Insights into dengue transmission modeling: index of memory, carriers, and vaccination dynamics explored via non-integer derivative

It is acknowledged that dengue infection has a signiﬁcant economic impact due to health care costs and lost productivity. Research can provide insights into the economic burden of the disease, guiding policymakers in allocating resources for prevention and control interventions. In this work, we structured a novel mathematical model that describe the spread of dengue with the eﬀect of carriers, index of memory and vaccination. To show the eﬀect of treatment on the dynamics of dengue, we incorporate treatment due to medication in the system. The proposed dynamics is represented through fractional derivative to capture the role of memory in the control of the infection. We introduce the fundamental principles and notions of non-integer derivatives for the analysis of the model, moreover, the existence and uniqueness results of the solution of the system are established with the help of mathematical skills. The theory of ﬁxed point is utilized for the analysis and examination of the system. We establish Ulam-Hyers stability for the recommended system of dengue infection. For numerical ﬁndings, a numerical method is presented to highlight the solution pathways of the system of dengue infection. Several simulations are performed to visualize the contribution of input parameters of the system for the prevention and control of the infection. The index of memory, vaccination and treatment are recommended to be attractive parameters which can reduce the level of infection while biting rate, asymptomatic carriers and transmission rate are critical which can increase the risk of the infection in society. Our ﬁndings not only provide information for the eﬀective management of the infection but also possess valuable insights for improving public health.


Introduction
Dengue fever, a renowned tropical disease provoked by dengue viruses and predominantly spread by female Aedes aegypti mosquitoes, has become a global health concern, affecting public health and economies in approximately 128 countries worldwide due to the impact of global warming [1].
tile and nuanced mathematical framework [19][20][21].In various research disciplines and engineering applications, many phenomena exhibit non-integer order behaviors that cannot be accurately described by traditional calculus [22][23][24][25].This mathematical approach finds successful application across various scientific fields, providing a robust models for representing a broad spectrum of practical challenges in areas such as economics, physics, mathematics, control systems and biology [26][27][28].In the theory of fractional calculus, the concept of two-scale provides a sound explanation [29].This innovative idea, recently developed, centers on the critical consideration of scale when analyzing practical problems [30].It has been reported that memory significantly influences the transmission dynamics of mosquito-borne infections, particularly in retaining information about their preceding stages [31,32].The application of fractional operators enhances the accuracy and precision of modeling these phenomena.In this study, our choice is to express the dynamics of dengue infection within a fractional framework, aiming to illustrate the influence of memory on the propagation and management of dengue infection.We establish a qualitative framework embedded in fractional calculus to investigate the dynamics of dengue transmission.Our focus extends to exploring aspects such as vaccination, memory effect, treatment and the presence of asymptotic carriers.
Summary of the article is given as follows: • Section 2: Presents essential concepts and outcomes of fractional calculus.
• Section 3: Formulates an epidemic model for studying dengue transmission, considering vaccination, index of memory, asymptotic fraction, and treatment to enhance realism.
• Section 4: Dedicates to the investigation of the proposed model.
• Section 6: Introduces a numerical approach to solving the model and examines dengue dynamics with various input factors.
• Section 7: Provides the article's conclusion and closing remarks.

Foundations of Fractional Calculus
This section outlines the essential terms and foundational principles of fractional theory to be applied in the analysis of the proposed model.The significant advantage of fractional calculus stems from its incorporation of the memory index, a pivotal factor shaping the transmission dynamics of dengue infection.The researchers specifically focused on fractional systems due to their broad applicability across various domains [22,33].Some of the basic definitions are given as: Definition 2.1.( [34]).Assume that f : R + → R is a function whose fractional integral is of order ξ > 0 as follows the function specified on the right part of the equation is defined on the real numbers in a pointwise manner, denoted by R + .In this article, the symbol Γ(.) represents the gamma function.
Definition 2.2.( [34]).The expression denoting the Caputo fractional derivative of the order ξ ∈ (m − 1, m) applied to a continuous function f can be stated as follows Specifically, for values where 0 < ξ < 1, we obtain the following result provides the function f which is differentiable in [0, +∞), with Γ representing the function.
Theorem 2.1.( [34]).If Re(ξ) > 0 and m equals [Re(ξ)] + 1, then 3 Mathematical framework for model formulation In this conceptual framework, we establish the interrelationships between female vectors denoted as N h and hosts denoted as N v , thereby elucidating the mechanism underlying the transmission dynamics of dengue fever.The host population is divided into distinct classes: S h for susceptible, V h for vaccinated, I Ah for asymptomatic infections, I h for infected individuals, and R h for those who have recovered.Meanwhile, the female mosquito population represented as N v is categorized into susceptible S v and infectious I v compartments.We make the presumption that the natural birth and death rates, denoted by mu h for the host population and mu v for the vector population, are constant throughout both populations.In this model, the incidence rates originating from the susceptible classes (S h and V h ) are given by ( bβ 1 N h I v ) and ( bβ 2 N h V h ) respectively.Additionally, the incidence rate from susceptible mosquitoes (S v ) to infectious mosquitoes (I v ) is represented by ( bβ 3 N h (I h + I hA )), where b denotes the mosquito biting rate.The asymptomatic fraction is symbolized by ψ and the vaccination rate is given by p.In addition to this, the rate of recovery is indicated by γ h whereas the recovery through treatment is denoted by η.Then, the dynamics of dengue is given by the following system of equations with the initial conditions where β 1 , β 2 and β 3 are transmission probability with the condition that β 1 ≥ β 2 .Furthermore, the we have and It is well-known that fractional calculus theory is rich in applications and produces more accurate results for the dynamics of biological phenomena.The two-scale fractal theory for population dynamics is a novel area of research that aims to understand the dynamics of population growth in closed systems [35].The two-scale fractal theory for population dynamics is based on the idea that populations exhibit fractal patterns at different scales.The theory considers the effects of nonlinear diffusion and fractional spatial diffusion on population growth.In this work, we will structure the dynamics of dengue in a fractional framework to know the importance of memory in the spread and control of the infection.Also, the transmission dynamics of dengue involve an associative learning mechanism, where knowledge of previous stages is retained.Host population memory, connected to individual awareness, reduces contact rates between vectors and hosts.Meanwhile, mosquitoes draw from past experiences on human location, blood preference, color, and defensive behaviors to choose suitable hosts.Integrating fractional-order systems into mathematical models of dengue infection effectively captures and represents these intricate phenomena.Thus, we represent our model ( 5) of dengue infection through fractional derivative with the effect of memory as: where C D ξ 0 + indicates Caputo fractional derivative with order ξ.Here, we focussed on timefractional epidemic model which plays an important role in advancing our understanding of infectious diseases, providing a more realistic framework for analysis and offering valuable insights into disease dynamics.The spatial diffusion of biological populations is also an important research area in ecology and population biology.The research on the spatial diffusion of biological populations has focused on developing models that take into account nonlinear diffusion effects and fractional spatial diffusion.These models have important implications for understanding the dynamics of biological populations and predicting their spatial patterns [35,36].In our future work, we will focus on fractional space diffusion system to investigate the dynamics of infectious diseases.The following theorem is on the non-negativity and boundedness of the solutions of our proposed system which can be easily proved through analytic skills.
Theorem 3.1.The solutions of our fractional model ( 8) are non-negative and bounded for non-negative initial values of state variables.
It is well-known that equilibrium points in epidemic models are essential for comprehending the dynamics of infectious diseases, guiding the development of effective control measures, and predicting the overall trajectory of an epidemic in a population.There are two meaningful equilibrium points of an epidemic model, that are, disease-free and endemic point.The disease-free equilibrium is crucial for assessing the potential success of prevention and control measures.It represents a stable state where the infection has been eliminated, providing insights into conditions that promote disease control.It is denoted by E 0 and is given by , 0, 0, 0, N v , 0).
On the other hand, the endemic equilibrium is vital for understanding the persistent existence of the disease in a population.Analysis of this equilibrium helps identify factors that contribute to the sustained transmission of the infection and informs strategies for long-term management and intervention.In this work, we focussed on solution behaviour and Ulam-Hyers stability of the system while other aspects of the system will be investigated in our future work.

Existence and uniqueness results
The existence and uniqueness of solutions govern fractional-order differential equation theory.Many researchers have recently been interested in the theory, we refer to [37,38] and their references for some of the recent growth.We will utilize fixed point theorems to explore that whether the solution of the suggested framework is real and unique.The proposed model ( 8) can be reformulated as follows: Where Therefore, model ( 8) can be summarized as under the circumstances that where (.) T signifies the transposition technique.In light of the Theorem in [39], obstacle ( 11) is stated by Consider F = G ([0, a]; R), denoting the Banach space comprising continuous functions mapping from [0, a] to R. This space is equipped with a norm given as where and Theorem 4.1.( [40]).Consider M = θ to represent a closed, bounded, convex subset of a Banach Space B. Assume P 1 and P 2 be two operators that satisfy the following relationships 2) P 1 is smooth and compact.
3) P 2 is a mapping of contractions.Subsequently, the existence of an element denoted as U , belonging to the set M, is confirmed, satisfying U = P 1 U + P 2 U .
Proof.The operator denoted as P : E → E and formulated by the following definition The well-defined nature of the operator P is obvious, and the fixed point of P corresponds to the model's (8) Consequently, the outcomes are derived.Moreover, considering any Ψ 1 , Ψ 2 ∈ E, we obtain the following thus indicating that Consequently, based on the Banach contraction principle, we can conclude that the suggested model ( 8) possess a unique solution.

Stability analysis
We conduct a stability analysis within the Ulam-Hyers and generalized Ulam-Hyers framework to assess the suggested model ( 8) in this section.Ulam [42,43] originally introduced the concept of Ulam stability.The aforementioned stability has been investigated in various research articles on classical fractional derivatives, such as [44,45].To ensure the stability of the approximated solutions, we employ nonlinear functional analysis to examine both the Ulam-Hyers stability and the generalized stability of the presented model (8).For this purpose, the following definitions are necessary.Consider the subsequent inequality, where ε represents a positive real value whereas ε = max (ε J ) T , J = 1, ..., 7.
Definition 5.1.If Cκ > 0, the model ( 8) exhibits Ulam-Hyers stability.For any positive value ε, and for a solution Ψ belonging to the set E that fulfills condition (5.1), there exists a unique solution Ψ ∈ E to Eq. ( 8).
Remark 5.1.The expression Ψ ∈ E meets the requirement of Eq. ( 21), if and only if a function g ∈ E exhibits the subsequent property: 1) g(t) ≤ ε, g = max (g j ) T , t ∈ J .
Theorem 5.1.Assuming that Ψ belongs to the set E and fulfills the inequality (21), then Ψ effects the integral inequality defined as follows in mathematical terms Proof.Utilizing (2) of Remark (5.1).
Proof.Assume that Ψ ∈ E fulfils the inequality (21) and that Ψ ∈ E is the only single solution to (8).As a result, for every ε > 0, t ∈ J , in accordance with Lemma (5.1), we obtain where, So, setting ψ κ (ε) = C κ (ε) such that ψ κ (0) = 0. Our analysis leads us to the perfect rephrasing that the stated problem (8) exhibits both Ulam-Hyers stability and generalized Ulam-Hyers stability.determine whether the fractional parameter could function as a viable control parameter.The findings revealed the fractional parameter's significance as an influential factor, serving as a valuable tool to modulate the extent of infection within the community.The results indicated a noteworthy trend: the infection level exhibited sensitivity to changes in the index of memory.Specifically, decreasing the memory index correlated with a reduction in the infection level.This observation underscores the potential effectiveness of interventions aimed at strategically managing the fractional parameter to curtail the spread of infection.As a result, we advocate for proactive policymaking and targeted actions to adjust the memory index, offering a promising avenue for infection control measures and public health management.

Time in days
In the second case illustrated in Figure 2, we explored the role of the transmission probability on the populations of infected hosts and vectors.This investigation was undertaken to elucidate the pivotal role of the transmission probability and its impact on the dynamics of the infection.The results highlighted the critical nature of the transmission probability, showcasing its substantial influence on the risk of infection.As this parameter is adjusted, we observed corresponding shifts in the populations of both infected hosts and vectors.Higher transmission probabilities were associated with an high risk of infection, emphasizing the need for careful consideration and strategic management of this parameter in disease control efforts.In the same way, Figure 3 illustrated the importance of the mosquito biting rate.Our observations indicate that these parameters hold considerable significance, possessing the capacity to heighten the risk of infection within the community.
In Figure 4, the impact of treatment on the infected classes is illustrated.Our observations indicate that the treatment rate exerts an effective control over the prevalence of dengue infection within the society, demonstrating its potential in mitigating the spread of the disease.Subsequently, in the most recent scenario portrayed in Figure 5, we have elucidated the variation in the prevalence of dengue infection with alterations in the carrier fraction.It is evident from the results that this factor assumes a pivotal role, possessing the potential to escalate the risk of infection within both endemic and non-endemic regions.It has been observed in these findings that the control of memory index can control the infection level of both the classes in the community.Therefore, we recommended that this parameter is attractive and are suggested to the public health officials for the control of dengue.
We believe that getting vaccinated and receiving treatment are important for the control of dengue infection.In addition to this, managing the infection effectively involves using things like bed nets and insecticide sprays, along with adjusting the memory index on purpose.

Concluding Remarks
The worldwide prevalence of dengue viral infection presents a substantial risk to public health, carrying the potential for life-threatening outcomes.At present, devising successful approaches to manage this viral malady emerges as a considerable hurdle for policymakers, researchers, and public health authorities.In this work, we structured the dynamics of dengue infection with the effect of different control measures for public health.We presented the proposed dynamics of dengue in fractional framework to capture the role of memory in the transmission of dengue infection.The basic concepts and ideas of fractional theory are introduced for the analysis of our model.It has been shown that the solutions of the model are non-negative and bounded for nonnegative initial values.The existence and uniqueness of the proposed dengue model's solution are examined using Banach's and Schaefer's frameworks, employing the fixed-point theorem.Furthermore, we have established adequate conditions for Ulam-Hyers stability of our system of dengue infection.To visualize the dynamical behaviour of dengue infection, we perform different simulations with the variation of input parameters of the system.We demonstrated the pivotal role of asymptomatic carriers, biting rate, and transmission probability as critical parameters that can exacerbate the difficulty of controlling dengue infection.On the other hand, the index of memory, vaccination, and treatment has the potential to effectively manage dengue infection.The results of our study emphasize the noteworthy influence of memory on the behavior of dengue, indicating its potential as a controlling factor for managing infections.In our future work, we intend to explore how the dynamics of dengue infection are affected by maturation and incubation delay.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :
Figure 1: Visual examination was performed to analyze the dynamic characteristics of the recommended system (8) of dengue infection with the variation of the fractional parameter ξ, i.e., ξ = 0.85, 0.90, 0.95 and 1.00.

Figure 5 :
Figure 5: The infected categories of dengue infection in our model (8) were subjected to time series analysis while altering the carrier fraction ψ across values of ψ = 0.40, 0.50, 0.60 and 0.70.
unique solution.To demonstrate this, consider the following approach sup t∈J ||κ(t, 0)|| = M 1 and k ≥ ||Ψ 0 ||+ M 1 .Consequently,it is enough to prove that P H k ⊂ H k , where the set H k = {Ψ ∈ E : ||Ψ|| ≤ k}, which possesses both closed and convex properties.For any given Ψ belonging to H k , we obtain