Qualitative and Numerical Study of a Zika Epidemic Model with Fractal–Fractional Operators ^†

Abstract

This paper studies a fractal–fractional model of Zika virus transmission using the Atangana–Baleanu and Caputo–Fabrizio operators. We first establish the existence and uniqueness of the model solution and then examine its Hyers–Ulam stability using fixed-point theory. Numerical simulations are presented to study the effects of the fractional and fractal orders on the spread of infection. The results show that the Atangana–Baleanu operator yields a slower decline in infected humans and mosquitoes, while the Caputo–Fabrizio operator yields a faster decline and earlier stabilization. The simulations also show that increasing the fractal order from $0.85$ to $0.95$ leads to a faster reduction in the infected human population. These results show how different memory kernels and fractal orders can influence predicted Zika virus transmission.

1. Introduction

The Zika virus [1] is an arthropod-borne disease transmitted mainly by infected Aedes mosquitoes. Researchers first isolated the virus from a rhesus monkey in the Zika Forest of Uganda in 1947 and later identified it in humans in Uganda and Tanzania. For many years, Zika infection was considered relatively mild and received limited attention compared with other mosquito-borne diseases. However, large outbreaks on Yap Island, French Polynesia, and Brazil revealed its potential for rapid transmission and serious health consequences. The clinical manifestations of Zika infection usually include fever, skin rash, conjunctivitis, headache, muscle pain, and joint pain [2,3,4]. In recent years, studies have shown that the virus can also be transmitted through sexual contact, blood transfusion, and from infected mothers to their infants. Of particular concern is the reported association of Zika virus infection with Guillain–Barré syndrome and congenital abnormalities such as microcephaly. Because of its widespread distribution and the lack of a specific treatment, the Zika virus remains a topic of considerable interest in epidemiological and mathematical modeling studies [2,5,6,7,8].

Fractal–fractional (FF) operators are employed to more accurately represent the complex dynamics of Zika virus transmission, including memory effects, population heterogeneity, and long-term trends. Integer-order models frequently overlook these features, whereas fractional models may not adequately capture the fractal structures and irregular temporal patterns characteristic of biological systems. Consequently, we analyze both the Caputo–Fabrizio (CF) and Atangana–Baleanu (AB) FF operators. The CF operator uses a nonsingular exponential kernel, which is well-suited to modeling processes with fading memory and enables efficient computation [9,10,11,12]. In contrast, the AB operator employs the Mittag-Leffler kernel (M-L) [13,14], which more effectively captures the nonlocal effects and long-term memory observed in disease transmission. Because these operators address memory differently, we must evaluate how these differences affect the predictions of Zika virus transmission. We compare the two operators using numerical solutions to assess their accuracy and computational efficiency. This assessment clarifies how different memory kernels affect modeling. It informs the selection of the most appropriate FF framework for the reliable simulation and prediction of Zika virus spread [15,16,17,18,19].

In [20,21], the authors analyzed a new fractal–fractional operational matrix based on orthonormal normalized ultraspherical polynomials to solve the FF Riccati differential equation involving a generalized Caputo derivative. By combining the operational matrix with the spectral Tau method. The authors reduced the nonlinear differential equation to a system of algebraic equations and solved it using Newton’s method. Several test problems and comparisons with existing methods demonstrated the proposed approach’s accuracy and efficiency.

Recent studies have highlighted the value of fractal–fractional methods for analyzing Zika virus dynamics. Khan et al. integrated a FF Zika transmission model with artificial intelligence and neural networks, using simulation data to train and test the network and assess prediction errors [22]. Rezapour et al. introduced an FF M-L model to study the co-infection dynamics of COVID-19 and the Zika virus [23]. They examined the stability of the disease-free equilibrium, proved the existence and uniqueness of solutions and Hyers–Ulam stability, and developed a numerical approximation scheme using Newton polynomials. These findings confirm the effectiveness of FF operators in modeling memory-dependent Zika transmission and support further comparisons of different memory kernels.

Although fractional and FF operators have been increasingly applied to epidemic models, the qualitative analysis of Zika transmission models and direct numerical comparisons of the CF and AB memory kernels remain underexplored. This study develops a FF Zika model to characterize disease transmission between human and mosquito populations. The primary contributions include establishing the positivity and uniform boundedness of the model solutions, identifying a biologically feasible region, and numerically investigating how the fractional order and fractal dimension affect the susceptible and infected populations. Additionally, we compare the FF–CF and FF–AB operators using identical parameter values and initial conditions to elucidate how their distinct memory kernels affect predicted epidemic dynamics. These findings provide a foundation for selecting appropriate FF operators for modeling memory-dependent infectious disease transmission.

The paper is organized as follows. Section 2 presents the preliminary concepts of the fractal–fractional calculus. Section 3 formulates the proposed Zika epidemic model. Section 4 discusses the model’s qualitative properties, and Section 5 presents numerical simulations and discusses the results. Finally, the Section 6 presents the concluding remarks.

2. Preliminaries

This section provides the basic definitions from fractional and fractal–fractional calculus that form the foundation for our model analysis. We include key operators in various senses and recall their integral and derivative forms for later use [17,18,24].

Definition 1

([17,25]). The fractal derivative (Hausdorff derivative) of a function $\mathsf{\Psi }\left(t\right)$ with respect to the fractal measure $t^{{\sigma }_2}$ is defined as

$$\begin{array}{c}\hfill {\displaystyle \frac{d\mathsf{\Psi }\left(t\right)}{d{t}^{{\sigma }_2}}}=\underset{t^{\prime }\to t}{lim}{\displaystyle \frac{\mathsf{\Psi }\left(t^{\prime }\right)-\mathsf{\Psi }\left(t\right)}{t^{\prime {\sigma }_2}-t^{{\sigma }_2}}}.\end{array}$$

(1)

Definition 2

([17,18]). Let $\mathsf{\Psi }\left(t\right)$ be continuous and fractal-differentiable on $(a,b)$. The fractal–fractional derivative of order ${\sigma }_1$, with fractal dimension ${\sigma }_2$, in the Riemann–Liouville sense with a power-law kernel, is given by

$$\begin{array}{c}\hfill {}^{FF-RL}D_{0,t}^{{\sigma }_1,{\sigma }_2}\mathsf{\Psi }\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(n-{\sigma }_1)}}{\displaystyle \frac{d}{d{t}^{{\sigma }_2}}}{\int }_0^t{(t-s)}^{n-{\sigma }_1-1}\mathsf{\Psi }\left(s\right)ds,\end{array}$$

(2)

where ${\sigma }_1,{\sigma }_2\in (n-1,n]$ and $n>0$. Moreover,

$${\displaystyle \frac{d\mathsf{\Psi }\left(s\right)}{d{s}^{{\sigma }_2}}}=\underset{t\to s}{lim}{\displaystyle \frac{\mathsf{\Psi }\left(t\right)-\mathsf{\Psi }\left(s\right)}{t^{{\sigma }_2}-s^{{\sigma }_2}}}.$$

Definition 3

([17]). Let $\mathsf{\Psi }\left(t\right)$ be continuous on $(a,b)$. The fractal–fractional derivative of order ${\sigma }_1$, with fractal dimension ${\sigma }_2$, in the Caputo sense with a power-law kernel, is defined as

$$\begin{array}{c}\hfill {}^{FF-C}D_{0,t}^{{\sigma }_1,{\sigma }_2}\mathsf{\Psi }\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(n-{\sigma }_1)}}{\int }_0^t{(t-s)}^{n-{\sigma }_1-1}{\displaystyle \frac{d\mathsf{\Psi }\left(s\right)}{d{s}^{{\sigma }_2}}}ds,\end{array}$$

(3)

where ${\sigma }_1,{\sigma }_2\in (n-1,n]$ and $n>0$.

Definition 4

([17]). The FF integral of a continuous function $\mathsf{\Psi }\left(t\right)$ of order ${\sigma }_1$ and fractal dimension ${\sigma }_2$ in the CF sense is defined by

$${}^{FF-CF}I_{0,t}^{{\sigma }_1,{\sigma }_2}\mathsf{\Psi }\left(t\right)={\displaystyle \frac{{\sigma }_2(1-{\sigma }_1)}{M\left({\sigma }_1\right)}}{t}^{{\sigma }_2-1}\mathsf{\Psi }\left(t\right)+{\displaystyle \frac{{\sigma }_1{\sigma }_2}{M\left({\sigma }_1\right)}}{\int }_0^t{s}^{{\sigma }_2-1}\mathsf{\Psi }\left(s\right)ds,$$

(4)

where $0<{\sigma }_1<1$, $0<{\sigma }_2\le 1$, and $M\left({\sigma }_1\right)$ is the normalization function satisfying $M\left(0\right)=M\left(1\right)=1$.

Definition 5

([18]). Let $\mathsf{\Psi }\left(t\right)$ be fractal differentiable on $(0,T)$. The FF derivative of order ${\sigma }_1$ and fractal dimension ${\sigma }_2$ in the CF sense is

$${}^{FF-CF}D_{0,t}^{{\sigma }_1,{\sigma }_2}\mathsf{\Psi }\left(t\right)={\displaystyle \frac{M\left({\sigma }_1\right)}{1-{\sigma }_1}}{\int }_0^t{\displaystyle \frac{d\mathsf{\Psi }\left(s\right)}{d{s}^{{\sigma }_2}}}exp\left[-{\displaystyle \frac{{\sigma }_1}{1-{\sigma }_1}}(t-s)\right]ds,$$

(5)

where

$${\displaystyle \frac{d\mathsf{\Psi }\left(s\right)}{d{s}^{{\sigma }_2}}}=\underset{t\to s}{lim}{\displaystyle \frac{\mathsf{\Psi }\left(t\right)-\mathsf{\Psi }\left(s\right)}{t^{{\sigma }_2}-s^{{\sigma }_2}}}.$$

(6)

Definition 6

([18]). Let $\mathsf{\Psi }\left(t\right)\in C\left(\right[0,T\left]\right)$ and suppose that $X(t,\mathsf{\Psi }(t\left)\right)$ is continuous. Consider the FF Cauchy problem

$${}^{FF-CF}D_{0,t}^{{\sigma }_1,{\sigma }_2}\mathsf{\Psi }\left(t\right)=X(t,\mathsf{\Psi }(t)),$$

(7)

$$\mathsf{\Psi }\left(0\right)={\mathsf{\Psi }}_0,$$

(8)

with $0<{\sigma }_1<1$ and $0<{\sigma }_2\le 1$.

Then the equivalent integral equation is

$$\mathsf{\Psi }\left(t\right)={\mathsf{\Psi }}_0+{\displaystyle \frac{{\sigma }_2(1-{\sigma }_1)}{M\left({\sigma }_1\right)}}{t}^{{\sigma }_2-1}X(t,\mathsf{\Psi }\left(t\right))+{\displaystyle \frac{{\sigma }_1{\sigma }_2}{M\left({\sigma }_1\right)}}{\int }_0^t{s}^{{\sigma }_2-1}X(s,\mathsf{\Psi }\left(s\right))ds.$$

(9)

Definition 7

([18]). Let $\mathsf{\Psi }\left(t\right)$ be fractal-differentiable on $(a,b)$, with order $0<{\sigma }_2\le 1$. The FF derivative of order ${\sigma }_1$ in the AB sense is defined by

$$\begin{array}{c}\hfill {}^{FF-AB}D_{0,t}^{{\sigma }_1,{\sigma }_2}\mathsf{\Psi }\left(t\right)={\displaystyle \frac{h\left({\sigma }_1\right)}{1-{\sigma }_1}}{\displaystyle \frac{d}{d{t}^{{\sigma }_2}}}{\int }_0^t\mathsf{\Psi }\left(s\right)E_{{\sigma }_1}\left(-\frac{{\sigma }_1}{1-{\sigma }_1}{(t-s)}^{{\sigma }_1}\right)ds,\end{array}$$

(10)

where

$$h\left({\sigma }_1\right)=1-{\sigma }_1+{\displaystyle \frac{{\sigma }_1}{\mathsf{\Gamma }\left({\sigma }_1\right)}},$$

and

$${\displaystyle \frac{d\mathsf{\Psi }\left(s\right)}{d{s}^{{\sigma }_2}}}=\underset{t\to s}{lim}{\displaystyle \frac{\mathsf{\Psi }\left(t\right)-\mathsf{\Psi }\left(s\right)}{t^{{\sigma }_2}-s^{{\sigma }_2}}}.$$

Definition 8

([18]). The FF integral in the AB setting is given by

$$\begin{array}{c}\hfill {}^{FF-AB}I_{0,t}^{{\sigma }_1,{\sigma }_2}\mathsf{\Psi }\left(t\right)={\displaystyle \frac{{\sigma }_1{\sigma }_2}{h\left({\sigma }_1\right)\mathsf{\Gamma }\left({\sigma }_1\right)}}{\int }_0^t{(t-s)}^{{\sigma }_1-1}{s}^{{\sigma }_2-1}\mathsf{\Psi }\left(s\right)ds+{\displaystyle \frac{{\sigma }_2(1-{\sigma }_1)}{h\left({\sigma }_1\right)}}{t}^{{\sigma }_2-1}\mathsf{\Psi }\left(t\right).\end{array}$$

(11)

Remark 1.

The AB operator is chosen for its nonsingular M-L, which effectively captures nonlocal and long-term memory effects in disease dynamics. This is important because an epidemic’s current state often depends on previous states. In contrast, the CF operator uses a nonsingular exponential kernel that models fading memory and is computationally simpler. Its integral form incorporates both local and historical terms, highlighting the influence of current and past conditions. Comparing these operators shows how different memory structures affect Zika transmission predictions.

3. C-F Fractional Zika Model

In this section, we formulate the fractal–fractional Zika epidemic model using the Caputo–Fabrizio (CF) operator. The model is given by [3,26]

$$\begin{array}{cc}\hfill {}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{S}_h\left(t\right)& ={\mathsf{\Lambda }}_h-{\beta }_1{S}_h{I}_h-{\beta }_2{S}_h{I}_n-k_1{S}_h,\hfill \\ \hfill {}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{I}_h\left(t\right)& ={\beta }_1{S}_h{I}_h+{\beta }_2{S}_h{I}_n-k_1{I}_h,\hfill \\ \hfill {}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{S}_n\left(t\right)& ={\mathsf{\Lambda }}_n-\mu S_n{I}_h-k_2{S}_n,\hfill \\ \hfill {}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{I}_n\left(t\right)& =\mu S_n{I}_h-k_2{I}_n.\hfill \end{array}$$

(12)

and $S_h\left(0\right)$, $I_h\left(0\right)$, $S_n\left(0\right)$, and $I_n\left(0\right)$ denote the initial conditions of the model [3].

Additional information regarding model (12) can be found in the chapter [27], in which the authors formulated an FF Zika virus model using the Atangana–Baleanu (FF–AB) derivative. The M-L kernel associated with this operator enables the model to capture memory effects and fractal behavior inherent in biological systems. The proposed FF–AB model is given by

$$\begin{array}{cc}\hfill {}^{\mathrm{FF}–\mathrm{AB}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{S}_h\left(t\right)& ={\mathsf{\Lambda }}_h-{\beta }_1{S}_h{I}_h-{\beta }_2{S}_h{I}_n-k_1{S}_h,\hfill \\ \hfill {}^{\mathrm{FF}–\mathrm{AB}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{I}_h\left(t\right)& ={\beta }_1{S}_h{I}_h+{\beta }_2{S}_h{I}_n-k_1{I}_h,\hfill \\ \hfill {}^{\mathrm{FF}–\mathrm{AB}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{S}_n\left(t\right)& ={\mathsf{\Lambda }}_n-\mu S_n{I}_h-k_2{S}_n,\hfill \\ \hfill {}^{\mathrm{FF}–\mathrm{AB}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{I}_n\left(t\right)& =\mu S_n{I}_h-k_2{I}_n.\hfill \end{array}$$

(13)

For this model, the authors established conditions that ensure the existence and uniqueness of solutions and investigated its Hyers–Ulam stability using fixed-point theory. Overall, this study extends the work in [27] by comparing the FF–AB and FF–CF operators. In particular, it provides a detailed mathematical analysis of the FF–CF formulation and investigates its application to the Zika virus transmission model.

In the proposed model (12), the human population consists of susceptible ($S_h$) and infected ($I_h$) individuals. The mosquito population is similarly divided into susceptible ($S_n$) and infected ($I_n$) groups within the same region.

The parameters ${\mathsf{\Lambda }}_h$ and ${\mathsf{\Lambda }}_n$ represent the recruitment rates of susceptible humans and mosquitoes, respectively. Zika virus is transmitted from infected mosquitoes to susceptible humans at the rate ${\beta }_2$, and from infected humans to susceptible mosquitoes at the rate $\mu$. The parameter ${\beta }_1$ denotes the infection rate among susceptible humans. The natural death rates of humans and mosquitoes are $k_1$ and $k_2$, respectively.

Thus, every term in system (12) has the same physical dimension $T^{1-{\sigma }_1-{\sigma }_2}$. We interpret the model parameters as effective epidemiological rates under the fractal–fractional time scale, following [15]. We assume that ${\sigma }_1+{\sigma }_2>1$. In the classical case $({\sigma }_1,{\sigma }_2)=(1,1)$, the parameters recover their usual epidemiological dimensions and interpretations.

In compact vector form,

$${}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}\mathbf{X}\left(t\right)=F\left(t,\mathbf{X}\left(t\right)\right),$$

(14)

where

$$\mathbf{X}\left(t\right)=\left(\begin{array}{c}{S}_h\left(t\right)\\ I_h\left(t\right)\\ S_n\left(t\right)\\ I_n\left(t\right)\end{array}\right).$$

(15)

For the system (14), the corresponding FF–CF integral representation is given by

$$\mathbf{X}\left(t\right)=X\left(0\right)+{\displaystyle \frac{{\sigma }_2(1-{\sigma }_1)}{M\left({\sigma }_1\right)}}{t}^{{\sigma }_2-1}F\left(t,\mathbf{X}\left(t\right)\right)+{\displaystyle \frac{{\sigma }_1{\sigma }_2}{M\left({\sigma }_1\right)}}{\int }_0^t{\tau }^{{\sigma }_2-1}F\left(\tau ,X\left(\tau \right)\right)d\tau .$$

(16)

4. Qualitative Properties of the Solution

Motivated by the approach described in [19], we establish the following result.

Theorem 1.

For the FF–CF Zika model (12), all solutions originating in ${\mathbb{R}}_{+}^4$ remain uniformly bounded.

Proof. 

Let $\mathcal{W}\left(t\right)=S_h\left(t\right)+I_h\left(t\right)+S_n\left(t\right)+I_n\left(t\right).$ Using the linearity of the FF–CF derivative operator [28], summing all equations of the FF–CF Zika model (12) yields

$$\begin{array}{cc}\hfill {}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}\mathcal{W}\left(t\right)& ={}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{S}_h\left(t\right)+{}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{I}_h\left(t\right)\hfill \\ \hfill & +{}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{S}_n\left(t\right)+{}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{I}_n\left(t\right)\hfill \\ \hfill & =({\mathsf{\Lambda }}_h-{\beta }_1{S}_h{I}_h-{\beta }_2{S}_h{I}_n-k_1{S}_h)\hfill \\ \hfill & +({\beta }_1{S}_h{I}_h+{\beta }_2{S}_h{I}_n-k_1{I}_h)\hfill \\ \hfill & +({\mathsf{\Lambda }}_n-\mu S_n{I}_h-k_2{S}_n)\hfill \\ \hfill & +(\mu S_n{I}_h-k_2{I}_n).\hfill \end{array}$$

(17)

The nonlinear terms cancel pairwise, such as

$$\begin{array}{c}\hfill -{\beta }_1{S}_h{I}_h+{\beta }_1{S}_h{I}_h=0,\\ \hfill -{\beta }_2{S}_h{I}_n+{\beta }_2{S}_h{I}_n=0,\\ \hfill -\mu S_n{I}_h+\mu S_n{I}_h=0.\end{array}$$

(18)

After cancellation of the nonlinear terms, we obtain

$${}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}\mathcal{W}\left(t\right)={\mathsf{\Lambda }}_h+{\mathsf{\Lambda }}_n-k_1(S_h+I_h)-k_2(S_n+I_n).$$

(19)

Since all state variables are nonnegative, we have $k_1(S_h+I_h)+k_2(S_n+I_n)\ge 0.$ Therefore,

$${}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}\mathcal{W}\left(t\right)={\mathsf{\Lambda }}_{h,n}-k_1\left(S_h\left(t\right)+I_h\left(t\right)\right)-k_2\left(S_n\left(t\right)+I_n\left(t\right)\right),$$

(20)

where ${\mathsf{\Lambda }}_{h,n}={\mathsf{\Lambda }}_h+{\mathsf{\Lambda }}_n\mathrm{and}\mathcal{W}\left(0\right)=S_h\left(0\right)+I_h\left(0\right)+S_n\left(0\right)+I_n\left(0\right).$

Taking $\sigma =min\{k_1,k_2\}>0$, we obtain

$${}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}\mathcal{W}\left(t\right)+\sigma \mathcal{W}\left(t\right)\le {\mathsf{\Lambda }}_{h,n}.$$

(21)

Applying the comparison principle stated in Lemma 3.2 of [29,30] to the FF–CF operator defined in [15], since $0<e^{-kt}\le 1,k>0$ for all $t\ge 0$, we obtain

$$0\le \mathcal{W}\left(t\right)\le {\displaystyle \frac{{\mathsf{\Lambda }}_{h,n}}{\sigma }}+{\left(\mathcal{W}\left(0\right)-{\displaystyle \frac{{\mathsf{\Lambda }}_{h,n}}{\sigma }}\right)}^{+}{e}^{-kt},k>0,$$

(22)

where ${\left(z\right)}^{+}:=max\{z,0\}$ denotes the positive part of z. Therefore,

$$\mathcal{W}\left(t\right)\le max\left\{\mathcal{W}\left(0\right),{\displaystyle \frac{{\mathsf{\Lambda }}_{h,n}}{\sigma }}\right\},$$

(23)

and

$$\underset{t\to \infty }{lim\; sup}\mathcal{W}\left(t\right)\le {\displaystyle \frac{{\mathsf{\Lambda }}_{h,n}}{\sigma }}.$$

(24)

Hence, the solutions are uniformly bounded in the region

$$\mathsf{\Sigma }=\left\{(S_h,I_h,S_n,I_n)\in {\mathbb{R}}_{+}^4:\mathcal{W}\left(t\right)\le max\left\{\mathcal{W}\left(0\right),{\displaystyle \frac{{\mathsf{\Lambda }}_{h,n}}{\sigma }}\right\}\right\}.$$

(25)

Moreover, for any $\epsilon >0$, every solution eventually enters and remains in the region

$${\mathsf{\Sigma }}_{\epsilon }=\left\{(S_h,I_h,S_n,I_n)\in {\mathbb{R}}_{+}^4:\mathcal{W}\left(t\right)\le {\displaystyle \frac{{\mathsf{\Lambda }}_{h,n}}{\sigma }}+\epsilon \right\}.$$

(26)

Thus, ${\mathsf{\Sigma }}_{\epsilon }$ is an ultimately bounded region.

Next, we verify positivity. On the boundary of the positive orthant, we have

$$\begin{array}{cc}\hfill {}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{S}_h\left(t\right){|}_{S_h=0}& ={\mathsf{\Lambda }}_h\ge 0,\hfill \\ \hfill {}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{I}_h\left(t\right){|}_{I_h=0}& ={\beta }_2{S}_h{I}_n\ge 0,\hfill \\ \hfill {}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{S}_n\left(t\right){|}_{S_n=0}& ={\mathsf{\Lambda }}_n\ge 0,\hfill \\ \hfill {}^{\mathrm{FF}–\mathrm{CF}}D_{0,t}^{{\sigma }_1,{\sigma }_2}{I}_n\left(t\right){|}_{I_n=0}& =\mu S_n{I}_h\ge 0.\hfill \end{array}$$

(27)

In Equation (21), all state variables are nonnegative. Therefore, ${\left.I_h^{\prime }\left(t\right)\right|}_{I_h=0}={\beta }_2{S}_h{I}_n\ge 0,$ ${\left.I_n^{\prime }\left(t\right)\right|}_{I_n=0}=\mu S_n{I}_h\ge 0.$

Therefore, every solution with nonnegative initial conditions remains in ${\mathbb{R}}_{+}^4$ for all $t\ge 0$. Consequently, $\mathsf{\Sigma }$ constitutes the biologically feasible region of the FF–CF Zika model (12). □

5. Numerical Results and Discussion

We ran numerical simulations of the FF Zika virus model using these parameter values: ${\mathsf{\Lambda }}_h=1.2,{\mathsf{\Lambda }}_n=0.3,{\beta }_1=0.4\times {10}^{-4},{\beta }_2=0.125\times {10}^{-4},k_1=0.004,$ $k_2=0.0014,$ and $\mu =0.475\times {10}^{-5}.$ The initial conditions were $\left(S_h\left(0\right),I_h\left(0\right),S_n\left(0\right),I_n\left(0\right)\right)=(800,200,600,300).$ Numerical simulations were performed using Python 3.11.11, NumPy 2.2.2, and the RK45 method in SciPy [31,32]; figures were generated with Matplotlib 3.10.0.

We calculated numerical solutions using schemes for the FF–AB [33] and FF–CF [15] operators. By changing the fractional order ${\sigma }_1$ and fractal order ${\sigma }_2$, we studied how memory and fractal effects influence Zika transmission. The simulations show that both orders affect the numbers of susceptible and infected humans and mosquitoes. We also compared the AB and CF operators to examine how their distinct memory kernels affect the predicted epidemic course.

Figure 1a–d presents the numerical comparison between the fractal–fractional AB and CF operators. The results show that the AB operator produces stronger memory effects and slower infection decay, whereas the CF operator leads to faster stabilization. Moreover, changes in fractal order ${\sigma }_2$ noticeably affect the human-infected population, highlighting the role of fractal memory in the dynamics of Zika virus transmission [3].

Figure 1. Comparative numerical simulations of the FF-AB and FF-CF operators for the Zika virus model [3]. The results illustrate how memory effects and fractal order influence disease transmission dynamics.

6. Conclusions

The results indicate that the proposed FF Zika virus model realistically describes transmission between humans and mosquitoes, highlighting the important role of memory effects in the spread and persistence of the disease. The comparative study further shows that the AB operator retains stronger memory and therefore predicts a slower decline in infections. In contrast, the CF operator leads to more rapid stabilization of the epidemic because of its fading-memory characteristics. These observations suggest that the choice of fractional operator can substantially influence the model’s predictions. Future studies can extend the considered Zika model to include variable-order and mixed FF derivatives, directional memory effects, and appropriate control interventions, which may offer a more realistic framework for managing Zika virus outbreaks.