Analysis of the dynamics of a vector-borne infection with the effect of imperfect vaccination from a fractional perspective

The burden of vector-borne infections is significant, particularly in low- and middle-income countries where vector populations are high and healthcare infrastructure may be inadequate. Further, studies are required to investigate the key factors of vector-borne infections to provide effective control measure. This study focuses on formulating a mathematical framework to characterize the spread of chikungunya infection in the presence of vaccines and treatments. The research is primarily dedicated to descriptive study and comprehension of dynamic behaviour of chikungunya dynamics. We use Banach’s and Schaefer’s fixed point theorems to investigate the existence and uniqueness of the suggested chikungunya framework resolution. Additionally, we confirm the Ulam–Hyers stability of the chikungunya system. To assess the impact of various parameters on the dynamics of chikungunya, we examine solution pathways using the Laplace-Adomian method of disintegration. Specifically, to visualise the impacts of fractional order, vaccination, bite rate and treatment computer algorithms are employed on the infection level of chikungunya. Our research identified the framework’s essential input settings for managing chikungunya infection. Notably, the intensity of chikungunya infection can be reduced by lowering mosquito bite rates in the affected area. On the other hand, vaccination, memory index or fractional order, and treatment could be used as efficient controlling variables.


Theory of fractional calculus
This section will go through the basic definitions and concepts of fractional calculus, which will be used to investigate our system.The ability to integrate memory effects, which are critical for comprehending vectorborne virus transmission.Furthermore, fractional calculus has many applications in diverse fields of study.We have covered the basics of the Caputo fractional operator, which we will utilise to analyse our proposed model.Definition 2.1 If we examine a function g(k) in the Lebesgue integrable space L 1 ([a, b], R) 23 , the fractional integration can be characterised as follows where ϑ is the fractional order in such a way that 0 < ϑ ≤ 1. Definition 2.2 Consider a function g(k) such that g(k) ∈ B n [a, b] 23 , and the fractional derivative in Caputo form is provided by with the condition that 0 < ϑ ≤ 1. Lemma 2.1 Assuming the fractional system stated as follows 23 , where C(k) ∈ B([0, υ]) , the response of the aforementioned fractional framework is then Definition 2.3 The Laplace transformation is presented for the Caputo Fractional operator in the following way 24 , In addition, consider be the standard deviation specified on Z = B([0, υ]) in which Z is a Banach's space.Theorem 2.1 Consider a Banach space Z in which F : Z → Z is smooth and confined 25 .If the following set is limited, then F is a fixed point.

Evaluation of the dynamics
We offer a computational framework that represents the propagation of chikungunya virus infection in this portion of the research.The entire population of both hosts and vectors at a particular moment is represented by N h (t) and N v (t) , accordingly.The vector community N v (t) is split into two categories: the susceptible class S v (t) and the infected class I v (t) .Similarly, the host community is separated into four groups: the susceptible S h (t) , the vaccinated V h (t) , the infected I h (t) , and the recovered R h (t) .The vector community N v (t) is a combination of the S v (t) and I v (t) classes, whereas the total host population is the sum of S h (t) , V h (t) , I h (t) and R h (t) .In this formulation, µ h and µ v represents the natural mortality rates for both kinds of individuals, accordingly.A fraction p of susceptible individuals are vaccinated and a fraction τ of the infected class move to the recover class after successful treatment.We represent the dynamics chikungunya infection with the effect of vaccination and vertical transmission is given by (1) In our formulation, the rate of recruitment of humans is represented by µ h N h where µ h represent the nativity and mortality rates of humans.In vector inhabitants, the infection outlay from susceptible class to infected class is given by β 3 .On the other hand, the host population is recruited by a rate µ v N v where µ v is the birth and death rate of vectors.The transmission rates from susceptible host to infected is indicated by β 1 and β 2 .One of the latest and most newly prominent disciplines of mathematics is fractional calculus, which works with derivatives and integrals of real and complex classes.In truth, although being as old as the original calculus, this type of calculus has piqued the interest of scholars from a variety of disciplines due to the astonishing results obtained when some of these scholars employed fractional operations to describe real-world difficulties.To better understand the spreading phenomenon more properly, we characterize the dynamics of chikungunya infection using a Caputo-derivative in the following manner: where B 0 D ξ k is the derivative of Liouville-Caputo.ξ represents the storage index in this approach.E 0 and is given by , 0, 0, N 0 v , 0) denotes the disease-free steady-state of our suggested fractional framework (9) of chikungunya illness.The aforementioned framework of chikungunya illness threshold parameter is offered by For our system of chikungunya infection, the existence and uniqueness of the solution will be investigated in the upcoming result.The below result can easily be determined through analytic skills.
Theorem 3.1 The solutions of the recommended system (9) of chikungunya virus infection are positive and bounded for positive initial condition of state-variable.

Sensitivity analysis.
Sensitivity analysis is a critical tool used in various scientific disciplines, including epidemiology, to assess the sensitivity of different interventions of a biological system.In the context of epidemic modeling, sensitivity analysis helps understand how uncertainties or variations in the model's parameters impact the predictions and conclusions drawn from the model.Local sensitivity analysis and global sensitivity analysis are two different approaches used in sensitivity analysis to evaluate the impact of input parameter variations on the output of an intervention 26 .In local sensitivity analysis, one parameter is varied at a time, while all others are kept fixed at their baseline values.This allows researchers to observe the individual impact of each parameter on the system output.Global sensitivity analysis considers the interactions between multiple parameters simultaneously.It assesses the collective influence of all parameters on the system's output and provides a more comprehensive understanding of their combined effects 27 . (8) .
Vol www.nature.com/scientificreports/ Global sensitivity analysis provides a more comprehensive assessment of parameter influences, especially when dealing with complex and nonlinear models or when interactions between parameters are essential for accurate predictions.Here, we will perform global sensitivity analysis of the basic reproduction number of our system.We will use the well-known partial rank correlation coefficient method (PRCC) to show the impact of input parameters on the output of R 0 .The significant test provides PRCC and p-values for each parameter, indicating that the parameter with the highest PRCC and lowest p-values is the most sensitive parameter in the system.From Fig. 1 and Table 1, we noticed that the input parameters β 3 , µ v , β 1 , β 2 and ξ are the most sensitive parameters in the basic reproduction number with PRCC 0.8448, −0.6858 , 0.6638, 0.5900 and 0.5807, respectively.Therefore, these parameters are recommended to the policy makers and health officials for the prevention and control of the infection.

Theory of existence
The suggested fractional algorithm's fundamental framework (9) will be described below.This is accomplished in the subsequent style: www.nature.com/scientificreports/ The aforementioned chikungunya viral infection model ( 10) could be represented in the following manner where Using Lemma (2.1), the corresponding integral form of the previous (11) is The following parameters are the major phases in analysing our suggested system: (A1) The variables K Q , M Q , and q ∈ [0, 1) could be found in the following way: (A2) The variables LQ > 0 , and all W , W ∈ Z could be found in the following way: In this portion, we define a map F on Z as follows If assumptions (A1) and (A2) are valid, then the framework (11) has a minimum of a single answer.This concept will be used to our suggested approach to chikungunya virus transmission.Proof 4.1.We shall utilise Schaefer's fixed point theorem to demonstrate the intended outcome.This theorem will be illustrated over four stages, outlined below: S1: In this stage, we shall demonstrate the consistency of F. In this case, we presume that W i is continuous for i = 1, 2, . . ., 9 , which means that Q(k, W(k)) is also continual.We select W j , W ∈ Z in such a way where W j → W , and FWj → FW is required.Next, we'll look at: Insofar as Q is persistent, FWj → FW ; as a consequence, the operator F is also continual.

S2:
The second stage tends to determine the boundedness of the F expression.If W ∈ Z , then the operator F must satisfy the subsequent scenario Vol www.nature.com/scientificreports/Our goal is therefore to demonstrate the boundedness of F(S) for a bounded subgroup S of Z. Consider arbitrary W ∈ S ; because the set S is limited, we may discover a K ≥ 0 in the following way As a result of extending the preceding criterion to any W ∈ S , we get Hence, F(S) is bounded.

S3:
In the third stage, we suppose k 1 , k 2 ∈ [0, υ] in such a way that k 1 ≥ k 2 to demonstrate the equi-continuity, and then As a result, Arzela-Ascoli's theorem suggests that F(S) is relatively concise.
Step 4: In the final phase, we select the subsequent set up To demonstrate the boundedness of the group E , let W ∈ E , than for any k ∈ [0, υ] , the argument that follows is true This indicates that the group E is finite.As a consequence of Schaefer's theorem, the variable F has a definite position; as a result, it provides a minimum of one remedy to our suggested scheme (11).
Remark 4.1 If the requirement (H1 ) for q = 1 is fulfilled, the conclusion of Theorem (4.1) is satisfied for Theorem 4.2 If the premise υ α K Q Ŵ(α+1) < 1 is valid, then our desired fractional framework (11) has a distinctive answer.
Proof 4.2.For the desired outcome, assuming W, W ∈ Z and using Banach's contraction theorem, we get As consequence, the operator F has a distinct fixed point, therefore the results of the proposed fractional system (11)  is exceptional.
Ulam-Hyers stability is a mathematical concept used to assess the behavior of a system.This stability notion is particularly relevant when dealing with fractional-order models because these equations can describe complex systems where memory effects and long-range interactions play a crucial role.In study of a fractional system, one of the main challenges is that exact analytical solutions are often difficult or even impossible to obtain.Instead, researchers commonly seek approximate or close-to-perfect solutions to gain insights into the system's behavior.Ulam-Hyers stability, along with the extended Ulam-Hyers-Rassias stability definition, provides a useful framework to study the stability of approximate solutions of fractional system.It helps researchers understand the robustness of these solutions under small perturbations, which is crucial for evaluating the reliability of the model's predictions.By employing Ulam-Hyers stability in the context of a system described by fractional differential equations, researchers can ascertain whether the approximate solutions obtained are reliable and meaningful.This stability analysis contributes to a deeper understanding of the system's dynamics and aids in developing accurate and reliable mathematical models, which, in turn, are valuable for decision-making in various fields, such as epidemiology for infectious disease control and management.

Ulam-Hyers stability
In this section, we will establish stability results for our recommended model of chikungunya introduced by Ulam 28 and Hyers 29 .Numerous researchers used the Ulam-Hyers stability concept in various fields of science and engineering [30][31][32] .Here are some fundamental definitions and notions: Consider a controller N : Z → Z in the following way Definition 5.1 Previously ( 25) is Ulam-Hyers stable (UHS) kind if for every scenario W ∈ Z and ǫ > 0 , we exhibit: The situation (25) has a distinctive answer W in which B q > 0 and the requirements listed below are met Definition 5. 2 The aforementioned ( 25) is a generalised UHS, thus we get the subsequent result for all possibilities W of ( 27) and any additional approach W of (25) while Y ∈ B(R, R) and the zero frame is zero.
Remark 5.1 If the subsequent conditions are satisfied, the solution W ∈ Z accomplishes ( 27) Considering a minor disruption, the framework (11) may be expressed in the following manner Ŵ(α+1) < 1 is met, the approach to the framework (11) is UHS and modified UHS on Lemma (5.1).Proof 5.2.For the needed evidence, we consider the system response W ∈ Z (29) and W ∈ F as an exceptional result (11), thus we get As a consequence, the chikungunya virus infection framework (11) indicated before is UHS and generalised UHS.25) is Ulam-Hyers-Rassias stable (UHRS) if each of the subsequent solutions W ∈ Z is Ulam-Hyers-Rassias stable (UHRS) One can discover an innovative approach W of the framework (25) that meets the constraint B q > 0 providing Definition 5.4 Assume that W be any solution of (32) and W is the unique solution of ( 25) in a manner that (31) www.nature.com/scientificreports/ in which ǫ > 0 and ω ∈ B[[0, υ], R] so that B q,ω .The aforementioned ( 25) is therefore generalised UHRS.

Remark 5.2
If the subsequent conditions are met, then the outcome of W ∈ X fulfils ( 27)

Lemma 5.2
The inequality stated underneath holds valid for the perturb system (5.1)provided by Proof.The needed consequence is simply shown utilising Lemma (2.1) and Remark (5.2).
Proof.If we presume a solution W ∈ X and pick the particular answer W ∈ X of the framework (11), we get the following result As a result, UHRS and expanded UHRS are the answers to (11).

Solution of our fractional system
In this section, we will utilise the transformation of Laplace to create an efficient scheme for our fractional system (9) of chikungunya infection.The Adomian Decomposition Method is an efficient and trustworthy numerical scheme, providing a dependable and effective approach for solving fractional models.This method exhibits excellent convergence properties, enabling researchers to obtain reliable approximations even when dealing with complex fractional models.
(41)  www.nature.com/scientificreports/simulation illustrated in Fig. 5, we emphasized the solution paths of the chikungunya virus infection model with biting rate variation.We discovered that when the bite rate boosted so did the degree of illness.As a result, this input component is crucial.
We demonstrated the influence of vaccination on the propagation dynamics of chikungunya viral infection in the third simulation depicted in Fig. 6.We discovered that immunization is a key element in reducing chikungunya infection in the community.The vaccine parameters are advised to health officials for infection Figure 2. Demonstration of the dynamical behaviour of our chikungunya framework (9) with varying memory index ξ values, i.e., ξ = 0.85, 0.90, 0.95, 1.00.control based on these findings.The impact of treatment has been visualized using numerical simulations in the final simulation shown in Fig. 7.We proposed that the index of memory, immunization, and treatment be utilized as control criteria for chikungunya viral infection avoidance.In our future work, the proposed model will be examined and validated using real-world data on the availability of data, enabling us to to predict the future course of the epidemic.

Concluding remarks
In this study, we developed a mathematical framework for the transmission of chikungunya infection with vaccination and therapy.The suggested chikungunya model is constructed in a fractional framework to demonstrate the influence of memory on the dynamics of chikungunya.We used the basic principles of fractional calculus to analyze our mathematical framework.In our study, we focused on qualitative analysis and the dynamical behavior of chikungunya viral infection.The uniqueness and existence of the solution of the provided chikungunya model are investigated using the fixed-point theorem within the context of Banach's and Schaefer's.Through these, we obtained the Ulam-Hyers stability criteria for our chikungunya viral infection system.The influence of various factors on the dynamics of chikungunya virus infection is investigated by employing the www.nature.com/scientificreports/Laplace Adomian reduction methodology to demonstrate the impact of many parameters on the dynamics of this viral infection.Numerical simulations, in particular, are employed to illustrate the impacts of fractional-order, immunization, and treatment.We have shown that the bite rate is an important metric that can render a more challenging controller.The biting rate of mosquitos is expected to be harmful, whereas vaccines and treatment www.nature.com/scientificreports/are promising characteristics for infection management.It has been proposed that reducing mosquito bite rates can reduce the severity of chikungunya virus illness.We demonstrated the role of memory in the dynamics of chikungunya infection and propose that it might be employed as a control measure for the prevention of infection.In addition to this, we hypothesized that chikungunya within society could be managed by reducing bite rates and enhancing vaccine and treatment.

Theorem 4 . 1
If assumptions A1 and A2 are valid, the suggested fractional model(9) of chikungunya viral infection has a minimum of a single solution.

Table 1 .
PRCC and p values of significant test for R 0 .