Worldwide, thousands of dengue cases are reported every year. The world's tropical and subtropical regions are affected by dengue infection, which is a mosquito-carrying disease. A high temperature and flu-like symptoms are signs of mild illness or asymptomatic to stern disease. DHF (Dengue Hemorrhagic Fever) or DSS (Dengue Show Syndromes Syndromes) is a highly infectious form of dengue fever that causes serious bleeding, shock, and death. Generally, it was noticed that only one out of four dengue contagions is symptomatic. Dengue virus occurs in four major types (DENV types 1, 2, 3, and 4), all of which can cause serious illness. The usual signs of DENV type 1 are like a common cold and mild fever, which will not lead directly to DHF; conversely, later DENV types can lead to DHF (1–3).

To understand the dynamical behavior of dengue transmission, we formulated a mathematical model, particularly focusing on vector-borne disease transmission from mosquitoes to humans. Esteva and Vargas (4, 5) pioneered the creation of a fundamental dengue model and explored numerous fundamental mathematical concepts and their accompanying numerical simulations. Feng et al. (6) presented a two-strain dengue infection model and examined competitive exclusion. Researchers have conducted numerous studies to better understand the transmission of dengue fever (7–9).

The importance of fractional-order models lies in their ability to capture the complex dynamics and long-term dependencies within the transmission process. By incorporating fractional derivatives, these models provide a more comprehensive understanding of disease spread, which is crucial for designing effective intervention strategies. The fractional-order models can accommodate the nuanced behavior of dengue transmission, offering insights that integer-order models may overlook, thereby enhancing the accuracy and effectiveness of disease control measures.

The fractional order model has been conclusively demonstrated by a recent study to be capable of controlling the trend of complex diffusion disorder (10–14). Many have emphasized various mathematical models for Dengue transmission and prevention (15–19). All cited references explain the transmission process of Dengue infection from different perspectives, including dynamic analysis, evaluation of vaccination, and optimal control measures (20–24). The most updated studies on Dengue with real-life data are presented in (25, 26). The mathematical description of Dengue is briefly described in Deterministic and Stochastic terms. The evolution of dengue with asymptomatic carriers using optimal control measures was investigated in (27).

Therefore, motivated by the aforementioned literature, we propose a computational framework for the dissemination of dengue at a given viremia level. We investigated whether symptom-free people were markedly more susceptible to mosquitoes than clinically symptom-positive patients. The new idea of a mathematical model to analyse the immune-boosting factor for asymptomatic infected cases and the waning immunity that cases re-infect is reported. To make practical applications and simulations easier, we utilize the Adams-Bash forth numerical scheme, which is renowned for its accuracy and stability. This choice ensures that our model reflects real-world scenarios while maintaining computational efficacy. A key highlight of this study is the incorporation of optimal control strategies into the ABC fractional order Dengue viremia model. These strategies are designed to explore how interventions, such as self-prevention and vector control, can be optimized to curtail disease spread. The analysis extends to investigating disease-free and endemic stability, providing crucial insights into the long-term behavior of the system under various control scenarios.

This article is prepared as follows: In portion 2, we review the fundamental definitions for the fractional-order operator and provide a list of mathematical properties that were used throughout the work. The dengue viral mathematical model with fractional order was presented in portion 3. Portion 4 examines the local as well as global consistency of the suggested model through the Routh-Hurwitz criteria and the Lyapunov function. An optimal control solution and discussion are present in portion 5. The final section focuses on numerical simulations and a comprehensive conclusion.

This section introduces fractional derivation and some of its properties, which will be used in the following components.

Definition 2.1.

Consider ψ ∈ ℍ′ (0, T) and η ∈ [0, ȶ], then Atangana-Baleanu fraction component in Caputo case is

The method yields a variation operator Caputo-Fabrizio that replaces

It's noteworthy that AℬC0𝔇ȶη[constant] = 0. Here N(η)¯ is the typical function and it is defined as N(0)¯=1 and N(1)¯=1.  N(η)¯ depict the familiar Mittag–Leffler operator, it also reflects the exponential function generality.

Definition 2.2.

The fractional integral of ABC with order η given by

Lemma 2.1.

Consider a fractional-order system

Where ηϵ(0, 1) in the initial case x(ȶ0).

If f(ȶ,x) fulfills the Lipschitz condition in relation to x, then system (Equation 3) exhibits a unique solution in the region [t₀, +∞) × φ and φ ⊆ ℝⁿ.

Lemma 2.2.

If x(ȶ)∈ℝ+ become an ongoing and attainable consequence. Then

Here t > t₀, ηϵ(0, 1) and x*∈ ℝ+.

In this section, we expand upon the previously described Dengue SIR-SI model (18) by incorporating additional factors and refining the classification of both human and mosquito populations. Our model includes viremia levels, an immune-boosting factor for asymptomatic infected cases, and clinical treatment for symptomatic infected cases.

To study the mode of spread of dengue sickness, the human species (N𝔥) is subdivided into four classes: susceptible (Ȿ_𝔥), symptomatic infectious (𝔗_Ȿ𝔥), asymptomatic infectious (𝔗A𝔥) and recovered human populations (𝔎_𝔥). We classified female mosquito species (N𝔪) into Susceptible (Ȿ_𝔪) and infective mosquitoes (𝔗_𝔪). A Susceptible individual among as one who is not infected and immune, infected humans are both asymptomatic and symptomatic are those who have acquired Dengue viremia from an infected mosquito populations and are all capable of spreading dengue virus to susceptible mosquitoes. Let we examines π_𝔥 and π_𝔪 the acquisition rates of humans and mosquitoes. The proposed model, illustrated in the flowchart, demonstrates the dengue transmission dynamics. Based on Figure 1, we developed the following differential equation.

Where

α_𝔪- individual mosquito‘s biting rate

χ_Ȿ𝔥- Dissemination to human by mosquitoes, which leads to a symptomatic infectious in humans

χA𝔥- Dissemination to human by mosquitoes, which leads to a asymptomatic infectious in humans

χ_𝔪- Viremia dissemination to mosquito by human species

δ_𝔥- Human Fatality rate

τ - Symptomatic infected human treatment rate

γ - Recovering rate.

θ - Transition rate at which a recovered person becomes defenseless due to loss of immunity

ϱ−Rates of immunosuppression for asymptomatic victims

δ_𝔪- Rate of mosquito natural mortality (an average mosquito life span)

From the basic cases (Ȿh , TⱾh, TAh,Kh,Ȿm, Tm)≥0.

In this approach, the aggregate human and mosquito population ratios are provided by

In addition, the area of biologically significance for the aforementioned dengue model is indicated and presented by the covered set

A fractional representation of the 𝔄𝔅ℭ model as

A fractional derivation of Atangana-Baleanu of order 0 < η < 1 is denoted AℬC0𝔇ȶη in Caputo notation.

This section examines the validity, singularity and positive variance of the solution of the SIR-SI type model. Additionally, a reliability estimate for Model (Equation 6) has also been developed.