A case study of fractional-order varicella virus model to nonlinear dynamics strategy for control and prevalence

Kottakkaran Sooppy Nisar; Muhammad Farman; Manal Ghannam; Evren Hincal; Aceng Sambas

doi:10.1515/nleng-2024-0072

Article Open Access

A case study of fractional-order varicella virus model to nonlinear dynamics strategy for control and prevalence

Kottakkaran Sooppy Nisar , Muhammad Farman , Manal Ghannam , Evren Hincal and Aceng Sambas

Published/Copyright: March 28, 2025

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From the journal Nonlinear Engineering Volume 14 Issue 1

Abstract

The purpose of this work is to construct and evaluate a dynamical susceptible–vaccinated–infected–recovered model for the propagation of the varicella virus in Jordan using existing epidemiological data. We use the fractal–fractional derivative in the Caputo sense to investigate the dynamical aspects of the suggested model. We investigate the model’s equilibria and evaluate the threshold parameter known as the reproductive number. A sensitivity analysis is also performed to detect the uncertainty of infection. Fixed point theorems and Arzela–Ascoli concepts are used to prove positivity, boundedness, existence, and uniqueness. The stability of the fractal–fractional model is examined in terms of Ulam–Hyers and generalized Ulam–Hyers types. Finally, using a two-step Newton polynomial technique, numerical simulations of the effects of various parameters on infection are used to explore the impact of the fractional operator on different conditions and population data. Chaos analysis and error analysis revealed the accuracy and precessions of solutions in the viable range. Several findings have been discussed by considering various fractal dimensions and arbitrary order. Overall, this study advances our understanding of disease progression and recurrence by establishing a mathematical model that can be used to replicate and evaluate varicella virus model behavior.

Keywords: varicella–zoster virus; fractal–fractional operator; qualitative analysis; Ulam–Hyers stability; error analysis

1 Introduction

The Varicella–Zoster virus (VZV), also called human herpesvirus, is a common alphaherpesvirus that infects people on its own. It mostly targets T lymphocytes, epithelial cells, and ganglia [1]. It has two different diseases: herpes zoster (shingles), a localized cutaneous manifestation led by neuralgic pain that is more frequent in elderly people, and varicella (chickenpox), a contagious severe illness that affects young children [2]. Primary varicella is a systemic rash that resembles a fever that mostly affects the trunk and face and goes away in 7–10 days. Serious illness and visceral invasion may result from it, and adverse effects such as encephalitis, pneumonitis, pancreatitis, and hepatitis are more likely in adults and small children. Hospitalization and mortality rates have decreased as a result of routine childhood vaccinations [3]. With peak occurrence in the winter and spring or during chilly, dry seasons, varicella exhibits a significant seasonal trend. Hospitals, daycare centers, schools, camps for refugees, and prisons frequently have outbreaks. Adults, children, and immune-deficient people are more likely to experience serious complications and mortality; in about 1% of afflicted pregnancies, the newborn has extensive congenital disorders [4]. Childhood immunization against varicella has had a major influence on epidemiology; since 1995, hospitalizations and deaths in the US have decreased by 95%. The demise of yearly epidemics and other collateral vaccination outcomes such as population immunity are accountable for this [5]. The incidence of varicella varies significantly from 13 to 16 cases per 1,000 persons each year. Over 90% of those infected before puberty in temperate climates had the highest incidence in preschool-aged children or early elementary school. The incidence is higher in age and has a higher percentage of adult cases in tropical settings. Varicella characteristics or exposure risk factors could be the cause of variances in varicella epidemiology [6]. Compared to other infectious diseases, including measles, pertussis, rotavirus, or invasive pneumococcal infection, varicella has lower specific fatality rates worldwide. Prior to vaccination, the death rate from varicella was 3 cases per 100,000 in high-income nations; in 2010, that number was 0.1 instances per 100,000. In comparison, the corresponding rates for measles and pertussis were 1.7 and 1.1. Both the monovalent and combination universal varicella vaccines are safe and effective at lowering the disease’s incidence, complications, and fatalities [7]. Mathematical approaches for analyzing communicable diseases have sparked a divide between mathematicians seeking comprehensive understanding and public health practitioners seeking practical disease management strategies. Mathematical modeling can assess the efficacy of vaccination campaigns, particularly in the complex dynamics of VZV transmission and the interplay between direct and indirect vaccine effects, with several models examining varicella immunization effects [8–10].

Because of memory and genetic characteristics, fractional calculus uses differentiation and integration with fractional order, which is more advantageous than standard integer order when describing real-world issues and modeling phenomena. As demonstrated by previous studies [11–16], fractional differential equations are crucial in a variety of fields, including engineering, physics, biology, and biomedical processes, since they may be used to represent real-world scenarios and have links to memory-based systems [11–16]. In order to overcome the constraints of ordinary differential equations, fractional mathematical models are being employed to properly and effectively explain biological processes. These models are well established in the study of infectious disease dynamics and provide a potent means of integrating biological systems’ memory and genetic features. Using fractional calculus, researchers have significantly advanced the study of infectious disease dynamics [17–26]. In order to solve fractional susceptible-infected-recovered models with unknown parameters, Kumar et al. [27] employed Bernstein wavelets. They combined matrices with collocation and evaluated the accuracy and application of the solutions using the Adams–Bashforth predictor corrector method. Using actual statistical data from tuberculosis (TB) cases in Yemen between 2000 and 2019 and comparing it to other nations, a new fractional TB model was created in the study by Shatanawi et al. [28]. An in-depth understanding of the spread of TB in the area was made possible by the model’s solutions, which were produced by graphical simulations and Adams–Bashforth iterative approaches. Using the fractional derivative with the Mittag–Leffler rule, Alderremy et al. [29] examined a fractional mathematical model of COVID-19 in a fuzzy environment. It shows how the model changes with various fractional exponents and how the contact rate affects infected individuals. Maayah et al. [30] used the Atangana–Baleanu–Caputo derivative and replicating kernel approach to study geometric attractors and numerical solutions for a fractional model of cancer immunity. In addition to presenting the visualization theory of solutions and associated findings, they also covered Hilbert spaces. A trustworthy analytical method for creating numerical solutions for fractional Lienard’s model with nonhomogeneous starting points has been devised by researchers [31]. This method, known as the replicating kernel Hilbert space method, produces a fractional solution with a fast converging form by optimizing numerical solutions utilizing the Fourier approximation principle. Researchers have shown interest in fractal–fractional differential and integral operators because of their unique capacity to recover classical fractional differential operators when the fractional-order is 1 and when the fractal dimension is set to 1 [32]. These differential operators can be regarded within fractal geometry as fractional differential operators that, in comparison with their comparable fractional derivatives, recreate more complex physical phenomena. A mathematical model for Alzheimer’s disease was created by Yadav et al. [33] using the Caputo sense fractal fractional operator. They proved that solutions existed and were unique by applying Banach fixed-point techniques and the Leray–Schaefer theorem. Using a fractional Adams–Bashforth method that required a two-step Lagrange polynomial, the model was numerically simulated. Outcomes were superior to a regular integer order with the new fractal fractional operator. Employing the fractal–fractional-order derivative in the Caputo sense, Shah et al. [34] investigated multiple sclerosis. They analyzed stability using the Hyers–Ulam idea, illustrated approximate solutions of different compartments numerically using the modified Euler technique, and applied the fixed point theorems of Banach and Krasnoselskii for existence theory results. To analyze the monkeypox outbreak, Kubra et al. [35] employed an Atangana–Baleanu fractal–fractional derivative. They determined feasible zones, computed reproduction numbers, and equilibrium points. Banach fixed-point theory and Picard’s sequential approximation technique were used to demonstrate the existence and stability of the model. The stability of endemic and disease-free equilibrium states as well as the consistency of the obsessive solution was also investigated in their study. Numerous studies [36–38] also looked at the mathematical depiction of the VZV using fractional differential operators. The Caputo operator is the best example of the short memory concept, which takes into account constant variable-order changes [39,40]. It employs a single kernel. Fractal–fractional derivatives are nonsingular and nonlocal, outperforming classical and fractional derivatives. They allow for the simultaneous study of the fractional operator and the fractal dimension, which has sparked considerable attention among researchers [41,42] due to its advanced characteristics.

The aim of this study is to use available epidemiological data to develop and assess a fractional-order susceptible–vaccinated–infected–recovered (SVIR) model for the varicella virus’s spread in Jordan. Between 2008 and 2021, 82,856 cases of the varicella virus were reported in Jordan. Due to the COVID-19 epidemic and the ensuing quarantine in Jordan, particularly as a result of schools and hospitals closing, a minimum of instances were recorded in 2020. The government will benefit from the model’s major parameters as it builds new policies and strategies for future cases. This research is necessary to address the continually increasing incidence of infections. With a power law kernel, the fractal–fractional derivative is a novel family of fractional derivatives that can be applied to practical issues. In emergent quantitative aspects of biodiversity, power laws characterize empirical scaling connections that enable extrapolation and prediction across a broad range of scales, producing patterns that resemble fractals or self-similarity. Considering a shortage of research on modeling varicella disease using fractal–fractional derivatives, we created a new model and employed a power law kernel in conjunction with fractal–fractional derivatives due to its distinct characteristics. The following are some illustrations of the remaining sections of this research: Section 2 provides a generalized model and a synopsis of the suggested model’s presentation. Additionally, a theoretical analysis of the proposed fractional operator is carried out. The proposed system’s qualitative investigation is covered in Section 3. Section 4 provides the numerical solutions for the proposed fractional-order model with power law kernel. The computer simulations, findings, and prospective perspectives are presented in Sections 5 and 6 (Table 1).

Table 1

Jordan’s experience with the varicella virus between 2008 and 2021 [44]

Year	Varicella virus-infected cases	Year	Varicella virus-infected cases
2008	11,356	2015	4,715
2009	6,906	2016	4,074
2010	9,362	2017	6,880
2011	6,181	2018	6,211
2012	6,435	2019	3,515
2013	6,706	2020	999
2014	7,888	2021	1,628

2 Materials and methods

2.1 Model description

Recent years have seen a progressive increase in the prevalence of chickenpox in Jordan, highlighting the need for a deeper comprehension of its epidemiology. There are four compartments in the model:

S ( t ) : susceptible individuals that are prone to VZV;
V ( t ) : vaccinated individuals that received vaccinations, giving them a lifetime immunity to the infection;
I ( t ) : infected individuals that carry the virus and is able to spread it to others;
R ( t ) : recovered populations that contracted the virus and recovered with a lifetime immunity to it.

The varicella virus in Jordan’s SVIR model theoretical framework provides a conceptual foundation for comprehending the dynamics of virus transmission as well as the variables and factors affecting it. The total population, which includes all compartments, is denoted by N ( t ) :

N ( t ) = S ( t ) + V ( t ) + I ( t ) + R ( t ) .

2.2 Assumptions

We have to make a number of assumptions in order to build our SVIR model, whereas these assumptions can be crucial to our research. These are as follows:

The idea that people could differ from one another in ways that are crucial for the spread of infection is not taken into consideration by the model.
Even after receiving immunizations, susceptible individuals may still be susceptible due to vaccination failure, or if the vaccine provides strong protection, they may be able to move to the recovery compartment without contracting an infection.
Following infection and recovery, permanent immunity is given.
The rate of natural death and the birth rate are equal.
Exponentially distributed infection duration: an individual spreads the infection immediately afterward as it happens.
The likelihood of contracting the virus is unaffected by age or gender.
All of the model’s parameters are assumed to be positive.

2.3 Dynamical system

Using the mass action law of infectious illnesses, we can describe the varicella outbreak in Jordan with the following system of nonlinear differential equations:

(1) d S d t = η N − α IS − ( κ + η ) S , d V d t = κ S − β IV − ( δ + η ) V , d I d t = α IS + β IV − ( ρ + η ) I , d R d t = ρ I + δ V − η R .

Table 2 shows the descriptions for all of the parameters utilized in the model.

Table 2

Model’s parameter values [44]

Parameter	Description	Value
α	The transmission rate, or contact rate	0.24
κ	The vaccination rate of susceptible individuals	0.05
η	Natural mortality rate equivalent to birth rate	0.03
β	The rate of susceptibility among vaccinated persons due to vaccination failure	0.08
δ	The effectiveness rate of a vaccination in preventing infection	0.92
ρ	The recovery rate	1 14

2.4 Fractional-order model of varicella virus

Using the mass action law of infectious illnesses, when compared to integer-order techniques, fractional-order derivatives and integrals produce superior findings, encouraging the development of control theory to generalize existing research techniques and result interpretation. Unlike typical models that use integer dimensions and smooth curves to represent complicated patterns seen in biological processes, fractal–fractional models provide a novel and sophisticated way to explore intricate and self-repeating variations in a variety of systems. we can describe the varicella outbreak in Jordan with the following system of nonlinear differential equations. An auxiliary parameter, ϖ , is used to reduce the model’s dimension to a time − 1 dimension:

(2) 1 ϖ ω − 1 × D t ω , υ 0 FFP [ S ( t ) ] = η N − α IS − ( κ + η ) S , 1 ϖ ω − 1 × D t ω , υ 0 FFP [ V ( t ) ] = κ S − β IV − ( δ + η ) V , 1 ϖ ω − 1 × D t ω , υ 0 FFP [ I ( t ) ] = α IS + β IV − ( ρ + η ) I , 1 ϖ ω − 1 × D t ω , υ 0 FFP [ R ( t ) ] = ρ I + δ V − η R .

In the case of D ω , υ FFP , it denotes the generalized form for power law kernel (Caputo) derivative with 0 < υ ≤ 1 and order 0 < ω ≤ 1 fractal dimensions and fractal order, respectively. According to the proposed system, the given initial conditions are

(3) S ( 0 ) ≥ 0 , V ( 0 ) ≥ 0 , I ( 0 ) ≥ 0 , R ( 0 ) ≥ 0 .

2.5 Basic notions

This subsection contains significant findings for later sections.

Definition 2.1

[45] Let us assume that λ ( t ) is differentiable on the open interval (a,b). The fractal–fractional derivative of λ of order ω in the Caputo perceive with a power law kernel will be as follows if λ is fractal differentiable on (a and b) with order υ :

(4) D t ω , υ 0 FFP [ λ ( t ) ] = 1 Γ ( l − ω ) d d t υ ∫ 0 t ( t − ς ) l − ω − 1 λ ( ς ) d ς , l − 1 < ω , υ ≤ l ∈ N .

The fractal derivative of λ ( t ) can be calculated as

(5) d λ ( t ) d t υ = lim t → υ λ ( t ) − λ ( ς ) t υ − ς υ .

Definition 2.2

[45] The expression for the fractal–fractional integral associated with (2.1) is given by

(6) I t ω , υ 0 FFP [ λ ( t ) ] = υ Γ ( ω ) ∫ 0 t ς υ − 1 ( t − ς ) ω − 1 λ ( ς ) d ς .

Lemma 2.3

[45] The fractal–fractional Cauchy problem for λ ∈ C ( [ 0 , T ] )

(7) D t ω , υ 0 FFP [ λ ( t ) ] = ϖ ( t , λ ( t ) ) , ω , υ ∈ ( 0 , 1 ] t ∈ [ 0 , T ] , λ ( 0 ) = λ 0 ,

has the solution

(8) λ ( t ) = λ ( 0 ) + υ Γ ( ω ) ∫ 0 t ς υ − 1 ( t − ς ) ω − 1 ϖ ( ς , λ ( ς ) ) d ς .

Remark 2.1

[46] The relation between the fractal and classical derivatives of a function, as determined by the implicit function theorem and chain rule, is as follows:

(9) d λ ( t ) d t υ = d λ ( t ) d t d t d t υ = d λ ( t ) d t 1 υ t υ − 1 ,

as long as the first-order derivative of λ ( t ) exists. Additionally, taking into account (2.3) and the differentiability aspect of the integral, the fractal–fractional differential equation’s right-hand side

(10) D t ω , υ 0 FFP [ λ ( t ) ] = ϖ ( t , λ ( t ) ) , t ∈ [ 0 , T ] ,

might additionally be stated as

(11) D t ω 0 FFP [ λ ( t ) ] = υ t υ − 1 ϖ ( t , λ ( t ) ) , t ∈ [ 0 , T ] .

Corollary 2.4

[47] With respect to λ ∈ C 1 ( a , b ) , the following relation holds:

(12) I t ω , υ 0 FFP [ D t ω 0 FFP λ ( t ) ] = λ ( t ) − λ ( 0 ) .

Remark 2.2

[45] The fractal–fractional initial value problem (7) can be reformulated after taking (11) into consideration.

(13) D t ω 0 FFP [ λ ( t ) ] = υ t υ − 1 ϖ ( t , λ ( t ) ) , ω , υ ∈ ( 0,1 ] t ∈ [ 0 , T ] , λ ( 0 ) = y 0 .

It is clear from both sides of (13) that using the fractal–fractional integral yields

(14) I t ω , υ 0 FFP [ D t ω 0 FFP λ ( t ) ] = I t ω , υ 0 FFP [ υ t υ − 1 ϖ ( t , λ ( t ) ) ] .

3 Model’s qualitative aspects

3.1 Non-negativity and boundedness

Let us define

(15) R + 4 = { ( S , V , I , R ) : S , V , I , R ≥ 0 } .

The following theorem is illustrated using a fractional comparison idea and a generalized mean value theorem [48].

Theorem 3.1

Let ( S ( 0 ) , V ( 0 ) , I ( 0 ) , R ( 0 ) ) represent the initial data of R + 4 , and ( S ( t ) , V ( t ) , I ( t ) , R ( t ) ) is the solution that contributes to the initial data. Then, the set R + 4 is positively invariant in model (2). Additionally, we have

(16) lim t → ∞ S ( t ) ≤ S ∞ ≔ η N α I ∞ κ + η , lim t → ∞ V ( t ) ≤ V ∞ ≔ η N ∞ β I ∞ + δ + η , lim t → ∞ I ( t ) ≤ I ∞ ≔ η N ∞ ρ + η − β I ∞ , lim t → ∞ R ( t ) ≤ R ∞ ≔ ρ I ∞ + δ V ∞ η .

Proof

We first prove that there is a unique solution for model (2) for the period ( 0 , ∞ ) . We have

(17) D t ω , υ 0 FFP [ S ( t ) ] ∣ S = 0 = η N ≥ 0 , D t ω , υ 0 FFP [ V ( t ) ] ∣ V = 0 = κ S ≥ 0 , D t ω , υ 0 FFP [ I ( t ) ] ∣ I = 0 = 0 , D t ω , υ 0 FFP [ R ( t ) ] ∣ R = 0 = ρ I + δ V ≥ 0 .

This demonstrates that the system outcome will stay in R + 4 for all times t ≥ 0 .

Now, using the system (2)’s first equation, we obtain

D t ω , υ 0 FFP [ S ( t ) ] = η N − α IS − ( κ + η ) S , ≤ η N − ( α I + κ + η ) S .

One can find

(18) lim t → ∞ S ( t ) ≤ S ∞ ≔ η N ∞ α I ∞ + κ + η ,

which implies the first estimate of (16). Next, we find

D t ω , υ 0 FFP [ S ( t ) + V ( t ) ] = η N − α IS − ( κ + η ) S + κ S − β IV − ( δ + η ) V , ≤ η N − ( β I + δ + η ) V

and

(19) lim t → ∞ ( S ( t ) + V ( t ) ) ≤ V ∞ ≔ η N ∞ β I ∞ + δ + η .

Likewise, we have

D t ω , υ 0 FFP [ S ( t ) + I ( t ) ] = η N − α IS − ( κ + η ) S + α IS + β IV − ( ρ + η ) I ≤ η N − ( ρ + η − β I ) V

and

(20) lim t → ∞ ( S ( t ) + I ( t ) ) ≤ I ∞ ≔ η N ∞ ρ + η − β I ∞ .

Finally, we obtain

D t ω , υ 0 FFP [ R ( t ) ] = ρ I + δ V − η R .

(21)□ lim t → ∞ R ( t ) ≤ R ∞ ≔ ρ I ∞ + δ V ∞ η .

3.2 Equilibrium points

Every compartment of system (2) can be rewritten as a ratio of the entire population as follows:

(22) P S ( t ) = S ( t ) N ; P V ( t ) = V ( t ) N ; P I ( t ) = I ( t ) N ; P R ( t ) = R ( t ) N ,

to obtain

(23) P N ( t ) = P S ( t ) + P V ( t ) + P I ( t ) + P R ( t ) = 1 .

The suggested system has two equilibrium points:

The disease-free equilibrium, E 0 , when P I = 0 ;
The endemic equilibrium, E * , when P I > 0 .

In the absence of infection, P I = P R = 0 , we find

(24) E 0 = { P S 0 , P V 0 , P I 0 , P R 0 } = η η + κ , η κ ( η + κ ) ( δ + η ) , 0 , 0 .

After simplifications of (2), we obtain for P S ,

(25) P S = η η + κ + α P I * ,

and, solving for P V , we find

(26) P V = η κ ( β P I * + δ + η ) ( η + κ + α P I * ) .

Now, substituting (35) and (36) into third equation of (2), we obtain

(27) η α η + κ + α P I * + η β κ ( β P I * + δ + η ) ( η + κ + α P I * ) − η − ρ = 0 ,

then, by putting Eq. (37) in the quadratic form, we obtain

(28) H 1 P I * 2 + H 2 P I * + H 3 ( 1 − q ) = 0 ,

where

H 1 = α β ( η + ρ ) > 0 , H 2 = ( β ( η + κ ) + α ( δ + η ) ) ( η + ρ ) − α β η > 0 , H 3 = ( η + ρ ) ( η + κ ) ( δ + η ) > 0 , q = α η ( η + ρ ) ( η + κ ) + η β κ ( η + ρ ) ( η + κ ) ( δ + η ) .

It is obvious that P I * > 0 , and q > 1 . Hence, the endemic equilibrium ( E * ) is

(29) E * = { P S * , P V * , P I * , P R * } = η η + κ + α P I * , η κ ( β P I * + δ + η ) ( η + κ + α P I * ) , P I * , 0 .

3.3 Reproductive number

To evaluate the stability of the system, the reproduction number is computed using the next-generation matrix technique [49]. Examine the system

(30) D t ω , υ 0 FFP [ I ( t ) ] = α IS + β IV − ( ρ + η ) I , D t ω , υ 0 FFP [ V ( t ) ] = κ S − β IV − ( δ + η ) V .

A Jacobian matrix is developed using the new infection aspects ( F ) and the remaining transferring terms ( V ) as follows:

F = α S + β V 0 0 0 , V = ρ + η 0 0 β P I * + δ + η ,

and

F ( E 0 ) = α P S 0 + β P V 0 0 0 0 and V − 1 = 1 ρ + η 0 0 1 β P I * + δ + η .

Then, det ∣ F ( E 0 ) V − 1 − λ I ∣ = 0 generates the reproductive number ( R 0 ) as:

(31) R 0 = α P S 0 + β P V 0 ρ + η = α η ( δ + η ) + η β κ ( η + ρ ) ( η + κ ) ( δ + η ) .

The basic reproductive number sheds light on how common the varicella virus is in a community and how it spreads. In the event that R 0 > 1 , the virus can begin to spread; if it is less than 1, it fails. Controlling the varicella virus’s spread in society becomes more difficult as the value of R 0 increases.

3.4 Sensitivity analysis

Sensitivity analysis is an important strategy for analyzing and handling complex systems because it helps establish control methods and direct future research by identifying the parameters that are most relevant in lowering disease levels. Sensitivity analysis finds important process factors and aids in determining a model’s stability, especially when dealing with uncertain data. One way to examine the sensitivity of R 0 is to compute its partial derivatives in relation to the pertinent parameters:

(32) ∂ R 0 ∂ α = η ( η + ρ ) ( η + κ ) > 0 , ∂ R 0 ∂ η = [ α ( δ + 2 η ) + β κ ] ( η + ρ ) ( η + κ ) ( δ + η ) − [ 3 η 2 + 2 η ( ρ + κ + δ ) + ρ δ + δ κ + ρ κ ] [ α η ( δ + η ) + η β κ ] [ ( η + ρ ) ( η + κ ) ( δ + η ) ] 2 > 0 , ∂ R 0 ∂ δ = − η β κ ( η + ρ ) ( η + κ ) ( δ + η ) 2 < 0 , ∂ R 0 ∂ β = η κ ( η + ρ ) ( η + κ ) ( δ + η ) > 0 , ∂ R 0 ∂ κ = − α η δ + η 2 ( α − β ) ( η + ρ ) ( η + κ ) 2 ( δ + η ) < 0 , ∂ R 0 ∂ ρ = − α η ( δ + η ) + η β κ ( η + ρ ) 2 ( η + κ ) ( δ + η ) < 0 .

It is clear that when we adjust the parameters, the value of R 0 is highly sensitive. The parameters α , η , and β indicate growth in our analysis, while the parameters δ , κ , and ρ show contraction. Using the parameter values in Eq. (32), we obtain values in Table 3.

Table 3

Sensitivity indices for each parameter in R 0

Parameter	Sensitivity index	Parameter	Sensitivity index
α	0.983	η	0.591
δ	− 0.017	β	0.020
ρ	− 0.704	κ	− 0.608

The previously described indices are helpful in identifying the factors that are critical in determining the ability for the infection to spread, as Figure 1 illustrates.

$Figure 1 Reproductive behavior of the proposed model: (a) α \alpha and κ \kappa , (b) α \alpha and η \eta , (c) α \alpha and β \beta , (d) α \alpha and ρ \rho , (e) κ \kappa and η \eta , (f) ρ \rho and η \eta , (g) η \eta and δ \delta , (h) δ \delta and β \beta , (i) ρ \rho and δ \delta .$

Figure 1

Reproductive behavior of the proposed model: (a) α and κ , (b) α and η , (c) α and β , (d) α and ρ , (e) κ and η , (f) ρ and η , (g) η and δ , (h) δ and β , (i) ρ and δ .

The basic reproduction number R 0 is shown in Figure 1(a)–(i) in terms of different parameters. Considering α and κ in Figure 1(a), R 0 is presented; the other coefficient values remain unchanged. As it can be seen, if α increases or κ decreases, R 0 grows. Similar behavior is shown in Figure 1(b), in which R 0 is generated in the context of parameters α and η . Next, as shown in Figure 1(c), R 0 is displayed in terms of parameters α and β , indicating that if either of them rise, R 0 grows. Subsequently, with ρ and ‘ α in Figure 1(d), R 0 is given; the remaining parameter values remain unchanged. R 0 rises when α rises or ρ lowers. Using parameters κ and η , R 0 is shown in Figure 1(e), indicating that R 0 increases as κ and η decrease. A comparable trend can be seen in Figure 1(f), where R 0 is simulated in terms of ρ and η . Figure 1(g) provides R 0 in terms of parameters η and δ , demonstrating that a decrease in the values of both of these parameters leads to an increase in R 0 . Figure 1(i) displays the same behavior of R 0 in terms of ρ and δ . Figure 1(h) shows R 0 in terms of δ and β . It shows that if β grows or δ drops, R 0 advances.

3.5 Solutions’ existence and uniqueness

This work investigates the fractal–fractional varicella outbreak model using a fixed point approach. Usually, this is accomplished by using normal nonlinear functional analysis results and by transforming the model into a fixed point problem that satisfies the same constraints as the original integral depiction. Suppose that M = [ 0 , T ] denotes this orientation. Let S , V , I , R ∈ C ( M , R ) and U ≔ C ( M , R 4 ) be a Banach space of continuous functions equipped with the norm:

(33) ‖ Z ( t ) ‖ = max t ∈ M ∣ Z ( t ) ∣

The following vector function is then defined:

(34) H ( t , Z ( t ) ) = H 1 ( t , S ( t ) ) H 2 ( t , V ( t ) ) H 3 ( t , I ( t ) ) H 4 ( t , R ( t ) ) = η N − φ IS − ( κ + η ) S κ S − β IV − ( δ + η ) V φ IS + β IV − ( ρ + η ) I ρ I + δ V − η R ,

where λ ( t ) = ( S ( t ) + V ( t ) + I ( t ) + R ( t ) ) T . Then, taking into account (13), the model under study (2) has the following abstract form:

(35) D t ω 0 FFP [ λ ( t ) ] = υ t υ − 1 H ( t , λ ( t ) ) , ω , υ ∈ ( 0,1 ] t ∈ [ 0 , T ] , λ ( 0 ) = λ 0 ≥ 0 ,

with the initial conditions being represented as a vector λ ( 0 ) = ( S ( 0 ) + V ( 0 ) + I ( 0 ) + R ( 0 ) ) T . Using Lemma (2.3) to apply the fractal–fractional integral to (35) yields the following nonlinear integral-type equation:

(36) λ ( t ) = λ ( 0 ) + υ Γ ( ω ) ∫ 0 t ς υ − 1 ( t − ς ) ω − 1 H ( ς , λ ( ς ) ) d ς .

Proving the same for the equivalent integral system (36) has clearly reduced the problem of establishing if system (2) has a unique solution. To do this, the self-map η : U → U is defined as follows:

(37) η [ λ ( t ) ] = λ ( 0 ) + υ Γ ( ω ) ∫ 0 t ς υ − 1 ( t − ς ) ω − 1 H ( ς , λ ( ς ) ) d ς .

Theorem 3.2

Assume that ϖ : M × R 6 → R is a continuous nonlinear function that satisfies the linear growth requirement:

(38) ∣ ϖ ( t , λ ( t ) ) ∣ ≤ ϒ ϖ ∣ λ ( t ) ∣ + ξ ϖ ,

t ∈ M and λ ∈ U , where ϒ ϖ and ξ ϖ are the positive constants. The integral equation (36), may then have a unique solution identified if

(39) υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) C ϖ < 1 ,

where β ( ω , υ ) represents the beta function. As a result, the suggested model (2) offers a unique solution.

Proof

Let the function η : U → U be defined in (37) and U has a closed convex subset called ℬ φ ≔ { λ ( t ) ∈ U : ‖ λ ‖ ≤ φ , φ > 0 } . The function ϖ is continuous, so η is continuous. The function ϖ satisfies condition (38), yielding

(40) ∣ η [ λ ( t ) ] ∣ = λ ( 0 ) + υ Γ ( ω ) ∫ 0 t ς υ − 1 ( t − ς ) ω − 1 ϖ ( ς , λ ( ς ) ) d ς , ≤ ∣ λ 0 ∣ + max t ∈ M υ Γ ( ω ) ∫ 0 t ς υ − 1 ( t − ς ) ω − 1 ∣ ϖ ( ς , λ ( ς ) ) ∣ d ς , ≤ ∣ λ 0 ∣ + υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) [ ϒ ϖ ∣ λ ( t ) ∣ + ξ ϖ ] , ≤ ∣ λ 0 ∣ + υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) [ ϒ ϖ φ + ξ ϖ ] , ≤ φ ,

∀ λ ∈ ℬ . We have

(41) φ > ∣ λ 0 ∣ + υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) ξ ϖ 1 − υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) ϒ ϖ .

Additionally, we have η B φ ⊂ B φ . This implies that U has uniform bounds for η .

Given t 1 < t 2 < T and λ ∈ U , we now have

(42) ∣ η [ λ ( t 1 ) ] − η [ λ ( t 2 ) ] ∣ = ∣ υ Γ ( ω ) ∫ 0 t 2 ς υ − 1 ( t 2 − ς ) ω − 1 ϖ ( ς , λ ( ς ) ) d ς − υ Γ ( ω ) ∫ 0 t 1 ς υ − 1 ( t 1 − ς ) ω − 1 ϖ ( ς , λ ( ς ) ) d ς ∣ , = ∣ υ Γ ( ω ) ∫ 0 t 1 ς υ − 1 { ( t 2 − ς ) ω − 1 − ( t 1 − ς ) ω − 1 } ϖ ( ς , λ ( ς ) ) d ς + υ Γ ( ω ) ∫ t 1 t 2 ς υ − 1 ( t 2 − ς ) ω − 1 ϖ ( ς , λ ( ς ) ) d ς ∣ , ≤ υ Γ ( ω ) [ ϒ ϖ ∣ λ ( t ) ∣ + ξ ϖ ] β ( ω , υ ) [ t 2 ω + υ − 1 − t 2 ω + υ − 1 ] .

When t 1 → t 2 , ∣ η [ λ ( t 2 ) ] − η [ λ ( t 2 ) ] ∣ → 0 . As a result, η is equicontinuous. Perfect continuity is obtained by guaranteeing the relative compactness of η by the Arzela–Ascoli theorem. It is implied by Schauder’s fixed theorem that there is a solution for the suggested model (2).□

Theorem 3.3

Let ϖ : M × R 4 → R be a continuous nonlinear function that fulfills the Lipschitz requirement:

(43) ∣ ϖ ( t , λ ¯ ( t ) ) − ϖ ( t , λ ˜ ( t ) ) ∣ ≤ Λ ϖ ∣ λ ¯ ( t ) − λ ˜ ( t ) ∣ ,

t ∈ M and λ ∈ U , where Λ ϖ is a positive constant. Then, the proposed model (2) has a unique solution if

(44) υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) Λ ϖ < 1 .

Proof

For λ ¯ , λ ˜ ∈ U and t ∈ M , we have

(45) ∥ η [ λ ¯ ( t ) ] − η [ λ ˜ ( t ) ] ∥ = max t ∈ M υ Γ ( ω ) ∫ 0 t ς υ − 1 ( t − ς ) ω − 1 × [ ϖ ( ς , λ ¯ ( ς ) ) − ϖ ( ς , λ ˜ ( ς ) ) ] d ς ∣ , ≤ υ Γ ( ω ) max t ∈ M ∫ 0 t ς υ − 1 ( t − ς ) ω − 1 ∣ ϖ ( ς , λ ¯ ( ς ) ) − ϖ ( ς , λ ˜ ( ς ) ) ∣ d ς , ≤ υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) Λ ϖ ∣ λ ¯ ( t ) − λ ˜ ( t ) ∣ .

⇒ η is a contraction map.

Therefore, the proposed system has a unique solution, according to the Banach contraction mapping theory.□

3.6 Ulam–Hyres and generalized Ulam–Hyres stability

The lack of assumptions makes stability analysis of nonlinear systems a challenging and complicated task. Systems can be unstable, which makes classification challenging for both basic and complicated systems. System designers are becoming more and more interested in novel approaches to the current research on nonlinear system stability analysis. Thus, one of the most important aspects of system design is making sure the system is stable. Ulam–Hyers stability and Ulam–Hyers–Rassias stability are two forms of stability that have drawn more attention from mathematicians. This has made studying fractional differential equations a crucial field of research. In consideration of the fractal–fractional starting value problem (35), Ulam–Hyres and generalized Ulam–Hyres stability-type results are discussed in this subsection. In this perspective, we assume that the following inequalities are true:

(46) ∣ D t ω , υ 0 FFP [ λ ¯ ( t ) ] − ϖ ( t , λ ¯ ( t ) ) ∣ ≤ ψ ,

(47) ∣ D t ω , υ 0 FFP [ λ ¯ ( t ) ] − ϖ ( t , λ ¯ ( t ) ) ∣ ≤ ξ ϖ ψ ,

t ∈ J , where ψ > 0 is a real number and ϖ : M → R + a continuous function with

ψ = max ( ψ 1 , ψ 2 , ψ 3 , ψ 4 )

and

ξ ϖ ≕ max ( ξ ϖ 1 , ξ ϖ 2 , ξ ϖ 3 , ξ ϖ 4 ) ,

respectively. In addition, we present the following definitions that are required and taken from the study of Rezapour et al. [50].

Definition 3.4

When Ω ϖ > 0 exists, the proposed model (35) is Ulam–Hyres stable. This implies that for some ψ > 0 and λ ¯ ∈ U satisfying (46), the system recognizes a unique solution λ ∈ U , such that

(48) ∣ λ ¯ ( t ) − λ ( t ) ∣ ≤ Ω ϖ ψ , ∀ t ∈ M ,

where Ω ϖ ≕ max ( Ω ϖ 1 , Ω ϖ 2 , Ω ϖ 3 , Ω ϖ 4 , Ω ϖ 5 , Ω ϖ 6 ) .

Definition 3.5

Suppose ∃ ξ ϖ ∈ C ( R + , R + ) , where ξ ϖ ( 0 ) = 0 . When Ω ϖ > 0 , the proposed model (35) is generalized Ulam–Hyres stable. This implies that for some ψ > 0 and λ ¯ ∈ U satisfying (47), the system recognizes a unique solution λ ∈ U , such that

(49) ∣ λ ¯ ( t ) − λ ( t ) ∣ ≤ ξ ϖ ( ψ ) , ∀ t ∈ M .

Remark 3.1

The function λ ¯ is believed to satisfy (46) if and only if h ( t ) ∈ C ( M , R + ) (which is dependent upon λ ¯ ), so that

(50) ( A ) . ∣ h ( t ) ∣ ≤ ψ ,

(51) ( B ) . D t ω , υ 0 FFP [ λ ¯ ( t ) ] = ϖ ( t , λ ¯ ( t ) ) + h ( t ) ,

∀ t ∈ M , where h ( t ) = max ( h 1 ( t ) , h 2 ( t ) , h 3 ( t ) , h 4 ( t ) ) T .

Theorem 3.6

If λ ¯ ∈ U is a solution of inequality (46), then λ ¯ for ω , υ ∈ ( 0 , 1 ] satisfies the subsequent integral inequality:

(52) ∣ λ ¯ ( t ) − λ ¯ ( 0 ) − I t ω , υ 0 FFP ϖ ( t , λ ¯ ( t ) ) ∣ ≤ υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) ψ .

Proof

Both Theorem (3.3) and Remark (3.1) assert that the perturbed problem

(53) D t ω 0 FFP [ λ ¯ ( t ) ] = ϖ ( t , λ ¯ ( t ) ) + h ( t ) , λ ¯ ( 0 ) = λ ( 0 ) ,

has a unique solution

(54) λ ¯ ( t ) = λ ¯ ( 0 ) + I t ω , υ 0 FFP ( ϖ ( t , λ ¯ ( t ) ) + h ( t ) ) .

Then, we have

(55)□ ∣ λ ¯ ( t ) − λ ¯ ( 0 ) − I t ω , υ 0 FFP ( ϖ ( t , λ ¯ ( t ) ) ) ∣ = ∣ h ( t ) ∣ , ≤ υ Γ ( ω ) ∫ 0 t ς υ − 1 ( t − ς ) ω − 1 ∣ h ( ς ) ∣ d ς , ≤ ψ υ Γ ( ω ) ∫ 0 t ς υ − 1 ( t − ς ) ω − 1 d ς , ≤ υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) ψ .

Theorem 3.7

The fractal–fractional initial value problem (35) is Ulam–Hyres stable on M , assuming (43) and (3.6).

(56) 1 − υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) Λ ϖ < 1 .

The proposed model (2) is hence Ulam–Hyres stable on M .

Proof

With λ ¯ ( 0 ) = λ ( 0 ) as the initial condition, let λ be a unique solution of (3.3). Let λ ¯ ∈ U satisfy inequality (46). Next, we derive Theorem (3.6) and Assumption (43) from which, we obtain

∣ λ ¯ ( t ) − λ ( t ) ∣ ≤ λ ¯ ( t ) − λ ¯ ( 0 ) − υ Γ ( ω ) × ∫ 0 t ς υ − 1 ( t − ς ) ω − 1 ϖ ( ς , λ ( ς ) ) d ς , ≤ λ ¯ ( t ) − λ ¯ ( 0 ) − υ Γ ( ω ) × ∫ 0 t ς υ − 1 ( t − ς ) ω − 1 ϖ ( ς , λ ¯ ( ς ) ) d ς + υ Γ ( ω ) ∫ 0 t ς υ − 1 ( t − ς ) ω − 1 ∣ ϖ ( ς , λ ¯ ( ς ) ) − ϖ ( ς ) , λ ( ς ) ∣ d ς ,

(57) ≤ υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) ψ + υ Γ ( ω ) × ∫ 0 t ( t − ς ) ω − 1 ∣ ϖ ( ς , λ ¯ ( ς ) ) − ϖ ( ς , λ ( ς ) ) ∣ d ς , ≤ υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) ψ + υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) Λ ϖ ∣ λ ¯ ( t ) − λ ( t ) ∣ .

(58) ⇒ ∣ λ ¯ ( t ) − λ ( t ) ∣ ≤ υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) 1 − υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) Λ ϖ ψ .

Now, assume that

Ω ϖ = υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) 1 − υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) Λ ϖ ,

with

1 − υ Γ ( ω ) T ω + υ − 1 β ( ω , υ ) Λ ϖ > 0 ,

then we have

(59) ∣ λ ¯ ( t ) − λ ( t ) ∣ ≤ Ω ϖ ψ .

Consequently, an Ulam–Hyres stable solution exists for the fractal–fractional initial value problem (35). It follows that the suggested model (2) is Ulam–Hyres stable.□

Theorem 3.8

The initial value problem (35) is generalized Ulam–Hyres stable on M if and only if ∃ ξ ϖ : R + → R + with Ω ϖ ( 0 ) = 0 , as per the conjectures of Theorem 3.7. As a result, on M , the suggested model (2) is generalized Ulam–Hyres stable.

Proof

Proceed by selecting ξ ϖ ( ψ ) = Ω ϖ ψ , and then Ω ϖ ( 0 ) = 0 . Consequently, we obtain utilizing Theorem (3.7):

(60) ∣ λ ¯ ( t ) − λ ( t ) ∣ ≤ ξ ϖ ( ψ ) .

Therefore, a generalized Ulam–Hyres stable solution exists for the fractal–fractional initial value problem (35). Thus, the suggested model (2) is generalized Ulam–Hyres stable.□

4 Computational algorithm for numerical results

The literature suggests that the power-law kernel-based Caputo derivative can be used to simulate power-law processes in real-life situations. For the proposed model (2), we provide an efficient numerical discretization technique based on Newton polynomial interpolation [51] for the numerical solution of the system:

(61) D t ω , υ 0 FFP [ S ( t ) ] = η N − α IS − ( κ + η ) S , D t ω , υ 0 FFP [ V ( t ) ] = κ S − β IV − ( δ + η ) V , D t ω , υ 0 FFP [ I ( t ) ] = α IS + β IV − ( ρ + η ) I , D t ω , υ 0 FFP [ R ( t ) ] = ρ I + δ V − η R .

In order to simplify the use of the previously mentioned system, we have

(62) ζ 1 ( t , S , V , I , R ) = η N − α IS − ( κ + η ) S , ζ 2 ( t , S , V , I , R ) = κ S − β IV − ( δ + η ) V , ζ 3 ( t , S , V , I , R ) = α IS + β IV − ( ρ + η ) I , ζ 4 ( t , S , V , I , R ) = ρ I + δ V − η R .

The following outcome is reached by applying the fractional integral:

(63) S ( t l + 1 ) = S ( 0 ) + 1 Γ ( ω ) ∑ i = 2 l ∫ t i t i + 1 × ς υ − 1 ( t l + 1 − ς ) ω − 1 ζ 1 ( ς , S , V , I , R ) d ς , V ( t l + 1 ) = V ( 0 ) + 1 Γ ( ω ) ∑ i = 2 l ∫ t i t i + 1 × ς υ − 1 ( t l + 1 − ς ) ω − 1 ζ 2 ( ς , S , V , I , R ) d ς , I ( t l + 1 ) = I ( 0 ) + 1 Γ ( ω ) ∑ i = 2 l ∫ t i t i + 1 × ς υ − 1 ( t l + 1 − ς ) ω − 1 ζ 3 ( ς , S , V , I , R ) d ς , R ( t l + 1 ) = R ( 0 ) + 1 Γ ( ω ) ∑ i = 2 l ∫ t i t i + 1 × ς υ − 1 ( t l + 1 − ς ) ω − 1 ζ 4 ( ς , S , V , I , R ) d ς .

Let us now discuss the Newton polynomial:

(64) P ( t , S , V , I , R ) ≃ P ( t l − 2 , S l − 2 , V l − 2 , I l − 2 , R l − 2 ) + 1 Δ t { P ( t l − 1 , S l − 1 , V l − 1 , I l − 1 , R l − 1 ) − P ( t l − 2 , S l − 2 , V l − 2 , I l − 2 , R l − 2 ) } × ( α − t l − 2 ) + 1 2 Δ t 2 { P ( t l , S l , V l , I l , R l ) − 2 P ( t l − 2 , S l − 1 , V l − 1 , I l − 1 , R l − 1 ) + P ( t l − 2 , S l − 2 , V l − 2 , I l − 2 , R l − 2 ) } × ( α − t l − 2 ) ( α − t l − 1 ) .

Upon substituting the Newton polynomial (64) into Eq. (63), we obtain

(65) S ( l + 1 ) = S ( 0 ) + 1 Γ ( ω ) ∑ i = 2 l t i − 2 1 − υ ζ 1 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) × ∫ t i t i + 1 ( t m + 1 − ς ) ω − 1 d ς + 1 Γ ( ω ) ∑ i = 2 l 1 Δ t { t i − 1 1 − υ ζ 1 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) − t i − 2 1 − υ ζ 1 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) } × ∫ t i t i + 1 ( ς − t i − 2 ) ( t m + 1 − ς ) ω − 1 d ς + 1 Γ ( ω ) ∑ i = 2 l 1 2 Δ t 2 { t i 1 − υ ζ 1 ( t i , S i , V i , I i , R i ) − 2 t i − 1 1 − υ ζ 1 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) + t i − 2 1 − υ ζ 1 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) } × ∫ t i t i + 1 ( ς − t i − 2 ) ( ς − t i − 1 ) ( t m + 1 − ς ) ω − 1 d ς ,

(66) V ( l + 1 ) = V ( 0 ) + 1 Γ ( ω ) ∑ i = 2 l t i − 2 1 − υ ζ 2 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) × ∫ t i t i + 1 ( t m + 1 − ς ) ω − 1 d ς + 1 Γ ( ω ) ∑ i = 2 l 1 Δ t { t i − 1 1 − υ ζ 2 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) − t i − 2 1 − υ ζ 2 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) } × ∫ t i t i + 1 ( ς − t i − 2 ) ( t m + 1 − ς ) ω − 1 d ς + 1 Γ ( ω ) ∑ i = 2 l 1 2 Δ t 2 { t i 1 − υ ζ 2 ( t i , S i , V i , I i , R i ) − 2 t i − 1 1 − υ ζ 2 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) + t i − 2 1 − υ ζ 2 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) } × ∫ t i t i + 1 ( ς − t i − 2 ) ( ς − t i − 1 ) ( t m + 1 − ς ) ω − 1 d ς ,

(67) I ( l + 1 ) = I ( 0 ) + 1 Γ ( ω ) ∑ i = 2 l t i − 2 1 − υ ζ 3 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) × ∫ t i t i + 1 ( t m + 1 − ς ) ω − 1 d ς + 1 Γ ( ω ) ∑ i = 2 l 1 Δ t × { t i − 1 1 − υ ζ 3 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) − t i − 2 1 − υ ζ 3 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) } × ∫ t i t i + 1 ( ς − t i − 2 ) ( t m + 1 − ς ) ω − 1 d ς + 1 Γ ( ω ) ∑ i = 2 l 1 2 Δ t 2 { t i 1 − υ ζ 3 ( t i , S i , V i , I i , R i ) − 2 t i − 1 1 − υ ζ 3 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) + t i − 2 1 − υ ζ 3 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) } × ∫ t i t i + 1 ( ς − t i − 2 ) ( ς − t i − 1 ) ( t m + 1 − ς ) ω − 1 d ς ,

(68) R ( l + 1 ) = R ( 0 ) + 1 Γ ( ω ) ∑ i = 2 l t i − 2 1 − υ ζ 4 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) × ∫ t i t i + 1 ( t m + 1 − ς ) ω − 1 d ς + 1 Γ ( ω ) ∑ i = 2 l 1 Δ t × { t i − 1 1 − υ ζ 4 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) − t i − 2 1 − υ ζ 4 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) } × ∫ t i t i + 1 ( ς − t i − 2 ) ( t m + 1 − ς ) ω − 1 d ς + 1 Γ ( ω ) ∑ i = 2 l 1 2 Δ t 2 { t i 1 − υ ζ 4 ( t i , S i , V i , I i , R i ) − 2 t i − 1 1 − υ ζ 4 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) + t i − 2 1 − υ ζ 4 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) } × ∫ t i t i + 1 ( ς − t i − 2 ) ( ς − t i − 1 ) ( t m + 1 − ς ) ω − 1 d ς .

The following formulas can be used to calculate the integral given by the previous equations:

(69) ∫ t i t i + 1 ( t l + 1 − ς ) ω − 1 d ς = ( Δ t ) ω ω [ ( l − i + 1 ) ω − ( l − i ) ω ] ,

(70) ∫ t i t i + 1 ( ς − t i − 2 ) ( t l + 1 − ς ) ω − 1 d ς = ( Δ t ) ω + 1 ω ( ω + 1 ) [ ( l − i + 1 ) ω ( l − i + 3 + 2 ω ) − ( l − i ) ω ( l − i + 3 + 3 ω ) ] ,

(71) ∫ t i t i + 1 ( ς − t i − 2 ) ( ς − t i − 1 ) ( t l + 1 − ς ) ω − 1 d ς = ( Δ t ) ω + 2 ω ( ω + 1 ) ( ω + 2 ) × [ ( l − i + 1 ) ω { 2 ( l − i ) 2 + ( 3 ω + 10 ) ( l − i ) + 2 ω 2 + 9 ω + 12 } − ( l − i ) ω { 2 ( l − i ) 2 + ( 5 ω + 10 ) ( l − i ) + 6 ω 2 + 18 ω + 12 } ] .

Thus, we ultimately obtain

(72) S ( t l + 1 ) = S ( 0 ) + ( Δ t ) ω Γ ( ω + 1 ) ∑ i = 2 l t i − 2 1 − υ × ζ 1 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) × G 1 + ( Δ t ) ω Γ ( ω + 2 ) ∑ i = 2 l [ t i − 1 1 − υ ζ 1 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) − t i − 2 1 − υ ζ 1 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) ] × G 2 + ω ( Δ t ) ω 2 Γ ( ω + 3 ) ∑ i = 2 l [ t i 1 − υ ζ 1 ( t i , S i , V i , I i , R i ) − 2 t i − 1 1 − υ ζ 1 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) + t i − 2 1 − υ ζ 1 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) ] × G 3 ,

where

(73) G 1 = ( l − i + 1 ) ω − ( l − i ) ω , G 2 = ( l − i + 1 ) ω ( l − i + 3 + 2 ω ) − ( l − i ) ω ( l − i + 3 + 3 ω ) , G 3 = ( l − i + 1 ) ω [ 2 ( l − i ) 2 + ( 3 ω + 10 ) ( l − i ) + 2 ω 2 + 9 ω + 12 ] , − ( l − i ) ω [ 2 ( l − i ) 2 + ( 5 ω + 10 ) ( l − i ) + 6 ω 2 + 18 ω + 12 ] .

Also, we find

(74) V ( t l + 1 ) = V ( 0 ) + ( Δ t ) ω Γ ( ω + 1 ) ∑ i = 2 l t i − 2 1 − υ ζ 2 × ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) × G 1 + ( Δ t ) ω Γ ( ω + 2 ) ∑ i = 2 l [ t i − 1 1 − υ ζ 2 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) − t i − 2 1 − υ ζ 2 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) ] × G 2 + ω ( Δ t ) ω 2 Γ ( ω + 3 ) ∑ i = 2 l [ t i 1 − υ ζ 2 ( t i , S i , V i , I i , R i ) − 2 t i − 1 1 − υ ζ 2 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) + t i − 2 1 − υ ζ 2 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) ] × G 3 .

(75) I ( t l + 1 ) = I ( 0 ) + ( Δ t ) ω Γ ( ω + 1 ) ∑ i = 2 l t i − 2 1 − υ ζ 3 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) × G 1 + ( Δ t ) ω Γ ( ω + 2 ) ∑ i = 2 l [ t i − 1 1 − υ ζ 3 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) − t i − 2 1 − υ ζ 3 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) ] × G 2 + ω ( Δ t ) ω 2 Γ ( ω + 3 ) ∑ i = 2 l [ t i 1 − υ ζ 3 ( t i , S i , V i , I i , R i ) − 2 t i − 1 1 − υ ζ 3 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) + t i − 2 1 − υ ζ 3 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) ] × G 3 .

(76) R ( t l + 1 ) = R ( 0 ) + ( Δ t ) ω Γ ( ω + 1 ) ∑ i = 2 l t i − 2 1 − υ ζ 4 × ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) × G 1 + ( Δ t ) ω Γ ( ω + 2 ) ∑ i = 2 l [ t i − 1 1 − υ ζ 4 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) − t i − 2 1 − υ ζ 4 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) ] × G 2 + ω ( Δ t ) ω 2 Γ ( ω + 3 ) ∑ i = 2 l [ t i 1 − υ ζ 4 ( t i , S i , V i , I i , R i ) − 2 t i − 1 1 − υ ζ 4 ( t i − 1 , S i − 1 , V i − 1 , I i − 1 , R i − 1 ) + t i − 2 1 − υ ζ 4 ( t i − 2 , S i − 2 , V i − 2 , I i − 2 , R i − 2 ) ] × G 3 .

4.1 Convergence and error analysis

While error analysis looks at the sources of error, establishes boundaries, and uses computations to track error propagation, understanding convergence types, criteria, and rates helps choose the best approaches for particular issues. Here, we conduct the suggested methodology’s error analysis for the fractal–fractional problem. When the fractal size is 1, the polynomial interpolation that was utilized to build this scheme inevitably results in a stable and convergent system. As described in the study by Khan and Atangana [52], we analyze a general Cauchy issue in which the derivative is of that particular Caputo type in this subsection:

(77) D t ω , υ 0 FFP [ λ ( t ) ] = ϖ ( t , λ ( t ) ) ,

where λ ( t ) = { S ( t ) , V ( t ) , L ( t ) , A ( t ) , R ( t ) , T ( t ) } . Using (2.2), we obtain

(78) λ ( t ) = υ Γ ( ω ) ∫ 0 t ς υ − 1 ( t − ς ) ω − 1 ϖ ( ς , λ ( ς ) ) d ς .

At t l + 1 = ( l + 1 ) Δ t , we have

(79) λ ( t l + 1 ) = υ Γ ( ω ) ∫ 0 t l + 1 ς υ − 1 ( t l + 1 − ς ) ω − 1 ϖ ( ς , λ ( ς ) ) d ς , = υ Γ ( ω ) ∑ i = 0 l ∫ t i t i + 1 ς υ − 1 ( t l + 1 − ς ) ω − 1 ϖ ( ς , λ ( ς ) ) d ς .

Using the Newton polynomial interpolation, we approximate the function Φ ( t ) = ς υ − 1 ϖ ( ς , λ ( ς ) ) within the interval [ t i , t i + 1 ] as:

(80) Q ( t ) = P i ( t ) = ς − t i − 1 Δ t Φ ( t i , λ i ) − ς − t i Δ t Φ ( t i − 1 , λ i − 1 ) .

Considering the error, we acquire

(81) λ ( t l + 1 ) = υ Γ ( ω ) ∑ i = 0 l ∫ t i t i + 1 × ς υ − 1 ( t l + 1 − ς ) ω − 1 [ P i ( ς ) + R i ( ς ) ] d ς , = υ Γ ( ω ) ∑ i = 0 l ∫ t i t i + 1 ς υ − 1 ( t l + 1 − ς ) ω − 1 P i ( ς ) d ς + υ Γ ( ω ) ∑ i = 0 l ∫ t i t i + 1 ς υ − 1 ( t l + 1 − ς ) ω − 1 R i ( ς ) d ς ,

(82) R ( ω , l , υ ) = υ Γ ( ω ) ∑ i = 0 l ∫ t i t i + 1 ( t l + 1 − ς ) ω − 1 R i ( ς ) d ς , = υ Γ ( ω ) ∑ i = 0 l ∫ t i t i + 1 ( ς − t i ) ( ς − t i − 1 ) 2 ! × ∂ 2 ∂ ς 2 ( Φ ( ς , y ( ς ) ) ∣ ς = σ ς ) × ( t l + 1 − ς ) ω − 1 d ς , = υ Γ ( ω ) ∑ i = 0 l ∫ t i t i + 1 ( ς − t i ) ( ς − t i − 1 ) 2 ! × ∂ 2 ∂ ς 2 ( ς υ − 1 ϖ ( ς , y ( ς ) ) ∣ ς = σ ς ) ( t l + 1 − ς ) ω − 1 d ς .

It is evident that the formula ( ς − t i − 1 ) ( t l + 1 − ς ) ω − 1 is positive in the interval [ t i , t i + 1 ] . Hence, it is possible to determine σ i ∈ [ t i , t i + 1 ] by verifying

(83) R ( ω , l , υ ) = ω υ Γ ( ω ) ∑ i = 0 l ∂ 2 ∂ ς 2 ( ς υ − 1 ϖ ( ς , y ( ς ) ) ∣ ς = σ i ) σ i − ς i 2 × ∫ t i t i + 1 ( ς − t i − 1 ) ( t l + 1 − ς ) ω − 1 d ς , = ω υ Γ ( ω ) ∑ i = 0 l ∂ 2 ∂ ς 2 ( ς υ − 1 ϖ ( ς , y ( ς ) ) ∣ ς = σ i ) σ i − ς i 2 × ( l + 1 − i ) ω ( l − i + ω + 2 ) − ( l − i ) ω ( l − i + 2 ω + 2 ) ω ( ω + 1 ) , = ω υ Γ ( ω + 2 ) ∑ i = 0 l ∂ 2 ∂ ς 2 ( ς υ − 1 ϖ ( ς , y ( ς ) ) ∣ ς = σ i ) σ i − ς i 2 × [ ( l + 1 − i ) ω ( l − i + ω + 2 ) − ( l − i ) ω ( l − i + 2 ω + 2 ) ] ,

where

(84) ∫ t i t i + 1 ( ς − t i − 1 ) ( t l + 1 − ς ) ω − 1 d ς = ( l + 1 − i ) ω ( l − i + ω + 2 ) − ( l − i ) ω ( l − i + 2 ω + 2 ) ω ( ω + 1 ) .

Consequently, applying the absolute value to both sides yields

(85) ‖ R ( ω , l , υ ) ‖ = ω υ ( Δ t ) ω + 2 2 Γ ( ω ) max ς ∈ [ 0 , t l + 1 ] ∂ 2 ∂ ς 2 ( ς υ − 1 ϖ ( ς , y ( ς ) ) ) × ∑ i = 0 l [ ( l + 1 − i ) ω ( l − i + ω + 2 ) − ( l − i ) ω ( l − i + 2 ω + 2 ) ] .

We have

(86) ( l − i + 1 ) ω − ω ( l − i ) ω ≤ ( l + 1 ) ω − ω l ω ,

and

(87) ∑ i = 0 l ( l − i + 2 ω + 2 ) = 1 2 l ( l + 2 ω + 4 ) .

This suggests that

(88) ‖ R ( ω , l , υ ) ‖ = ω υ ( Δ t ) ω + 2 2 Γ ( ω ) max ς ∈ [ 0 , t l + 1 ] × ∂ 2 ∂ ς 2 ( ς υ − 1 ϖ ( ς , y ( ς ) ) ) × 1 2 l ( l + 2 ω + 4 ) [ ( l + 1 ) ω − ω l ω ] .

5 Simulations and discussion

The primary goals of the simulations are to clarify the dynamic behavior of the suggested model’s compartments and to validate the analytical conclusions of this work. Numerical simulations provide visual representations and are useful tools for scenario assessment, making it easy to see how different parameters affect outbreak dynamics. For numerical results, we used α = 0.24 , κ = 0.05 , η = 0.03 , β = 0.08 , δ = 0.92 , and ρ = 0.071428571 . To validate the analytical results, parameter values are used to investigate the effects of different fractional orders on model solutions.

5.1 Effect of fractional order ( ω ) and fractal dimension ( υ )

We run simulations with various fractional orders ( ω = 0.95,0.90,0.80 ) and fractal dimensions ( υ = 0.7,1.0 ) . Figure 2 depicts the dynamic behavior of the proposed model’s compartments at fractal dimension υ = 1 and different fractional orders, while Figure 3 depicts the dynamic behavior of the compartments at fractal dimension υ = 0.7 and different fractional orders. The study demonstrates that the number of susceptible people decreases rapidly at large fractional orders, emphasizing the impact of vaccinations on those who are susceptible. The drop in immunization rates is due to an insufficient number of recipients contracting the illness after close contact with active cases. The susceptible and immunized population members swiftly spread the infection by joining the infected class. The non-locality of the suggested operator aptly illustrates the part memory plays in disease transmission. Given that a decrease in the number of infected patients occurs with an increase in fractional order, the non-local nature of the suggested operator illustrates the impact of memory on disease transmission. Vaccination rates are effective, as evidenced by the number of people who entered the recovered class after completing the necessary vaccinations. Moreover, we observe that the fractal dimension υ is a scale-independent measure of irregularity, roughness, or variation of the proposed model that calculates the number of points in a set to determine the complexity of a self-similar figure.

$Figure 2 Simulation of model at various fractional orders ( ω ) \left(\omega ) and fractal dimension ( υ = 1 ) \left(\upsilon =1) .$

Figure 2

Simulation of model at various fractional orders ( ω ) and fractal dimension ( υ = 1 ) .

$Figure 3 Simulation of model at various fractional orders ( ω ) \left(\omega ) and fractal dimension ( υ = 0.7 ) \left(\upsilon =0.7) .$

Figure 3

Simulation of model at various fractional orders ( ω ) and fractal dimension ( υ = 0.7 ) .

5.2 Effect of varying parameter values and population sizes

Figure 4 illustrates the dynamical behavior of the compartments in the proposed model when parameter values are increased by 10% at fractal dimension υ = 1 and different fractional orders. Figure 5 illustrates the dynamical behavior of the compartments when parameter values are increased by 10% at fractal dimension υ = 0.7 and different fractional orders. The dynamic behavior of the model with varying population sizes is depicted in Figure 6. On the other hand, Figure 7 displays the model’s dynamical behavior when the population size is changed and the parameter values are increased by 10%. These figures show that the population size and parameter values for various scenarios yield comparable results.

$Figure 4 Simulation of the model (at υ = 1 \upsilon =1 ) with a 10% increase in parameter values.$

Figure 4

Simulation of the model (at υ = 1 ) with a 10% increase in parameter values.

$Figure 5 Simulation of the model (at υ = 0.7 \upsilon =0.7 ) with a 10% increase in parameter values.$

Figure 5

Simulation of the model (at υ = 0.7 ) with a 10% increase in parameter values.

Figure 6

Simulation with different population sizes for the suggested model.

Figure 7

Simulation of the suggested model with different population sizes and a 10% increase in parameter values.

5.3 Chaotic behavior

Many scientific and engineering applications heavily rely on the chaotic behavior of the system as depicted in Figure 8. It is well recognized that there is a strong inclination to conceptualize and describe chaotic system behavior. Chaotic modeling makes the proposed mathematical model realistic and scalable, which may be applied to the research of new chaos systems. We can see that the solutions are bounded in the feasible domain in this instance.

Figure 8

Chaotic behavior of the proposed model.

5.4 Comparison to classical integer-order model

We have the solution of the classical integer-order model at ω = 1 , which should not be forgotten. This enables us to correlate the outcomes of our simulation with the integer-order model’s classical results. The curves for ω = 0.95 , 0.90, and 0.85 exhibit a significantly slower rise/fall over extended periods of time when compared to curves produced from a classical model with a ω = 1 .

6 Conclusion

A fractal–fractional model was used to examine the dynamics of varicella virus populations in Jordan. This study examines the dynamics of the SVIR model and includes a numerical simulation. The model’s equilibrium points were determined, along with the fundamental reproduction number R 0 . The presence and uniqueness of the model’s solution demonstrate that the model’s predictions are accurate and useful for decision-making. We talked about the model’s characteristics, such as reproductive number, positivity, boundedness, biological viability, equilibrium points, and sensitivity analysis. The study demonstrated the stability of a unique solution for a fractal–fractional model in Ulam–Hyers and used fixed point arguments to generalize Ulam–Hyers types. The impact of the fractional operator on the power law kernel was investigated using numerical simulations and a two-step Newton polynomial method. The study found that increasing vaccination rates and limiting contact with sick people reduces the incidence of chickenpox in Jordan significantly. Varicella has a significant impact on the country’s healthcare system and can be managed by the Jordanian Ministry of Health by implementing universal vaccination programs that provide long-term protection, prevent outbreaks, and foster a healthier community. To account for complex relationships and memory effects in disease spread, the model employs a fractional operator. Fractal-fractional derivatives can help slow down disease spread and improve control measures. Real data can be used to study optimal control, while a deterministic version can be used to investigate dynamics and the impact of white noise. The varicella virus strains Jordan’s healthcare system, necessitating immunization campaigns. This research can help governments and medical organizations develop effective plans to mitigate the virus’s effects and protect vulnerable people. Using this data, effective control measures can be developed.

Funding information: This study was supported via funding from Prince Sattam bin Abdulaziz University Project Number (PSAU/2024/R/1445).
Author contributions: K.S.N.: conceptualization, methodology, formal analysis, writing – original draft, writing – review, editing. M.F.: conceptualization, methodology, formal analysis, software, visualization, writing – original draft., M.G: methodology, formal analysis, software, formatting, writing review, editing. E.H.: formal analysis, supervision, visualization, writing – review, editing. A.S.: formal analysis, software, visualization, writing – review editing. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: All data generated or analysed during this study are included in this published article.

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Received: 2024-09-05

Revised: 2024-10-27

Accepted: 2024-11-22

Published Online: 2025-03-28

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2024-0072

Keywords for this article

varicella–zoster virus; fractal–fractional operator; qualitative analysis; Ulam–Hyers stability; error analysis

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