Fractional Order Mathematical Model for Predicting and Controlling Dengue Fever Spread Based on Awareness Dynamics

Abstract

Dengue fever (DF) is considered one of the most rapidly spreading infectious diseases, which is primarily transmitted to humans by bites from infected Aedes mosquitoes. The current investigation considers the spread patterns of dengue disease with and without host population awareness. It is assumed that some individuals decrease their contact with infected mosquitoes by adopting precautionary behaviors due to their awareness of the disease. Certain susceptible groups actively prevent mosquito bites, and a few infected are isolated to reduce further infections. The basic reproduction number and population dynamics are modeled by a system of fractional-order differential equations. The system of equations is solved using the Adomian Decomposition Method (ADM) since it converges rapidly to the exact solution and can give explicit analytical solutions. Solutions derived are analyzed and plotted for different fractional orders, providing useful insights into population dynamics and contributing to a better understanding of the initiation and control of disease.

1. Introduction

During these days, the COVID-19 pandemic has intimately linked the world and brought along many negative changes. It used to be presumed that the coronavirus was an endemic illness, but it quickly ballooned out of control and turned into a worldwide pandemic. There have been suspicions lately that some individuals who are diagnosed display the symptoms of dengue fever, which are very much like the ones of dengue fever [1]. Dengue fever [2,3] refers to the flu caused by the virus a person gets through the bites of mosquitoes, which are carriers. The production of more already infected mosquitoes allows viruses to survive and spread in case of warm weather and rainfall. The first known cases of the onset of dengue fever occurred in 1779 in Cairo, Egypt, and Batavia, Indonesia [4]. By the year 1818, the number of dengue fever cases that were reported had reached 50,000. The disease has symptoms like high temperature and body pain, in addition to such symptoms as a skin rash; in more serious cases, vision loss and muscle pains are also noted. Normally, it can cause internal bleeding and shock if the patient is in a critical condition. However, symptomatic treatment can relieve the side effects since, as of now, there is no specific cure for the disease in question.

Mathematical models are the backbone of natural sciences, as they can be utilized to study human interactions and the surrounding environment [5]. Monitoring the spread dynamics of infectious diseases is important to estimate outbreaks and formulate policies to stop them [5,6]. To analyze the biological model behavior, several advanced methods have been applied, such as the meshless method [7], the Jacobi elliptic equation method [8], the variational iterative method [9], the Sine–Gordon expansion method [10], the finite difference method [9], the generalized Kudryashov method [11], the Sine–Cosine method [12], and others [13,14,15,16].

Several researchers have investigated the dynamics of dengue fever (DF) using mathematical and statistical models. The differential transform method [17] was applied to obtain the series solution of the DF model. Numerical simulations conducted in these studies analyzed the long-term behavior of the DF model and demonstrated that treatment plays a crucial role in reducing the disease burden within a population. Additionally, the Adams–Bashforth–Moulton predictor–corrector scheme was utilized to approximate the solution of the DF model [18]. Furthermore, a mathematical DF model was proposed and analyzed [19] to study the transmission dynamics of the virus in a human population while accounting for the effects of treatment. Previous studies [20,21,22] significantly enhanced the mathematical modeling of dengue control via mosquito population management and Wolbachia-mediated treatment. Wan and Xu [20] constructed a deterministic model that incorporated Wolbachia infection dynamics and dengue transmission, applying optimal control theory to determine cost-minimizing release strategies in cases of high mortality among infected immature mosquitoes. Their findings indicated that increased larval mortality hindered Wolbachia establishment and hence diminished its efficacy for the treatment of dengue. Wang et al. [21] introduced a stochastic vector–host model with environmental stochasticity in the form of white noise and demonstrated the dynamics of mosquito and human infection levels as a function of changes in death rates and life stage switching. Their analysis verified that controlling mosquito mortality rates significantly lowered the prevalence of dengue, as a key finding also obtained by numerical simulations with Fuzhou, China, data. Zhang et al. [22] set up an impulsive differential system to investigate Wolbachia-based population suppression and replacement strategies, including sex ratios, release time, and cytoplasmic incompatibility. Their bifurcation and sensitivity analyses demonstrated that the performance of these techniques was significantly influenced by initial mosquito densities, release parameters, and selection of Wolbachia strain. These studies collectively provide a comprehensive comprehension of the operational and ecological processes required to control dengue effectively. Jan et al. [23] reported a study in which they exemplified the very important role played by memory and non-Markovian dynamics in the propagation of dengue. They developed a fractional-order model accounting for asymptomatic carriers with the use of Caputo–Fabrizio (CF) and Atangana–Baleanu (AB) derivatives. They quantified the basic reproduction number as well as the sensitivity analysis using the PRCC method. Simulations revealed that reducing the fractional-order parameter had a significant effect on the infected population number, and the AB model played a more important role in this sense than the CF model. Moreover, they discovered that decreasing the mosquito bite rate could significantly lower the disease rate. Jajarmi et al. [24] formulated a new model of dengue epidemics based on the fractional differential equation. The new model was established and confirmed from Cape Verde’s 2009 genuine epidemic data. Equilibrium states and system stability were discussed. Simulation outcomes determined that the use of the exponential kernel adequately delineated the start of the epidemic but not the Mittag–Leffler kernel for the remaining stages. The application of the combined kernel was accurate in simulating the entire disease dynamics. The authors suggested two control strategies that efficiently stabilized the disease-free equilibrium in simulations. Fatmawati et al. [25] established a fractional-order model of dengue transmission involving carrier and partially immune classes. They applied fractional calculus to study the system and estimated the basic reproduction number based on the next-generation method. Memory effects and vector biting rate were crucial to studying and controlling dengue transmission, as demonstrated by the work. Srivastava et al. [26] presented a new fractional-order model of dengue with asymptomatic carriers and reinfection, studied through the Liouville–Caputo and Atangana–Baleanu methods. Existence, uniqueness, and boundedness of the solutions were confirmed, and the basic reproduction number was approximated using the next-generation approach. Sensitivity analysis identified the most critical determinants of the dynamics of illness. Their finding indicated that input parameters and the fractional order had significant roles in infection patterns. Boulaaras et al. [27] introduced a fractional-order complex dengue model with vaccination, reinfection, and carrier dynamics based on the Caputo–Fabrizio framework. They estimated the basic reproduction number using the next-generation matrix and examined the local stability at the disease-free equilibrium. Numerical experiments demonstrated the influence of altering the fractional order and input parameters on disease behavior.

It identified the most important parameters that would be applied to eliminate dengue and gave some suggestions on how disease modeling can be enhanced through the use of fractional calculus.

The novelty of this study is in its application of a fractional-order mathematical model to analyze the transmission dynamics of dengue fever, considering awareness-based prevention strategies. This is a departure from previous models that mainly rely on integer-order differential equations; in this work, fractional calculus is employed to describe the memory effects and long-range effects of disease progression in a better manner. In addition, the Adomian Decomposition Method (ADM) is utilized to analytically solve the system and provides a highly convergent analytical approximation that is more computationally efficient as opposed to numerical methods. Furthermore, by explicitly incorporating public awareness and behavioral responses into the model, this study provides a more realistic representation of disease control dynamics, thereby making it different from other studies performed previously, which focused only on interactions between the vector and the host. The organization of the manuscript is as follows: Section 2 describes the mathematical equation of the dengue fever model. Section 3 explains the mathematical approach and then applies it. Section 4 gives a case study along with the analysis of the model. Finally, Section 5 reports the conclusion of the research.

2. Mathematical Model Formulation of Dengue Fever

The mathematical model of $DF$ transmission divides the populations into two main parts, namely, human hosts and mosquito vectors. The host population ($N_h$) consists of five distinct compartments: those who are fully susceptible ($S_h$), susceptible with partial immunity ($S_{hk}$), individuals with symptomatic infections ($I_h$), those with asymptomatic infections ($I_{hA}$), and recovered individuals ($R_h$). The mosquito population ($N_v$) is divided into two categories, the susceptible ($S_v$) and infectious ($I_v$) groups. The model considers the dynamics of natural population through birth and death rates, denoted by ${\mu }_h$ for humans and ${\mu }_v$ for mosquitoes. While the dengue-related death rate in humans is considered negligible, mosquitoes are modeled with equal birth and death rates (${\mu }_v$). For simplicity, different dengue strains are considered the same in the mathematical model. The transmission of DF occurs when infected mosquitoes bite susceptible humans. The rate of disease spread depends on multiple factors, including the mosquito biting rate ($b$), transmission probabilities, and the numbers of susceptible and infected individuals in both populations. The transmission probability from humans to mosquitoes is denoted as ${\beta }_h$ for fully susceptible individuals and ${\beta }_{hk}$ for partially immune individuals. The model assumes that fully susceptible individuals have a higher transmission probability than those with partial immunity (${\beta }_h$ > ${\beta }_{hk}$). The relation ${\displaystyle \frac{b}{N_h}}$ is known as the number of mosquito bites a human receives from each one, and then ${\displaystyle \frac{b{\beta }_h}{N_h}}{I}_v, {\displaystyle \frac{b{\beta }_{hk}}{N_h}}{I}_v,{\displaystyle \frac{b{\beta }_v}{N_h}}{I}_v$ represent the infection rates per susceptible human and each susceptible vector. Therefore, the initial infection from susceptible $S_h$ and the second infection from susceptible $S_{hk}$ are both included in the infected class ($I_h$). A percentage v of the recovered persons ($R_h$) lose immunity and join the susceptible class with just partial immunity ($S_{hk}$). These dynamics are mathematically formalized through a system of fractional order differential equations based on these assumptions and relationships [28,29,30].

$$\left\{\begin{array}{c}{\displaystyle \frac{d{S}_h}{dt}}={\mu }_h{N}_h-{\displaystyle \frac{{\beta }_{h}b}{N_h}} S_h{I}_v-{\mu }_h{S}_h,\\ {\displaystyle \frac{d{S}_{hk}}{dt}}=-{\displaystyle \frac{{\beta }_{hk}b}{N_h}} S_{hk}{I}_v-{\mu }_h{S}_{hk}+\upsilon R_h,\\ {\displaystyle \frac{d{I}_h}{dt}}=\left(1-\psi \right){\displaystyle \frac{{\beta }_{h}b}{N_h}} S_h{I}_v-\left({\mu }_h+\tau +\gamma \right)I_h+{\displaystyle \frac{{\beta }_{hk}b}{N_h}} S_{hk}{I}_v,\\ {\displaystyle \frac{d{I}_{hA}}{dt}}=\psi {\displaystyle \frac{{\beta }_{h}b}{N_h}}{S}_h{I}_v-\left({\mu }_h+\gamma \right)I_{hA}\\ {\displaystyle \frac{d{R}_h}{dt}}=\gamma \left(I_h+I_{hA}\right)+\tau I_h-\left(v+{\mu }_h\right)R_h,\\ {\displaystyle \frac{d{S}_v}{dt}}={\mu }_v{N}_v-{\displaystyle \frac{{\beta }_{h}b}{N_h}} \left(I_h+I_{hA}\right)S_v-{\mu }_v{S}_v,\\ {\displaystyle \frac{d{I}_v}{dt}}={\displaystyle \frac{{\beta }_{v}b}{N_h}} I_h+I_{hA}{S}_v-{\mu }_v{I}_v\end{array}\right..$$

(1)

For better and generalized modeling, the fractional derivatives are employed, which are inherently nonlocal, meaning they depend not just on the value at a single time point but on the entire history of the function. In epidemics, the future state of the system does not depend only on the current number of infections but also on past exposures, recoveries, or immunity loss. Accordingly, and from a mathematical point of view, a study on the effect of different fractional orders, $\alpha ,$ is introduced to secure different models, which could describe different situations.

$$\left\{\begin{array}{c}{D}_t^{\alpha }{S}_h={\mu }_h{N}_h-{\displaystyle \frac{{\beta }_{h}b}{N_h}} S_h{I}_v-{\mu }_h{S}_h,\\ D_t^{\alpha }{S}_{hk}=-{\displaystyle \frac{{\beta }_{hk}b}{N_h}} S_{hk}{I}_v-{\mu }_h{S}_{hk}+\upsilon R_h,\\ D_t^{\alpha }{I}_h=\left(1-\psi \right){\displaystyle \frac{{\beta }_{h}b}{N_h}} S_h{I}_v-\left({\mu }_h+\tau +\gamma \right)I_h+{\displaystyle \frac{{\beta }_{hk}b}{N_h}} S_{hk}{I}_v,\\ D_t^{\alpha }{I}_{hA}=\psi {\displaystyle \frac{{\beta }_{h}b}{N_h}}{S}_h{I}_v-\left({\mu }_h+\gamma \right)I_{hA}\\ D_t^{\alpha }{R}_h=\gamma \left(I_h+I_{hA}\right)+\tau I_h-\left(v+{\mu }_h\right)R_h,\\ D_t^{\alpha }{S}_v={\mu }_v{N}_v-{\displaystyle \frac{{\beta }_{h}b}{N_h}} \left(I_h+I_{hA}\right)S_v-{\mu }_v{S}_v,\\ D_t^{\alpha }{I}_v={\displaystyle \frac{{\beta }_{v}b}{N_h}} I_h+I_{hA}{S}_v-{\mu }_v{I}_v\end{array}\right.,$$

(2)

with an initial condition,

$$S_h\left(0\right)=n_1,S_{hk}\left(0\right)=n_2,I_h\left(0\right)=n_3,I_{hA}\left(0\right)=n_4,R_h\left(0\right)=n_5,S_v\left(0\right)=n_6,I_v\left(0\right)=n_7,$$

(3)

where $D_t^{\alpha }$ satisfies the definition of the Caputo derivative, with order $\alpha$ defined as follows:

$$D^{\alpha }f\left(s\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(\mathrm{n}-\alpha )}}{\int }_0^s{\left(s-t\right)}^{\mathrm{n}-\alpha -1}\left(f^{\left(n\right)}\left(t\right)\right)dt, n-1<\alpha \le n,n\in \mathrm{N},s>0.$$

(4)

3. The Adomian Decomposition Method

To find the solution of the system (2), the Adomian Decomposition Method (ADM) [31,32] starts with deeming this system in an operator form and then evaluating the inverse operator for the system.

$$\left\{\begin{array}{c}{S}_h=D_t^{-\alpha }{\mu }_h{N}_h-D_t^{-\alpha }{\displaystyle \frac{{\beta }_{h}b}{N_h}} S_h{I}_v-D_t^{-\alpha }{\mu }_h{S}_h,\\ S_{hk}=-D_t^{-\alpha }{\displaystyle \frac{{\beta }_{hk}b}{N_h}} S_{hk}{I}_v-D_t^{-\alpha }{\mu }_h{S}_{hk}+D_t^{-\alpha }\upsilon R_h,\\ I_h=D_t^{-\alpha }\left(1-\psi \right){\displaystyle \frac{{\beta }_{h}b}{N_h}} S_h{I}_v-D_t^{-\alpha }\left({\mu }_h+\tau +\gamma \right)I_h+D_t^{-\alpha }{\displaystyle \frac{{\beta }_{hk}b}{N_h}} S_{hk}{I}_v,\\ I_{hA}=D_t^{-\alpha }\psi {\displaystyle \frac{{\beta }_{h}b}{N_h}}{S}_h{I}_v-D_t^{-\alpha }\left({\mu }_h+\gamma \right)I_{hA}\\ R_h=D_t^{-\alpha }\gamma \left(I_h+I_{hA}\right)+D_t^{-\alpha }\tau I_h-D_t^{-\alpha }\left(v+{\mu }_h\right)R_h,\\ S_v=D_t^{-\alpha }{\mu }_v{N}_v-D_t^{-\alpha }{\displaystyle \frac{{\beta }_{h}b}{N_h}} \left(I_h+I_{hA}\right)S_v-D_t^{-\alpha }{\mu }_v{S}_v,\\ I_v=D_t^{-\alpha }{\displaystyle \frac{{\beta }_{v}b}{N_h}} I_h+D_t^{-\alpha }{I}_{hA}{S}_v-D_t^{-\alpha }{\mu }_v{I}_v\end{array}\right..$$

(5)

According to ADM assumptions, the following series are introduced:

$$S_h={\sum }_{n=0}^{\mathrm{\infty }}{S}_{hn},S_{hk}={\sum }_{n=0}^{\mathrm{\infty }}{S}_{hkn},I_h={\sum }_{n=0}^{\mathrm{\infty }}{I}_{hn},I_{hA}={\sum }_{n=0}^{\mathrm{\infty }}{I}_{hAn},R_h={\sum }_{n=0}^{\mathrm{\infty }}{R}_{hn},S_v={\sum }_{n=0}^{\mathrm{\infty }}{S}_{vn},I_v={\sum }_{n=0}^{\mathrm{\infty }}{I}_{vn}.$$

(6)

The sum of nonlinear components can be described as follows:

$$\left\{\begin{array}{c}{S}_h{I}_v={\sum }_{n=0}^{\mathrm{\infty }}{A}_n={\sum }_{m=0}^{\mathrm{\infty }}{S}_{hm}{I}_{v(n-m)},\\ S_{hk}{I}_v={\sum }_{n=0}^{\mathrm{\infty }}{B}_n={\sum }_{m=0}^{\mathrm{\infty }}{S}_{hkm}{I}_{v(m-k)},\\ I_{hA}{S}_v={\sum }_{n=0}^{\mathrm{\infty }}{C}_n={\sum }_{k=0}^{\mathrm{\infty }}{I}_{hAm}{S}_{v(m-k)},\\ I_h{S}_v={\sum }_{n=0}^{\mathrm{\infty }}{D}_n={\sum }_{k=0}^{\mathrm{\infty }}{I}_{hm}{S}_{v(m-k)}.\end{array}\right.,$$

(7)

where n = 0, 1, 2, …, n.

The Adomian polynomials A_n, B_n, C_n, and D_n are obtained as follows:

$$\left\{\begin{array}{c}{A}_0=S_{h0}{I}_{v0}\\ A_1=S_{h0}{I}_{v1}+S_{h1}{I}_{v0}\\ A_2=S_{h0}{I}_{v2}+S_{h1}{I}_{v1}+S_{h2}{I}_{v0}\\ A_3=S_{h0}{I}_{v3}+S_{h1}{I}_{v2}+S_{h2}{I}_{v1}+S_{h3}{I}_{v0}\\ \dots \dots .\end{array}\right.,$$

(8)

$$\left\{\begin{array}{c}{B}_0=S_{hk0}{I}_{v0}\\ B_1=S_{hk0}{I}_{v1}+S_{hk1}{I}_{v0}\\ B_2=S_{hk0}{I}_{v2}+S_{hk1}{I}_{v1}+S_{hk2}{I}_{v0}\\ B_3=S_{hk0}{I}_{v3}+S_{hk1}{I}_{v2}+S_{hk2}{I}_{v1}+S_{hk3}{I}_{v0}\\ \dots \dots .\end{array}\right.,$$

(9)

$$\left\{\begin{array}{c}{C}_0=I_{hA0}{S}_{v0}\\ C_1=I_{hA0}{S}_{v1}+I_{hA1}{S}_{v0}\\ C_2=I_{hA0}{S}_{v2}+I_{hA1}{S}_{v1}+I_{hA2}{S}_{v0}\\ C_3=I_{hA0}{S}_{v3}+I_{hA1}{S}_{v2}+I_{hA2}{S}_{v1}+I_{hA3}{S}_{v0}\\ \dots \dots .\end{array}\right.,$$

(10)

$$\left\{\begin{array}{c}{D}_0=I_{h0}{S}_{v0}\\ D_1=I_{h0}{S}_{v1}+I_{h1}{S}_{v0}\\ D_2=I_{h0}{S}_{v2}+I_{h1}{S}_{v1}+I_{h2}{S}_{v0}\\ D_3=I_{h0}{S}_{v3}+I_{h1}{S}_{v2}+I_{h2}{S}_{v1}+I_{h3}{S}_{v0}\\ \dots \dots .\end{array}\right..$$

(11)

Then, substituting Equalities (6)–(11) into System (5), yields the following results:

$${\sum }_{n=0}^{\infty }{S}_{hn}=n_1+D_t^{-\alpha }{\mu }_h{N}_h-D_t^{-\alpha }{\displaystyle \frac{{\beta }_{h}b}{N_h}} {\sum }_{n=0}^{\infty }{A}_n-D_t^{-\alpha }{\mu }_h{\sum }_{n=0}^{\infty }{S}_{hn},$$

(12)

$${\sum }_{n=0}^{\mathrm{\infty }}{S}_{hkn}=n_2-D_t^{-\alpha }{\displaystyle \frac{{\beta }_{hk}b}{N_h}}{\sum }_{n=0}^{\mathrm{\infty }}{B}_n-D_t^{-\alpha }{\mu }_h{\sum }_{n=0}^{\mathrm{\infty }}{S}_{hkn}+D_t^{-\alpha }\upsilon {\sum }_{n=0}^{\mathrm{\infty }}{R}_{hn},$$

(13)

$${\sum }_{n=0}^{\mathrm{\infty }}{I}_{hn}=n_3+D_t^{-\alpha }\left(1-\psi \right){\displaystyle \frac{{\beta }_{h}b}{N_h}}{\sum }_{n=0}^{\mathrm{\infty }}{A}_n-D_t^{-\alpha }\left({\mu }_h+\tau +\gamma \right){\sum }_{n=0}^{\mathrm{\infty }}{I}_{hn}+D_t^{-\alpha }{\displaystyle \frac{{\beta }_{hk}b}{N_h}}{\sum }_{n=0}^{\mathrm{\infty }}{B}_n,$$

(14)

$${\sum }_{n=0}^{\mathrm{\infty }}{I}_{hAn}=n_4+D_t^{-\alpha }\psi {\displaystyle \frac{{\beta }_{h}b}{N_h}}{\sum }_{n=0}^{\mathrm{\infty }}{A}_n-D_t^{-\alpha }\left({\mu }_h+\gamma \right){\sum }_{n=0}^{\mathrm{\infty }}{I}_{hAn},$$

(15)

$${\sum }_{n=0}^{\mathrm{\infty }}{R}_{hn}=n_5+D_t^{-\alpha }\gamma {\sum }_{n=0}^{\mathrm{\infty }}{I}_{hn}+D_t^{-\alpha }\gamma {\sum }_{n=0}^{\mathrm{\infty }}{I}_{hAn}+D_t^{-\alpha }\tau {\sum }_{n=0}^{\mathrm{\infty }}{I}_{hn}-D_t^{-\alpha }\left(v+{\mu }_h\right){\sum }_{n=0}^{\mathrm{\infty }}{R}_{hn},$$

(16)

$${\sum }_{n=0}^{\mathrm{\infty }}{S}_{vn}=n_6+D_t^{-\alpha }{\mu }_v{N}_v-D_t^{-\alpha }{\displaystyle \frac{{\beta }_{h}b}{N_h}}\mathrm{ }{\sum }_{n=0}^{\mathrm{\infty }}{D}_n-D_t^{-\alpha }{\displaystyle \frac{{\beta }_{h}b}{N_h}}{\sum }_{n=0}^{\mathrm{\infty }}{C}_n-D_t^{-\alpha }{\mu }_v{\sum }_{n=0}^{\mathrm{\infty }}{S}_{vn},$$

(17)

$${\sum }_{n=0}^{\mathrm{\infty }}{I}_{vn}=n_7+D_t^{-\alpha }{\displaystyle \frac{{\beta }_{v}b}{N_h}}{\sum }_{n=0}^{\mathrm{\infty }}{I}_{hn}+D_t^{-\alpha }{\sum }_{n=0}^{\mathrm{\infty }}{C}_n-D_t^{-\alpha }{\mu }_v{\sum }_{n=0}^{\mathrm{\infty }}{I}_{vn}.$$

(18)

Considering the Caputo integral properties, yields the following:

$$S_{h0}=n_1, S_{hk0}=n_2,I_{h0}=n_3,I_{hA0}=n_4,R_{h0}=n_5,S_{v0}=n_6,I_{v0}=n_7,$$

(19)

$$S_{h(m+1)}={\displaystyle \frac{{\mu }_h{N}_h}{\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }-{\displaystyle \frac{{\beta }_{h}b}{N_h\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{A}_m-{\displaystyle \frac{{\mu }_h}{\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{S}_{hm}, m=0,1,2,\dots ,$$

(20)

$$S_{hk(m+1)}=-{\displaystyle \frac{{\beta }_{hk}b}{N_h\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{B}_m-{\displaystyle \frac{{\mu }_h}{\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{S}_{hkm}+{\displaystyle \frac{\upsilon }{\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{R}_{hm},$$

(21)

$$I_{h(m+1)}={\displaystyle \frac{\left(1-\psi \right){\beta }_{h}b}{N_h\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{A}_m-{\displaystyle \frac{\left({\mu }_h+\tau +\gamma \right)}{\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{I}_{hm}+{\displaystyle \frac{{\beta }_{hk}b}{N_h\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{B}_m,$$

(22)

$$I_{hA(m+1)}=\psi {\displaystyle \frac{{\beta }_{h}b}{N_h\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{A}_m-{\displaystyle \frac{\left({\mu }_h+\gamma \right)}{\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{I}_{hAm},$$

(23)

$$R_{h(m+1)}={\displaystyle \frac{\gamma }{\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{I}_{hm}+{\displaystyle \frac{\gamma }{\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{I}_{hAm}+{\displaystyle \frac{\tau }{\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{I}_{hm}-{\displaystyle \frac{\left(v+{\mu }_h\right)}{\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{R}_{hm},$$

(24)

$$S_{v(m+1)}={\displaystyle \frac{{\mu }_v{N}_v}{\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }-{\displaystyle \frac{{\beta }_{h}b}{N_h\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{D}_m-{\displaystyle \frac{{\beta }_{h}b}{N_h\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{C}_m-{\displaystyle \frac{{\mu }_v{N}_v}{\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{S}_{vm},$$

(25)

$$I_{v(m+1)}={\displaystyle \frac{{\beta }_{v}b}{N_h\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{I}_{hm}+{\displaystyle \frac{t^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}}{C}_m-{\displaystyle \frac{{\mu }_v}{\mathrm{\Gamma }\left(\alpha +1\right)}}{t}^{\alpha }{I}_{vm}.$$

(26)

4. Results and Discussion

The numerical simulations are carried out using the parameter values in

Table 1. Parameter information.

Symbol	Physical Meaning	Value	References
${\mathit{N}}_{\mathit{h}}$	Constant host population size	8,266,000	[33]
${\mathit{s}}_{\mathit{h}}$	Susceptible people	8,065,518	[33]
${\mathit{s}}_{\mathit{h}\mathit{k}}$	Susceptible people with partial immunity	200,000	[33]
${\mathit{I}}_{\mathit{h}}$	Symptomatic infected people	200	[33]
${\mathit{I}}_{\mathit{h}\mathit{A}}$	Asymptomatic infected people	282	[33]
${\mathit{R}}_{\mathit{h}}$	Recovered people	0	[33]
${\mathit{N}}_{\mathit{v}}$	Vector (mosquito) population size	50,000	Assumed
${\mathit{s}}_{\mathit{v}}$	Susceptible mosquitoes	49,000	Assumed
${\mathit{I}}_{\mathit{v}}$	Infected mosquitoes	1000	Assumed
${\mathit{\mu }}_{\mathit{h}}$	Death rates of humans per capita	0.0000457/day	[3,28,29,30]
${\mathit{\mu }}_{\mathit{v}}$	Death rates of mosquitoes per capita	0.03/day	[3,28,29,30]
${\mathit{\beta }}_{\mathit{h}}$	The transmission probability from vector to human (effective contact rate)	0.75	[3,28,29,30]
${\mathit{\beta }}_{\mathit{h}\mathit{k}}$	The transmission probability from vector to human with partial immunity (effective contact rate)	1	Assumed
${\mathit{\beta }}_{\mathit{v}}$	The transmission probability from human to vector (effective contact rate)	0.75	[3,28,29,30]
b	Average bite per mosquito per day	0.3/day	[3,28,29,30]
$\mathit{b}{\mathit{\beta }}_{\mathit{v}}$	Effective contact rate, human to vector	0.3750000	[28]
${\mathit{b}\mathit{\beta }}_{\mathit{h}}$	Effective contact rate, vector to human	0.75	[28]
$\mathit{\gamma }$	Recovery rate in the host population, time −1	0.14286/day	[3,28,29,30]
$\mathit{\psi }$	Proportion of asymptomatic infection rate of people	0.009	[34]
${\mathit{n}}_{\mathbf{1}}$	$S_h\left(0\right)$	0.9757	${\displaystyle \frac{s_h}{N_h}}$
${\mathit{n}}_{\mathbf{2}}$	$S_{hk}\left(0\right)$	0.0242	${\displaystyle \frac{s_{hk}}{N_h}}$
${\mathit{n}}_{\mathbf{3}}$	$I_h\left(0\right)$	0.0000242	${\displaystyle \frac{I_h}{N_h}}$
${\mathit{n}}_{\mathbf{4}}$	$I_{hA0}$	0.000034	${\displaystyle \frac{I_{hA}}{N_h}}$
${\mathit{n}}_{\mathbf{5}}$	$R_{h0}$	0	${\displaystyle \frac{R_h}{N_h}}$
${\mathit{n}}_{\mathbf{6}}$	$S_v\left(0\right)$	0.98	${\displaystyle \frac{s_V}{N_V}}$
${\mathit{n}}_{\mathbf{7}}$	$I_v\left(0\right)$	0.02	${\displaystyle \frac{I_V}{N_V}}$

.

At $\mathrm{\alpha }=1$, the results can be obtained in the following form:

$$S_h = 0.9757+1511.0248t-0.0519t^2+1.5902\ast {10}^{-6}{t}^3+1.5074\ast {10}^{-6}{t}^4+\cdots ,$$

(27)

$$S_{hk} = 0.0242-1.106\ast {10}^{-6}t+5.10427\ast {10}^{-11}{t}^2-89.5916\ast {10}^{-12}{t}^3+6.7197\ast {10}^{-8}{t}^4+\cdots ,$$

(28)

$$I_h = 0.0000242-3.4578\ast {10}^{-6}t+1.091929\ast {10}^{-6}{t}^2-135.44\ast {10}^{-9}{t}^3-1.546\ast {10}^{-6}{t}^4+\cdots ,$$

(29)

$$I_{hA} = 0.000034-4.85726\ast {10}^{-6}t+699.68182\ast {10}^{-9}{t}^2-99.27595\ast {10}^{-9}{t}^3-2.778\ast {10}^{-8}{t}^4+\cdots ,$$

(30)

$$R_h=1.07344\ast {10}^{-5}t+229.668\ast {10}^{-9}{t}^2-136.752\ast {10}^{-9}{t}^3-3.9443\ast {10}^{-8}{t}^4+\cdots ,$$

(31)

$$S_v = 0.98+6000t-6.75\ast {10}^6{t}^2+6.75\ast {10}^9{t}^3-5.0625\ast {10}^{12}{t}^4,+\cdots ,$$

(32)

$$I_v = 0.02-5.6668\ast {10}^{-4}t+ 0.153t^2-153.017t^3+114.76\ast {10}^3{t}^4+\cdots .$$

(33)

For $\mathrm{\alpha }={\displaystyle \frac{1}{2}}$, the results are given hereafter.

$$S_h = 0.9757+1.705009\ast {10}^3{t}^{0.5}-21.980781\ast {10}^{-3}t+15.14401281\ast {10}^{-6}{t}^{{\displaystyle \frac{3}{2}}}+1.70092t^2+\cdots ,$$

(34)

$$S_{hk}= 0.0242-1.247939\ast {10}^{-6}{t}^{0.5}+64.934982\ast {10}^{-12}t-128.74597\ast {10}^{-12}{t}^{{\displaystyle \frac{3}{2}}}+7.5823695\ast {10}^{-8}{t}^2+\cdots .$$

(35)

$$I_h = 0.0000242-3.9017948\ast {10}^{-6}{t}^{0.5}+1.378081386\ast {10}^{-6}t-6.559415133\ast {10}^{-6}{t}^{{\displaystyle \frac{3}{2}}}-1.744474\ast {10}^{-6}{t}^2+\cdots .$$

(36)

$$I_{hA} = 0.000034-5.480831\ast {10}^{-6}{t}^{0.5}+892.59524487\ast {10}^{-9}t-200.917186\ast {10}^{-9}{t}^{{\displaystyle \frac{3}{2}}}-3.1346373\ast {10}^{-8}{t}^2+\cdots .$$

(37)

$$R_h=12.112473\ast {10}^{-6}{t}^{0.5}-1.952752\ast {10}^{-6}t+43.9112769\ast {10}^{-8}{t}^{{\displaystyle \frac{3}{2}}}-4.45066595\ast {10}^{-8}{t}^2+\cdots .$$

(38)

$$S_v = 0.98+6.770276\ast {10}^3{t}^{0.5}-8.594367\ast {10}^{6}t+9.6977046\ast {10}^9{t}^{{\displaystyle \frac{3}{2}}}-5.712419\ast {10}^{12}{t}^2+\cdots .$$

(39)

$$I_v = 0.02-639.429906\ast {10}^{-6}{t}^{0.5}+194.8056\ast {10}^{-3}t-219.839968t^{{\displaystyle \frac{3}{2}}}+129.49279\ast {10}^3{t}^2+\cdots .$$

(40)

In the same way, other results can be obtained for different values of $\alpha$. The fractional order in the mathematical model is used to describe memory effects, hereditary properties, and anomalous dynamics in systems in physics, epidemiology, and engineering. The fractional order $\alpha (0<\alpha \le 1)$ includes a concept of history dependence and non-locality, which cannot be achieved with the standard integer-order models.

To verify the existing model, the obtained results are compared against results published by Z. Ullah [35] for significant epidemiological compartments, such as susceptible and infected populations. From this comparison, the accuracy of the ADM model versus the Laplace adomian method in simulating the transmission of dengue can be established, as well as deviations or improvements from previous results. Figure 1 and Figure 2 show high agreement between the presented model and literature.

Figure 3 illustrates the time evolution of the susceptible population $S_h$ as a function of time (days) with various values of the fractional order parameter $\alpha$. The fractional order derivative causes memory effects within the model, one of the most significant characteristics of fractional calculus. This figure indicates how the memory parameter α controls the susceptibility dynamics in a fractional-order epidemic model. For small α values (e.g., $\alpha =0.1$), the buildup of susceptibility is far slower. This behavior illustrates the extremely strong memory effect of the fractional model: the system “remembers” more of its past states, which can be interpreted in epidemiological terms as long-term immunity, delayed exposure, or environmental survival that slows the progression of individuals into the susceptible class. These delays could simulate some scenarios where elder infections impart long-term partial immunity or where environmental conditions reduce immediate re-exposure. As $\alpha$ increases, the curve steepens, with a faster accumulation of susceptibles. This corresponds to weaker memory effects, where individuals lose immunity or become re-exposed to the disease vector more quickly. Biologically, this can be attributed to waning immunity, rapid mosquito–human transmission cycles, or environmental feedbacks that reinforce the rate at which susceptibility accumulates, such as mosquito breeding increases due to climatic changes or human migration into endemic areas. At the largest considered value of $\alpha =0.5$, the susceptibility increases nearly exponentially with time and reaches significantly larger values than the other curves. This is the behavior of a system in which past states have little effect, and the disease dynamics are more like a typical first-order system. The steep increase in susceptibility may be due to hastened disease transmission cycles, perhaps fueled by urbanization, waning immunity at the population level, or disruption of vector control policies. The monotonicity and increasing convexity of the curves with increasing α support the reasoning that memory decay leads to faster replenishment of the susceptible pool. Thus, higher values of α weaken the dampening effect of the past disease dynamics, pointing to the importance of temporal memory in disease control and recurrence.

Figure 4 illustrates the temporal evolution of the partially immune susceptible population $S_{hk}$ for different values of the fractional-order parameter $\alpha$. The graph shows a steady decrease in $S_{hk}$ across all values of $\alpha$, suggesting a progressive loss of partial immunity and movement towards alternative disease states, including full susceptibility or infection. This trend illustrates the immunological dynamics in diseases such as dengue, where partial immunity, typically acquired through previous exposure to various serotypes, diminishes over time. The fractional-order derivative $\alpha$ significantly influences this behavior by incorporating memory effects into the system. At reduced values of $\alpha$, the system demonstrates enhanced memory, indicating that previous states exert a more extended impact on current dynamics. The slower decrease in $S_{hk}$ indicates that individuals maintain partial immunity for an extended period, which consequently delays susceptibility and slows the epidemic’s progression. As $\alpha$ increases, the memory effect diminishes, resulting in a more rapid decline in $S_{hk}$. This suggests that immunity diminishes more rapidly, enabling quicker shifts between disease states and promoting faster epidemic spread. This may indicate a situation in which protective immunity is transient or where environmental and immunological factors facilitate a more rapid decline in protection. The fractional-order model’s capacity to represent nuanced behavior underscores its significance in modeling real-world infectious disease dynamics, particularly in scenarios where immune memory, reinfection risk, and cross-protection are essential for comprehending long-term disease propagation patterns.

Figure 5 depicts the symptomatic infected human population. The rate and amount of this reduction depend significantly on the selection of α, the memory or inherited characteristics of the system. In fractional calculus models, the parameter $\alpha \in \left(\mathrm{0,1}\right]$ determines the amount to which previous states affect current behavior, with smaller values giving more weight to historical influence and larger values approximating conventional models that assume nearly instantaneous dynamics. At lower values of α (say, α = 0.05 and 0.1), the memory effect is stronger in the sense that the spread of infection is more inertial due to the persistent effect of earlier infection rates. This causes the decline of $I_h$ to be slower, as the system retains past symptomatic loads, potentially indicating responses delayed to intervention strategies or prolonged durations of illness and recovery. In contrast, large values for α (e.g., α = 0.2 and 0.3) are indicative of low memory systems that facilitate rapid state change. This results in a more rapid reduction in symptomatic disease, probably indicating more effective public health measures, such as a rapid diagnosis, treatment, effective vaccination programs, or mosquito control causing rapid reduction in transmission. Continuous decrease in $I_h$ with increased α might reflect enhanced host immunity or behavioral reduction in contact with vectors. This result highlights the role of dynamical memory in predicting the efficiency of control measures from an epidemiological perspective. The capability of the fractional model to account for various levels of system memory enables a more general and realistic modeling of disease progression, especially for vector-borne infections such as dengue, where immune response, onset of treatment, and delay of vector exposure are quantities that decide the outcome. Consequently, the determination of α provides a robust modeling method to simulate varied public health scenarios and guide best intervention practices.

Figure 6 displays the temporal behavior of the population of asymptomatic infected individuals $I_{hA}$ in a fractional-order model of dengue fever, showing how this population evolves over time with different values of the fractional-order parameter α. The fractional-order parameter α, determining the effect of memory in fractional calculus, plays an important role in how fast asymptomatic patients decline. At lower α values, the decline is slower, suggesting a longer duration of asymptomatic carriers within the community. Conversely, large values of α result in a steeper decline, suggesting rapid elimination of the silent infections. This trend illustrates that more powerful memory effects in systems (more negative α) keep past infection processes going for extended time intervals, possibly because of late immune response or persistence of the virus in the environment. Epidemiologically, the faster disappearance of $I_{hA}$ with larger α might indicate better public health interventions, e.g., increased diagnosis and quarantine of asymptomatic carriers, greater access to healthcare, or greater community immunity. The asymptomatic course in $I_{hA}$ is nearly indistinguishable from that of symptomatic disease $I_{hA}$, but the fraction of asymptomatic disease begins at slightly elevated levels, consistent with known dengue epidemiology, in which most patients are subclinical but still facilitate transmission. The figure highlights the advantages of fractional-order models in describing the complexity of infectious disease dynamics and the parameter α as a quantitative metric of biological processes and the effectiveness of interventions.

Figure 7 illustrates the recovery time course of recovered population $R_h\left(t\right)$ in a fractional-order dengue disease model as illustrates the continued increase in recoveries by time over the range of the fractional-order parameter α. This rise is consistent with earlier-found reductions in both symptomatic ($I_h$) and asymptomatic ($I_{hA}$) infective individuals and would indicate ongoing clearing of infection from within the population. The growth rate of recovery is significantly affected by the value of α, with large fractional orders (e.g., α = 0.3) producing a more rapid and higher rate of recovery accumulation. It indicates that as the influence of memory in the system diminishes, reaching the classical case, sick individuals recover more quickly. This may indicate stronger immune responses and effective healthcare infrastructures.

Smaller values of α indicate a slow recovery, in agreement with long-term memory effects that may be due to biological or logistical delays, e.g., delayed diagnosis, limited access to treatment, or inherently longer durations of recovery. From the modeling perspective, fractional calculus provides a more in-depth explanation of epidemic disease processes in real life by considering memory and genetic characteristics of infectious disease development. As such, parameter α is a valuable instrument to study the role of systemic and biological delays in recovery outcomes and provides a congenial framework to evaluate the contribution of public health interventions and individual recovery paths in dengue transmission.

Figure 8 displays the time dynamics of the infected mosquito population $I_v\left(t\right)$ of the fractional-order dengue model, which indicates an explicit correlation with the fractional-order parameter α. As time goes on, the infected mosquitoes grow in number, with the amount of this growth depending to a large extent on the value of α. For higher values, e.g., α = 0.3, $I_v$ growth is rapid and roughly linear, echoing increased transmission dynamics between humans and mosquito vectors. This could simulate situations with minimal memory effects, perhaps because of high vector biting rates, favorable weather conditions, or inefficient vector control measures. In contrast, lower values of α (e.g., 0.05 and 0.1) result in a much more gradual rise in mosquito infections, suggesting better suppression of transmission cycles. These scenarios may represent significant environmental limits on mosquito populations, effective public health measures, or behavioral changes that limit human–vector contact. The synchrony between infected mosquito trends and the foregoing reductions in human infections is indicative of the coupled dynamics of host–vector transmission in the model. The fractional-order derivative effectively captures memory effects in mosquito infection dynamics, including incubation lag and infection accumulation slowness. As a result, α is a critical parameter in the modeling and control of vector-borne diseases such as dengue. These results emphasize the importance of early and sustained vector control interventions, particularly in scenarios with high α values, where transmission may rapidly increase and saturate public health systems.

5. Conclusions

This paper has investigated the analytical solutions of the mathematical model of dengue fever by applying an approximate method called the Adomian Decomposition Method. The solutions obtained using this method are approximate series solutions. The following remarks have been drawn:

• The study verifies that the use of a fractional-order differential model offers a more realistic representation of dengue fever transmission dynamics than integer-order models. The inclusion of memory effects and hereditary properties enhances prediction accuracy and supports improved control strategies.
• The simulations reveal that smaller fractional orders (α) lead to longer disease persistence due to stronger memory effects, while larger α values result in faster infection reduction. This demonstrates that interventions altering the transmission rate significantly influence disease development.
• The study confirms that infected mosquito populations closely follow the patterns of human infections. High values of α lead to explosive vector transmission, validating the need for mosquito control measures, such as the use of insecticides and the elimination of breeding sites.
• The outcomes show that an increase in recovery corresponds to a decline in both symptomatic and asymptomatic infections.

The Government of India published National Guidelines for Clinical Management of Dengue Fever [36] to highlight key public health suggestions for the control and management of dengue fever. These suggestions included early case detection and categorization for early treatment, public education about warning signs and mosquito control, and enhancement of primary healthcare systems for early diagnosis and referral. The guidelines endorse careful fluid management, especially during the critical period, and discourage the use of nonsteroidal anti-inflammatory drugs (NSAIDs) due to the threat of bleeding. Special caution is exercised in vulnerable groups such as children, pregnant women, and those with the presence of comorbidities. Intensive surveillance, vector control, and public awareness campaigns are also essential components of India’s dengue control program.