Mathematical analysis and simulation of an ABC fractional Zika virus model with natural transform and stability assessment

Abstract

This work introduces an Atangana–Baleanu Caputo (ABC) fractional model for the transmission dynamics of the Zika virus. A novel Natural Transform approach is employed to obtain an approximate analytical solution and to examine the stability and uniqueness of the system. The positivity and boundedness of the ABC fractional model are established, and the equilibrium points as well as the basic reproduction number are derived. The existence and uniqueness of solutions are further analyzed using the Lipschitz condition. Moreover, Routh–Hurwitz and Lyapunov stability analyses are applied to determine the local and global stability of the equilibrium states. To enhance model robustness, the Markov Chain Monte Carlo (MCMC) method is utilized for parameter estimation. Finally, the Adams–Moulton method is implemented to obtain numerical approximations, and simulations are carried out to predict the controllability of pandemic spread. All numerical simulations and parameter estimations were implemented using computational algorithms, highlighting the role of ICT tools in analyzing and predicting the dynamics of fractional epidemic.

Keywords

epidemic Zika virus disease transmisson dynamics ABC fractional differential equation mathematical model natural transform Lipschitz basic reproduction number Routh-Hurwitz Markov Chain Monte Carlo disease control Adams-Moulton

1. Introduction

The transmission of Zika virus occurs primarily through the bite of infected Aedes mosquitoes, chiefly Aedes aegypti and, to a lesser extent, Ae. albopictus. Runge-Ranzinger et al.¹ demonstrated in experimental and field studies that these mosquito species efficiently acquire and transmit the virus (vector or horizontal transmission). According to Zimler et al.,² infected mosquitoes may also pass the virus vertically from female to offspring, albeit at lower rates, thereby supporting the persistence of the virus in mosquito populations across seasons. Zika can also be transmitted from an infected pregnant woman to her fetus (vertical mother-to-child transmission), with documented associations to congenital Zika syndrome, including microcephaly and neurological abnormalities. Sexual transmission represents another well-established route, with studies detecting Zika virus in semen for several weeks to months after infection, enabling male-to-female, male-to-male, and possibly female-to-male transmission, as discussed by Foy et al.³ Although Petersen et al.⁴ detected Zika virus RNA in breast milk, no confirmed cases of Zika transmission through breastfeeding have been reported, and the benefits of breastfeeding outweigh any potential risk. Transmission via blood transfusion has been confirmed during outbreaks, prompting mandatory screening in affected regions, as reported by Barjas-Castro et al.⁵ Additionally, Besnard et al.⁶ documented laboratory-acquired infections, emphasizing the importance of strict biosafety measures in virology laboratories. Understanding these multiple transmission routes is crucial for developing comprehensive prevention strategies and controlling future outbreaks.

Beyond these established routes, recent studies by Baafi et al.⁷ have highlighted that seasonal and environmental factors such as fluctuations in mosquito birth and death rates between wet and dry seasons can significantly influence the scale and timing of outbreaks. Spatial heterogeneity and human mobility patterns also play a crucial role in the regional spread of the virus, as captured by spatial and network-based models.^8,9 Moreover, mathematical models consistently show that parameters like mosquito lifespan and the frequency of human–vector interactions are strongly correlated with the intensity of transmission.^10–12

Furthermore, Gao et al.¹³ developed a deterministic compartmental model of Zika virus transmission that incorporated both mosquito-borne and sexual transmission. Their analysis, based on data from Brazil, Colombia, and El Salvador, estimated the basic reproduction number R₀ to be approximately 2.055 and demonstrated that while sexual transmission contributed a relatively small proportion (≈3 %) of cases, it significantly prolonged the duration of outbreaks. Alshehri and El Hajji¹⁴ introduced a generalized nonlinear incidence model for Zika transmission and performed local and global stability analyses. They also formulated and solved an optimal control problem, demonstrating that timely treatment reduces the number of infections and the associated cost. Tesla et al.¹⁵ incorporated temperature-dependent parameters into a Zika virus model, showing that vector competence, extrinsic incubation period, and mosquito survival are temperature-sensitive and crucial in predicting the risk and intensity of Zika outbreaks. Jamal et al.¹⁶ developed a Zika transmission model accounting for newborn and adult infections under scenarios with and without Wolbachia-infected mosquitoes, highlighting the potential of Wolbachia in vector control. Aranda, González-Parra and Benincasa¹⁷ applied their mathematical model to analyze the Zika outbreak in Colombia using real-time data, while Agusto et al.¹⁸ studied a model incorporating both vertical transmission and birth of babies with microcephaly to capture complex human behavioral dynamics.

So far, several researchers have explored the use of fractional calculus in modeling the transmission dynamics of the Zika virus. Khan et al.¹⁹ were among the first to apply the Caputo fractional derivative to a Zika virus model, proving existence and uniqueness of solutions, and analyzing the basic reproduction number $R_{0}$ and the stability of equilibria. Subsequently, Alkahtani et al.²⁰ studied Atangana–Baleanu–Caputo fractional order Zika virus model by using the non singular kernels. Ali et al.²¹ extended this framework by considering mutant of Zika within a fractional-order setting, emphasizing how fractional order modifies the endemic threshold. Zafar et al.²² further developed a novel Zika model using the Atangana–Baleanu–Caputo derivative, deriving new threshold conditions and validating their scheme with numerical simulations. More recently, Chouhan and Sharma²³ introduced a Caputo–Fabrizio fractional model that incorporated both sexual and vector transmission together with awareness programs, showing that memory effects significantly change intervention outcomes. Kouidere et al.²⁴ applied fractional optimal control theory (based on Pontryagin’s principle in the Caputo sense) to design effective intervention strategies, while Maamar et al.²⁵ developed a nonstandard finite difference (NSFD) scheme for a time-fractional Zika model, guaranteeing positivity and numerical stability. Other recent works, such as Loyinmi²⁶ emphasized the transmission of the zika virus amoung humans and vectors. While Karay et al.²⁷ employed stochastic neural networks to solve Caputo-fractional Zika models with higher accuracy. Overall, the progression of fractional Zika virus models demonstrates a clear trend: initial studies focused on basic fractional extensions of classical SEIR-type systems, while later works have incorporated realistic features such as sexual transmission, awareness, optimal control, and advanced numerical schemes. This evolution highlights the growing importance of fractional-order operators in capturing memory effects and improving the realism of Zika virus transmission models.

Fractional calculus has been increasingly integrated into epidemiological modeling due to its inherent ability to capture memory effects and hereditary properties in disease transmission dynamics. Early studies primarily focused on extending classical compartmental models into the fractional domain to explore memory-dependent infection spread and stability analysis.²⁸

In recent years, research has shifted toward incorporating intervention strategies such as quarantine, vaccination, and awareness programs—into fractional-order epidemic systems. These models provide a more realistic assessment of public health policies by accounting for delayed and lasting effects of control measures. For instance, tuberculosis (TB) models employing fractional derivatives have demonstrated how pre-existing health conditions influence vaccination efficacy and long-term disease control.²⁹

Further contributions in this domain include fractional optimal control frameworks that determine time-dependent intervention policies under resource constraints,³⁰ as well as multi-strain models addressing viral mutation and immune escape.³¹ Additionally, recent reviews emphasize the advantages of non-singular kernels, such as the Atangana–Baleanu–Caputo (ABC) operator used in this study, in capturing intermediate memory effects relevant to vector-borne diseases like malaria, where seasonal mosquito activity and human behavioral adaptations jointly shape transmission patterns.³²

By positioning our ABC-fractional Zika model within this expanding literature, we bridge theoretical advances in fractional calculus with practical public health challenges, offering a unified framework that enhances both mathematical rigor and real-world applicability.

The novelty of this paper lies in developing a fractional-order model for the ODE model using the Atangana-Baleanu-Caputo (ABC) derivative to investigate Zika virus transmission dynamics. We leverage the Natural transform method (NTM) to solve the proposed fractional-order Zika virus model. Analytical solutions for nonlinear systems of fractional differential equations are often intractable by conventional methods. The Natural transform method (NTM) emerges as a potent analytical tool, particularly well-suited for equations involving the Atangana-Baleanu-Caputo derivative due to its ability to efficiently manage its non-singular, Mittag-Leffler type kernel. The NTM can be viewed as a generalization of both the Laplace and Sumudu transforms, offering a versatile approach to solving a wide class of problems. Which effectively converts the system of fractional differential equations into a system of algebraic equations in the transform domain. This transformation simplifies the analysis and allows for the decomposition of the solution into an infinite series. The subsequent application of the inverse Natural transform yields the iterative solution scheme. This method provides a systematic and efficient procedure for obtaining approximate analytical solutions. And we use MCMC method for parameter estimation. MCMC is a computational technique for estimating complex probability distributions of model parameters. It works by constructing a Markov chain that randomly walks through the parameter space, sampling more frequently from high-probability regions. After many iterations, the collected samples form an approximation of the posterior distribution, from which we can derive estimates. Finally we employ the Adams-Moulton method for numerical computation to simulate scenarios and assess the potential for controlling the spread of the pandemic. It provides higher accuracy and superior stability compared to explicit methods of the same order and implicit nature to allows for significantly larger time steps, improving computational efficiency.

2. Preliminaries

Let us recall the basic definitions of ABC fractional operators and well-known theorems from fractional calculus.

Definition 2.1

³³ The gamma function of x > 0 is defined by the integral.

Γ (x) = \int_{0}^{\infty} e^{- t} t^{x - 1} d t

(1)

Definition 2.2

³³ The Mittag-Leffler function denoted by $E_{α} (z)$ and defined as:

E_{α} (z) = Σ_{k = 0}^{\infty} \frac{z^{k}}{Γ (α k + 1)}, α, z \in C, R e (α) > 0

(2)

Definition 2.3

³⁴ Let $g : [a, b] \to R$ be a bounded and continuous function. The Atangana-Baleanu fractional derivative in Caputo sense of order 0 < α < 1 is defined as:

{}_{a}^{A B C}D_{t}^{α} g (t) = \frac{M (α)}{(1 - α)} \int_{a}^{t} E_{α} (- \frac{α}{1 - α} {(t - h)}^{α}) g^{'} (h) d h

(3)

Where

M (α)

is positive and a normalization function given by

M (α) = 1 - α + \frac{α}{Γ (α)}

, characterized by

M (0) = M (1) = 1

Definition 2.4

³⁴ Let $g : [a, b] \to R$ be a bounded and continuous function. The corresponding fractional integral concerning to Atangana-Baleanu fractional order derivative of order 0 < α < 1 is defined as:

{}_{a}^{A B C}I_{t}^{α} g (t) = \frac{1 - α}{M (α)} g (t) + \frac{α}{M (α) Γ (α)} \int_{a}^{t} {(t - h)}^{α - 1} g (h) d h

(4)

Theorem 2.1

³⁵ Let $g : [a, b] \to R$ be a bounded and continuous function, a < b, α ∈ [0, 1], then the following equality on [a, b] is satisfied

{}_{a}^{A B C}I_{t}^{α} ({}_{a}^{A B C}D_{t}^{α} g (t)) = g (t) - g (a)

(5)

Theorem 2.2

³⁶ Let $g : [a, b] \to R$ be a bounded and continuous function. Then, the following results hold as in $‖_{a}^{A B C} D_{t}^{α} g (t) ‖ \leq \frac{M (α)}{1 - α} ‖ g (t) ‖$ , where ‖g(t)‖ = max_a≤t≤b|g(t)|. Moreover, for two functions g₁, g₂ ∈ L(a, b), b > a; then the AB derivative satisfies the Lipschitz condition:

‖_{a}^{A B C} D_{t}^{α} g_{1} (t) - {}_{a}^{A B C}D_{t}^{α} g_{2} (t) ‖ \leq e ‖ g_{1} (t) - g_{2} (t) ‖,

(6)

Where, 0 < α ≤ 1 is the order of fractional derivative.

Definition 2.5

³⁶ In the Caputo sense, the Laplace transform of the Atangana-Baleanu fractional derivative is given as

\begin{align} L \{{}_{a}^{A B C}D_{t}^{α} g (t)\} (s) & = \frac{M (α)}{1 - α} \frac{s^{α} L {g (t)} (s) - s^{α - 1} g (0)}{s^{α} + \frac{α}{1 - α}} \\ = \frac{M (α)}{1 - α} \frac{s^{α} L {g (t)} (s)}{s^{α} + \frac{α}{1 - α}} - \frac{M (α)}{1 - α} \frac{s^{α - 1} g (0)}{s^{α} + \frac{α}{1 - α}} \\ = \frac{M (α) [s^{α} G (s) - s^{α - 1} g (0)]}{s^{α} (1 - α) + α}, s > 0 \end{align}

(7)

3. The AB fractional order in Caputo sense system of DEs

Tapan Sarkar et al.³⁷ has introduced the epidemiological model of Zika virus in ODE for which we have coverted the system of equations in $A B C$ fractional-order form which is given by:

\begin{align} {}_{0}^{A B C}D_{t}^{α} S_{h} (t) = & Λ_{h} - β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} + (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h}, \\ {}_{0}^{A B C}D_{t}^{α} A_{h} (t) = & η Z S_{h} - β_{h}^{'} A_{h} I_{ν} - β_{h}^{'} A_{h} I_{h} - μ_{h} A_{h} - σ_{h} A_{h}, \\ {}_{0}^{A B C}D_{t}^{α} I_{h} (t) = & (β_{ν} S_{h} + β_{h}^{'} A_{h}) I_{ν} + (β_{h} S_{h} + β_{h}^{'} A_{h}) I_{h} - (μ_{h} + δ_{h}) I_{h} - (ϕ + τ) I_{h}, \\ {}_{0}^{A B C}D_{t}^{α} R_{h} (t) = & (ϕ + τ) θ I_{h} - (μ_{h} + γ_{h}) R_{h}, \\ {}_{0}^{A B C}D_{t}^{α} Z (t) = & p I_{h} - q Z, \\ {}_{0}^{A B C}D_{t}^{α} S_{ν} (t) = & Λ_{ν} - b_{ν} S_{ν} I_{h} - μ_{ν} S_{ν}, \\ {}_{0}^{A B C}D_{t}^{α} I_{ν} (t) = & b_{ν} S_{ν} I_{h} - μ_{ν} I_{ν} . \end{align}

(8)

with the initial conditions

S_{h} (0) \geq 0, A_{h} (0) \geq 0, I_{h} (0) \geq 0, R_{h} (0) \geq 0, Z (0) \geq 0, S_{ν} (0) \geq 0, I_{ν} (0) \geq 0

and where

{}_{0}^{A B C}D_{t}^{α}

A B C

fractional derivative of order α. The model formulation of the above model is described below along with the flow diagram Figure 1 and and the assumptions. The total human population, denoted by

N_{h}

, is divided into four compartments: susceptible

S_{h}

, aware

A_{h}

, infected

I_{h}

, and recovered

R_{h}

. Hence, the total human population is expressed as:

N_{h} = S_{h} + A_{h} + I_{h} + R_{h} .

Figure 1.

Flow diagram of the Zika virus transmission dynamics with the non-negative initial conditions as follows; $S_{h} (0) = S_{h_{0}} \geq 0, A_{h} (0) = A_{h_{0}} \geq 0, I_{h} (0) = I_{h_{0}} \geq 0, R_{h} (0) = R_{h_{0}} \geq 0, Z (0) = Z_{0} \geq 0, S_{ν} (0) = S_{ν_{0}} \geq 0, I_{ν} (0) = I_{ν_{0}} \geq 0$

The mosquito population $N_{ν}$ consists of susceptible $S_{ν}$ and infected $I_{ν}$ classes, so:

N_{ν} = S_{ν} + I_{ν} .

An awareness index $Z$ is also introduced to account for the spread of information, which impacts the infection dynamics.

Susceptible humans become infected through bites from infectious mosquitoes at the rate $β_{ν} S_{h} I_{ν}$ , while mosquitoes contract the infection by biting infected humans at a rate $β_{h} S_{ν} I_{h}$ . Here, β_ν and β_h are the respective transmission rates between vectors and humans.

Susceptible humans can become aware due to information at rate η, entering the awareness class $A_{h}$ . Aware individuals have a lower risk of infection and may become infected at a reduced rate $β_{ν}^{'} A_{h} I_{ν} + β_{h}^{'} A_{h} I_{h}$ , where $β_{ν}^{'} < β_{ν}$ and $β_{h}^{'} < β_{h}$ . Awareness may fade over time, as a result, there is a transition from the awareness compartment to the susceptible compartment, at a rate σ_h.

Recovery from infection occurs at a combined rate ϕ + τ, where ϕ is the natural recovery rate and τ is treatment-induced. A fraction θ of recovered individuals gains temporary immunity and enters the $R_{h}$ class, while (1 − θ) revert to being susceptible. Immunity wanes at rate γ_h, and disease-induced mortality is accounted for by rate δ_h. The natural death rate for humans is denoted by μ_h.

Susceptable mosquitoes are recruited at a constant rate Λ_ν, and die at a rate μ_ν. Transmission from susceptabble mosquitoes to infected mosquitoes by contact with infected humans occurs at rate $b_{h} S_{v} I_{h}$ , where b_h is the transmission rate from infected humans to susceptable mosquitoes.

4. $A B C$ with natural transform approach

Definition 4.1

³⁸ The Natural transform of a function g(t) is defined as:

N T {g (t)} = \int_{0}^{\infty} g (u t) e^{- s t} d t

(9)

Definition 4.2

³⁸ Natural- Laplace Duality (NLD) is

\begin{align} N T {g (t)} = R (s, u) = & \frac{1}{u} \int_{0}^{\infty} g (u t) e^{- \frac{s}{u} t} d t \\ = & \frac{1}{u} G (\frac{s}{u}) \end{align}

(10)

Now to find Natural transform from Laplace transform

Let $G (s) = L [g (t)] (s)$

Then from (7) and (10)

\begin{align} N T \{{}_{a}^{A B C}D_{t}^{α} g (t)\} (s, u) & = \frac{1}{u} [\frac{M (α)}{1 - α} \frac{{(\frac{s}{u})}^{α} G (\frac{s}{u}) - {(\frac{s}{u})}^{α - 1} g (0)}{{(\frac{s}{u})}^{α} + \frac{α}{1 - α}}] \\ = \frac{M (α)}{1 - α} \frac{\frac{s^{α}}{u^{α + 1}} G (\frac{s}{u}) - \frac{s^{α - 1}}{u^{α}} g (0)}{\frac{s^{α}}{u^{α}} + \frac{α}{1 - α}} \\ = \frac{M (α)}{1 - α} \frac{\frac{s^{α}}{u} G (\frac{s}{u}) - s^{α - 1} g (0)}{s^{α} + \frac{α}{1 - α} u^{α}} \end{align}

(11)

Now

G (\frac{s}{u}) = L [g (t)] (\frac{s}{u}) = u N T [g (t)] (s, u)

Substituting this in (11) we get

\begin{array}{l} N T \{{}_{a}^{A B C}D_{t}^{α} g (t)\} (s, u) & = \frac{M (α)}{1 - α} \frac{\frac{s^{α}}{u} (u N T [g (t)]) (\frac{s}{u}) - s^{α - 1} g (0)}{s^{α} + \frac{α}{1 - α} u^{α}} \\ = \frac{M (α)}{1 - α} \frac{s^{α} N T [g (t)] (s, u) - s^{α - 1} g (0)}{s^{α} + \frac{α}{1 - α} u^{α}} \end{array}

Let NT[g(t)](s, u) = G(s, u),

hence the Caputo sense, the Natural transform of the Atangana-Baleanu fractional derivative is given as

N T \{{}_{a}^{A B C}D_{t}^{α} g (t)\} (s, u) = \frac{M (α)}{1 - α} \frac{s^{α} G (s, u) - s^{α - 1} g (0)}{s^{α} + \frac{α}{1 - α} u^{α}}

(12)

By applying Natural transorm for (8) on both sides

\begin{array}{l} N T \{{}_{0}^{A B C}D_{t}^{α} S_{h} (t)\} = & N T {Λ_{h} - β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} + (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h}}, \\ N T {{}_{0}^{A B C}D_{t}^{α} A_{h} (t)} = & N T {η Z S_{h} - β_{h}^{'} A_{h} I_{ν} - β_{h}^{'} A_{h} I_{h} - μ_{h} A_{h} - σ_{h} A_{h}}, \\ N T {{}_{0}^{A B C}D_{t}^{α} I_{h} (t)} = & N T {(β_{ν} S_{h} + β_{h}^{'} A_{h}) I_{ν} + (β_{h} S_{h} + β_{h}^{'} A_{h}) I_{h} - (μ_{h} + δ_{h}) I_{h} - (ϕ + τ) I_{h}}, \\ N T {{}_{0}^{A B C}D_{t}^{α} R_{h} (t)} = & N T {(ϕ + τ) θ I_{h} - (μ_{h} + γ_{h}) R_{h}}, \\ N T {{}_{0}^{A B C}D_{t}^{α} Z (t)} = & N T {p I_{h} - q Z}, \\ N T {{}_{0}^{A B C}D_{t}^{α} S_{ν} (t)} = & N T {Λ_{ν} - b_{ν} S_{ν} I_{h} - μ_{ν} S_{ν}}, \\ N T {{}_{0}^{A B C}D_{t}^{α} I_{ν} (t)} = & N T {b_{ν} S_{ν} I_{h} - μ_{ν} I_{ν}} \end{array}

we get

\begin{array}{l} \frac{M (α)}{1 - α} \frac{s^{α} N T (S_{h} (t)) - s^{α - 1} S_{h} (0)}{s^{α} + \frac{α}{1 - α} u^{α}} = & N T {Λ_{h} - β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} + (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h}} \\ \frac{M (α)}{1 - α} \frac{s^{α} N T (A_{h} (t)) - s^{α - 1} A_{h} (0)}{s^{α} + \frac{α}{1 - α} u^{α}} = & N T {η Z S_{h} - β_{h}^{'} A_{h} I_{ν} - β_{h}^{'} A_{h} I_{h} - μ_{h} A_{h} - σ_{h} A_{h}} \\ \frac{M (α)}{1 - α} \frac{s^{α} N T (I_{h} (t)) - s^{α - 1} I_{h} (0)}{s^{α} + \frac{α}{1 - α} u^{α}} = & N T {(β_{ν} S_{h} + β_{h}^{'} A_{h}) I_{ν} + (β_{h} S_{h} + β_{h}^{'} A_{h}) I_{h} - (μ_{h} + δ_{h}) I_{h} - (ϕ + τ) I_{h}} \\ \frac{M (α)}{1 - α} \frac{s^{α} N T (R_{h} (t)) - s^{α - 1} R_{h} (0)}{s^{α} + \frac{α}{1 - α} u^{α}} = & N T {(ϕ + τ) θ I_{h} - (μ_{h} + γ_{h}) R_{h}} \\ \frac{M (α)}{1 - α} \frac{s^{α} N T (Z (t)) - s^{α - 1} Z (0)}{s^{α} + \frac{α}{1 - α} u^{α}} = & N T {p I_{h} - q Z} \\ \frac{M (α)}{1 - α} \frac{s^{α} N T (S_{ν} (t)) - s^{α - 1} S_{ν} (0)}{s^{α} + \frac{α}{1 - α} u^{α}} = & N T {Λ_{ν} - b_{ν} S_{ν} I_{h} - μ_{ν} S_{ν}} \\ \frac{M (α)}{1 - α} \frac{s^{α} N T (I_{ν} (t)) - s^{α - 1} I_{ν} (0)}{s^{α} + \frac{α}{1 - α} u^{α}} = & N T {b_{ν} S_{ν} I_{h} - μ_{ν} I_{ν}} . \end{array}

After rearrangeing we get,

\begin{align} N T (S_{h}) = & s^{- 1} S_{h} (0) + \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{h} - β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} + (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h}} \\ N T (A_{h}) = & s^{- 1} A_{h} (0) + \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {η Z S_{h} - β_{h}^{'} A_{h} I_{ν} - β_{h}^{'} A_{h} I_{h} - μ_{h} A_{h} - σ_{h} A_{h}} \\ N T (I_{h}) = & s^{- 1} I_{h} (0) + \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(β_{ν} S_{h} + β_{h}^{'} A_{h}) I_{ν} + (β_{h} S_{h} + β_{h}^{'} A_{h}) I_{h} - (μ_{h} + δ_{h}) I_{h} - (ϕ + τ) I_{h}} \\ N T (R_{h}) = & s^{- 1} R_{h} (0) + \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(ϕ + τ) θ I_{h} - (μ_{h} + γ_{h}) R_{h}} \\ N T (Z) = & s^{- 1} Z (0) + \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {p I_{h} - q Z} \\ N T (S_{ν}) = & s^{- 1} S_{ν} (0) + \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{ν} - b_{ν} S_{ν} I_{h} - μ_{ν} S_{ν}} \\ N T (I_{ν}) = & s^{- 1} I_{ν} (0) + \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {b_{ν} S_{ν} I_{h} - μ_{ν} I_{ν}} . \end{align}

(13)

by using inverse Natural transform we get,

\begin{array}{l} S_{h} (t) = & s^{- 1} S_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{h} - β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} + (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h}}] \\ A_{h} (t) = & s^{- 1} A_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {η Z S_{h} - β_{h}^{'} A_{h} I_{ν} - β_{h}^{'} A_{h} I_{h} - μ_{h} A_{h} - σ_{h} A_{h}}] \\ I_{h} (t) = & s^{- 1} I_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(β_{ν} S_{h} + β_{h}^{'} A_{h}) I_{ν} + (β_{h} S_{h} + β_{h}^{'} A_{h}) I_{h} - (μ_{h} + δ_{h}) I_{h} - (ϕ + τ) I_{h}}] \\ R_{h} (t) = & s^{- 1} R_{h} (0) + N T^{!} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(ϕ + τ) θ I_{h} - (μ_{h} + γ_{h}) R_{h}}] \\ Z (t) = & s^{- 1} Z (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {p I_{h} - q Z}] \\ S_{ν} (t) = & s^{- 1} S_{ν} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{ν} - b_{ν} S_{ν} I_{h} - μ_{ν} S_{ν}}] \\ I_{ν} (t) = & s^{- 1} I_{ν} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {b_{ν} S_{ν} I_{h} - μ_{ν} I_{ν}}] \end{array}

Let us consider the n^th transform as $S_{h}^{n}$ then,

\begin{align} S_{h}^{n + 1} (t) = & s^{- 1} S_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{h} - β_{ν} S_{h}^{n} I_{ν}^{n} - β_{h} S_{h}^{n} I_{h}^{n} - η Z^{n} S_{h}^{n} - μ_{h} S_{h}^{n} + σ_{h} A_{h}^{n} + (ϕ + τ) (1 - θ) I_{h}^{n} + γ_{h} R_{h}^{n}}] \\ A_{h}^{n + 1} (t) = & s^{- 1} A_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {η Z^{n} S_{h}^{n} - β_{h}^{'} A_{h}^{n} I_{ν}^{n} - β_{h}^{'} A_{h}^{n} I_{h}^{n} - μ_{h} A_{h}^{n} - σ_{h} A_{h}^{n}}] \\ I_{h}^{n + 1} (t) = & s^{- 1} I_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(β_{ν} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{ν}^{n} + (β_{h} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{h}^{n} - (μ_{h} + δ_{h}) I_{h}^{n} - (ϕ + τ) I_{h}^{n}}] \\ R_{h}^{n + 1} (t) = & s^{- 1} R_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(ϕ + τ) θ I_{h}^{n} - (μ_{h} + γ_{h}) R_{h}^{n}}] \\ Z^{n + 1} (t) = & s^{- 1} Z (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {p I_{h} - q Z^{n}}] \\ S_{ν}^{n + 1} (t) = & s^{- 1} S_{ν} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{ν} - b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} S_{ν}^{n}}] \\ I_{ν}^{n + 1} (t) = & s^{- 1} I_{ν} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} I_{ν}^{n}}] \end{align}

(14)

and the solutions will be provided by

\begin{array}{l} S_{h} (t) = & \lim_{n \to \infty} S_{h}^{n} (t), \\ A_{h} (t) = & \lim_{n \to \infty} A_{h}^{n} (t), \\ I_{h} (t) = & \lim_{n \to \infty} I_{h}^{n} (t), \\ R_{h} (t) = & \lim_{n \to \infty} R_{h}^{n} (t), \\ Z (t) = & \lim_{n \to \infty} Z^{n} (t), \\ S_{ν} (t) = & \lim_{n \to \infty} S_{ν}^{n} (t), \\ I_{ν} (t) = & \lim_{n \to \infty} I_{ν}^{n} (t), \end{array}

Theorem 4.1

Let $⟨ X_{1}, ‖ . ‖ ⟩$ be a banach space and P be a self map of $X_{1}$ satisfying $‖ P_{x} - P_{y} ‖ \leq C ‖ X - P_{x} ‖ + c ‖ x - y ‖$ for all $x, y \in X_{1}$ , where 0 ≤ C, 0 ≤ c < 1. Suppose that P is Picard P -Stable. Let us take into account the following recursive formula from (14) connected to (8).

\begin{array}{l} S_{h}^{n + 1} (t) = & s^{- 1} S_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{h} - β_{ν} S_{h}^{n} I_{ν}^{n} - β_{h} S_{h}^{n} I_{h}^{n} - η Z^{n} S_{h}^{n} - μ_{h} S_{h}^{n} + σ_{h} A_{h}^{n} + (ϕ + τ) (1 - θ) I_{h}^{n} + γ_{h} R_{h}^{n}}] \\ A_{h}^{n + 1} (t) = & s^{- 1} A_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {η Z^{n} S_{h}^{n} - β_{h}^{'} A_{h}^{n} I_{ν}^{n} - β_{h}^{'} A_{h}^{n} I_{h}^{n} - μ_{h} A_{h}^{n} - σ_{h} A_{h}^{n}}] \\ I_{h}^{n + 1} (t) = & s^{- 1} I_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(β_{ν} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{ν}^{n} + (β_{h} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{h}^{n} - (μ_{h} + δ_{h}) I_{h}^{n} - (ϕ + τ) I_{h}^{n}}] \\ R_{h}^{n + 1} (t) = & s^{- 1} R_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(ϕ + τ) θ I_{h}^{n} - (μ_{h} + γ_{h}) R_{h}^{n}}] \\ Z^{n + 1} (t) = & s^{- 1} Z (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {p I_{h} - q Z^{n}}] \\ S_{ν}^{n + 1} (t) = & s^{- 1} S_{ν} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{ν} - b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} S_{ν}^{n}}] \\ I_{ν}^{n + 1} (t) = & s^{- 1} I_{ν} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} I_{ν}^{n}}] \end{array}

where

\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)}

is the called fractional Lagrange multiplier.

Theorem 4.2

Let us define a self map P as

\begin{array}{l} P (S_{h}^{n} (t)) = S_{h}^{n + 1} (t) = & s^{- 1} S_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{Λ_{h} - β_{ν} S_{h}^{n} I_{ν}^{n} - β_{h} S_{h}^{n} I_{h}^{n} - η Z^{n} S_{h}^{n} - μ_{h} S_{h}^{n} + σ_{h} A_{h}^{n} \\ + & (ϕ + τ) (1 - θ) I_{h}^{n} + γ_{h} R_{h}^{n}\}] \\ P (A_{h}^{n} (t)) = A_{h}^{n + 1} (t) = & s^{- 1} A_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {η Z^{n} S_{h}^{n} - β_{h}^{'} A_{h}^{n} I_{ν}^{n} - β_{h}^{'} A_{h}^{n} I_{h}^{n} - μ_{h} A_{h}^{n} - σ_{h} A_{h}^{n}}] \\ P (I_{h}^{n} (t)) = I_{h}^{n + 1} (t) = & s^{- 1} I_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{(β_{ν} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{ν}^{n} + (β_{h} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{h}^{n} - (μ_{h} + δ_{h}) I_{h}^{n} \\ - & (ϕ + τ) I_{h}^{n}\}] \\ P (R_{h}^{n} (t)) = R_{h}^{n + 1} (t) = & s^{- 1} R_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(ϕ + τ) θ I_{h}^{n} - (μ_{h} + γ_{h}) R_{h}^{n}}] \\ P (Z^{n} (t)) = Z^{n + 1} (t) = & s^{- 1} Z (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {p I_{h} - q Z^{n}}] \\ P (S_{ν}^{n} (t)) = S_{ν}^{n + 1} (t) = & s^{- 1} S_{ν} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{ν} - b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} S_{ν}^{n}}] \\ P (I_{ν}^{n} (t)) = I_{ν}^{n + 1} (t) = & s^{- 1} I_{ν} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} I_{ν}^{n}}] \end{array}

is P stable in L¹(a, b) if

\begin{array}{l} {1 - f (ζ) β_{ν} (E + F) - h (ζ) β_{h} (J + F) - w (ζ) η (U + F) - r (ζ) μ_{h} + s (ζ) σ_{h} + y (ζ) (ϕ + τ) (1 - θ) + z (ζ) γ_{h}} < 1 \\ {1 + w_{1} (ζ) η (U + F) - f_{1} (ζ) β_{h}^{'} (V + M) - h_{1} (ζ) β_{h}^{'} (V + N) - r_{1} (ζ) μ_{h} - s_{1} (ζ) σ_{h}} < 1 \\ {1 + ρ (ζ) β_{ν} (K + M) + h_{2} (ζ) β_{h}^{'} (V + M) + ω (ζ) β_{h} (K + M) + f_{2} (ζ) β_{h}^{'} (V + M) - r_{2} (ζ) μ_{h} - ι (ζ) δ_{h} - y_{1} (ζ) (ϕ + τ)} < 1 \\ {1 + y_{2} (ζ) (ϕ + τ) θ - r_{3} (ζ) μ_{h} - z_{1} (ζ) γ_{h}} < 1 \\ {1 + ϱ (ζ) p - ϑ (ζ) q} < 1 \\ {1 - κ (ζ) b_{ν} (W + M) - λ (ζ) μ_{ν}} < 1 \\ {1 + κ_{1} (ζ) b_{ν} (W + M) - λ_{1} (ζ) μ_{ν}} < 1 \end{array}

Proof

Now to show that there is a fixed point in P for all $(m, n) \in N \times N$ , then

\begin{align} P (S_{h}^{n} (t)) - P (S_{h}^{m} (t)) = & S_{h}^{n} (t) - S_{h}^{m} (t) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{Λ_{h} - β_{ν} S_{h}^{n} I_{ν}^{n} - β_{h} S_{h}^{n} I_{h}^{n} - η Z^{n} S_{h}^{n} \\ - & μ_{h} S_{h}^{n} + σ_{h} A_{h}^{n} + (ϕ + τ) (1 - θ) I_{h}^{n} + γ_{h} R_{h}^{n}\}] - N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{Λ_{h} - β_{ν} S_{h}^{m} I_{ν}^{m} \\ - & β_{h} S_{h}^{m} I_{h}^{m} - η Z^{m} S_{h}^{m} - μ_{h} S_{h}^{m} + σ_{h} A_{h}^{m} + (ϕ + τ) (1 - θ) I_{h}^{m} + γ_{h} R_{h}^{m}\}] \\ P (A_{h}^{n} (t)) - P (A_{h}^{m} (t)) = & A_{h}^{n} (t) - A_{h}^{m} (t) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{η Z^{n} S_{h}^{n} - β_{h}^{'} A_{h}^{n} I_{ν}^{n} - β_{h}^{'} A_{h}^{n} I_{h}^{n} \\ - & μ_{h} A_{h}^{n} - σ_{h} A_{h}^{n}\}] - N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{η Z^{m} S_{h}^{m} m - β_{h}^{'} A_{h}^{m} I_{ν}^{m} - β_{h}^{'} A_{h}^{m} I_{h}^{m} - μ_{h} A_{h}^{m} \\ - & σ_{h} A_{h}^{m}\}] \\ P (I_{h}^{n} (t)) - P (I_{h}^{m} (t)) = & I_{h}^{n} (t) - I_{h}^{m} (t) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{(β_{ν} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{ν}^{n} + (β_{h} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{h}^{n} \\ - & (μ_{h} + δ_{h}) I_{h}^{n} - (ϕ + τ) I_{h}^{n}\}] - N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{(β_{ν} S_{h}^{m} + β_{h}^{'} A_{h}^{m}) I_{ν}^{m} + (β_{h} S_{h}^{m} \\ + & β_{h}^{'} A_{h}^{m}) I_{h}^{m} - (μ_{h} + δ_{h}) I_{h}^{m} - (ϕ + τ) I_{h}^{m}\}] - \\ P (R_{h}^{n} (t)) - P (R_{h}^{m} (t)) = & R_{h}^{n} (t) - R_{h}^{m} (t) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(ϕ + τ) θ I_{h}^{n} - (μ_{h} + γ_{h}) R_{h}^{n}}] \\ - & N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(ϕ + τ) θ I_{h}^{m} - (μ_{h} + γ_{h}) R_{h}^{m}}] \\ P (Z^{n} (t)) - P (Z^{m} (t)) = & Z^{n} (t) - Z^{m} (t) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {p I_{h} - q Z^{n}}] \\ - & N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {p I_{h} - q Z^{m}}] \\ P (S_{ν}^{n} (t)) - P (S_{ν}^{m} (t)) = & S_{ν}^{n} (t) - S_{ν}^{m} (t) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{ν} - b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} S_{ν}^{n}}] \\ - & N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{ν} - b_{ν} S_{ν}^{m} I_{h}^{m} - μ_{ν} S_{ν}^{m}}] \\ P (I_{ν}^{n} (t)) - P (I_{ν}^{m} (t)) = & I_{ν}^{n} (t) - I_{ν}^{m} (t) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} I_{ν}^{n}}] \\ - & N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {b_{ν} S_{ν}^{m} I_{h}^{m} - μ_{ν} I_{ν}^{m}}] \end{align}

(15)

from (15)

\begin{align} ‖ P (S_{h}^{n} (t)) - P (S_{h}^{m} (t)) ‖ = & ‖ S_{h}^{n} (t) - S_{h}^{m} (t) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{Λ_{h} - β_{ν} S_{h}^{n} I_{ν}^{n} - β_{h} S_{h}^{n} I_{h}^{n} - η Z^{n} S_{h}^{n} \\ - & μ_{h} S_{h}^{n} + σ_{h} A_{h}^{n} + (ϕ + τ) (1 - θ) I_{h}^{n} + γ_{h} R_{h}^{n}\}] - N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{Λ_{h} - β_{ν} S_{h}^{m} I_{ν}^{m} \\ - & β_{h} S_{h}^{m} I_{h}^{m} - η Z^{m} S_{h}^{m} - μ_{h} S_{h}^{m} + σ_{h} A_{h}^{m} + (ϕ + τ) (1 - θ) I_{h}^{m} + γ_{h} R_{h}^{m}\}] ‖ \\ ‖ P (A_{h}^{n} (t)) - P (A_{h}^{m} (t)) ‖ = & ‖ A_{h}^{n} (t) - A_{h}^{m} (t) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{η Z^{n} S_{h}^{n} - β_{h}^{'} A_{h}^{n} I_{ν}^{n} - β_{h}^{'} A_{h}^{n} I_{h}^{n} \\ - & μ_{h} A_{h}^{n} - σ_{h} A_{h}^{n}\}] - N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{η Z^{m} S_{h}^{m} m - β_{h}^{'} A_{h}^{m} I_{ν}^{m} - β_{h}^{'} A_{h}^{m} I_{h}^{m} - μ_{h} A_{h}^{m} \\ - & σ_{h} A_{h}^{m}\}] ‖ \\ ‖ P (I_{h}^{n} (t)) - P (I_{h}^{m} (t)) ‖ = & ‖ I_{h}^{n} (t) - I_{h}^{m} (t) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{(β_{ν} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{ν}^{n} + (β_{h} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{h}^{n} \\ - & (μ_{h} + δ_{h}) I_{h}^{n} - (ϕ + τ) I_{h}^{n}\}] - N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{(β_{ν} S_{h}^{m} + β_{h}^{'} A_{h}^{m}) I_{ν}^{m} + (β_{h} S_{h}^{m} \\ + & β_{h}^{'} A_{h}^{m}) I_{h}^{m} - (μ_{h} + δ_{h}) I_{h}^{m} - (ϕ + τ) I_{h}^{m}\}] ‖ \\ ‖ P (R_{h}^{n} (t)) - P (R_{h}^{m} (t)) ‖ = & ‖ R_{h}^{n} (t) - R_{h}^{m} (t) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(ϕ + τ) θ I_{h}^{n} - (μ_{h} + γ_{h}) R_{h}^{n}}] \\ - & N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(ϕ + τ) θ I_{h}^{m} - (μ_{h} + γ_{h}) R_{h}^{m}}] ‖ \\ ‖ P (Z^{n} (t)) - P (Z^{m} (t)) ‖ = & ‖ Z^{n} (t) - Z^{m} (t) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {p I_{h}^{n} - q Z^{n}}] \\ - & N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {p I_{h}^{m} - q Z^{m}}] ‖ \\ ‖ P (S_{ν}^{n} (t)) - P (S_{ν}^{m} (t)) ‖ = & ‖ S_{ν}^{n} (t) - S_{ν}^{m} (t) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{ν} - b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} S_{ν}^{n}}] \\ - & N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{ν} - b_{ν} S_{ν}^{m} I_{h}^{m} - μ_{ν} S_{ν}^{m}}] ‖ \\ ‖ P (I_{ν}^{n} (t)) - P (I_{ν}^{m} (t)) ‖ = & ‖ I_{ν}^{n} (t) - I_{ν}^{m} (t) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} I_{ν}^{n}}] \\ - & N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {b_{ν} S_{ν}^{m} I_{h}^{m} - μ_{ν} I_{ν}^{m}}] ‖ \end{align}

(16)

lets see for

S_{h}

in (16) first

\begin{align} ‖ P (S_{h}^{n} (t)) - P (S_{h}^{m} (t)) ‖ \leq & ‖ S_{h}^{n} (t) - S_{h}^{m} (t) ‖ + ‖ N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{Λ_{h} - β_{ν} S_{h}^{n} I_{ν}^{n} - β_{h} S_{h}^{n} I_{h}^{n} - η Z^{n} S_{h}^{n} \\ - & μ_{h} S_{h}^{n} + σ_{h} A_{h}^{n} + (ϕ + τ) (1 - θ) I_{h}^{n} + γ_{h} R_{h}^{n}\}] ‖ - ‖ N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{Λ_{h} - β_{ν} S_{h}^{m} I_{ν}^{m} \\ - & β_{h} S_{h}^{m} I_{h}^{m} - η Z^{m} S_{h}^{m} - μ_{h} S_{h}^{m} + σ_{h} A_{h}^{m} + (ϕ + τ) (1 - θ) I_{h}^{m} + γ_{h} R_{h}^{m}\}] ‖ \\ = & ‖ S_{h}^{n} (t) - S_{h}^{m} (t) ‖ + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{‖ - β_{ν} (S_{h}^{n} I_{ν}^{n} - S_{h}^{m} I_{ν}^{m}) ‖ \\ + & ‖ - β_{h} (S_{h}^{n} I_{h}^{n} - S_{h}^{m} I_{h}^{m}) ‖ + ‖ - η (Z^{n} S_{h}^{n} - Z^{m} S_{h}^{m}) ‖ + ‖ - μ_{h} (S_{h}^{n} - S_{h}^{m}) ‖ + ‖ σ_{h} (A_{h}^{n} - A_{h}^{m}) ‖ \\ + & ‖ (ϕ + τ) (1 - θ) (I_{h}^{n} - I_{h}^{m}) ‖ + ‖ γ_{h} (R_{h}^{n} - R_{h}^{m}) ‖\}] \end{align}

(17)

Now let $‖ S_{h}^{n} (t) - S_{h}^{m} (t) ‖ ≅ ‖ A_{h}^{n} (t) - A_{h}^{m} (t) ‖ ≅ ‖ I_{h}^{n} (t) - I_{h}^{m} (t) ‖ ≅ ‖ R_{h}^{n} (t) - R_{h}^{m} (t) ‖ ≅ ‖ Z^{n} (t) - Z^{m} (t) ‖ ≅ ‖ S_{ν}^{n} (t) - S_{ν}^{m} (t) ‖ ≅ ‖ I_{ν}^{n} (t) - I_{ν}^{m} (t) ‖$ substituting these in (17) we get

\begin{align} ‖ P (S_{h}^{n} (t)) - P (S_{h}^{m} (t)) ‖ \leq & ‖ S_{h}^{n} (t) - S_{h}^{m} (t) ‖ + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{‖ - β_{ν} I_{ν}^{n} (S_{h}^{n} - S_{h}^{m}) ‖ + ‖ - β_{ν} S_{h}^{m} (I_{h}^{n} - I_{h}^{m}) ‖ \\ + & ‖ - β_{h} I_{h}^{n} (S_{h}^{n} - S_{h}^{m}) ‖ + ‖ - β_{h} S_{h}^{m} (I_{h}^{n} - I_{h}^{m}) ‖ + ‖ - η Z^{n} (S_{h}^{n} - S_{h}^{m}) ‖ + ‖ - η S_{h}^{m} (Z^{n} - Z^{m}) ‖ \\ + & ‖ - μ_{h} (S_{h}^{n} - S_{h}^{m}) ‖ + ‖ σ_{h} (A_{h}^{n} - A_{h}^{m}) ‖ + ‖ (ϕ + τ) (1 - θ) (I_{h}^{n} - I_{h}^{m}) ‖ + ‖ γ_{h} (R_{h}^{n} - R_{h}^{m}) ‖\}] \\ = & ‖ S_{h}^{n} (t) - S_{h}^{m} (t) ‖ + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{‖ - β_{ν} I_{ν}^{n} (S_{h}^{n} - S_{h}^{m}) ‖ + ‖ - β_{ν} S_{h}^{m} (S_{h}^{n} - S_{h}^{m}) ‖ \\ + & ‖ - β_{h} I_{h}^{n} (S_{h}^{n} - S_{h}^{m}) ‖ + ‖ - β_{h} S_{h}^{m} (S_{h}^{n} - S_{h}^{m}) ‖ + ‖ - η Z^{n} (S_{h}^{n} - S_{h}^{m}) ‖ + ‖ - η S_{h}^{m} (S_{h}^{n} - S_{h}^{m}) ‖ \\ + & ‖ - μ_{h} (S_{h}^{n} - S_{h}^{m}) ‖ + ‖ σ_{h} (S_{h}^{n} - S_{h}^{m}) ‖ + ‖ (ϕ + τ) (1 - θ) (S_{h}^{n} - S_{h}^{m}) ‖ + ‖ γ_{h} (S_{h}^{n} - S_{h}^{m}) ‖\}] \\ = & ‖ S_{h}^{n} - S_{h}^{m} ‖ [1 + N T^{- 1} \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T ‖ - β_{ν} (I_{ν}^{n} + S_{h}^{m}) ‖ + N T^{- 1} \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \\ ‖ - β_{h} (I_{h}^{n} + S_{h}^{m}) ‖ + N T^{- 1} \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T ‖ - η (Z^{n} + S_{h}^{m}) ‖ + N T^{- 1} \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \\ ‖ - μ_{h} ‖ + N T^{- 1} \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T ‖ σ_{h} ‖ + N T^{- 1} \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T ‖ (ϕ + τ) (1 - θ) ‖ \\ + & N T^{- 1} \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T ‖ γ_{h} ‖] \end{align}

(18)

Since

S_{h}^{m}, I_{h}^{n}, I_{ν}^{n}, Z^{n}

are convergent and bounded we can choose different constants for that F, J, E, U, V, M N, K for all t such that

‖ S_{h}^{m} ‖ < F, ‖ I_{h}^{n} ‖ < J, ‖ I_{ν}^{n} ‖ < E, ‖ Z^{n} ‖ < U, ‖ I_{ν}^{m} ‖ < N, ‖ I_{h}^{m} ‖ < M, ‖ A_{h}^{n} ‖ < V, ‖ S_{h}^{n} ‖ < K

Then (18) becomes

\begin{align} ‖ P (S_{h}^{n} (t)) - P (S_{h}^{m} (t)) ‖ = & \{1 - f (ζ) β_{ν} (E + F) - h (ζ) β_{h} (J + F) - w (ζ) η (U + F) - r (ζ) μ_{h} + s (ζ) σ_{h} + y (ζ) (ϕ + τ) (1 - θ) \\ + & z (ζ) γ_{h}\} ‖ S_{h}^{n} - S_{h}^{m} ‖ \end{align}

(19)

where f, h, w, r, s, y, z, ρ, ω, ι, κ, λ are functions from

N T^{- 1} (N T \frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)})

similarly we have

\begin{align} ‖ P (A_{h}^{n} (t)) - P (A_{h}^{m} (t)) ‖ = & {1 + w_{1} (ζ) η (U + F) - f_{1} (ζ) β_{h}^{'} (V + M) - h_{1} (ζ) β_{h}^{'} (V + N) - r_{1} (ζ) μ_{h} - s_{1} (ζ) σ_{h}} ‖ A_{h}^{n} - A_{h}^{m} ‖ \\ ‖ P (I_{h}^{n} (t)) - P (I_{h}^{m} (t)) ‖ = & \{1 + ρ (ζ) β_{ν} (K + M) + h_{2} (ζ) β_{h}^{'} (V + M) + ω (ζ) β_{h} (K + M) + f_{2} (ζ) β_{h}^{'} (V + M) - r_{2} (ζ) μ_{h} \\ - & ι (ζ) δ_{h} - y_{1} (ζ) (ϕ + τ)\} ‖ I_{h}^{n} - I_{h}^{m} ‖ \\ ‖ P (R_{h}^{n} (t)) - P (R_{h}^{m} (t)) ‖ = & {1 + y_{2} (ζ) (ϕ + τ) θ - r_{3} (ζ) μ_{h} - z_{1} (ζ) γ_{h}} ‖ R_{h}^{n} - R_{h}^{m} ‖ \\ ‖ P (Z^{n} (t)) - P (Z^{m} (t)) ‖ = & {1 + ϱ (ζ) p - ϑ (ζ) q} ‖ Z^{n} - Z^{m} ‖ \\ | | P (S_{ν}^{n} (t)) - P (S_{ν}^{m} (t)) ‖ = & {1 - κ (ζ) b_{ν} (W + M) - λ (ζ) μ_{ν}} ‖ S_{ν}^{n} - S_{ν}^{m} ‖ \\ | | P (I_{ν}^{n} (t)) - P (I_{ν}^{m} (t)) ‖ = & {1 + κ_{1} (ζ) b_{ν} (W + M) - λ_{1} (ζ) μ_{ν}} ‖ I_{ν}^{n} - I_{ν}^{m} ‖ \end{align}

(20)

where

\begin{array}{l} {1 - f (ζ) β_{ν} (E + F) - h (ζ) β_{h} (J + F) - w (ζ) η (U + F) - r (ζ) μ_{h} + s (ζ) σ_{h} + y (ζ) (ϕ + τ) (1 - θ) + z (ζ) γ_{h}} < 1 \\ {1 + w_{1} (ζ) η (U + F) - f_{1} (ζ) β_{h}^{'} (V + M) - h_{1} (ζ) β_{h}^{'} (V + N) - r_{1} (ζ) μ_{h} - s_{1} (ζ) σ_{h}} < 1 \\ {1 + ρ (ζ) β_{ν} (K + M) + h_{2} (ζ) β_{h}^{'} (V + M) + ω (ζ) β_{h} (K + M) + f_{2} (ζ) β_{h}^{'} (V + M) - r_{2} (ζ) μ_{h} - ι (ζ) δ_{h} - y_{1} (ζ) (ϕ + τ)} < 1 \\ {1 + y_{2} (ζ) (ϕ + τ) θ - r_{3} (ζ) μ_{h} - z_{1} (ζ) γ_{h}} < 1 \\ {1 + ϱ (ζ) p - ϑ (ζ) q} < 1 \\ {1 - κ (ζ) b_{ν} (W + M) - λ (ζ) μ_{ν}} < 1 \\ {1 + κ_{1} (ζ) b_{ν} (W + M) - λ_{1} (ζ) μ_{ν}} < 1 \end{array}

Hence the theorem holds and P is stable.

Theorem 4.3

The special solution of equation (8) using the iteration method is unique.

Proof

Let us define a Hilbert space H = L((a, b) × (0, T)) as

y : (a, b) \times (0, T) \to R, \iint u y d u d y < \infty

Considering

P (S_{h}, A_{h}, I_{h}, R_{h}, Z, S_{ν}, I_{ν}) = \{\begin{cases} Λ_{h} - β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} + (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h}, \\ η Z S_{h} - β_{h}^{'} A_{h} I_{ν} - β_{h}^{'} A_{h} I_{h} - μ_{h} A_{h} - σ_{h} A_{h}, \\ (β_{ν} S_{h} + β_{h}^{'} A_{h}) I_{ν} + (β_{h} S_{h} + β_{h}^{'} A_{h}) I_{h} - (μ_{h} + δ_{h}) I_{h} - (ϕ + τ) I_{h}, \\ (ϕ + τ) θ I_{h} - (μ_{h} + γ_{h}) R_{h}, \\ p I_{h} - q Z, \\ Λ_{ν} - b_{ν} S_{ν} I_{h} - μ_{ν} S_{ν}, \\ b_{ν} S_{ν} I_{h} - μ_{ν} I_{ν} . \end{cases}

P ((X_{11} - X_{12}, X_{21} - X_{22}, X_{31} - X_{32}, X_{41} - X_{42}, X_{51} - X_{52}, X_{61} - X_{62}, X_{71} - X_{72}), (O_{1}, O_{2}, O_{3}, O_{4}, O_{5}, O_{6}, O_{7}))

(21)

then

\begin{array}{l} \{- β_{ν} (X_{11} - X_{12}) (X_{71} - X_{72}) - β_{h} (X_{11} - X_{12}) (X_{31} - X_{32}) - η (X_{51} - X_{52}) (X_{11} - X_{12}) - μ_{h} (X_{11} - X_{12}) + σ_{h} (X_{21} - X_{22}) \\ + (ϕ + τ) (1 - θ) (X_{31} - X_{32}) + γ_{h} (X_{41} - X_{42}), O_{1}\}, \\ {η (X_{51} - X_{52}) (X_{11} - X_{12}) - β_{h}^{'} (X_{21} - X_{22}) (X_{71} - X_{72}) - β_{h}^{'} (X_{21} - X_{22}) (X_{31} - X_{32}) - μ_{h} (X_{21} - X_{22}) - σ_{h} (X_{21} - X_{22}), O_{2}}, \\ \{(β_{ν} (X_{11} - X_{12}) + β_{h}^{'} (X_{21} - X_{22}) (X_{71} - X_{72}) + (β_{h} (X_{11} - X_{12}) + β_{h}^{'} (X_{21} - X_{22})) (X_{31} - X_{32}) - (μ_{h} + δ_{h}) (X_{31} - X_{32}) \\ - (ϕ + τ) (X_{31} - X_{32}), O_{3}, \\ (ϕ + τ) θ (X_{31} - X_{32}) - (μ_{h} + γ_{h}) (X_{41} - X_{42}), O_{4}\}, \\ {p (X_{31} - X_{32}) - q (X_{51} - X_{52}), O_{5}}, \\ {- b_{ν} (X_{61} - X_{62}) (X_{31} - X_{32}) - μ_{ν} (X_{61} - X_{62}), O_{6}}, \\ {b_{ν} (X_{61} - X_{62}) (X_{31} - X_{32}) - μ_{ν} (X_{71} - X_{72}), O_{7}} . \end{array}

Now consider

\begin{align} \{- β_{ν} (X_{11} - X_{12}) (X_{71} - X_{72}) - β_{h} (X_{11} - X_{12}) (X_{31} - X_{32}) - η (X_{51} - X_{52}) (X_{11} - X_{12}) - μ_{h} (X_{11} - X_{12}) + σ_{h} (X_{21} - X_{22}) \\ + & (ϕ + τ) (1 - θ) (X_{31} - X_{32}) + γ_{h} (X_{41} - X_{42}), O_{1}\} \\ ≅ & {- β_{ν} (X_{11} - X_{12}) (X_{71} - X_{72}), O_{1}} + {- β_{h} (X_{11} - X_{12}) (X_{31} - X_{32}), O_{1}} + {- η (X_{51} - X_{52}) (X_{11} - X_{12}), O_{1}} \\ + & {- μ_{h} (X_{11} - X_{12}), O_{1}} + {σ_{h} (X_{21} - X_{22}), O_{1}} + {(ϕ + τ) (1 - θ) (X_{31} - X_{32}), O_{1}} + {γ_{h} (X_{41} - X_{42}), O_{1}} \end{align}

(22)

Let

(X_{11} - X_{12}) ≅ (X_{21} - X_{22}) ≅ (X_{31} - X_{32}) ≅ (X_{41} - X_{42}) ≅ (X_{51} - X_{52}) ≅ (X_{61} - X_{62}) ≅ (X_{71} - X_{72})

Then (22) becomes

\begin{align} ≅ & {- β_{ν} {(X_{11} - X_{12})}^{2}, O_{1}} + {- β_{h} {(X_{11} - X_{12})}^{2}, O_{1}} + {- η {(X_{11} - X_{12})}^{2}, O_{1}} + {- μ_{h} (X_{11} - X_{12}), O_{1}} \\ + & {σ_{h} (X_{21} - X_{22}), O_{1}} + {(ϕ + τ) (1 - θ) (X_{31} - X_{32}), O_{1}} + {γ_{h} (X_{41} - X_{42}), O_{1}} \\ \leq & β_{ν} ‖ {(X_{11} - X_{12})}^{2} ‖ ‖ O_{1} ‖ + β_{h} ‖ {(X_{11} - X_{12})}^{2} ‖ ‖ O_{1} ‖ + η ‖ {(X_{11} - X_{12})}^{2} ‖ ‖ O_{1} ‖ + μ_{h} ‖ (X_{11} - X_{12}) ‖ ‖ O_{1} ‖ \\ + & σ_{h} ‖ (X_{11} - X_{12}) ‖ ‖ O_{1} ‖ + (ϕ + τ) (1 - θ) ‖ (X_{11} - X_{12}) ‖ ‖ O_{1} ‖ + γ_{h} ‖ (X_{11} - X_{12}) ‖ ‖ O_{1} ‖ \\ = & (β_{ν} ψ_{1} + β_{h} ψ_{1} + η + μ_{h} + σ_{h} + (ϕ + τ) (1 - θ) + γ_{h}) ‖ (X_{11} - X_{12}) ‖ ‖ O_{1} ‖ \end{align}

(23)

Similarly we get

\begin{align} {η (X_{51} - X_{52}) (X_{11} - X_{12}) - β_{h}^{'} (X_{21} - X_{22}) (X_{71} - X_{72}) - β_{h}^{'} (X_{21} - X_{22}) (X_{31} - X_{32}) - μ_{h} (X_{21} - X_{22}) - σ_{h} (X_{21} - X_{22}), O_{2}} \\ \leq (η ψ_{2} + 2 β_{h}^{'} ψ_{2} + μ_{h} + σ_{h}) ‖ (X_{21} - X_{22}) ‖ ‖ O_{2} ‖, \\ \{(β_{ν} (X_{11} - X_{12}) + β_{h}^{'} (X_{21} - X_{22}) (X_{71} - X_{72}) + (β_{h} (X_{11} - X_{12}) + β_{h}^{'} (X_{21} - X_{22})) (X_{31} - X_{32}) - (μ_{h} + δ_{h}) (X_{31} - X_{32}) \\ - (ϕ + τ) (X_{31} - X_{32}), O_{3}\} \leq (β_{ν} + 2 β_{h}^{'} ψ_{3} + β_{h} + μ_{h} + δ_{h} + (ϕ + τ)) ‖ (X_{31} - X_{32}) ‖ ‖ O_{3} ‖ \\ (ϕ + τ) θ (X_{31} - X_{32}) - (μ_{h} + γ_{h}) (X_{41} - X_{42}), O_{4}\} \leq ((ϕ + τ) θ + μ_{h} + γ_{h}) ‖ (X_{41} - X_{42}) ‖ ‖ O_{4} ‖ \\ {p (X_{31} - X_{32}) - q (X_{51} - X_{52}), O_{5}} \leq (p + q) ‖ (X_{51} - X_{52}) ‖ ‖ O_{5} ‖ \\ {- b_{ν} (X_{61} - X_{62}) (X_{31} - X_{32}) - μ_{ν} (X_{61} - X_{62}), O_{6}} \leq (b_{ν} ψ_{6} + μ_{ν}) ‖ (X_{61} - X_{62}) ‖ ‖ O_{6} ‖ \\ {b_{ν} (X_{61} - X_{62}) (X_{31} - X_{32}) - μ_{ν} (X_{71} - X_{72}), O_{7}} \leq (b_{ν} ψ_{7} + μ_{ν}) ‖ (X_{71} - X_{72}) ‖ ‖ O_{7} ‖ . \end{align}

(24)

Substituting (23), (24) in (21) we get

\begin{align} P ((X_{11} - X_{12}, X_{21} - X_{22}, X_{31} - X_{32}, X_{41} - X_{42}, X_{51} - X_{52}, X_{61} - X_{62}, X_{71} - X_{72}), (O_{1}, O_{2}, O_{3}, O_{4}, O_{5}, O_{6}, O_{7})) \\ \leq \{\begin{cases} (β_{ν} ψ_{1} + β_{h} ψ_{1} + η + μ_{h} + σ_{h} + (ϕ + τ) (1 - θ) + γ_{h}) ‖ (X_{11} - X_{12}) ‖ ‖ O_{1} ‖, \\ (η ψ_{2} + 2 β_{h}^{'} ψ_{2} + μ_{h} + σ_{h}) ‖ (X_{21} - X_{22}) ‖ ‖ O_{2} ‖, \\ (β_{ν} + 2 β_{h}^{'} ψ_{3} + β_{h} + μ_{h} + δ_{h} + (ϕ + τ)) ‖ (X_{31} - X_{32}) ‖ ‖ O_{3} ‖, \\ ((ϕ + τ) θ + μ_{h} + γ_{h}) ‖ (X_{41} - X_{42}) ‖ ‖ O_{4} ‖, \\ (p + q) ‖ (X_{51} - X_{52}) ‖ ‖ O_{5} ‖, \\ (b_{ν} ψ_{6} + μ_{ν}) ‖ (X_{61} - X_{62}) ‖ ‖ O_{6} ‖, \\ (b_{ν} ψ_{7} + μ_{ν}) ‖ (X_{71} - X_{72}) ‖ ‖ O_{7} ‖ . \end{cases} \end{align}

(25)

But, for large values of m_i; with i = 1, 2, 3, 4, 5, 6, 7, hence solution exits for all small positive parameters $l_{m_{1}}, l_{m_{2}}, l_{m_{3}}, l_{m_{4}}, l_{m_{5}}, l_{m_{6}}, l_{m_{7}},$ such that

\begin{array}{l} ‖ (S_{h} - X_{11}) ‖, ‖ (S_{h} - X_{12}) ‖ < & \frac{l_{m_{1}}}{7 (β_{ν} ψ_{1} + β_{h} ψ_{1} + η + μ_{h} + σ_{h} + (ϕ + τ) (1 - θ) + γ_{h}) ‖ (X_{11} - X_{12}) ‖ ‖ O_{1} ‖} \\ ‖ (A_{h} - X_{21}) ‖, ‖ (A_{h} - X_{22}) ‖ < & \frac{l_{m_{2}}}{7 (η ψ_{2} + 2 β_{h}^{'} ψ_{2} + μ_{h} + σ_{h}) ‖ (X_{21} - X_{22}) ‖ ‖ O_{2} ‖} \\ ‖ (I_{h} - X_{31}) ‖, ‖ (I_{h} - X_{32}) ‖ < & \frac{l_{m_{3}}}{7 (β_{ν} + 2 β_{h}^{'} ψ_{3} + β_{h} + μ_{h} + δ_{h} + (ϕ + τ)) ‖ (X_{31} - X_{32}) ‖ ‖ O_{3} ‖} \\ ‖ (R_{h} - X_{41}) ‖, ‖ (R_{h} - X_{42}) ‖ < & \frac{l_{m_{4}}}{7 ((ϕ + τ) θ + μ_{h} + γ_{h}) ‖ (X_{41} - X_{42}) ‖ ‖ O_{4} ‖} \\ ‖ (Z - X_{51}) ‖, ‖ (Z - X_{52}) ‖ < & \frac{l_{m_{5}}}{7 (p + q) ‖ (X_{51} - X_{52}) ‖ ‖ O_{5} ‖} \\ ‖ (S_{ν} - X_{61}) ‖, ‖ (S_{ν} - X_{62}) ‖ < & \frac{l_{m_{6}}}{7 (b_{ν} ψ_{6} + μ_{ν}) ‖ (X_{61} - X_{62}) ‖ ‖ O_{6} ‖} \\ ‖ (I_{ν} - X_{71}) ‖, ‖ (I_{ν} - X_{72}) ‖ < & \frac{l_{m_{7}}}{7 (b_{ν} ψ_{7} + μ_{ν}) ‖ (X_{71} - X_{72}) ‖ ‖ O_{7} ‖} \end{array}

Thus, plugging the exact solution in the right-hand side of the (25) and applying the triangular inequality by taking M = max(m₁, m₂, m₃, m₄, m₅, m₆, m₇), l = max(l₁, l₂, l₃, l₄, l₅, l₆, l₇) we obtain

\{\begin{matrix} (β_{ν} ψ_{1} + β_{h} ψ_{1} + η + μ_{h} + σ_{h} + (ϕ + τ) (1 - θ) + γ_{h}) ‖ (X_{11} - X_{12}) ‖ ‖ O_{1} ‖, \\ (η ψ_{2} + 2 β_{h}^{'} ψ_{2} + μ_{h} + σ_{h}) ‖ (X_{21} - X_{22}) ‖ ‖ O_{2} ‖, \\ (β_{ν} + 2 β_{h}^{'} ψ_{3} + β_{h} + μ_{h} + δ_{h} + (ϕ + τ)) ‖ (X_{31} - X_{32}) ‖ ‖ O_{3} ‖, \\ ((ϕ + τ) θ + μ_{h} + γ_{h}) ‖ (X_{41} - X_{42}) ‖ ‖ O_{4} ‖, \\ (p + q) ‖ (X_{51} - X_{52}) ‖ ‖ O_{5} ‖, \\ (b_{ν} ψ_{6} + μ_{ν}) ‖ (X_{61} - X_{62}) ‖ ‖ O_{6} ‖, \\ (b_{ν} ψ_{7} + μ_{ν}) ‖ (X_{71} - X_{72}) ‖ ‖ O_{7} ‖ . \end{matrix} < \{\begin{matrix} l, \\ l, \\ l, \\ l, \\ l, \\ l, \\ l . \end{matrix}

\{\begin{matrix} (β_{ν} ψ_{1} + β_{h} ψ_{1} + η + μ_{h} + σ_{h} + (ϕ + τ) (1 - θ) + γ_{h}) ‖ (X_{11} - X_{12}) ‖ ‖ O_{1} ‖, \\ (η ψ_{2} + 2 β_{h}^{'} ψ_{2} + μ_{h} + σ_{h}) ‖ (X_{21} - X_{22}) ‖ ‖ O_{2} ‖, \\ (β_{ν} + 2 β_{h}^{'} ψ_{3} + β_{h} + μ_{h} + δ_{h} + (ϕ + τ)) ‖ (X_{31} - X_{32}) ‖ ‖ O_{3} ‖, \\ ((ϕ + τ) θ + μ_{h} + γ_{h}) ‖ (X_{41} - X_{42}) ‖ ‖ O_{4} ‖, \\ (p + q) ‖ (X_{51} - X_{52}) ‖ ‖ O_{5} ‖, \\ (b_{ν} ψ_{6} + μ_{ν}) ‖ (X_{61} - X_{62}) ‖ ‖ O_{6} ‖, \\ (b_{ν} ψ_{7} + μ_{ν}) ‖ (X_{71} - X_{72}) ‖ ‖ O_{7} ‖ . \end{matrix} < \{\begin{matrix} 0, \\ 0, \\ 0, \\ 0, \\ 0, \\ 0, \\ 0 . \end{matrix}

But it is obvious that

\begin{array}{l} (β_{ν} ψ_{1} + β_{h} ψ_{1} + η + μ_{h} + σ_{h} + (ϕ + τ) (1 - θ) + γ_{h}) \neq 0, (η ψ_{2} + 2 β_{h}^{'} ψ_{2} + μ_{h} + σ_{h}) \neq 0, (β_{ν} + 2 β_{h}^{'} ψ_{3} + β_{h} + μ_{h} + δ_{h} + (ϕ + τ)) \neq 0, \\ ((ϕ + τ) θ + μ_{h} + γ_{h}) \neq 0, (p + q) \neq 0, (b_{ν} ψ_{6} + μ_{ν}) \neq 0, (b_{ν} ψ_{7} + μ_{ν}) \neq 0 \end{array}

Therefore we have

‖ X_{11} - X_{12} ‖ = 0, ‖ X_{21} - X_{22} ‖ = 0, ‖ X_{31} - X_{32} ‖ = 0, ‖ X_{41} - X_{42} ‖ = 0, ‖ X_{51} - X_{52} ‖ = 0, ‖ X_{61} - X_{62} ‖ = 0, ‖ X_{71} - X_{72} ‖ = 0

which implies X₁₁ = X₁₂, X₂₁ = X₂₂, X₃₁ = X₃₂, X₄₁ = X₄₂, X₅₁ = X₅₂, X₆₁ = X₆₂, X₇₁ = X₇₂

Hence the theorem.

4.1. Numerical analysis

In this part, we use the two-step Newton interpolation approach to obtain the approximate solution of our model (8). We apply the new integral that the Natural transform yielded:

\begin{array}{l} S_{h} (t) & = s^{- 1} S_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{Λ_{h} - β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} + (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h}\}], \\ A_{h} (t) & = s^{- 1} A_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{η Z S_{h} - β_{h}^{'} A_{h} I_{ν} - β_{h}^{'} A_{h} I_{h} - μ_{h} A_{h} - σ_{h} A_{h}\}], \\ I_{h} (t) & = s^{- 1} I_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{(β_{ν} S_{h} + β_{h}^{'} A_{h}) I_{ν} + (β_{h} S_{h} + β_{h}^{'} A_{h}) I_{h} - (μ_{h} + δ_{h}) I_{h} - (ϕ + τ) I_{h}\}], \\ R_{h} (t) & = s^{- 1} R_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{(ϕ + τ) θ I_{h} - (μ_{h} + γ_{h}) R_{h}\}], \\ Z (t) & = s^{- 1} Z (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{p I_{h} - q Z\}], \\ S_{ν} (t) & = s^{- 1} S_{ν} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{Λ_{ν} - b_{ν} S_{ν} I_{h} - μ_{ν} S_{ν}\}], \\ I_{ν} (t) & = s^{- 1} I_{ν} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T \{b_{ν} S_{ν} I_{h} - μ_{ν} I_{ν}\}] . \end{array}

We now describe the Newton interpolation polynomial approach at t_n+1. Then our model becomes:

\begin{array}{l} S_{h}^{n + 1} (t) = & s^{- 1} S_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{h} - β_{ν} S_{h}^{n} I_{ν}^{n} - β_{h} S_{h}^{n} I_{h}^{n} - η Z^{n} S_{h}^{n} - μ_{h} S_{h}^{n} + σ_{h} A_{h}^{n} + (ϕ + τ) (1 - θ) I_{h}^{n} + γ_{h} R_{h}^{n}}], \\ A_{h}^{n + 1} (t) = & s^{- 1} A_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {η Z^{n} S_{h}^{n} - β_{h}^{'} A_{h}^{n} I_{ν}^{n} - β_{h}^{'} A_{h}^{n} I_{h}^{n} - μ_{h} A_{h}^{n} - σ_{h} A_{h}^{n}}], \\ I_{h}^{n + 1} (t) = & s^{- 1} I_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(β_{ν} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{ν}^{n} + (β_{h} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{h}^{n} - (μ_{h} + δ_{h}) I_{h}^{n} - (ϕ + τ) I_{h}^{n}}], \\ R_{h}^{n + 1} (t) = & s^{- 1} R_{h} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {(ϕ + τ) θ I_{h}^{n} - (μ_{h} + γ_{h}) R_{h}^{n}}], \\ Z^{n + 1} (t) = & s^{- 1} Z (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {p I_{h} - q Z^{n}}], \\ S_{ν}^{n + 1} (t) = & s^{- 1} S_{ν} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {Λ_{ν} - b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} S_{ν}^{n}}], \\ I_{ν}^{n + 1} (t) = & s^{- 1} I_{ν} (0) + N T^{- 1} [\frac{(1 - α) + α {(\frac{u}{s})}^{α}}{M (α)} N T {b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} I_{ν}^{n}}] . \end{array}

This gives,

\begin{array}{l} S_{h} {(t)}^{n + 1} & = S_{h} (0) + N T^{- 1} [\frac{(1 - α)}{M (α)} N T \{Λ_{h} - β_{ν} S_{h}^{n} I_{ν}^{n} - β_{h} S_{h}^{n} I_{h}^{n} - η Z^{n} S_{h}^{n} - μ_{h} S_{h}^{n} + σ_{h} A_{h}^{n} + (ϕ + τ) (1 - θ) I_{h}^{n} + γ_{h} R_{h}^{n}\}] \\ + N T^{- 1} [\frac{α {(\frac{u}{s})}^{α}}{M (α)} N T \{Λ_{h} - β_{ν} S_{h}^{n} I_{ν}^{n} - β_{h} S_{h}^{n} I_{h}^{n} - η Z^{n} S_{h}^{n} - μ_{h} S_{h}^{n} + σ_{h} A_{h}^{n} + (ϕ + τ) (1 - θ) I_{h}^{n} + γ_{h} R_{h}^{n}\}], \\ A_{h} {(t)}^{n + 1} & = A_{h} (0) + N T^{- 1} [\frac{(1 - α)}{M (α)} N T \{η Z^{n} S_{h}^{n} - β_{h}^{'} A_{h}^{n} I_{ν}^{n} - β_{h}^{'} A_{h}^{n} I_{h}^{n} - μ_{h} A_{h}^{n} - σ_{h} A_{h}^{n}\}] \\ + N T^{- 1} [\frac{α {(\frac{u}{s})}^{α}}{M (α)} N T \{η Z^{n} S_{h}^{n} - β_{h}^{'} A_{h}^{n} I_{ν}^{n} - β_{h}^{'} A_{h}^{n} I_{h}^{n} - μ_{h} A_{h}^{n} - σ_{h} A_{h}^{n}\}], \\ I_{h} {(t)}^{n + 1} & = I_{h} (0) + N T^{- 1} [\frac{(1 - α)}{M (α)} N T \{(β_{ν} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{ν}^{n} + (β_{h} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{h}^{n} - (μ_{h} + δ_{h}) I_{h}^{n} - (ϕ + τ) I_{h}^{n}\}] \\ + N T^{- 1} [\frac{α {(\frac{u}{s})}^{α}}{M (α)} N T \{(β_{ν} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{ν}^{n} + (β_{h} S_{h}^{n} + β_{h}^{'} A_{h}^{n}) I_{h}^{n} - (μ_{h} + δ_{h}) I_{h}^{n} - (ϕ + τ) I_{h}^{n}\}], \\ R_{h} {(t)}^{n + 1} & = R_{h} (0) + N T^{- 1} [\frac{(1 - α)}{M (α)} N T \{(ϕ + τ) θ I_{h}^{n} - (μ_{h} + γ_{h}) R_{h}^{n}\}] \\ + N T^{- 1} [\frac{α {(\frac{u}{s})}^{α}}{M (α)} N T \{(ϕ + τ) θ I_{h}^{n} - (μ_{h} + γ_{h}) R_{h}^{n}\}], \\ Z {(t)}^{n + 1} & = Z (0) + N T^{- 1} [\frac{(1 - α)}{M (α)} N T \{p I_{h}^{n} - q Z^{n}\}] + N T^{- 1} [\frac{α {(\frac{u}{s})}^{α}}{M (α)} N T \{p I_{h}^{n} - q Z^{n}\}], \\ S_{ν} {(t)}^{n + 1} & = S_{ν} (0) + N T^{- 1} [\frac{(1 - α)}{M (α)} N T \{Λ_{ν} - b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} S_{ν}^{n}\}] \\ + N T^{- 1} [\frac{α {(\frac{u}{s})}^{α}}{M (α)} N T \{Λ_{ν} - b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} S_{ν}^{n}\}], \\ I_{ν} {(t)}^{n + 1} & = I_{ν} (0) + N T^{- 1} [\frac{(1 - α)}{M (α)} N T \{b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} I_{ν}^{n}\}] + N T^{- 1} [\frac{α {(\frac{u}{s})}^{α}}{M (α)} N T \{b_{ν} S_{ν}^{n} I_{h}^{n} - μ_{ν} I_{ν}^{n}\}] . \end{array}

Let

\begin{array}{l} {}_{0}^{A B C}D_{t}^{α} S_{h} & = K_{1} (t, S_{h} (t)), \\ {}_{0}^{A B C}D_{t}^{α} A_{h} & = K_{2} (t, A_{h} (t)), \\ {}_{0}^{A B C}D_{t}^{α} I_{h} & = K_{3} (t, I_{h} (t)), \\ {}_{0}^{A B C}D_{t}^{α} R_{h} & = K_{4} (t, R_{h} (t)), \\ {}_{0}^{A B C}D_{t}^{α} Z & = K_{5} (t, Z (t)), \\ {}_{0}^{A B C}D_{t}^{α} S_{ν} & = K_{6} (t, S_{ν} (t)), \\ {}_{0}^{A B C}D_{t}^{α} I_{ν} & = K_{7} (t, I_{ν} (t)) . \end{array}

Now, approximating the function $K_{i}$ for i = 1, 2, 3, 4, 5, 6, 7 in the above system over the interval [t_j, t_j+1] through the Lagrangian piece-wise interpolation, we get

\begin{array}{l} P_{j_{1}}^{h} & = \frac{ξ - t_{j - 1}}{t_{j} - t_{j - 1}} K_{1} (ξ_{j}, S_{h}^{j}) - \frac{ξ - t_{j}}{t_{j} - t_{j - 1}} K_{1} (ξ_{j - 1}, S_{h}^{j - 1}), \\ Q_{j_{2}}^{h} & = \frac{ξ - t_{j - 1}}{t_{j} - t_{j - 1}} K_{2} (ξ_{j}, A_{h}^{j}) - \frac{ξ - t_{j}}{t_{j} - t_{j - 1}} K_{2} (ξ_{j - 1}, A_{h}^{j - 1}), \\ W_{j_{3}}^{h} & = \frac{ξ - t_{j - 1}}{t_{j} - t_{j - 1}} K_{3} (ξ_{j}, I_{h}^{j}) - \frac{ξ - t_{j}}{t_{j} - t_{j - 1}} K_{3} (ξ_{j - 1}, I_{h}^{j - 1}), \\ X_{j_{4}}^{h} & = \frac{ξ - t_{j - 1}}{t_{j} - t_{j - 1}} K_{4} (ξ_{j}, R_{h}^{j}) - \frac{ξ - t_{j}}{t_{j} - t_{j - 1}} K_{4} (ξ_{j - 1}, R_{h}^{j - 1}), \\ Y_{j_{5}}^{h} & = \frac{ξ - t_{j - 1}}{t_{j} - t_{j - 1}} K_{5} (ξ_{j}, Z^{j}) - \frac{ξ - t_{j}}{t_{j} - t_{j - 1}} K_{5} (ξ_{j - 1}, Z^{j - 1}), \\ L_{j_{6}}^{ν} & = \frac{ξ - t_{j - 1}}{t_{j} - t_{j - 1}} K_{6} (ξ_{j}, S_{ν}^{j}) - \frac{ξ - t_{j}}{t_{j} - t_{j - 1}} K_{6} (ξ_{j - 1}, S_{ν}^{j - 1}), \\ M_{j_{7}}^{ν} & = \frac{ξ - t_{j - 1}}{t_{j} - t_{j - 1}} K_{7} (ξ_{j}, I_{ν}^{j}) - \frac{ξ - t_{j}}{t_{j} - t_{j - 1}} K_{7} (ξ_{j - 1}, I_{ν}^{j - 1}) . \end{array}

The Atangana-Baleanu fractional integral of a function $K (t)$ is defined as:

{}_{0}^{A B C}I_{t}^{α} K (t) = \frac{1 - α}{M (α)} K (t) + \frac{α}{M (α) Γ (α)} \int_{0}^{t} K (ξ) {(t - ξ)}^{α - 1} d ξ

By applying the Natural transform property to our system, we obtain:

N T^{- 1} {(\frac{u}{s})}^{α} (N T [K_{i} (ξ_{i}, X^{i})]) = \frac{1}{Γ (α)} \int_{0}^{t} K_{i} (ξ_{i}, X^{i}) {(t - ξ)}^{α - 1} d ξ

Using the property of the Natural transform for the ABC fractional derivative, this corresponds to the Riemann-Liouville fractional integral:

N T^{- 1} \{N T [K_{ρ_{i}} (ξ_{ρ_{i}}, X^{ρ_{i}})]\} = \frac{1}{Γ (α)} \int_{0}^{t} K_{ρ_{i}} (ξ_{ρ_{i}}, X^{ρ_{i}}) {(t - ξ)}^{α - 1} d ξ .

By applying this property to our system we get

\begin{array}{l} S_{h}^{(n + 1)} (t) & = S_{h}^{(n)} (0) + (\frac{1 - α}{M (α)} [K_{1} (ξ_{1}, S_{h}^{1 (n)})]) + \frac{α}{M (α) Γ (α)} \sum_{j = 0}^{n} \int_{t_{j}}^{t_{j + 1}} K_{1} (ξ_{1}, S_{h}^{1}) {(t - ξ_{1})}^{α - 1} d ξ_{1}, \\ A_{h}^{(n + 1)} (t) & = A_{h}^{(n)} (0) + (\frac{1 - α}{M (α)} [K_{2} (ξ_{2}, A_{h}^{2 (n)})]) + \frac{α}{M (α) Γ (α)} \sum_{j = 0}^{n} \int_{t_{j}}^{t_{j + 1}} K_{2} (ξ_{2}, A_{h}^{2}) {(t - ξ_{2})}^{α - 1} d ξ_{2}, \\ I_{h}^{(n + 1)} (t) & = I_{h}^{(n)} (0) + (\frac{1 - α}{M (α)} [K_{3} (ξ_{3}, I_{h}^{3 (n)})]) + \frac{α}{M (α) Γ (α)} \sum_{j = 0}^{n} \int_{t_{j}}^{t_{j + 1}} K_{3} (ξ_{3}, I_{h}^{3}) {(t - ξ_{3})}^{α - 1} d ξ_{3}, \\ R_{h}^{(n + 1)} (t) & = R_{h}^{(n)} (0) + (\frac{1 - α}{M (α)} [K_{4} (ξ_{4}, R_{h}^{4 (n)})]) + \frac{α}{M (α) Γ (α)} \sum_{j = 0}^{n} \int_{t_{j}}^{t_{j + 1}} K_{4} (ξ_{4}, R_{h}^{4}) {(t - ξ_{4})}^{α - 1} d ξ_{4}, \\ Z^{(n + 1)} (t) & = Z^{(n)} (0) + (\frac{1 - α}{M (α)} [K_{5} (ξ_{5}, Z^{5 (n)})]) + \frac{α}{M (α) Γ (α)} \sum_{j = 0}^{n} \int_{t_{j}}^{t_{j + 1}} K_{5} (ξ_{5}, Z^{5}) {(t - ξ_{5})}^{α - 1} d ξ_{5}, \\ S_{ν}^{(n + 1)} (t) & = S_{ν}^{(n)} (0) + (\frac{1 - α}{M (α)} [K_{6} (ξ_{6}, S_{ν}^{6 (n)})]) + \frac{α}{M (α) Γ (α)} \sum_{j = 0}^{n} \int_{t_{j}}^{t_{j + 1}} K_{6} (ξ_{6}, S_{ν}^{6}) {(t - ξ_{6})}^{α - 1} d ξ_{6}, \\ I_{ν}^{(n + 1)} (t) & = I_{ν}^{(n)} (0) + (\frac{1 - α}{M (α)} [K_{7} (ξ_{7}, I_{ν}^{7 (n)})]) + \frac{α}{M (α) Γ (α)} \sum_{j = 0}^{n} \int_{t_{j}}^{t_{j + 1}} K_{7} (ξ_{7}, I_{ν}^{7}) {(t - ξ_{7})}^{α - 1} d ξ_{7} . \end{array}

We approximate the kernel $K_{i} (ξ_{i}, X^{i})$ using two-step Newton interpolation on each interval [t_j, t_j+1]:

K (ξ) \approx K (t_{j}) + \frac{K (t_{j + 1}) - K (t_{j})}{Δ t} (ξ - t_{j}), ξ \in [t_{j}, t_{j + 1}] .

By discretizing the fractional integral, we have

\frac{1}{Γ (α)} \sum_{j = 0}^{n} \int_{t_{j}}^{t_{j + 1}} K (ξ) {(t_{j + 1} - ξ)}^{α - 1} d ξ \approx \frac{1}{Γ (α)} \sum_{j = 0}^{n} (K (t_{j}) F_{1}^{j} + K (t_{j + 1}) F_{2}^{j}),

where

F_{1}^{j} = \int_{t_{j}}^{t_{j + 1}} [1 - \frac{ξ - t_{j}}{Δ t}] {(t_{j + 1} - ξ)}^{α - 1} d ξ,

F_{2}^{j} = \int_{t_{j}}^{t_{j + 1}} [\frac{ξ - t_{j}}{Δ t}] {(t_{j + 1} - ξ)}^{α - 1} d ξ .

After evaluation, we obtain

F_{1}^{j} = \frac{{(Δ t)}^{α}}{Γ (α + 2)} [{(n - j + 1)}^{α} (n - j + 2 + α) - {(n - j)}^{α} (n - j + 2 + 2 α)],

F_{2}^{j} = \frac{{(Δ t)}^{α}}{Γ (α + 2)} [{(n - j + 1)}^{α + 1} - {(n - j)}^{α} (n - j + 1 + α)] .

Hence, the final numerical scheme for our model is given by:

\begin{array}{l} S_{h}^{(n + 1)} (t) & = S_{h}^{(n)} (0) + (\frac{1 - α}{M (α)} [K_{1} (ξ_{1}, S_{h}^{1 (n)})]) \\ + \frac{α}{M (α) Γ (α + 2)} \sum_{j = 0}^{n} (K_{1} (ξ_{j}, S_{h}^{j}) [{(n - j + 1)}^{α} (n - j + 2 + α) - {(n - j)}^{α} (n - j + 2 + 2 α)] \\ - K_{1} (ξ_{j - 1}, S_{h}^{j - 1}) [{(n - j + 1)}^{α + 1} - {(n - j)}^{α} (n - j + 1 + α)]), \\ A_{h}^{(n + 1)} (t) & = A_{h}^{(n)} (0) + (\frac{1 - α}{M (α)} [K_{2} (ξ_{2}, A_{h}^{2 (n)})]) \\ + \frac{α}{M (α) Γ (α + 2)} \sum_{j = 0}^{n} (K_{2} (ξ_{j}, A_{h}^{j}) [{(n - j + 1)}^{α} (n - j + 2 + α) - {(n - j)}^{α} (n - j + 2 + 2 α)] \\ - K_{2} (ξ_{j - 1}, A_{h}^{j - 1}) [{(n - j + 1)}^{α + 1} - {(n - j)}^{α} (n - j + 1 + α)]), \\ I_{h}^{(n + 1)} (t) & = I_{h}^{(n)} (0) + (\frac{1 - α}{M (α)} [K_{3} (ξ_{3}, I_{h}^{3 (n)})]) \\ + \frac{α}{M (α) Γ (α + 2)} \sum_{j = 0}^{n} (K_{3} (ξ_{j}, I_{h}^{j}) [{(n - j + 1)}^{α} (n - j + 2 + α) - {(n - j)}^{α} (n - j + 2 + 2 α)] \\ - K_{3} (ξ_{j - 1}, I_{h}^{j - 1}) [{(n - j + 1)}^{α + 1} - {(n - j)}^{α} (n - j + 1 + α)]), \\ R_{h}^{(n + 1)} (t) & = R_{h}^{(n)} (0) + (\frac{1 - α}{M (α)} [K_{4} (ξ_{4}, R_{h}^{4 (n)})]) \\ + \frac{α}{M (α) Γ (α + 2)} \sum_{j = 0}^{n} (K_{4} (ξ_{j}, R_{h}^{j}) [{(n - j + 1)}^{α} (n - j + 2 + α) - {(n - j)}^{α} (n - j + 2 + 2 α)] \\ - K_{4} (ξ_{j - 1}, R_{h}^{j - 1}) [{(n - j + 1)}^{α + 1} - {(n - j)}^{α} (n - j + 1 + α)]), \\ Z^{(n + 1)} (t) & = Z^{(n)} (0) + (\frac{1 - α}{M (α)} [K_{5} (ξ_{5}, Z^{5 (n)})]) \\ + \frac{α}{M (α) Γ (α + 2)} \sum_{j = 0}^{n} (K_{5} (ξ_{j}, Z^{j}) [{(n - j + 1)}^{α} (n - j + 2 + α) - {(n - j)}^{α} (n - j + 2 + 2 α)] \\ - K_{5} (ξ_{j - 1}, Z^{j - 1}) [{(n - j + 1)}^{α + 1} - {(n - j)}^{α} (n - j + 1 + α)]), \\ S_{ν}^{(n + 1)} (t) & = S_{ν}^{(n)} (0) + (\frac{1 - α}{M (α)} [K_{6} (ξ_{6}, S_{ν}^{6 (n)})]) \\ + \frac{α}{M (α) Γ (α + 2)} \sum_{j = 0}^{n} (K_{6} (ξ_{j}, S_{ν}^{j}) [{(n - j + 1)}^{α} (n - j + 2 + α) - {(n - j)}^{α} (n - j + 2 + 2 α)] \\ - K_{6} (ξ_{j - 1}, S_{ν}^{j - 1}) [{(n - j + 1)}^{α + 1} - {(n - j)}^{α} (n - j + 1 + α)]), \\ I_{ν}^{(n + 1)} (t) & = I_{ν}^{(n)} (0) + (\frac{1 - α}{M (α)} [K_{7} (ξ_{7}, I_{ν}^{7 (n)})]) \\ + \frac{α}{M (α) Γ (α + 2)} \sum_{j = 0}^{n} (K_{7} (ξ_{j}, I_{ν}^{j}) [{(n - j + 1)}^{α} (n - j + 2 + α) - {(n - j)}^{α} (n - j + 2 + 2 α)] \\ - K_{7} (ξ_{j - 1}, I_{ν}^{j - 1}) [{(n - j + 1)}^{α + 1} - {(n - j)}^{α} (n - j + 1 + α)]) . \end{array}

4.2. Numerical results

After substituting the parameter values from the 2016 Argentina Zika outbreak into the NT-ABC fractional model and solving using the two-step Newton interpolation scheme, we obtain the graphical results shown in Figure 2. From these plots, we observe that decreasing the fractional order α from 1.0 to 0.7 introduces a memory effect that significantly alters the disease dynamics—peak values are delayed and reduced, and the system takes longer to reach steady state. This demonstrates that fractional-order modeling captures the long-term dependency and historical effects inherent in real-world epidemiological systems, providing a more realistic representation of disease transmission compared to classical integer-order models.

Figure 2.

Natural transform solution of fractional Zika model.

5. Positivity and boundedness

From (8) the following inequality hold

\begin{align} {}_{0}^{A B C}D_{t}^{α} S_{h} (t) |_{S_{h} = 0} = & Λ_{h} + σ_{h} A_{h} + (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h} > 0, \\ {}_{0}^{A B C}D_{t}^{α} A_{h} (t) |_{A_{h} = 0} = & η Z S_{h} \geq 0, \\ {}_{0}^{A B C}D_{t}^{α} I_{h} (t) |_{I_{h} = 0} = & (β_{ν} S_{h} + β_{h}^{'} A_{h}) I_{ν} \geq 0, \\ {}_{0}^{A B C}D_{t}^{α} R_{h} (t) |_{R_{h} = 0} = & (ϕ + τ) θ I_{h} \geq 0, \\ {}_{0}^{A B C}D_{t}^{α} Z (t) |_{Z = 0} = & p I_{h} \geq 0, \\ {}_{0}^{A B C}D_{t}^{α} S_{ν} (t) |_{S_{ν} = 0} = & Λ_{ν} \geq 0, \\ {}_{0}^{A B C}D_{t}^{α} I_{ν} (t) |_{I_{ν} = 0} = & b_{ν} S_{ν} I_{h} \geq 0 . \end{align}

(26)

From, (26) the outcome remains confined within the hyper planes. Consequently, the solutions of the system (8) are guaranteed to be non-negative.

The biologically-feasible region for the fractional Zika virus model (8) defined as $Φ \in R^{+}$ . Such that

\begin{array}{l} Φ & = \{S_{h} (t), A_{h} (t), I_{h} (t), R_{h} (t), S_{ν} (t), I_{ν} (t), Z \in R^{+} \\ : S_{h} (t) + A_{h} (t) + I_{h} (t) + R_{h} (t) \leq 1, S_{ν} (t) + I_{ν} (t) \leq 1, Z \leq 1\} . \end{array}

Lemma 5.1

The model 8 with non-negative initial conditions in region $R^{+}$ is positively invariant.

Proof

The boundedness of the fractional model solutions is established for the total human population as:

\begin{align} {}_{0}^{A B C}D_{t}^{α} N_{h} (t) = & {}_{0}^{A B C}D_{t}^{α} S_{h} (t) + {}_{0}^{A B C}D_{t}^{α} A_{h} (t) + {}_{0}^{A B C}D_{t}^{α} I_{h} (t) + {}_{0}^{A B C}D_{t}^{α} R_{h} (t) \\ = & Λ_{h} - μ_{h} N_{h} - S_{h} I_{h} \end{align}

(27)

By applying the Laplace transform,

\begin{array}{l} L {{}_{0}^{A B C}D_{t}^{α} N_{h} (t)} & = L {Λ_{h} - μ_{h} N_{h} - S_{h} I_{h}} \\ s N_{h} (s) - N_{h} (0) & = \frac{Λ_{h}}{s} - μ_{h} N_{h} (s) - S_{h} I_{h} (s) \\ N_{h} (s) & = \frac{\frac{Λ_{h}}{s} - S_{h} I_{h} (s) + N_{h} (0)}{s + μ_{h}} \end{array}

The Laplace inverse gives,

N_{h} (t) = \frac{Λ_{h}}{μ_{h}} (1 - e^{- μ_{h} t}) + N_{h} (0) e^{- μ_{h} t} - S_{h} \int_{0}^{t} I_{h} (τ) e^{- μ_{h} (t - τ)} d τ

and the rate of change of total population of mosquitoes is,

\begin{align} {}_{0}^{A B C}D_{t}^{α} N_{ν} (t) = & {}_{0}^{A B C}D_{t}^{α} S_{ν} (t) + {}_{0}^{A B C}D_{t}^{α} I_{ν} (t) \\ = & Λ_{ν} - μ_{ν} N_{ν} \end{align}

(28)

By applying Laplace transformation,

\begin{array}{l} L {{}_{0}^{A B C}D_{t}^{α} N_{ν} (t)} & = L {Λ_{ν} - μ_{ν} N_{ν}} \\ s N_{ν} (s) - N_{ν} (0) & = \frac{Λ_{ν}}{s} - μ_{ν} N_{ν} (s) \\ N_{ν} (s) & = \frac{\frac{Λ_{ν}}{s} + N_{ν} (0)}{s + μ_{h}} \end{array}

The Laplace inverse is

N_{ν} (t) = \frac{Λ_{ν}}{μ_{ν}} (1 - e^{- μ_{ν} t}) + N_{ν} (0) e^{- μ_{ν} t}

For information rate

{}_{0}^{A B C}D_{t}^{α} Z (t) = p I_{h} - q Z

(29)

By applying Laplace transformation,

\begin{array}{l} L {{}_{0}^{A B C}D_{t}^{α} Z (t)} = & L {p I_{h} - q Z} \\ s Z (s) - Z (0) = & p I_{h} (s) - q Z (s) \\ Z (s) = & \frac{p I_{h} (s) + Z (0)}{s + q} \end{array}

and its inverse transform is,

Z (t) = Z (0) e^{- q t} + p \int_{0}^{t} I_{h} (τ) e^{- q (t - τ)}

As a result, $N_{h} (t)$ , $N_{ν} (t)$ and $Z$ converges for t → ∞. So the region Φ is positively invariant.

6. Equilibria

We determine the equilibria of the system (8) by assuming the state variables are constant and by putting the right side equal to zero. Eq. (8) admits two types of equilibria as follows:

The disease-free steady state of the Zika carrier model is referred to as E0 and determined by Eq. (8) in the absence of disease, i.e., I_h = 0 = I_ν as follows

\begin{array}{l} E_{0} = & (S_{h_{0}}, 0,0,0,0, S_{ν_{0}}, 0) \\ = & (\frac{Λ_{h}}{μ_{h}}, 0,0,0,0, \frac{Λ_{ν}}{μ_{ν}}, 0) . \end{array}

In the endemic equilibrium (EE) state, the disease persists within the population without being completely eradicated. At this stage, all compartments representing vaccinated, infected, susceptible, and exposed individuals must have values greater than zero. Let the endemic equilibrium point be:

E^{*} = (S_{h}^{*}, A_{h}^{*}, I_{h}^{*}, R_{h}^{*}, Z^{*}, S_{ν}^{*}, I_{ν}^{*}),

where each variable represents a positive constant value at equilibrium.

The system of equations at equilibrium becomes:

\begin{array}{l} 0 & = Λ_{h} - β_{ν} S_{h}^{*} I_{ν}^{*} - β_{h} S_{h}^{*} I_{h}^{*} - η Z^{*} S_{h}^{*} - μ_{h} S_{h}^{*} + σ_{h} A_{h}^{*} + (ϕ + τ) (1 - θ) I_{h}^{*} + γ_{h} R_{h}^{*}, \\ 0 & = η Z^{*} S_{h}^{*} - β_{h}^{'} A_{h}^{*} I_{ν}^{*} - β_{h}^{'} A_{h}^{*} I_{h}^{*} - μ_{h} A_{h}^{*} - σ_{h} A_{h}^{*}, \\ 0 & = (β_{ν} S_{h}^{*} + β_{h}^{'} A_{h}^{*}) I_{ν}^{*} + (β_{h} S_{h}^{*} + β_{h}^{'} A_{h}^{*}) I_{h}^{*} - (μ_{h} + δ_{h}) I_{h}^{*} - (ϕ + τ) I_{h}^{*}, \\ 0 & = (ϕ + τ) θ I_{h}^{*} - (μ_{h} + γ_{h}) R_{h}^{*}, \\ 0 & = p I_{h}^{*} - q Z^{*}, \\ 0 & = Λ_{ν} - b_{ν} S_{ν}^{*} I_{h}^{*} - μ_{ν} S_{ν}^{*}, \\ 0 & = b_{ν} S_{ν}^{*} I_{h}^{*} - μ_{ν} I_{ν}^{*} . \end{array}

\begin{array}{l} S_{h}^{*} & = \frac{Λ_{h} + σ_{h} A_{h}^{*} + [(ϕ + τ) (1 - θ) + \frac{γ_{h} (ϕ + τ) θ}{μ_{h} + γ_{h}}] I_{h}^{*}}{β_{ν} (\frac{b_{ν} Λ_{ν} I_{h}^{*}}{μ_{ν} (b_{ν} I_{h}^{*} + μ_{ν})}) + β_{h} I_{h}^{*} + \frac{η p}{q} I_{h}^{*} + μ_{h}}, \\ A_{h}^{*} & = \frac{\frac{η p}{q} S_{h}^{*} I_{h}^{*}}{β_{ν}^{'} (\frac{b_{ν} Λ_{ν} I_{h}^{*}}{μ_{ν} (b_{ν} I_{h}^{*} + μ_{ν})}) + β_{h}^{'} I_{h}^{*} + (μ_{h} + σ_{h})}, \\ R_{h}^{*} & = \frac{(ϕ + τ) θ}{μ_{h} + γ_{h}} I_{h}^{*}, \\ Z^{*} & = \frac{p}{q} I_{h}^{*}, \\ S_{ν}^{*} & = \frac{Λ_{ν}}{b_{ν} I_{h}^{*} + μ_{ν}}, \\ I_{ν}^{*} & = \frac{b_{ν} Λ_{ν}}{μ_{ν} (b_{ν} I_{h}^{*} + μ_{ν})} I_{h}^{*} . \end{array}

where

I_{h}^{*}

is obtained by solving:

(β_{ν} S_{h}^{*} + β_{ν}^{'} A_{h}^{*}) (\frac{b_{ν} Λ_{ν}}{μ_{ν} (b_{ν} I_{h}^{*} + μ_{ν})}) + (β_{h} S_{h}^{*} + β_{h}^{'} A_{h}^{*}) = μ_{h} + δ_{h} + ϕ + τ .

7. Basic reproduction number

The basic reproduction number, denoted R₀, is a fundamental metric in epidemiology that quantifies the average number of secondary infections produced by a single infected individual in a completely susceptible population. From³⁷ The new infection matrix $F$ is given by

F = [\begin{matrix} \frac{β_{h} Λ_{h}}{μ_{h}} & \frac{Λ_{h} β_{ν}}{μ_{h}} \\ \frac{b_{ν} Λ_{ν}}{μ_{ν}} & 0 \end{matrix}]

The transition matrix $V$ is

V = [\begin{matrix} δ_{h} + μ_{h} + τ + ϕ & 0 \\ 0 & μ_{ν} \end{matrix}]

Then,

F V^{- 1} = [\begin{matrix} \frac{β_{h} Λ_{h}}{μ_{h} (δ_{h} + μ_{h} + τ + ϕ)} & \frac{Λ_{h} β_{ν}}{μ_{h} μ_{ν}} \\ \frac{b_{ν} Λ_{ν}}{μ_{ν} (δ_{h} + μ_{h} + τ + ϕ)} & 0 \end{matrix}] = [\begin{matrix} R_{h h} & R_{ν h} \\ R_{h ν} & 0 \end{matrix}]

where

R_{h h} = \frac{β_{h} Λ_{h}}{μ_{h} (δ_{h} + μ_{h} + τ + ϕ)}

R_{ν h} = \frac{Λ_{h} β_{ν}}{μ_{h} μ_{ν}}

and

R_{h ν} = \frac{b_{ν} Λ_{ν}}{μ_{ν} (δ_{h} + μ_{h} + τ + ϕ)}

Let the eigen value be λ, and $κ = [\begin{matrix} R_{h h} & R_{ν h} \\ R_{h ν} & 0 \end{matrix}]$ then

d e t [κ - λ I] = 0

d e t [\begin{matrix} R_{h h} - λ & R_{ν h} \\ R_{h ν} & - λ \end{matrix}] = 0

\Rightarrow (R_{h h} - λ) (- λ) - (R_{ν h}) (R_{h ν}) = 0

\Rightarrow λ^{2} - λ R_{h h} - R_{ν h} R_{h ν} = 0

\begin{array}{l} λ = & \frac{R_{h h} \pm \sqrt{R_{h h}^{2} + 4 R_{ν h} R_{h ν}}}{2} \\ = & \frac{R_{h h}}{2} + \frac{1}{2} \sqrt{R_{h h}^{2} + 4 R_{ν h} R_{h ν}} \end{array}

which is expressed as the basic reproduction number by:

R_{0} = \frac{R_{h h}}{2} + \frac{1}{2} \sqrt{R_{h h}^{2} + 4 R_{ν h} R_{h ν}}

From Figure 3 the set of surface plots shows the effect of varying human-to-vector transmission (R_hν) while examining the relationship between R_hh and R_νh. When R_hν is low (0.5), R₀ remains relatively small even as the other parameters increase. At medium (2.0) and high (4.0) values of R_hν, the surfaces rise sharply, indicating that stronger human-to-vector transmission greatly amplifies R₀. This highlights that while all transmission routes contribute, the human-to-vector pathway has the greatest influence on driving R₀ and thus the potential for disease spread.

Figure 3.

R₀ transmission.

From Figure 4 the 3D scatter plot illustrates how the basic reproduction number (R₀) varies with different combinations of transmission parameters: human-to-human (R_hh), vector-to-human (R_νh), and human-to-vector (R_hν). Each point represents a possible parameter set, and the color indicates the corresponding R₀value. We observe that higher R₀values (red regions) generally cluster where transmission parameters are larger, while lower values (blue regions) appear when the parameters are small. This shows that the combined effect of all three transmission routes plays a significant role in determining outbreak potential.

Figure 4.

R₀ transmission.

7.1. Sensitivity analysis of R₀

The sensitivity index of R₀ with respect to a parameter ϕ is defined as:

ϒ_{ψ} = \frac{\partial R_{0}}{\partial ψ} \times \frac{ψ}{R_{0}} .

Where

R_{0} = \frac{\frac{β_{h} Λ_{h}}{μ_{h} (δ_{h} + μ_{h} + τ + ϕ)}}{2} + \frac{1}{2} \sqrt{{(\frac{β_{h} Λ_{h}}{μ_{h} (δ_{h} + μ_{h} + τ + ϕ)})}^{2} + 4 (\frac{Λ_{h} β_{ν}}{μ_{h} μ_{ν}}) (\frac{b_{ν} Λ_{ν}}{μ_{ν} (δ_{h} + μ_{h} + τ + ϕ)})}

7.1.1. Sensitivity index for β_h

The partial derivative of R₀ with respect to β_h is:

\begin{array}{l} \frac{\partial R_{0}}{\partial β_{h}} & = \frac{1}{2} \frac{\partial R_{h h}}{\partial β_{h}} + \frac{1}{4} {(R_{h h}^{2} + 4 R_{ν h} R_{h ν})}^{- 1 / 2} \cdot 2 R_{h h} \frac{\partial R_{h h}}{\partial β_{h}} \\ = \frac{Λ_{h}}{2 μ_{h} (δ_{h} + μ_{h} + τ + ϕ)} (1 + \frac{R_{h h}}{\sqrt{R_{h h}^{2} + 4 R_{ν h} R_{h ν}}}) \end{array}

where

\frac{\partial R_{h h}}{\partial β_{h}} = \frac{Λ_{h}}{μ_{h} (δ_{h} + μ_{h} + τ + ϕ)}

Therefore, the sensitivity index for β_h is:

\begin{array}{l} ϒ_{β_{h}} & = \frac{\partial R_{0}}{\partial β_{h}} \times \frac{β_{h}}{R_{0}} \\ = (\frac{Λ_{h}}{2 μ_{h} (δ_{h} + μ_{h} + τ + ϕ)} (1 + \frac{R_{h h}}{\sqrt{R_{h h}^{2} + 4 R_{ν h} R_{h ν}}})) \times \frac{β_{h}}{\frac{R_{h h}}{2} + \frac{1}{2} \sqrt{R_{h h}^{2} + 4 R_{ν h} R_{h ν}}} \end{array}

7.1.2. Sensitivity index for β_ν

The partial derivative of R₀ with respect to β_ν is:

\begin{array}{l} \frac{\partial R_{0}}{\partial β_{ν}} & = \frac{1}{4} {(R_{h h}^{2} + 4 R_{ν h} R_{h ν})}^{- 1 / 2} \cdot 4 R_{h ν} \frac{\partial R_{ν h}}{\partial β_{ν}} \\ = \frac{Λ_{h} R_{h ν}}{μ_{h} μ_{ν} \sqrt{R_{h h}^{2} + 4 R_{ν h} R_{h ν}}} \end{array}

where

\frac{\partial R_{ν h}}{\partial β_{ν}} = \frac{Λ_{h}}{μ_{h} μ_{ν}}

The sensitivity index for β_ν is:

\begin{array}{l} ϒ_{β_{ν}} & = \frac{\partial R_{0}}{\partial β_{ν}} \times \frac{β_{ν}}{R_{0}} \\ = (\frac{Λ_{h} R_{h ν}}{μ_{h} μ_{ν} \sqrt{R_{h h}^{2} + 4 R_{ν h} R_{h ν}}}) \times \frac{β_{ν}}{\frac{R_{h h}}{2} + \frac{1}{2} \sqrt{R_{h h}^{2} + 4 R_{ν h} R_{h ν}}} \end{array}

7.1.3. Sensitivity Index for b_ν

The partial derivative of R₀ with respect to b_ν is:

\begin{array}{l} \frac{\partial R_{0}}{\partial b_{ν}} & = \frac{1}{4} {(R_{h h}^{2} + 4 R_{ν h} R_{h ν})}^{- 1 / 2} 4 R_{ν h} \frac{\partial R_{h ν}}{\partial b_{ν}} \\ = \frac{Λ_{ν} R_{ν h}}{μ_{ν} (δ_{h} + μ_{h} + τ + ϕ) \sqrt{R_{h h}^{2} + 4 R_{ν h} R_{h ν}}} \end{array}

where

\frac{\partial R_{h ν}}{\partial b_{ν}} = \frac{Λ_{ν}}{μ_{ν} (δ_{h} + μ_{h} + τ + ϕ)}

The sensitivity index for b_ν is:

\begin{array}{l} ϒ_{b_{ν}} & = \frac{\partial R_{0}}{\partial b_{ν}} \times \frac{b_{ν}}{R_{0}} \\ = (\frac{Λ_{ν} R_{ν h}}{μ_{ν} (δ_{h} + μ_{h} + τ + ϕ) \sqrt{R_{h h}^{2} + 4 R_{ν h} R_{h ν}}}) \times \frac{b_{ν}}{\frac{R_{h h}}{2} + \frac{1}{2} \sqrt{R_{h h}^{2} + 4 R_{ν h} R_{h ν}}} \end{array}

Similarly, we can calculate the sensitivity index for each of the parameters by using the same method. Following the implementation of the parameter values, we get the following outcomes:

The partial rank correlation coefficient (PRCC) analysis reveals the sensitivity of the basic reproduction number R₀ to variations in each model parameter, as illustrated in Figure 5. The results demonstrate that Λ_h (human recruitment rate) exhibits the strongest positive correlation with R₀ (PRCC = 0.95), indicating that an increase in human birth or immigration rate would significantly elevate disease transmission. This is followed by the mosquito mortality rate μ_ν (PRCC = 0.90) and treatment rate τ (PRCC = 0.85), both showing strong positive influences. The mosquito recruitment rate Λ_ν also demonstrates a substantial positive correlation (PRCC = 0.80).

Figure 5.

The sensitivity of model parameters with respect to the basic reproduction number R₀.

Conversely, several parameters exhibit negative correlations with R₀, meaning they act as control parameters. The mosquito birth rate b_ν shows the strongest negative correlation (PRCC = -0.70), followed by the mosquito transmission rate β_ν (PRCC = -0.65), the natural recovery rate ϕ (PRCC = -0.60), human mortality rate μ_h (PRCC = -0.55), and disease-induced death rate δ_h (PRCC = -0.50).

These negative correlations suggest that interventions targeting mosquito birth rate, mosquito-to-human transmission rate, and natural recovery processes would be effective in reducing R₀. The high magnitude of these PRCC values (all above |0.50|) indicates that all parameters included in the analysis have a substantial impact on R₀, with human demographic parameters (Λ_h, μ_ν) being the most influential drivers of disease spread, while mosquito-related parameters (b_ν, β_ν) and the natural recovery rate (ϕ) serve as the most effective targets for control strategies.

8. Existence and uniqueness of fractional $A B C$ model

To establish the existence and uniqueness of solutions for the given system of differential equations, we adopt a standard method similar to that used in 39. We begin by verifying that the system meets the Lipschitz condition, followed by the application of the Banach fixed-point theorem to confirm both existence and uniqueness. The system is then reformulated in terms of kernel functions as follows:

\begin{array}{l} K_{1} (t, S_{h} (t)) = & Λ_{h} - β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} + (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h}, \\ K_{2} (t, A_{h} (t)) = & η Z S_{h} - β_{h}^{'} A_{h} I_{ν} - β_{h}^{'} A_{h} I_{h} - μ_{h} A_{h} - σ_{h} A_{h}, \\ K_{3} (t, I_{h} (t)) = & (β_{ν} S_{h} + β_{h}^{'} A_{h}) I_{ν} + (β_{h} S_{h} + β_{h}^{'} A_{h}) I_{h} - (μ_{h} + δ_{h}) I_{h} - (ϕ + τ) I_{h}, \\ K_{4} (t, R_{h} (t)) = & (ϕ + τ) θ I_{h} - (μ_{h} + γ_{h}) R_{h}, \\ K_{5} (t, Z (t)) = & p I_{h} - q Z, \\ K_{6} (t, S_{ν} (t)) = & Λ_{ν} - b_{ν} S_{ν} I_{h} - μ_{ν} S_{ν}, \\ K_{7} (t, I_{ν} (t)) = & b_{ν} S_{ν} I_{h} - μ_{ν} I_{ν} . \end{array}

Let

W^{*} = (S_{h}, A_{h}, I_{h}, R_{h}, Z, S_{ν}, I_{ν})

and

W^{*} = (S_{h}^{*}, A_{h}^{*}, I_{h}^{*}, R_{h}^{*}, Z^{*}, S_{ν}^{*}, I_{ν}^{*})

be a bounded solution for the

A B C

model system (8). Suppose ‖I_h‖ ≤ J₁ where J₁ is a positive constant and

‖ I_{ν} ‖ \leq E_{1}, ‖ Z ‖ \leq U_{1}, ‖ A_{n} ‖ \leq V_{1}

,. For the kernel

K_{1}

with respect to the variable

S_{h}

and

S_{h}^{*}

we obtain:

\begin{array}{l} ‖ K_{1} (t, S_{h}) - K_{1} (t, S_{h}^{*}) ‖ & = ‖ Λ_{h} - β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} + (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h} \\ - (Λ_{h} - β_{ν} S_{h}^{*} I_{ν}^{*} - β_{h} S_{h}^{*} I_{h}^{*} - η Z^{*} S_{h}^{*} - μ_{h} S_{h}^{*} + σ_{h} A_{h}^{*} + (ϕ + τ) (1 - θ) I_{h}^{*} + γ_{h} R_{h}^{*}) ‖ \\ = ‖ (β_{ν} I_{ν} + β_{h} I_{h} + η Z + μ_{h}) (S_{h} - S_{h}^{*}) ‖ \\ \leq (β_{ν} ‖ I_{ν} ‖ + β_{h} ‖ I_{h} ‖ + η ‖ Z ‖ + μ_{h}) ‖ S_{h} - S_{h}^{*} ‖ \\ \leq (β_{ν} E_{1} + β_{h} J_{1} + η U_{1} + μ_{h}) ‖ S_{h} - S_{h}^{*} ‖ \\ \leq e_{1} ‖ S_{h} - S_{h}^{*} ‖ \end{array}

Where e₁ = β_νE₁ + β_hJ₁ + ηU₁ + μ_h. Hence

K_{1}

satisfies the Lipschitz condition. Similarly we can show that:

\begin{array}{l} ‖ K_{2} (t, A_{h}) - K_{2} (t, A_{h}^{*}) ‖ & \leq e_{2} ‖ A_{h} - A_{h}^{*} ‖ \\ ‖ K_{3} (t, I_{h}) - K_{3} (t, I_{h}^{*}) ‖ & \leq e_{3} ‖ I_{h} - I_{h}^{*} ‖ \\ ‖ K_{4} (t, R_{h}) - K_{4} (t, R_{h}^{*}) ‖ & \leq e_{4} ‖ R_{h} - R_{h}^{*} ‖ \\ ‖ K_{5} (t, Z) - K_{5} (t, Z^{*}) ‖ & \leq e_{5} ‖ Z - Z^{*} ‖ \\ ‖ K_{6} (t, S_{ν}) - K_{6} (t, S_{ν}^{*}) ‖ & \leq e_{6} ‖ S_{ν} - S_{ν}^{*} ‖ \\ ‖ K_{7} (t, I_{ν}) - K_{7} (t, I_{ν}^{*}) ‖ & \leq e_{7} ‖ I_{ν} - I_{ν}^{*} ‖ \end{array}

where

e_{2} = β_{h}^{'} I_{ν} + β_{h}^{'} J + μ_{h} + σ_{h}

e_{3} = β_{h} S_{h} + β_{h}^{'} A_{h} - μ_{h} - δ_{h} - ϕ - τ

, e₄ = μ_h + γ_h, e₅ = q,

e_{6} = b_{ν} J + μ_{ν}

and e₇ = μ_ν. Hence,

K_{i = 2,3,4, \dots, 7}

satifies the Lipschitz condition. By definition (2.4) (4) the model (8) can be expressed in Volterra function:

W_{i} (t) = W_{i} (0) + \frac{1 - α}{M (α)} K_{i} (t, W_{i}) + \frac{α}{M (α) Γ (α)} \int_{0}^{t} {(t - h)}^{α - 1} K_{i} (h, W_{i}) d h,

(30)

for i = 1, 2, …, 7. Then (30) yields:

W_{i, n} (t) = W_{i} (0) + \frac{1 - α}{M (α)} K_{i} (t, W_{i, n - 1}) + \frac{α}{M (α) Γ (α)} \int_{0}^{t} {(t - h)}^{α - 1} K_{i} (h, W_{i, n - 1}) d h,

(31)

for i = 1, 2, …, 7.

Importantly, as n → ∞, the solution to model (8) can be approximated through scheme (31), using the following set of initial values: $W_{1,0} = S_{h} (0), W_{2,0} = A_{h} (0), W_{3,0} = I_{h} (0), W_{4,0} = R_{h} (0), W_{5,0} = Z (0), W_{6,0} = S_{ν} (0), W_{7,0} = I_{ν} (0)$ . The variation between consecutive terms can be expressed as:

\begin{array}{l} Π_{i, n} & = W_{i, n} - W_{i, n - 1} \\ = \frac{1 - α}{M (α)} [K_{i} (t, W_{i, n - 1}) - K_{i} (t, W_{i, n - 2})] + \frac{α}{M (α) Γ (α)} \int_{0}^{t} {(t - h)}^{α - 1} [K_{i} (h, W_{i, n - 1}) - K_{i} (h, W_{i, n - 2})] d h \end{array}

As each $K_{j}$ meets the Lipschitz condition for j = 1, 2, …, 7, it follows that:

‖ Π_{i, n} ‖ \leq \frac{1 - α}{M (α)} e_{i} ‖ Π_{i, n} ‖ + \frac{α}{M (α) Γ (α)} \int_{0}^{t} ‖ Π_{i, n - 1} ‖ {(t - h)}^{α - 1} d h

(32)

for i = 1, 2, …, 7. By estimating the variation between consecutive terms in (32) through an iterative method, we obtain:

‖ Π_{i, n} ‖ \leq W_{i, 0} {(\frac{1 - α}{M (α)} e_{i} + \frac{t^{α}}{M (α) Γ (α)} e_{i})}^{n},

for i = 1, 2, …, 7. Assuming that there exists t = t* such that

(\frac{1 - α}{M (α)} e_{i} + \frac{t^{α}}{M (α) Γ (α)} e_{i}) < 1

for i = 1, 2, …, 7. We deduce that ‖Π_j,n‖ approaches zero as n → ∞ for each j = 1, 2, …, 8. Consequently, under the given assumption, model (8) admits a solution represented by (30).

Now, we show that the solution to model (8) is unique. Suppose, for contradiction, that there exists another solution to model (8). Let it be $\vec{H} = (H_{1}^{*}, H_{2}^{*}, H_{3}^{*}, H_{4}^{*}, H_{5}^{*}, H_{6}^{*}, H_{7}^{*}) = (S_{h}^{*}, A_{h}^{*}, I_{h}^{*}, R_{h}^{*}, Z^{*}, S_{ν}^{*}, I_{ν}^{*})$ . Then for i = 1, 2, …, 7 yields:

\begin{array}{l} ‖ E_{i} - H_{i}^{*} ‖ = & \frac{1 - α}{M (α)} [K_{i} (t, W_{i}) - K_{i} (t, H_{i}^{*})] + \frac{α}{M (α) Γ (α)} \int_{0}^{t} {(t - h)}^{α - 1} [K_{i} (h, W_{i}) - K_{i} (h, H_{i}^{*})] d h \\ \leq & \frac{1 - α}{M (α)} e_{i} ‖ K_{i} - H_{i}^{*} ‖ + \frac{α}{M (α) Γ (α)} e_{i} \int_{0}^{t} ‖ K_{i} - H_{i}^{*} ‖ {(t - h)}^{α - 1} d s \\ \leq & ‖ K_{i} - H_{i}^{*} ‖ {(\frac{1 - α}{M (α)} e_{i} + \frac{t^{α}}{M (α) Γ (α)} e_{i})}^{n} \end{array}

At t = t* we obtain

‖ K_{i} - H_{i}^{*} ‖ \leq 0

for i = 1, 2, …, 7. This leads to

‖ K_{i} - H_{i}^{*} ‖ = 0

, which implies that

K_{i} = H_{i}^{*}

for i = 1, 2, .., 7. This results in a contriduction. Thus the model (8) has a unique solution, provided the assumption holds.

9. Local stability of Equilibria

Theorem 9.1

The DFE E₀ is locally asymptotically stable if R₀ < 1.

Proof

The Jacobian matrix of Zika virus system at E₀ is given by

J (E_{0}) = (\begin{matrix} - μ_{h} & σ_{h} & (1 - θ) (τ + ϕ) - \frac{β_{h} Λ_{h}}{μ_{h}} & γ_{h} & - \frac{η Λ_{h}}{μ_{h}} & 0 & - \frac{Λ_{h} β_{ν}}{μ_{h}} \\ 0 & - μ_{h} - σ_{h} & 0 & 0 & \frac{η Λ_{h}}{μ_{h}} & 0 & 0 \\ 0 & 0 & \frac{β_{h} Λ_{h}}{μ_{h}} - δ_{h} - μ_{h} - τ - ϕ & 0 & 0 & 0 & \frac{Λ_{h} β_{ν}}{μ_{h}} \\ 0 & 0 & θ (τ + ϕ) & - γ_{h} - μ_{h} & 0 & 0 & 0 \\ 0 & 0 & p & 0 & - q & 0 & 0 \\ 0 & 0 & - \frac{b_{ν} Λ_{ν}}{μ_{ν}} & 0 & 0 & - μ_{ν} & 0 \\ 0 & 0 & \frac{b_{ν} Λ_{ν}}{μ_{ν}} & 0 & 0 & 0 & - μ_{ν} \end{matrix}) .

The local stability is guaranteed if all eigenvalues of J(E₀) have negative real parts. From the structure of the Jacobian matrix, it is evident that several diagonal entries such as −μ_h, −μ_h − σ_h, −γ_h − μ_h, −q, and −μ_ν are strictly negative. The critical term appears in the third row, where the expression $\frac{β_{h} Λ_{h}}{μ_{h}} - (δ_{h} + μ_{h} + τ + ϕ)$ must be negative to ensure local stability. This condition is satisfied when $\frac{β_{h} Λ_{h}}{μ_{h}} < δ_{h} + μ_{h} + τ + ϕ$ , which coincides with the basic reproduction number $R_{0}$ being less than one. Therefore, the DFE E₀ is locally asymptotically stable if $R_{0} < 1$ . □

Theorem 9.2

The endemic equilibrium E* is locally asymptotically stable if R₀ > 1.

Proof

The Jacobian matrix evaluated at the endemic equilibrium is given by:

J (E^{*}) = [\begin{matrix} - β_{ν} I_{ν}^{*} - β_{h} I_{h}^{*} - η Z^{*} - μ_{h} & σ_{h} & - β_{h} S_{h}^{*} + (ϕ + τ) (1 - θ) & γ_{h} & - η S_{h}^{*} & 0 & - β_{ν} S_{h}^{*} \\ η Z^{*} & - β_{ν}^{'} I_{ν}^{*} - β_{h}^{'} I_{h}^{*} - μ_{h} - σ_{h} & - β_{h}^{'} A_{h}^{*} & 0 & η S_{h}^{*} & 0 & - β_{ν}^{'} A_{h}^{*} \\ β_{ν} I_{ν}^{*} + β_{h} I_{h}^{*} & β_{ν}^{'} I_{ν}^{*} + β_{h}^{'} I_{h}^{*} & β_{h} S_{h}^{*} + β_{h}^{'} A_{h}^{*} - (μ_{h} + δ_{h} + ϕ + τ) & 0 & 0 & 0 & β_{ν} S_{h}^{*} + β_{ν}^{'} A_{h}^{*} \\ 0 & 0 & (ϕ + τ) θ & - (μ_{h} + γ_{h}) & 0 & 0 & 0 \\ 0 & 0 & p & 0 & - q & 0 & 0 \\ 0 & 0 & - b_{ν} S_{ν}^{*} & 0 & 0 & - b_{ν} I_{h}^{*} - μ_{ν} & 0 \\ 0 & 0 & b_{ν} S_{ν}^{*} & 0 & 0 & b_{ν} I_{h}^{*} & - μ_{ν} \end{matrix}]

Then the characteristic polynomial is λ⁷ + a₁λ⁶ + a₂λ⁵ + a₃λ⁴ + a₄λ³ + a₅λ² + a₆λ + a₇ = 0,where coefficients of the characteristic polynomial are:

\begin{array}{l} a_{0} & = 1 \\ a_{1} & = - trace (J) = - \sum_{i = 1}^{7} J_{i i}, \\ a_{2} & = \sum_{1 \leq i < j \leq 7} |\begin{matrix} J_{i i} & J_{i j} \\ J_{j i} & J_{j j} \end{matrix}|, \\ a_{3} & = - \sum_{1 \leq i < j < k \leq 7} |\begin{matrix} J_{i i} & J_{i j} & J_{i k} \\ J_{j i} & J_{j j} & J_{j k} \\ J_{k i} & J_{k j} & J_{k k} \end{matrix}|, \\ a_{4} & = \sum_{1 \leq i < j < k < l \leq 7} |\begin{matrix} J_{i i} & J_{i j} & J_{i k} & J_{i l} \\ J_{j i} & J_{j j} & J_{j k} & J_{j l} \\ J_{k i} & J_{k j} & J_{k k} & J_{k l} \\ J_{l i} & J_{l j} & J_{l k} & J_{l l} \end{matrix}|, \\ a_{5} & = - \sum_{1 \leq i < j < k < l < m \leq 7} |\begin{matrix} J_{i i} & J_{i j} & J_{i k} & J_{i l} & J_{i m} \\ J_{j i} & J_{j j} & J_{j k} & J_{j l} & J_{j m} \\ J_{k i} & J_{k j} & J_{k k} & J_{k l} & J_{k m} \\ J_{l i} & J_{l j} & J_{l k} & J_{l l} & J_{l m} \\ J_{m i} & J_{m j} & J_{m k} & J_{m l} & J_{m m} \end{matrix}|, \\ a_{6} & = \sum_{1 \leq i < j < k < l < m < n \leq 7} |\begin{matrix} J_{i i} & J_{i j} & J_{i k} & J_{i l} & J_{i m} & J_{i n} \\ J_{j i} & J_{j j} & J_{j k} & J_{j l} & J_{j m} & J_{j n} \\ J_{k i} & J_{k j} & J_{k k} & J_{k l} & J_{k m} & J_{k n} \\ J_{l i} & J_{l j} & J_{l k} & J_{l l} & J_{l m} & J_{l n} \\ J_{m i} & J_{m j} & J_{m k} & J_{m l} & J_{m m} & J_{m n} \\ J_{n i} & J_{n j} & J_{n k} & J_{n l} & J_{n m} & J_{n n} \end{matrix}|, \\ a_{7} & = - \det (J), \end{array}

then we get:

\begin{array}{l} a_{0} & = 1, \\ a_{1} & = - J_{11} - J_{22} - J_{33} - J_{44} - J_{55} - J_{66} - J_{77} \\ = β_{ν} I_{ν}^{*} + β_{h} I h^{*} + η Z^{*} + μ_{h} + β^{'} ν I_{ν}^{*} + β_{h}^{'} I_{h}^{*} + μ_{h} + σ_{h} \\ - [β_{h} S_{h}^{*} + β^{'} h A h^{*} - (μ_{h} + δ_{h} + ϕ + τ)] + (μ_{h} + γ_{h}) + q + (b ν I h^{*} + μ ν) + μ ν \\ a_{2} & = - A_{h}^{*} I_{h}^{*} b_{ν} β_{h}^{'} - A_{h}^{*} I_{h}^{*} β_{h} β_{h}^{'} - A_{h}^{*} I_{ν}^{*} β_{h}^{'} β_{ν} - A_{h}^{*} S_{ν}^{*} b_{ν} β_{ν}^{'} - A_{h}^{*} Z^{*} β_{h}^{'} η - A_{h}^{*} q β_{h}^{'} \\ - A_{h}^{*} β_{h}^{'} γ_{h} - 3 A_{h}^{*} β_{h}^{'} μ_{h} - 2 A_{h}^{*} β_{h}^{'} μ_{ν} - A_{h}^{*} β_{h}^{'} σ_{h} + {(I_{h}^{*})}^{2} b_{ν} β_{h} + {(I_{h}^{*})}^{2} b_{ν} β_{h}^{'} + {(I_{h}^{*})}^{2} β_{h} β_{h}^{'} \\ + I_{h}^{*} I_{ν}^{*} b_{ν} β_{ν} + I_{h}^{*} I_{ν}^{*} b_{ν} β_{ν}^{'} + I_{h}^{*} I_{ν}^{*} β_{h} β_{ν}^{'} + I_{h}^{*} I_{ν}^{*} β_{h}^{'} β_{ν} - I_{h}^{*} S_{h}^{*} b_{ν} β_{h} - I_{h}^{*} S_{h}^{*} β_{h} β_{h}^{'} \\ + I_{h}^{*} Z^{*} b_{ν} η + I_{h}^{*} Z^{*} β_{h}^{'} η + I_{h}^{*} b_{ν} q + I_{h}^{*} b_{ν} γ_{h} + I_{h}^{*} b_{ν} δ_{h} + 4 I_{h}^{*} b_{ν} μ_{h} + I_{h}^{*} b_{ν} μ_{ν} \\ + I_{h}^{*} b_{ν} σ_{h} + I_{h}^{*} b_{ν} τ + I_{h}^{*} b_{ν} φ + I_{h}^{*} q β_{h} + I_{h}^{*} q β_{h}^{'} + I_{h}^{*} β_{h} γ_{h} + I_{h}^{*} β_{h} δ_{h} + I_{h}^{*} β_{h} θ τ \\ + I_{h}^{*} β_{h} θ φ + 3 I_{h}^{*} β_{h} μ_{h} + 2 I_{h}^{*} β_{h} μ_{ν} + I_{h}^{*} β_{h} σ_{h} + I_{h}^{*} β_{h}^{'} γ_{h} + I_{h}^{*} β_{h}^{'} δ_{h} + 3 I_{h}^{*} β_{h}^{'} μ_{h} + 2 I_{h}^{*} β_{h}^{'} μ_{ν} \\ + I_{h}^{*} β_{h}^{'} τ + I_{h}^{*} β_{h}^{'} φ + {(I_{ν}^{*})}^{2} β_{ν} β_{ν}^{'} - I_{ν}^{*} S_{h}^{*} β_{h} β_{ν}^{'} + I_{ν}^{*} Z^{*} β_{ν}^{'} η + I_{ν}^{*} q β_{ν} + I_{ν}^{*} q β_{ν}^{'} + I_{ν}^{*} β_{ν} γ_{h} \\ + I_{ν}^{*} β_{ν} δ_{h} + I_{ν}^{*} β_{ν} θ τ + I_{ν}^{*} β_{ν} θ φ + 3 I_{ν}^{*} β_{ν} μ_{h} + 2 I_{ν}^{*} β_{ν} μ_{ν} + I_{ν}^{*} β_{ν} σ_{h} + I_{ν}^{*} β_{ν}^{'} γ_{h} + I_{ν}^{*} β_{ν}^{'} δ_{h} \\ + 3 I_{ν}^{*} β_{ν}^{'} μ_{h} + 2 I_{ν}^{*} β_{ν}^{'} μ_{ν} + I_{ν}^{*} β_{ν}^{'} τ + I_{ν}^{*} β_{ν}^{'} φ - S_{h}^{*} S_{ν}^{*} b_{ν} β_{ν} - S_{h}^{*} Z^{*} β_{h} η - S_{h}^{*} q β_{h} \\ - S_{h}^{*} β_{h} γ_{h} - 3 S_{h}^{*} β_{h} μ_{h} - 2 S_{h}^{*} β_{h} μ_{ν} - S_{h}^{*} β_{h} σ_{h} + Z^{*} q η + Z^{*} γ_{h} η + Z^{*} δ_{h} η + 3 Z^{*} η μ_{h} \\ + 2 Z^{*} η μ_{ν} + Z^{*} η τ + Z^{*} η φ + q γ_{h} + q δ_{h} + 4 q μ_{h} + 2 q μ_{ν} + q σ_{h} + q τ + q φ + γ_{h} δ_{h} \\ + 3 γ_{h} μ_{h} + 2 γ_{h} μ_{ν} + γ_{h} σ_{h} + γ_{h} τ + γ_{h} φ + 3 δ_{h} μ_{h} + 2 δ_{h} μ_{ν} + δ_{h} σ_{h} + 6 μ_{h}^{2} + 8 μ_{h} μ_{ν} \\ + 3 μ_{h} σ_{h} + 3 μ_{h} τ + 3 μ_{h} φ + μ_{ν}^{2} + 2 μ_{ν} σ_{h} + 2 μ_{ν} τ + 2 μ_{ν} φ + σ_{h} τ + σ_{h} φ, \end{array}

then the other derived coefficients of the characteristic polynomial are given in the Appendix section16.

For a seventh-degree polynomial, the Routh array is constructed as follows:

\begin{matrix} λ^{7} & 1 & a_{2} & a_{4} & a_{6} \\ λ^{6} & a_{1} & a_{3} & a_{5} & a_{7} \\ λ^{5} & b_{1} & b_{2} & b_{3} & 0 \\ λ^{4} & c_{1} & c_{2} & c_{3} & 0 \\ λ^{3} & d_{1} & d_{2} & 0 & 0 \\ λ^{2} & e_{1} & e_{2} & 0 & 0 \\ λ^{1} & f_{1} & 0 & 0 & 0 \\ λ^{0} & g_{1} & 0 & 0 & 0 \end{matrix}

where:

\begin{array}{l} b_{1} & = \frac{a_{1} a_{2} - a_{3}}{a_{1}}, b_{2} = \frac{a_{1} a_{4} - a_{5}}{a_{1}}, b_{3} = \frac{a_{1} a_{6} - a_{7}}{a_{1}}, \\ c_{1} & = \frac{b_{1} a_{3} - a_{1} b_{2}}{b_{1}}, c_{2} = \frac{b_{1} a_{5} - a_{1} b_{3}}{b_{1}}, c_{3} = a_{7}, \\ d_{1} & = \frac{c_{1} b_{2} - b_{1} c_{2}}{c_{1}}, d_{2} = \frac{c_{1} b_{3} - b_{1} a_{7}}{c_{1}}, \\ e_{1} & = \frac{d_{1} c_{2} - c_{1} d_{2}}{d_{1}}, e_{2} = a_{7}, \\ f_{1} & = \frac{e_{1} d_{2} - d_{1} a_{7}}{e_{1}}, g_{1} = a_{7} . \end{array}

The endemic equilibrium E* is locally asymptotically stable if and only if all the following conditions hold:

All coefficients are positive: a_i > 0 for i = 1, 2, …, 7.

All elements in the first column of the Routh array are positive:

a_{1} > 0, b_{1} > 0, c_{1} > 0, d_{1} > 0, e_{1} > 0, f_{1} > 0, g_{1} > 0 .

Hence, the Routh–Hurwitz conditions are applied to confirm that all roots of the characteristic equation have negative real parts, thereby establishing the local stability of the endemic equilibrium.

10. Global stability of Equilibria

Using Lyapunov stability theory, we examine the stability of both the disease-free equilibrium and the endemic equilibrium. Based on the Jacobian matrix 1 and matrix 2, it is evident that all eigenvalues possess negative real parts. Therefore, the system is globally asymptotically stable.

Lemma 10.1

⁴⁷Suppose $g (t) \in R_{+}$ be a continuous and derivable function. Then, ${}_{0}^{A B C}D_{t}^{α} [g (t) - g^{*} - g^{*} l o g \frac{g (t)}{g^{*}}] \leq (1 - \frac{g^{*}}{g (t)}) {}_{0}^{A B C}D_{t}^{α} g (t), g^{*} \in R_{+}, \forall α \in (0,1] .$

Theorem 10.2

The sysytem at disease free equilibrium is globally asymptotically stable if R₀ < 1

Proof

We define the Lyapunov function as:

\begin{array}{l} L = & (S_{h} + S_{ν} + (A_{h} + I_{h} + R_{h} + Z + I_{ν})) - (S_{h_{0}} + S_{ν_{0}}) - (S_{h_{0}} + S_{ν_{0}}) l o g \frac{(S_{h} + S_{ν})}{(S_{h_{0}} + S_{ν_{0}})} \\ = & ((S_{h} + S_{ν}) - (S_{h_{0}} + S_{ν_{0}}) l o g \frac{(S_{h} + S_{ν})}{(S_{h_{0}} + S_{ν_{0}})}) + (A_{h} + I_{h} + R_{h} + Z + I_{ν}) \end{array}

Applying $A B C$ fractonal derivative on both sides,

\begin{array}{l} {}_{0}^{A B C}D_{t}^{α} L = & {}_{0}^{A B C}D_{t}^{α} ((S_{h} + S_{ν}) - (S_{h_{0}} + S_{ν_{0}}) l o g \frac{(S_{h} + S_{ν})}{(S_{h_{0}} + S_{ν_{0}})}) + ({}_{0}^{A B C}D_{t}^{α} A_{h} + {}_{0}^{A B C}D_{t}^{α} I_{h} + {}_{0}^{A B C}D_{t}^{α} R_{h} + {}_{0}^{A B C}D_{t}^{α} Z + {}_{0}^{A B C}D_{t}^{α} I_{ν}) \\ \leq & (1 - \frac{(S_{h_{0}} + S_{ν_{0}})}{(S_{h} + S_{ν})} ({}_{0}^{A B C}D_{t}^{α} S_{h} + {}_{0}^{A B C}D_{t}^{α} S_{ν})) + {}_{0}^{A B C}D_{t}^{α} A_{h} + {}_{0}^{A B C}D_{t}^{α} I_{h} + {}_{0}^{A B C}D_{t}^{α} R_{h} + {}_{0}^{A B C}D_{t}^{α} Z + {}_{0}^{A B C}D_{t}^{α} I_{ν} (b y 10.1) \\ = & (1 - \frac{(S_{h_{0}} + S_{ν_{0}})}{(S_{h} + S_{ν})} (Λ_{h} - β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} + (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h} + Λ_{ν} - b_{ν} S_{ν} I_{h} - μ_{ν} S_{ν})) \\ + & (η Z S_{h} - β_{h}^{'} A_{h} I_{ν} - β_{h}^{'} A_{h} I_{h} - μ_{h} A_{h} - σ_{h} A_{h} + (β_{ν} S_{h} + β_{h}^{'} A_{h}) I_{ν} + (β_{h} S_{h} + β_{h}^{'} A_{h}) I_{h} - (μ_{h} + δ_{h}) I_{h} - (ϕ + τ) I_{h} \\ + & (ϕ + τ) θ I_{h} - (μ_{h} + γ_{h}) R_{h} + p I_{h} - q Z + b_{ν} S_{ν} I_{h} - μ_{ν} I_{ν}) \\ = & Λ_{h} + Λ_{ν} - μ_{h} N_{h} - μ_{ν} N_{ν} - I_{h} (δ_{h} + p) - q Z - \frac{(S_{h_{0}} + S_{ν_{0}})}{(S_{h} + S_{ν})} (Λ_{h} - β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} \\ + & (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h} + Λ_{ν} - b_{ν} S_{ν} I_{h} - μ_{ν} S_{ν}) \\ \leq & - μ_{h} N_{h} - μ_{ν} N_{ν} - I_{h} (δ_{h} + p) - q Z - \frac{(S_{h_{0}} + S_{ν_{0}})}{(S_{h} + S_{ν})} (σ_{h} A_{h} + (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h}) \end{array}

Since all parameters of the system are positive, then

{}_{0}^{A B C}D_{t}^{α} L \leq 0

if R₀ ≤ 0 and system are positive, then

{}_{0}^{A B C}D_{t}^{α} L = 0

S_{h} = S_{h_{0}} = \frac{Λ_{h}}{μ_{h}}

S_{ν} = S_{ν} = \frac{Λ_{ν}}{μ_{ν_{0}}}

, I_h = 0 = I_ν □

Theorem 10.3

The sysytem at endemic equilibrium is globally asymptotically stable if R₀ > 1

Proof

We define the Lyapunov function as:

\begin{array}{l} L = & ((S_{h} + S_{ν}) - (S_{h}^{*} + S_{ν}^{*}) l o g \frac{(S_{h} + S_{ν})}{(S_{h}^{*} + S_{ν}^{*})}) + ((I_{h} + I_{ν}) - (I_{h}^{*} + I_{ν}^{*}) l o g \frac{(I_{h} + I_{ν})}{(I_{h}^{*} + I_{ν}^{*})}) + (A_{h} - A_{h}^{*} l o g \frac{A_{h}}{A_{h}^{*}}) \\ + & (R_{h} - R_{h}^{*} l o g \frac{R_{h}}{R_{h}^{*}}) + (Z - Z^{*} l o g \frac{Z}{Z^{*}}) \\ {}_{0}^{A B C}D_{t}^{α} L = & {}_{0}^{A B C}D_{t}^{α} ((S_{h} + S_{ν}) - (S_{h}^{*} + S_{ν}^{*}) l o g \frac{(S_{h} + S_{ν})}{(S_{h}^{*} + S_{ν}^{*})}) + {}_{0}^{A B C}D_{t}^{α} ((I_{h} + I_{ν}) - (I_{h}^{*} + I_{ν}^{*}) l o g \frac{(I_{h} + I_{ν})}{(I_{h}^{*} + I_{ν}^{*})}) \\ + & {}_{0}^{A B C}D_{t}^{α} (A_{h} - A_{h}^{*} l o g \frac{A_{h}}{A_{h}^{*}}) + {}_{0}^{A B C}D_{t}^{α} (R_{h} - R_{h}^{*} l o g \frac{R_{h}}{R_{h}^{*}}) + {}_{0}^{A B C}D_{t}^{α} (Z - Z^{*} l o g \frac{Z}{Z^{*}}) \\ \leq & (\frac{S_{h} - S_{h}^{*}}{S_{h}}) {}_{0}^{A B C}D_{t}^{α} S_{h} + (\frac{S_{ν} - S_{ν}^{*}}{S_{ν}}) {}_{0}^{A B C}D_{t}^{α} S_{ν} + (\frac{I_{h} - I_{h}^{*}}{I_{h}}) {}_{0}^{A B C}D_{t}^{α} I_{h} + (\frac{I_{ν} - I_{ν}^{*}}{I_{ν}}) {}_{0}^{A B C}D_{t}^{α} I_{ν} \\ + & (\frac{A_{h} - A_{h}^{*}}{A_{h}}) {}_{0}^{A B C}D_{t}^{α} A_{h} + (\frac{R_{h} - R_{h}^{*}}{R_{h}}) {}_{0}^{A B C}D_{t}^{α} R_{h} + (\frac{Z - Z^{*}}{Z}) {}_{0}^{A B C}D_{t}^{α} Z (b y 10.1) \\ = & (\frac{S_{h} - S_{h}^{*}}{S_{h}} (Λ_{h} - β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} + (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h}) \\ + & (\frac{S_{ν} - S_{ν}^{*}}{S_{ν}}) (Λ_{ν} - b_{ν} S_{ν} I_{h} - μ_{ν} S_{ν}) + (\frac{I_{h} - I_{h}^{*}}{I_{h}}) ((β_{ν} S_{h} + β_{h}^{'} A_{h}) I_{ν} + (β_{h} S_{h} + β_{h}^{'} A_{h}) I_{h} - (μ_{h} + δ_{h}) I_{h} - (ϕ + τ) I_{h}) \\ + & (\frac{I_{ν} - I_{ν}^{*}}{I_{ν}}) (b_{ν} S_{ν} I_{h} - μ_{ν} I_{ν}) + (\frac{A_{h} - A_{h}^{*}}{A_{h}}) (η Z S_{h} - β_{h}^{'} A_{h} I_{ν} - β_{h}^{'} A_{h} I_{h} - μ_{h} A_{h} - σ_{h} A_{h}) \\ + & (\frac{R_{h} - R_{h}^{*}}{R_{h}}) ((ϕ + τ) θ I_{h} - (μ_{h} + γ_{h}) R_{h}) + (\frac{Z - Z^{*}}{Z}) (p I_{h} - q Z) \\ = & Λ_{h} + Λ_{ν} - μ_{h} N_{h} - μ_{ν} N_{ν} - I_{h} (δ_{h} + p) - q Z - \frac{S_{h}^{*}}{S_{h}} (Λ_{h} - β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} \\ + & (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h}) - \frac{S_{ν}^{*}}{S_{ν}} (Λ_{ν} - b_{ν} S_{ν} I_{h} - μ_{ν} S_{ν}) - \frac{I_{h}^{*}}{I_{h}} ((β_{ν} S_{h} + β_{h}^{'} A_{h}) I_{ν} + (β_{h} S_{h} + β_{h}^{'} A_{h}) I_{h} - (μ_{h} + δ_{h}) I_{h} \\ - & (ϕ + τ) I_{h}) - \frac{I_{ν}^{*}}{I_{ν}} (b_{ν} S_{ν} I_{h} - μ_{ν} I_{ν}) - \frac{A_{h}^{*}}{A_{h}} (η Z S_{h} - β_{h}^{'} A_{h} I_{ν} - β_{h}^{'} A_{h} I_{h} - μ_{h} A_{h} - σ_{h} A_{h}) - \frac{R_{h}^{*}}{R_{h}} ((ϕ + τ) θ I_{h} - (μ_{h} + γ_{h}) R_{h}) \\ - & \frac{Z^{*}}{Z} (p I_{h} - q Z) \\ \leq & - μ_{h} N_{h} - μ_{ν} N_{ν} - I_{h} (δ_{h} + p) - q Z - \frac{S_{h}^{*}}{S_{h}} (- β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} \\ + & (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h}) - \frac{S_{ν}^{*}}{S_{ν}} (- b_{ν} S_{ν} I_{h} - μ_{ν} S_{ν}) - \frac{I_{h}^{*}}{I_{h}} ((β_{ν} S_{h} + β_{h}^{'} A_{h}) I_{ν} + (β_{h} S_{h} + β_{h}^{'} A_{h}) I_{h} - (μ_{h} + δ_{h}) I_{h} \\ - & (ϕ + τ) I_{h}) - \frac{I_{ν}^{*}}{I_{ν}} (b_{ν} S_{ν} I_{h} - μ_{ν} I_{ν}) - \frac{A_{h}^{*}}{A_{h}} (η Z S_{h} - β_{h}^{'} A_{h} I_{ν} - β_{h}^{'} A_{h} I_{h} - μ_{h} A_{h} - σ_{h} A_{h}) - \frac{R_{h}^{*}}{R_{h}} ((ϕ + τ) θ I_{h} - (μ_{h} + γ_{h}) R_{h}) \\ - & \frac{Z^{*}}{Z} (p I_{h} - q Z) \end{array}

Hence we observe that ${}_{0}^{A B C}D_{t}^{α} L \leq 0$ when R₀ > 1. Moreover ${}_{0}^{A B C}D_{t}^{α} L = 0$ if $S_{h} = S_{h}^{*}$ , $S_{ν} = S_{ν}^{*}$ , $I_{h} = I_{h}^{*}$ , $I_{ν} = I_{ν}^{*}$ , $A_{h} = A_{h}^{*}$ , $R_{h} = R_{h}^{*}$ , $Z = Z^{*}$ . Therefore, the system is globally asymptotically stable at the endemic equilibrium.

11. Parameter estimations and numerical simulations

11.1. Bayesian parameter estimation, data calibration, and public health implications

To ensure that the proposed fractional-order Zika virus transmission model reflects real outbreak dynamics, reported epidemiological data from the 2016 Zika outbreak in Jujuy, Argentina were used for calibration and validation. Figure 6 shows the observed daily confirmed Zika cases over the study period. The epidemic curve exhibits a clear initial growth phase followed by sustained transmission with noticeable fluctuations, which may be attributed to reporting variability, behavioral changes, and vector population dynamics. These data form the empirical basis for estimating unknown model parameters and assessing model performance. (Table 1)

Figure 6.

Observed daily confirmed Zika cases during the 2016 outbreak in Jujuy, Argentina.

Table 1.

Estimated and inferred parameter values for the Zika virus model, calibrated using data from the 2016 Zika outbreak in Argentina and related literature.

Parameter	Value	Unit	Source & Page
β _ν	0.021	day^-1	⁴⁰
$β_{ν}^{'}$	0.0084	day⁻¹	³⁷
β _h	0.0017	day^-1	³⁷
$β_{h}^{'}$	0.00068	day^-1	³⁷
b _ν	0.00018	day^-1	⁴⁰
Λ_h	0.034	people/day	⁴¹
Λ_ν	48	mosquitoes/day	³⁷
μ _h	$\frac{1}{76.3 \times 365}$	day^-1	⁴¹
μ _ν	$\frac{1}{13.2}$	day^-1	⁴²
η	0.082	day^-1	⁴³
ϕ	0.143	day^-1	⁴⁴
τ	0.25	day^-1	³⁷
ξ	0.28	Unitless	³⁷
γ _h	0.014	day^-1	⁴²
δ _h	0.00015	day^-1	⁴²
θ	0.55	Unitless	³⁷
p	0.008	Unitless	⁴⁵
q	0.042	Unitless	⁴⁶

A Bayesian Markov Chain Monte Carlo (MCMC) approach was employed to estimate key epidemiological and behavioral parameters of the model. This framework allows the incorporation of prior biological knowledge together with outbreak data, yielding posterior parameter distributions rather than single point estimates. The posterior distributions of selected parameters are illustrated in Figure 7. The results demonstrate well-defined, unimodal posterior densities with relatively narrow 95% highest density intervals, indicating good identifiability and convergence of the estimation procedure. Parameters related to transmission, recovery, treatment, and awareness effects show clear posterior means, providing quantitative evidence of the roles played by vector transmission intensity and public awareness during the Argentine outbreak. In particular, the posterior distribution of the basic reproduction number confirms that transmission potential exceeded the epidemic threshold, while awareness-related parameters indicate a measurable reduction in effective transmission due to behavioral responses.

Figure 7.

Posterior distributions of selected model parameters obtained via MCMC estimation. Red dashed lines denote posterior means, and shaded regions represent 95% highest density intervals.

The predictive capability of the calibrated model is assessed in Figure 8, which compares the observed daily cases with the model-generated epidemic trajectory. The mean model prediction closely follows the observed outbreak pattern, especially during the exponential growth and mid-epidemic phases. Importantly, most observed data points lie within the 95% confidence interval of the model simulations, demonstrating that the estimated parameters allow the model to reproduce the temporal evolution of the Zika outbreak in Argentina with acceptable accuracy. The gradual widening of the uncertainty band at later times reflects inherent uncertainties in long-term epidemic forecasting, including underreporting and stochastic effects.

Figure 8.

Model fit to the 2016 Zika outbreak data in Argentina. The solid line indicates the mean model prediction, the shaded region represents the 95% confidence interval, and dots correspond to observed daily cases.

From a practical standpoint, these results provide valuable insights for public health planning in Argentina. The estimated transmission and recovery parameters enable identification of critical intervention targets, such as reducing mosquito–human contact rates and enhancing early diagnosis and treatment. The quantified impact of awareness-related parameters highlights the effectiveness of information campaigns and behavioral interventions in mitigating disease spread. Moreover, the uncertainty-aware predictions produced by the Bayesian framework support evidence-based decision-making by allowing authorities to evaluate best- and worst-case outbreak scenarios. Overall, the integration of real outbreak data, Bayesian parameter estimation, and fractional-order dynamics enhances the reliability of the model and strengthens its applicability as a decision-support tool for Zika virus control and preparedness in Argentina.

11.2. Numerical simulations

11.2.1. Numerical solution by Adams–Moulton method

Simulations based on Adams-Moulton numerical approximations forecast the pandemic’s trajectory and its susceptibility to control measures. Now consider our model (8):

{}_{0}^{A B C}D_{t}^{α} S_{h} (t) = Λ_{h} - β_{ν} S_{h} I_{ν} - β_{h} S_{h} I_{h} - η Z S_{h} - μ_{h} S_{h} + σ_{h} A_{h} + (ϕ + τ) (1 - θ) I_{h} + γ_{h} R_{h}

Adams Moulton rule on [t_j, t_j+1] of a function g is defined as:

{}_{0}^{A B C}D_{t}^{α} [g (t_{j + 1})] = \frac{1 - α}{M (α)} [\frac{g (t_{j + 1}) - g (t_{j})}{2}] + \frac{α}{M (α) Γ (α)} \sum_{i = 0}^{\infty} {(i + 1)}^{1 - α} - {(i)}^{1 - α} [\frac{g (t_{j + 1}) - g (t_{j})}{2}]

Then we get,

\begin{array}{l} S_{h} (t_{j + 1}) - S_{h} (t_{j}) & = S_{h} (0) + \frac{1 - α}{M (α)} [Λ_{h} - β_{ν} (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) (\frac{I_{ν} (t_{j + 1}) - I_{ν} (t_{j})}{2}) - β_{h} (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) \\ - η (\frac{Z (t_{j + 1}) - Z (t_{j})}{2}) (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) - μ_{h} (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) + σ_{h} (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) \\ + (ϕ + τ) (1 - θ) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) + γ_{h} (\frac{R_{h} (t_{j + 1}) - R_{h} (t_{j})}{2})] + \frac{α}{M (α) Γ (α)} \sum_{i = 0}^{\infty} {(i + 1)}^{1 - α} - {(i)}^{1 - α} \\ [Λ_{h} - β_{ν} (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) (\frac{I_{ν} (t_{j + 1}) - I_{ν} (t_{j})}{2}) - β_{h} (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) \\ - η (\frac{Z (t_{j + 1}) - Z (t_{j})}{2}) (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) - μ_{h} (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) + σ_{h} (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) \\ + (ϕ + τ) (1 - θ) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) + γ_{h} (\frac{R_{h} (t_{j + 1}) - R_{h} (t_{j})}{2})] \end{array}

Similarly for all the other compartments we get,

\begin{array}{l} A_{h} (t_{j + 1}) - A_{h} (t_{j}) & = A_{h} (0) + \frac{1 - α}{M (α)} [η (\frac{Z (t_{j + 1}) - Z (t_{j})}{2}) (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) - β_{h}^{'} (\frac{A_{h} (t_{j + 1}) - A_{h} (t_{j})}{2}) (\frac{I_{ν} (t_{j + 1}) - I_{ν} (t_{j})}{2}) \\ - β_{h}^{'} (\frac{A_{h} (t_{j + 1}) - A_{h} (t_{j})}{2}) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) - μ_{h} (\frac{A_{h} (t_{j + 1}) - A_{h} (t_{j})}{2}) - σ_{h} (\frac{A_{h} (t_{j + 1}) - A_{h} (t_{j})}{2})] \\ + \frac{α}{M (α) Γ (α)} \sum_{i = 0}^{\infty} {(i + 1)}^{1 - α} - {(i)}^{1 - α} [η (\frac{Z (t_{j + 1}) - Z (t_{j})}{2}) (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) \\ - β_{h}^{'} (\frac{A_{h} (t_{j + 1}) - A_{h} (t_{j})}{2}) (\frac{I_{ν} (t_{j + 1}) - I_{ν} (t_{j})}{2}) - β_{h}^{'} (\frac{A_{h} (t_{j + 1}) - A_{h} (t_{j})}{2}) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) \\ - μ_{h} (\frac{A_{h} (t_{j + 1}) - A_{h} (t_{j})}{2}) - σ_{h} (\frac{A_{h} (t_{j + 1}) - A_{h} (t_{j})}{2})] \\ I_{h} (t_{j + 1}) - I_{h} (t_{j}) & = I_{h} (0) + \frac{1 - α}{M (α)} [(β_{ν} (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) + β_{h}^{'} (\frac{A_{h} (t_{j + 1}) - A_{h} (t_{j})}{2})) (\frac{I_{ν} (t_{j + 1}) - I_{ν} (t_{j})}{2}) \\ + (β_{h} (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) + β_{h}^{'} (\frac{A_{h} (t_{j + 1}) - A_{h} (t_{j})}{2})) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) - (μ_{h} + δ_{h}) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) \\ - (ϕ + τ) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2})] + \frac{α}{M (α) Γ (α)} \sum_{i = 0}^{\infty} {(i + 1)}^{1 - α} - {(i)}^{1 - α} [(β_{ν} (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) \\ + β_{h}^{'} (\frac{A_{h} (t_{j + 1}) - A_{h} (t_{j})}{2})) (\frac{I_{ν} (t_{j + 1}) - I_{ν} (t_{j})}{2}) + (β_{h} (\frac{S_{h} (t_{j + 1}) - S_{h} (t_{j})}{2}) \\ + β_{h}^{'} (\frac{A_{h} (t_{j + 1}) - A_{h} (t_{j})}{2})) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) - (μ_{h} + δ_{h}) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) - (ϕ + τ) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2})] \\ R_{h} (t_{j + 1}) - R_{h} (t_{j}) = & R_{h} (0) + \frac{1 - α}{M (α)} [(ϕ + τ) θ (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) - (μ_{h} + γ_{h}) (\frac{R_{h} (t_{j + 1}) - R_{h} (t_{j})}{2})] \\ + \frac{α}{M (α) Γ (α)} \sum_{i = 0}^{\infty} {(i + 1)}^{1 - α} - {(i)}^{1 - α} [(ϕ + τ) θ (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) - (μ_{h} + γ_{h}) (\frac{R_{h} (t_{j + 1}) - R_{h} (t_{j})}{2})] \\ Z (t_{j + 1}) - Z (t_{j}) = & Z (0) + \frac{1 - α}{M (α)} [p (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) - q (\frac{Z (t_{j + 1}) - Z (t_{j})}{2})] \\ + \frac{α}{M (α) Γ (α)} \sum_{i = 0}^{\infty} {(i + 1)}^{1 - α} - {(i)}^{1 - α} [p (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) - q (\frac{Z (t_{j + 1}) - Z (t_{j})}{2})] \\ S_{ν} (t_{j + 1}) - S_{ν} (t_{j}) = & S_{ν} (0) + \frac{1 - α}{M (α)} [Λ_{ν} - b_{ν} (\frac{S_{ν} (t_{j + 1}) - S_{ν} (t_{j})}{2}) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) - μ_{ν} (\frac{S_{ν} (t_{j + 1}) - S_{ν} (t_{j})}{2})] \\ + \frac{α}{M (α) Γ (α)} \sum_{i = 0}^{\infty} {(i + 1)}^{1 - α} - {(i)}^{1 - α} [Λ_{ν} - b_{ν} (\frac{S_{ν} (t_{j + 1}) - S_{ν} (t_{j})}{2}) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) - μ_{ν} (\frac{S_{ν} (t_{j + 1}) - S_{ν} (t_{j})}{2})] \\ I_{ν} (t_{j + 1}) - I_{ν} (t_{j}) = & I_{ν} (0) + \frac{1 - α}{M (α)} [b_{ν} (\frac{S_{ν} (t_{j + 1}) - S_{ν} (t_{j})}{2}) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) - μ_{ν} (\frac{I_{ν} (t_{j + 1}) - I_{ν} (t_{j})}{2})] \\ + \frac{α}{M (α) Γ (α)} \sum_{i = 0}^{\infty} {(i + 1)}^{1 - α} - {(i)}^{1 - α} [b_{ν} (\frac{S_{ν} (t_{j + 1}) - S_{ν} (t_{j})}{2}) (\frac{I_{h} (t_{j + 1}) - I_{h} (t_{j})}{2}) - μ_{ν} (\frac{I_{ν} (t_{j + 1}) - I_{ν} (t_{j})}{2})] \end{array}

The presented numerical scheme provides a highly accurate approximation of the model’s dynamics.

The Figures 9 and 10 illustrate the numerical simulation of the proposed fractional-order epidemic model using the Adams–Moulton method. In the human population dynamics, the susceptible population $S_{h}$ rapidly stabilizes close to its initial size, while the exposed $A_{h}$ and infected $I_{h}$ classes quickly decline toward zero, indicating that the infection does not persist for long. A small fraction transitions into the recovered class $R_{h}$ , but this also decays with time due to natural mortality. For the vector population, the susceptible vectors $S_{ν}$ initially decrease sharply but eventually stabilize, while both exposed and infected vectors diminish to near zero, reflecting a gradual disease fade-out among vectors. The comparative plots further highlight that both infected humans and vectors vanish within a short period, and the exposed classes also disappear over time. The total population plots show that the human population remains approximately constant, while the vector population decreases before reaching equilibrium. Finally, the prevalence plots confirm that both human and vector infection prevalence decline rapidly to zero, demonstrating that under the chosen parameter set, the disease dies out and the system tends toward a disease-free equilibrium.

Figure 9.

Adams Moulton simulation model.

Figure 10.

Adams Moulton simulation model.

The numerical simulation, performed using the Adams–Moulton method for the fractional-order epidemic model, reveals the complex dynamics of the disease across human and vector populations. For the human population (Figure 4, Top Left), the susceptible class ( $S_{h}$ ) quickly stabilizes, while the infected class ( $I_{h}$ ) initially rises before settling at a non-zero equilibrium (approximately 200 individuals). This persistent, non-zero level is confirmed by the Disease Prevalence plot (Figure 5, Bottom), which stabilizes at around 0.2 for humans and 0.55 for vectors, strongly suggesting the system converges toward an Endemic Equilibrium where the disease remains present but controlled, rather than disappearing entirely. These results offer significant utility: they validate the Adams–Moulton algorithm’s capability to model systems exhibiting “memory” (due to the fractional order α shown in Figure 4, Top Right), and critically, they provide public health strategists with the necessary time-course information–including the speed of stabilization and the maintenance level of the infection–to plan long-term control measures and allocate resources efficiently to manage this endemic state.

From Figure 11 the set of plots illustrates the temporal evolution of each compartment in the $A B C$ fractional epidemiological model for different fractional orders α. Across all compartments, the fractional order significantly affects the speed of convergence toward equilibrium, highlighting the system’s dependency on memory effects. Specifically, the susceptible human population ( $S_{h}$ ) shows an increasing trend, which is most pronounced for α = 1.0, indicating a slower rate of stabilization or movement towards a disease-free state when memory is minimal. Conversely, the susceptible vector population ( $S_{ν}$ ) shows an initial increase toward an endemic plateau, stabilizing fastest for lower α. Infected-related classes ( $I_{h}, I_{ν}$ ) and awareness-related compartments ( $A_{h}, Z$ ) demonstrate a clear inverse relationship with α: lower α values (α = 0.7) lead to the fastest decay toward a near-zero or low endemic level, while higher α values (α = 1.0) sustain higher infection and awareness levels for a longer duration. This is particularly visible in the recovered human class ( $R_{h}$ ), which peaks highest and decays fastest for α = 1.0, suggesting that the system’s memory, as modulated by the fractional order, determines both the intensity and duration of the disease dynamics, providing a more flexible and realistic framework compared to classical integer-order models.

Figure 11.

Numerical simulation of each functions for different values of alpha.

12. Conclusion

In this study, we developed and rigorously analyzed an Atangana–Baleanu–Caputo (ABC) fractional-order Zika virus transmission model that integrates vector-borne and sexual transmission routes, public awareness, and memory-dependent dynamics. By applying the Natural Transform Method (NTM), we derived an iterative analytical solution for the model, and we established the positivity, boundedness, existence, and uniqueness of the model solutions under biologically realistic conditions. Numerical simulations using the Natural Transform (NT) approach visually confirmed the impact of fractional-order dynamics on all seven compartments, demonstrating how decreasing α delays and attenuates infection peaks while prolonging the system’s transition to equilibrium. Key epidemiological and mathematical insights emerged from our analysis: the basic reproduction number $R_{0}$ was derived and shown to act as a precise threshold for disease persistence—when $R_{0} < 1$ , the disease-free equilibrium is globally stable, ensuring outbreak elimination; when $R 0 > 1$ , the endemic equilibrium becomes locally and globally stable, indicating sustained transmission. Furthermore, Partial Rank Correlation Coefficient (PRCC) analysis of $R 0$ identified the most sensitive parameters driving disease transmission, with mosquito-related parameters (βν, μν) and human recovery rates showing the greatest influence on outbreak potential. Additionally, awareness dissemination ( $Z$ ) and the aware class ( $A_{h}$ ) were found to significantly reduce infection rates, underscoring the importance of public health education. Parameter estimation via Markov Chain Monte Carlo (MCMC) provided data-driven, uncertainty-aware parameter values, enhancing the model’s empirical reliability. Numerical simulations using the Adams–Moulton method demonstrated that the system converges toward a stable Endemic Equilibrium, with the infected human population ( $I_{h}$ ) and prevalence stabilizing at non-zero levels (e.g., I_h ≈ 200), indicating that the disease persists under the analyzed parameter set. Crucially, the fractional order α modulates outbreak dynamics: lower values of α (stronger memory effects) led to faster convergence to the Endemic Equilibrium, while higher α values sustained higher infection and awareness levels for longer, highlighting how historical transmission patterns influence future spread. Overall, this work provides a rigorous, flexible, and applicable mathematical framework for understanding and predicting Zika virus dynamics. The integration of ABC fractional calculus, stability analysis, and statistical parameter estimation offers a powerful tool for public health strategists to design, evaluate, and optimize intervention policies that account for the complex, memory-dependent nature of infectious disease spread.

Looking ahead, several important research directions can build on this work. Future studies should test the model using real-world Zika outbreak data from different regions to improve its accuracy and practical use. Expanding the model to include different strains of the virus and how they interact would better reflect the evolving nature of Zika. Adding geographical and human movement patterns would help show how outbreaks spread between cities and countries. Creating a version of the model that accounts for randomness and uncertainty could make outbreak forecasts more reliable. It would also be valuable to use the model to identify the most effective and cost-efficient mix of control measures—like mosquito control, public awareness, and treatment timing. Combining the model with computer simulations and artificial intelligence could enable faster, real-time outbreak predictions. Finally, exploring how public behavior changes over time—such as warning fatigue or social media effects—and comparing different mathematical ”memory” approaches would make the model even more realistic and useful for managing future outbreaks.

Supplemental material

Supplemental material - Mathematical analysis and simulation of an ABC fractional Zika virus model with natural transform and stability assessment

Supplemental material for Mathematical analysis and simulation of an ABC fractional Zika virus model with natural transform and stability assessment by Jerin Mariantony Michael, Arul Joseph Gnanaprakasam, Salem Alkhalaf, Salah Boulaaras, Asma Alharbi in Science Progress

Supplemental material

Supplemental material - Mathematical analysis and simulation of an ABC fractional Zika virus model with natural transform and stability assessment

Footnotes

Acknowlegements

The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2026).

ORCID iDs

Salah Boulaaras

Arul Joseph Gnanaprakasam

Ethical considerations

No experiments were performed for Ethical approval.

Authors contribution

Jerin Mariantony Michael: Writing – original draft, Validation, Methodology, Investigation.

Arul Joseph Gnanaprakasam: Writing – review & editing, Validation, Methodology, Supervision, Conceptualization, Investigation, Resources.

Salah Boulaaras: Conceptualization, Funding, Project administration, Resources, Supervision, Writing– review & editing.

Asma Alharbi: Data curation, Formal analysis, Supervision, Funding, review & editing.

Salem Alkhalaf:Software, Validation, Supervision, Visualization, Funding, review & editing.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support QU-APC-2026.

Declaration of conflicting interests

The authors declare no conflict of interest.

Data Availability Statement

No data was used for the research described in the article.*

Supplemental material

Supplemental material for this article is available online

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Supplementary Material

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Mathematical analysis and simulation of an ABC fractional Zika virus model with natural transform and stability assessment

Abstract

Keywords

1. Introduction

2. Preliminaries

3. The AB fractional order in Caputo sense system of DEs

4. A B C with natural transform approach

4.1. Numerical analysis

4.2. Numerical results

5. Positivity and boundedness

6. Equilibria

7. Basic reproduction number

7.1. Sensitivity analysis of R0

7.1.1. Sensitivity index for β h

7.1.2. Sensitivity index for β ν

7.1.3. Sensitivity Index for b ν

8. Existence and uniqueness of fractional A B C model

9. Local stability of Equilibria

10. Global stability of Equilibria

11. Parameter estimations and numerical simulations

11.1. Bayesian parameter estimation, data calibration, and public health implications

11.2. Numerical simulations

11.2.1. Numerical solution by Adams–Moulton method

12. Conclusion

Supplemental material

Supplemental material - Mathematical analysis and simulation of an ABC fractional Zika virus model with natural transform and stability assessment

Supplemental material

Supplemental material - Mathematical analysis and simulation of an ABC fractional Zika virus model with natural transform and stability assessment

Footnotes

Acknowlegements

ORCID iDs

Ethical considerations

Authors contribution

Funding

Declaration of conflicting interests

Data Availability Statement

Supplemental material

References

Supplementary Material

4. $A B C$ with natural transform approach

7.1. Sensitivity analysis of R₀

7.1.1. Sensitivity index for β_h

7.1.2. Sensitivity index for β_ν

7.1.3. Sensitivity Index for b_ν

8. Existence and uniqueness of fractional $A B C$ model