ(p+ 1) η1 )n+1 , (20) to be limn→∞ ∥ΞSh,n(u)∥ = 0, limn→∞ ∥ΞEh,n(u)∥ = 0, limn→∞ ∥ΞIh,n(u)∥ = 0, limn→∞ ∥ΞRh,n(u)∥ = 0, limn→∞ ∥ΞSv ,n(u)∥ = 0, limn→∞ ∥ΞEv ,n(u)∥ = 0, limn→∞ ∥ΞIv ,n(u)∥ = 0. This ensures that the proposed (4) will have a solution. We follow these steps to prove the uniqueness of the answer. Please allow Sh1(u), Eh1(u), Ih1(u), Rh1(u), Sv1(u), Ev1(u), Iv1(u) to be the solution system for (4). ∥Sh − Sh1∥ = ∥∥∥∥ 1 Γ(p) ∫ u 0 (u− y)p−1ψ1(y, Sh(u))− ψ1(y, Sh1(u)) dy ∥∥∥∥ , ≤ ( Up Γ(p+ 1) η1 ) ∥Sh − Sh1∥ . (21) Provided that ∥Sh − Sh1∥ ( 1− Up Γ(p+ 1) η1 ) ≤ 0. (22) So, ∥Sh − Sh1∥ = 0 It’s implied.Sh(u) → Sh1(u). We can get Eh(u), Ih(u), Rh(u), Sv(u), Ev(u), and Iv(u) by using the same procedure. This complete the proof of the (1). The positivity and limit of solutions is an important part of an epidemiological model because of several other features. For this paper, we prove that for every u > 0 all state N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 14 of 25 variables are non-negative. So from system(4) we get: CDp 0,uSh(u) ∣∣∣ Sh=0 = Λh ≥ 0, CDp 0,uEh(u) ∣∣∣ Eh=0 = β1Sh Iv Nv ≥ 0, CDp 0,uIh(u) ∣∣∣ Ih=0 = σhEh ≥ 0, CDp 0,uRh(u) ∣∣∣ Rh=0 = γIh ≥ 0, CDp 0,uSv(u) ∣∣∣p Sv=0 = Λv ≥ 0, CDp 0,uEv(u) ∣∣∣p Ev=0 = β2Sv Ih Nh ≥ 0, CDp 0,uIv(u) ∣∣∣p Iv=0 = σvEv ≥ 0. (23) Theorem 2. The region Υh = { (Sh, Eh, Ih, Rh) ∈ υh+ : 0 < Nh(u) = (Sh(u) + Eh(u) + Ih(u) +Rh(u)) ≤ Λh µh } , and Υv = { (Sv, Ev, Iv) ∈ υv+ : 0 < Nv(u) = (Sv(u) + Ev(u) + Iv(u)) ≤ Λv µv } , are positively invariant for all u ≥ 0. Proof. To add the equations of (4) of human and vector compartment, we get CDpNh(u) = Λh − µhNh, CDpNv(u) = Λv − µvNv, (24) use of standard comparison theorem [36], provides Nh(u) ≤ ( Nh(0)− Λh µh ) Fp (−µhtp) + Λh µh , ∀u ∈ [0,∞), Nv(u) ≤ ( Nv(0)− Λv µv ) Fp (−µvtp) + Λv µv , ∀u ∈ [0,∞), (25) This implies that Nh(u) → Λh µh t→ ∞ and Nv(u) → Λv µv t→ ∞. Next, let S = (Sh(u), Eh(u), Ih(u), Rh(u), Sv(u), Ev(u), Iv(u)), then Υ = { S ∈ R7 + : 0 < Nh(u) ≤ Λh µh , 0 < Nv(u) ≤ Λv µv } , (26) is the biologically applicable region of the model (4) of the infection. N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 15 of 25 3.2. Disease-free Steady State Here, we denote the disease-free steady state of our system by E0 and is determined from the following system in the absence of infection: Λh − β1Sh Iv Nv − µhSh = 0, β1Sh Iv Nv − (σh + µh)Eh = 0, σhEh − (γ + µh)Ih = 0, γIh − µhRh = 0, Λv − β2Sv Ih Nh − µvSv = 0, β2Sv Ih Nh − (σv + µv)Ev = 0, σvEv − µvIv = 0. (27) From the first equation of (27), we have Λh − µhSh = 0 =⇒ S0 h = Λh µh , (28) and from the fifth equation of (27), we have Λv − µvSv = 0 =⇒ S0 v = Λv µv . (29) Thus, the DFE of our system is E0(S0 h, E 0 h, I 0 h, R 0 h, S 0 v , E 0 v , I 0 v ) = ( Λh µh , 0, 0, 0, Λv µv , 0, 0). (30) 3.3. Basic Reproduction Number The basic reproduction number, R0, is a key metric in epidemiology, representing the average number of secondary infections caused by one infected individual in a fully susceptible population. It is an important parameter which helps in designing effective control measures, such as vaccination or social distancing, to reduce disease spread. To calculate R0, we apply the next-generation matrix method, which includes the matrices F and V [17, 18] as F =  0 0 0 β1 S0 h S0 v 0 0 0 0 0 β2 S0 v S0 h 0 0 0 0 0 0  and V =  σh + µh 0 0 0 −σh γ + µh 0 0 0 0 σv + µv 0 0 0 −σv µv  N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 16 of 25 this implies that FV −1 =  0 0 β1σvS0 h µv(µv+σv)S0 v β1S0 h µvS0 v 0 0 0 0 β2σhS 0 v (σh+µh)(γ+µh)S 0 h β2S0 v (γ+µh)S 0 h 0 0 0 0 0 0  (31) After simplification, R0 is obtained by ρ ( FV−1 ) , as R0 = √ β1β2 σhσv (σh + µh)(γ + µh)µv(σv + µv) (32) Theorem 3. The DFE of the model (4) is locally asymptotically stable if R0 < 1, in any other case it is unstable. The endemic equilibrium points of our system is denoted by E∗ and is given by the following E∗ = (S∗ h, E ∗ h, I ∗ h, R ∗ h, S ∗ v , E ∗ v , I ∗ v ) = ( Λh µh 1 R0 , Λh(R0 − 1)σh + µh (β1Nv + µhβ1S ∗ h)σh , Λh(R0 − 1) β1 Nv Iv + µh , γΛh(R0 − 1) (β1Nv + µhβ1S ∗ h)µh , Λv µv 1 R0 , µvΛv(R0 − 1) (β2Nh + µvβ2S∗ v)σv , Λv(R0 − 1) β2 Nh Ih + µv ) . (33) 4. Sensitivity Index with Respect to R0 The section below uses a direct sensitivity analysis to determine the significance of specific biological parameters that are linked to the basic reproductive number R0 in decreasing the spread of vector-borne illnesses [19, 21]: ΠR0 ζ = ζ R0 × ∂R0 ∂ζ , (34) is called the sensitivity index of the variable ζ ∈ {β1, β2σh, σv}, µh, µv, γ , which depends on the value R0 . To ensure the accuracy of model predictions for parameter values, it is frequently utilized as errors and changes in expected parameters are likely to occur during data collection[21]. In the SEIR model for vector-borne diseases, several key parameters related to disease dynamics are listed in Table 2: β1 is the human infectious rate with value 0.4, Eβ1 = N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 17 of 25 0.50. The elasticity index indicate that an increase in β1 increases the number of R0. β2 the vector rate of infection having value 0.3, it gives elasticity Value Eβ2 = 0.50, it shows that with increase of β2 the R0 would be increases.Instead, the human death rate µh was set to 0.01 and the vector death rate µv was set to 0.05. These parameters have negative elasticities of -0.15 and -0.20, respectively, indicating that the death rate decreases to R0 because fewer people live in the population and participate in death. The human recovery rate σh and the vector recovery rate σv are set to 0.5 and 0.4, respectively, and the elasticity indices are 0.08 and 0.10 increase the sensitivity and γ is set 0.1, decrease to -0.42. This means that R0 decreases with higher recovery rates because survivors no longer die.In general, these parameters and elasticity indices are important for understanding the epidemiology of vector-borne diseases and the effect of various factors on the basic reproduction number R0. Now below from figure 1: β1, β2, Λh, and Λv, it Parameter Value Elasticity Index Effect β1 0.4 0.50 Increase β2 0.3 0.50 Increase γ 0.1 -0.42 Decrease µh 0.02 -0.17 Decrease µv 0.03 -0.60 Decrease σh 0.2 0.08 Increase σv 0.2 0.10 Increase Table 2: Parameter values, elasticity indices, and their effects on R0 follows that increasing these parameters increases R0 while for parameters µh, µv, σh, and σv, increasing these parameters results in a decrease of R0. N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 18 of 25 Figure 1: Sensitivity analysis of the basic reproduction number 5. Graphical Analysis and Discussion To study the dynamic behavior of the model, we may use an efficient numerical model that is stable and convergent, see [24]. The numerical scheme for (4) is as follows: CSh(k+1) = b0 + rl Γ(l + 1) k∑ m=0 ( (k −m+ 1)l − (k −m)l ) (1− p)(Λh − (β1 Iv Nv + µh)Sh) CEh(k+1) = c0 + rl Γ(l + 1) k∑ m=0 ( (k −m+ 1)l − (k −m)l ) (β1Sh Iv Nv − (σh + µh)Eh) CIh(k+1) = d0 + rl Γ(l + 1) k∑ m=0 ( (k −m+ 1)l − (k −m)l ) (σhEh − (γ + µh)Ih) CRh(k+1) = e0 + rl Γ(l + 1) k∑ m=0 ( (k −m+ 1)l − (k −m)l ) (γIh − µhRh) CSv(k+1) = f0 + rl Γ(l + 1) k∑ m=0 ( (k −m+ 1)l − (k −m)l ) (Λv − (β2 Ih Nh + µv)Sv) CEv(k+1) = g0 + rl Γ(l + 1) k∑ m=0 ( (k −m+ 1)l − (k −m)l ) (β2Sv Ih Nh − (σv + µv)Ev) CIv(k+1) = h0 + rl Γ(l + 1) k∑ m=0 ( (k −m+ 1)l − (k −m)l ) (σvEv − µvIv). (35) N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 19 of 25 Understanding the dynamical behavior of an epidemic model is crucial for predicting and time t 0 10 20 30 40 50 60 70 80 S u s c e p t ib le p o p u la t io n o f h o s t 0 1 2 3 4 5 6 7 8 9 10 0.55 0.60 0.65 0.70 (a) time t 0 10 20 30 40 50 60 70 80 E x p o s e d p o p u la t io n o f h o s t 0 1 2 3 4 5 6 0.55 0.60 0.65 0.70 (b) time t 0 10 20 30 40 50 60 70 80 I n f e c t e d p o p u la t io n o f h o s t 0 2 4 6 8 10 12 0.55 0.60 0.65 0.70 (c) time t 0 10 20 30 40 50 60 70 80 S u s c e p t ib le p o p u la t io n o f v ir u s 0 1 2 3 4 5 6 7 8 9 10 0.55 0.60 0.65 0.70 (d) time t 0 10 20 30 40 50 60 70 80 E x p o s e d p o p u la t io n o f v ir u s 0 0.5 1 1.5 2 2.5 3 0.55 0.60 0.65 0.70 (e) time t (Days) 0 10 20 30 40 50 60 70 80 I n f e c t e d p o p u la t io n o f v ir u s 0 1 2 3 4 5 6 7 8 0.55 0.60 0.65 0.70 (f) Figure 2: Illustration of human and vector classes for l = 0.55, 0.6, 0.65, 0.7. controlling the spread of infectious diseases. Dynamical behavior refers to how the disease N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 20 of 25 time t 0 10 20 30 40 50 60 70 80 S u s c e p t ib le p o p u la t io n o f h o s t 0 1 2 3 4 5 6 7 8 9 10 0.75 0.80 0.85 0.90 (a) time t 0 10 20 30 40 50 60 70 80 E x p o s e d p o p u la t io n o f h o s t 0 1 2 3 4 5 6 0.75 0.80 0.85 0.90 (b) time t 0 10 20 30 40 50 60 70 80 I n f e c t e d p o p u la t io n o f h o s t 0 2 4 6 8 10 12 0.75 0.80 0.85 0.90 (c) time t 0 10 20 30 40 50 60 70 80 S u s c e p t ib le p o p u la t io n o f v ir u s 0 1 2 3 4 5 6 7 8 9 10 0.75 0.80 0.85 0.90 (d) time t 0 10 20 30 40 50 60 70 80 E x p o s e d p o p u la t io n o f v ir u s 0 0.5 1 1.5 2 2.5 3 0.75 0.80 0.85 0.90 (e) time t (Days) 0 10 20 30 40 50 60 70 80 I n f e c t e d p o p u la t io n o f v ir u s 1 2 3 4 5 6 7 8 0.75 0.80 0.85 0.90 (f) Figure 3: Illustration of human and vector classes for l = 0.75, 0.8, 0.85, 0.9. progresses over time, including the rates of infection, recovery, and mortality, as well as the interplay between susceptible, infected, and recovered populations. This understanding provides a foundation for designing interventions like vaccination campaigns, quarantine N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 21 of 25 time t 0 10 20 30 40 50 60 70 80 S u s c e p t ib le p o p u la t io n o f h o s t 0 1 2 3 4 5 6 7 8 9 10 0.92 0.95 0.98 1.0 (a) time t 0 10 20 30 40 50 60 70 80 E x p o s e d p o p u la t io n o f h o s t 0 1 2 3 4 5 6 0.92 0.95 0.98 1.0 (b) time t 0 10 20 30 40 50 60 70 80 I n f e c t e d p o p u la t io n o f h o s t 1 2 3 4 5 6 7 8 9 10 11 0.92 0.95 0.98 1.0 (c) time t 0 10 20 30 40 50 60 70 80 S u s c e p t ib le p o p u la t io n o f v ir u s 0 1 2 3 4 5 6 7 8 9 10 0.92 0.95 0.98 1.0 (d) time t 0 10 20 30 40 50 60 70 80 E x p o s e d p o p u la t io n o f v ir u s 0 0.5 1 1.5 2 2.5 3 0.92 0.95 0.98 1.0 (e) time t 0 10 20 30 40 50 60 70 80 R e c o v e r e d c la s s o f h o s t 2 3 4 5 6 7 8 9 10 0.92 0.95 0.98 1.0 (f) Figure 4: Illustration of human and vector classes for l = 0.92, 0.95, 0.98, 1. measures, or public health policies to mitigate the impact of the disease. Numerical simulations of epidemic models help visualize and evaluate how changes in parameters, such as transmission rates or intervention strategies, affect the course of the outbreak. N. Ullah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5687 22 of 25 For instance, simulations can reveal the timing and magnitude of infection peaks, helping allocate health-care resources effectively. They also allow researchers to assess the long- term outcomes of an epidemic, such as whether the disease will become endemic or be eradicated. Ultimately, studying the dynamical behavior of epidemic models empowers decision-makers to implement timely and effective control measures, reducing morbidity and mortality while minimizing economic and social disruptions. Here, we perform some simulation to show the impact of fractional parameter on the dynamics of the infection We numerically solved the proposed model (4) and computed the corresponding solutions for both integer and non-integer derivatives, specifically for fractional orders l = 0.55, 0.6, 0.65, 0.7, 0.75, 0.80, 0.85, 0.9, 0.92, 0.95, 0.98, 1. The results, as illustrated in Figures 2, 3, and 4, provide insights into the dynamic behavior of the CF fractional-order model. By systematically varying the order fractional order l, we explored the model’s intricate dynamics, highlighting deviations from the classical integer-order model. Adjusting the fractional order l revealed significant changes in the population dynamics across different compartments, as visualized in Figures 2, 3, and 4. These adjustments demonstrated an increase or decrease in the number of individuals in each compartment, underscoring the impact of fractional derivatives on the system’s behavior. Notably, this approach differs from the classical integer-order model and provides more nuanced control measures. The observed phenomenon offers practical implications for government agencies and medical personnel to better manage and control the spread of infections. Furthermore, our results indicate that as the order l decreases (0 < l < 1), the convergence of the system slows, particularly over shorter time scales, emphasizing the sensitivity of the model to fractional dynamics. 6. Conclusions In this study, a mathematical model for vector-borne infections was developed within a fractional-order framework, utilizing the Caputo-Fabrizio fractional derivative to enhance the analysis. The steady states of the system were examined, and the basic reproduction number R0 was derived using the next-generation matrix method. The existence and uniqueness of solutions were established through the application of the Banach fixed-point theorem. 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