Novel Investigation of Stochastic Fractional Differential Equations Measles Model via the White Noise and Global Derivative Operator Depending on Mittag-Leffler Kernel

1  Introduction

Measles is among the highly contagious airborne infections in humans, and it can result in significant sickness, life-long problems, and even fatality [1]. Paramyxovirus causes measles, an abrupt and deadly infectious infection. This infection can be transferred mostly through atmospheric spraying to mucosa in the pulmonary system, and it can survive in the phlegm of a contaminated person’s nasal passages. When an infectious individual coughs or has respiratory secretions, it can be communicated via exposure to a contaminated nasopharynx. Only individuals are intermediate victims of the measles infection. It is split into four rounds of disease, including implantation, prodrome, erythema, and recuperation [2], see Fig. 1.

Figure 1: Measles virus cycle

Measles complications seem to be particularly likely in children under the age of five and people over the age of twenty. Tuberculosis, eardrum, and nostril problems, ulcerations, chronic diarrhea, jaundice, conjunctivitis, starvation, and cognitive impairment are just a few of them [3].

Measles is highly contagious, with an infection rate of more than 90 percent among those who are susceptible. It is, indeed, a global health issue. Several underdeveloped nations, notably Asian countries, are affected. According to the WHO, measles affects over 20 million people each year, with developing countries accounting for more than 95 percent of fatalities [4], particularly in Sub-Saharan Africa, where the disease accounts for 15 percent of all fatalities. The simplest method to avoid acquiring measles is to be immunized. It is risk-free, productive, and affordable. Youngsters who have not been inoculated and expectant mothers are more vulnerable to measles and its ramifications, which can include mortality. Immunization from measles acquired by inoculation has been proven to last about two decades and is widely considered to be a reality among healthy humans. At 9–11 months after birth, vaccine effectiveness is estimated to be 85 percent, increasing to 97 percent with a one-year intravenous infusion [4].

The goal of medication is to relieve anxiety unless the ailment is cleared by the adaptive defensive mechanism. The diagnosis of acute is not killed by any therapeutic intervention. Meditation and basic fever-reduction treatments are usually all that is required for a swift comeback. Measles patients require hospitalization, hydration, temperature and anxiety relief, skin rash, as well as medicine [2], see Fig. 2. The technique of depicting important considerations employing quantitative concepts and formulas is known as numerical simulation. On the basis of foresight, mathematical formulation can be divided into stochastic and deterministic systems. Since the beginning of the nineteenth century, numerical simulations have been used to investigate contagious infections. Various deterministic and probabilistic epidemiological approaches have been used to comprehend contagious illnesses, including measles [5], diarrhea [6], the SIRC model [7], chikungunya spread [8], and henceforth.

Figure 2: Measles impact on immune system

In the numerical techniques of microbiological contamination, the deterministic methodology has had significant disadvantages. They are easily interpretative, however, supply fewer insights and are difficult to estimate, so they are not stochastic and generally require modeling from very similar modeling outcomes around one manifestation. The stochastic description of a dynamic reflects the system’s unpredictability in evolution. Variation, which is key to the development of the multiverse, and carelessness, which is a hallmark of people, both contribute to unpredictability. As a consequence, the random variable should specifically contain both levels of variation in order to depict ambiguity in a straightforward fashion [9]. Stochastic approaches are important in many interdisciplinary fields, notably measles prevalence and distribution, as they convey a higher sense of authenticity than deterministic approaches [10].

Owing to the robust computational formulas including index law, decay, and inversion from growth rate to index law, which can encapsulate perhaps concealed intricacies of existence, the concepts of fractional formulations have lately been coupled to generate innovative differentiation operators in several serious challenges. These novel formulations have the distinctive bonus of being able to describe phenomena that satisfy the index law, exponential decay kernel, and generalized Mittag-Leffler (M-L) kernels at the same time [11–13]. Because of their exceptional capabilities, these novel operators are well suited to modeling a wide range of complicated real-world concerns. Researchers have devised a framework known as Brownian motion, or stochastic elements, to express unpredictability [14]. This concept has been successfully implemented in a variety of disciplines in recent years, including neuroscience, automation, and epidemiology. While both have shown efficiency in simulating dynamic behavior on their own, it is important to note that differential operators, having mentioned kernels, as well as Brownian movements or stochastic ideas, never account for index law, fading memory, or overlapping influences [15–17]. However, we must keep in mind that several situations in existence are capable of exhibiting both mechanisms. Consequently, neither fractional differential operators nor stochastic techniques adequately explain them. The propagation of any contagious disease could be fully comprehended using straightforward mathematical formulae since multiple aspects influence its transmission among individuals [18–21]. Several researchers proposed various investigations for controlling the epidemics and their eradication. For example, Qureshi et al. [5] discussed the monotonic reduction of measles spread via Atangana-Baleanu-Caputo derivative operator, Qureshi et al. [6] represented the nonlinear dynamics of diarrhoea transmission via fractal-fractional operator, Rihan et al. [7] contemplated the SIRC model via Caputo fractional derivative operator, and Rashid et al. [22] investigated the oncolytic effectiveness model with M1 virus via Atangana-Baleanu fractional derivative operator.

Adopting the above propensity, we consider the transmission of the measles infection via stochastic fractional derivative operators in the Antagana-Baleanu and Caputo-Fabrizio sense. In the past decade, only a handful of important studies on the prevalence and distribution of measles have been conducted. A SIR mathematical formulation of measles incorporating inoculation and two periods of transmissibility was used in the investigation. Their research discovered that if all vulnerable people were inoculated, the ailment would be eradicated. They proposed that the measles vaccine be implemented immediately, as no child should be permitted to join a classroom lacking proof of at least two doses of measles immunization. Stochastic modeling of measles emergence and spread involving inoculation intervention was investigated in [23,24]. In comparison to a deterministic approach, stochastic interpretation proved to be more productive in investigating the evolution of measles. Furthermore, Edward et al. [25] developed a quantitative framework for controlling and eliminating measles disease propagation. Measles eradication necessitates keeping the efficient reproductive count below 1 and establishing moderate concentrations of vulnerability. In this analysis, we aim to generate stochastic and deterministic differential equation (DE) systems of measles outbreaks while taking the overall community and immunization regime into account via the fractional derivative operator techniques. For small sensitive community densities, simulated findings indicate that the probabilistic model’s responses will have considerable stochastic features. For greater vulnerable population levels, the deterministic framework’s result is a restriction of the stochastic counterpart’s alternatives. The presence, originality, consistency, and simulation studies of a mathematical framework for measles transmission are also investigated. We verified the model’s consistency, as well as the existence and uniqueness of the model’s findings via the linear growth and Lipschitiz conditions. The system is numerically solved using the Newton interpolating technique. All of the aforementioned investigations have created deterministic and stochastic mathematical formulas for measles propagation and prevention. To the best of the researchers’ expertise, no investigation has been performed on a stochastic framework of measles prevalence and distribution with dual dosage vaccination by segregating first and second treatment immunized groups. However, we must be certain of the interval specified.

2  Preliminaries

In this part, we will review several essential concepts for fractional calculus involving singular and non-singular kernels.

Definition 2.1. ([26]) For ϕ>0, then the Caputo fractional derivative of f1:(0,∞)↦R is continuous and differentiable, presented as

 cDξϕf1(ξ)=1Γ(1−ϕ)∫0ξ(ξ−x)−ϕddxf1(x)dx,0<ϕ≤1.(1)

Definition 2.2. ([26]) For ϕ>0, then the Riemann-Liouville fractional integral of f1:(0,∞)↦R is presented as

Iξϕf1(ξ)=1Γ(ϕ)∫0ξ(ξ−x)ϕ−1f1(x)dx,0<ϕ≤1.(2)

Definition 2.3. ([27]) Suppose there be a function f1∈H1(u,v), u<v, ϕ∈(0,1), then the Caputo-Fabrizio fractional derivative is defined as

 CFDξϕf1(ξ)=M(ϕ)(1−ϕ)∫uξexp⁡(−ϕξ−x1−ϕ)f1 ′(x)dx,0<ϕ≤1,(3)

where M(ϕ) is a normalization function such that M(0)=M(1)=1.

If f1∉H1(u,v), then the derivative operator is redefined as

 CFDξϕf1(ξ)=ϕM(ϕ)(1−ϕ)∫uξexp⁡(−ϕξ−x1−ϕ)(f1(ξ)−f1(x))dx,0<ϕ≤1,(4)

Theorem 2.4. ([27]) For ϕ∈(0,1), then the following ordinary DE

 0CFDξϕf1(ξ)=Ψ(ξ)(5)

has a unique solution by implementing the inverse Laplace transform and convolution theorem described as

f1(ξ)=2(1−ϕ)(2−ϕ)M(ϕ)Ψ(ξ)+2ϕ(2−ϕ)M(ϕ)∫0ξΨ(s1)ds1,ξ≥0.(6)

Definition 2.5. ([28]) Suppose there be a function f1∈H1(u,v), u<v, ϕ∈(0,1), then the Atangana-Baleanu fractional derivative in the Caputo context is defined as

 ABCDξϕf1(ξ)=B(ϕ)(1−ϕ)∫uξEϕ(−ϕ(ξ−x)ϕ1−ϕ)f1 ′(x)dx,0<ϕ≤1,(7)

where Eϕ is the M-L kernel and B(ϕ)=ϕΓ(ϕ)+1−ϕ denotes the normalized function.

Definition 2.6. ([28]) Suppose there be a function f1∈H1(u,v), u<v, ϕ∈(0,1), is not differentiable, then the Atangana-Baleanu fractional derivative in the Riemann context is defined as

 ABRDξϕf1(ξ)=B(ϕ)(1−ϕ)ddξ∫uξf1(x)Eϕ(−ϕ(ξ−x)ϕ1−ϕ)dx.(8)

Definition 2.7. ([28]) For ϕ∈(0,1), then the Atanagana-Baleanu fractional integral is stated as:

 uABIξϕf1(ξ)=1−ϕB(ϕ)f1(ξ)+ϕB(ϕ)Γ(ϕ)∫uξf1(x)(ξ−x)ϕ−1dx.(9)

Theorem 2.8. For ϕ∈(0,1), then the following ordinary DE

 0ABCDξϕf1(ξ)=Ψ(ξ)(10)

has a unique solution by implementing the inverse Laplace transform and convolution theorem described as

f1(ξ)=1−ϕB(ϕ)Ψ(ξ)+ϕB(ϕ)Γ(ϕ)∫0ξΨ(x)(ξ−x)ϕ−1dx.(11)

It is worth noting that Atangana established the aforementioned concepts with the global notion quite early. In his study [29], he offered a description of the global derivative. Let us have a glance at a few different variations of it.

Definition 2.9. ([29]) For ϕ∈(0,1], then there be a continuous mapping f1(ξ) and an increasing positive mapping 𝒢(ξ) such that there be a singular/non-singular kernel 𝒦(ξ), then the fractional global derivative (GD) in Caputo context is stated as follows:

 0cD𝒢ϕf1(ξ)=D𝒢f1(ξ)∗𝒦(ξ),(12)

where ∗ denotes the convolution operator.

Next, we present the concept of fractional global derivative (GD) in Caputo form, Riemann-Liouville form, Caputo-Fabrizio form, and Atanagana-Baleanu form, respectively, which is mainly by Atangana [29].

Definition 2.10. ([29]) For 0<ϕ≤1, then the GD in the Caputo sense is defined as

 0cD𝒢ϕf1(ξ)=1Γ(1−ϕ)∫0ξ(ξ−x)−ϕD𝒢f1(x)dx.(13)

Definition 2.11. ([29]) For 0<ϕ≤1, then the GD in the Riemann-Liouville sense is defined as

 0RLD𝒢ϕf1(ξ)=1Γ(1−ϕ)D𝒢∫0ξ(ξ−x)−ϕf1(x)dx.(14)

Definition 2.12. ([29]) For 0<ϕ≤1, then the GD in the Caputo-Fabrizio sense is defined as

 0CFD𝒢ϕf1(ξ)=M(ϕ)(1−ϕ)∫0ξD𝒢f1(x)exp(−ϕξ−x1−ϕ)dx.(15)

Definition 2.13. ([29]) For 0<ϕ≤1, then the GD in the Atangana-Baleanu in the Caputo sense is defined as

 0ABCD𝒢ϕf1(ξ)=B(ϕ)(1−ϕ)∫0ξD𝒢f1(x)Eϕ(−ϕ(ξ−x)ϕ1−ϕ)dx.(16)

Definition 2.14. ([29]) For 0<ϕ≤1, then the GD in the Atangana-Baleanu in the Riemann sense is defined as

 0ABRD𝒢ϕf1(ξ)=B(ϕ)(1−ϕ)D𝒢∫0ξf1(x)Eϕ(−ϕ(ξ−x)ϕ1−ϕ)dx.(17)

Integral representations of the derivatives here are employed in numerical demonstrations, hence integral operators with GD in the Riemann-Liouville form are supplied as:

Definition 2.15. ([29]) For 0<ϕ≤1, then the integral version of RiemannLiouville in GD form is defined as

 0I𝒢ϕf1(ξ)=1Γ(ϕ)∫0ξ𝒢 ′(ξ)(ξ−x)ϕ−1f1(x)dx.(18)

Definition 2.16. ([29]) For 0<ϕ≤1, then the integral version of Caputo-Fabrizio in GD form is defined as

 0CFI𝒢ϕf1(ξ)=(1−ϕ)M(ϕ)𝒢 ′(x)f1(x)+ϕM(ϕ)∫0ξ𝒢 ′(x)f1(x)dx.(19)

Definition 2.17. ([29]) For 0<ϕ≤1, then the integral version of Atanagana-Baleanu in GD form is defined as

 0ABCI𝒢ϕf1(ξ)=(1−ϕ)B(ϕ)𝒢 ′(x)f1(x)+ϕB(ϕ)Γ(ϕ)∫0ξ𝒢 ′(x)(ξ−x)ϕ−1f1(x)dx.(20)

Definition 2.18. ([29]) For 0<ϕ≤1, then the GD in the Atangana-Baleanu in the Caputo sense is defined as

 0ABCD𝒢ϕf1(ξ)=B(ϕ)(1−ϕ)∫0ξD𝒢f1(x)Eϕ(−ϕ(ξ−x)ϕ1−ϕ)dx.(21)

3  Configuration of Stochastic Measles Epidemic Model

Despite the fact that deterministic DEs have been frequently employed to simulate the transmission of several contagious ailments, daily information gathering revealed that their distribution occasionally follows non-locality and unpredictability. This shows that neither fractional DEs nor stochastic differential equations (SDEs) can duplicate such a dispersion. However, SDEs are appropriate for simulating complex issues if the dissemination maintains random noise. Some examples of studies conducted that involve SDEs are [30–33]. The following is a brief description of how to differentiate when considering unpredictability:

dx=f1(ξ,x,y)dξ+f2(ξ,x,y)dρ1,(22)

where ρj=[ρ1,...,ρn],forj=1,...,n represents the independent Wiener process. In the measles epidemic model, the whole population is divided into five cohorts as follows: susceptible S(ξ), infected I(ξ), V1(ξ) initial prescription of vaccination, V2(ξ) second prescription of vaccination and recovered R(ξ), respectively are presented as follows:

{S˙(ξ)=Ξ+ϑV1(ξ)−ωS(ξ)I(ξ)−χ1S(ξ)−φS(ξ),V1˙(ξ)=Ω+χ1S(ξ)−ϑV1(ξ)−χ3V1(ξ)−φV1(ξ),V2˙(ξ)=χ3V1(ξ)−χ2V2(ξ)−φV2(ξ),I˙(ξ)=ωS(ξ)I(ξ)−δI(ξ)−ηI(ξ)−φI(ξ),R˙(ξ)=δI(ξ)+χ2V2(ξ)−φR(ξ).(23)

Therefore, the aforesaid (23) can be transformed into Itôs type SDEs, by inserting of noise environment.

{dS(ξ)=[Ξ+ϑV1(ξ)−ωS(ξ)I(ξ)−χ1S(ξ)−φS(ξ)]dξ+ρ1S(ξ)dB1(ξ),dV1(ξ)=[Ω+χ1S(ξ)−ϑV1(ξ)−χ3V1(ξ)−φV1(ξ)]dξ+ρ2V1(ξ)dB2(ξ),dV2(ξ)=[χ3V1(ξ)−χ2V2(ξ)−φV2(ξ)]dξ+ρ3V2(ξ)dB3(ξ),dI(ξ)=[ωS(ξ)I(ξ)−δI(ξ)−ηI(ξ)−φI(ξ)]dξ+ρ4I(ξ)dB4(ξ),dR(ξ)=[δI(ξ)+χ2V2(ξ)−φR(ξ)]dξ+ρ5R(ξ)dB5(ξ),(24)

supplemented with initial conditions (ICs) (S(0),V1(0),V2(0),I(0),R(0))T=(S0,V10,V20,I0,R0)T∈R+5 while (ρj)j=1,2,…,5 represents the densities of unpredictability and Bj(ξ)j=1,…,5 are Brownian motions of every cohort.

Susceptible class S(ξ) is grown at frequency Ξ, and diminishing for first dosage of immune at speed ϑV1, and lowered at rate ωSI, those who acquire first prescription of vaccine to vulnerable at rate χ1S. Interaction involving the vulnerable group increases the contaminated category by incidence ωSI, while the infectious group recovers at rate δI. The healed group is boosted because the contaminated group survived at a rate of δI and added a new treatment of vaccination to recuperate at a rate of χ2V2. The Immunizations attracted of newborns at a rate Ω, get initial dose of immune to susceptible at a rate χ1S, and diminished owing to dwindling for initial dosage of immune to susceptible at rate ϑV1, acquire first dosage of vaccine to intravenous infusion of vaccine at a rate χ3V1. The immunized subsequent dose group is enhanced by receiving the initial prescription of immune and decreasing by receiving the subsequent prescription χ3V1 of immune and healing at a rate χ2V2. Background extinction rate φ and infectious mortality rate η for the contaminated group only fell in all sub-classes. N(ξ) represents the entire population count at time ξ, where N(ξ)=S(ξ)+V1(ξ)+V2(ξ)+I(ξ)+R(ξ) is consistent and combines adequately. Because the incubation phase is not important for the highly vulnerable contact, the unprotected cohort is excluded. Recruitment of new infants who have completed their first dosage of vaccine is placed in the immunocompetent group, while anyone who has not gotten their initial dosage of vaccine is placed in the vulnerable category. When an infectious agent comes into contact with another person, the virus can be transmitted. There is no breakdown of therapy. A person will either recuperate or perish after receiving the initial and subsequent doses of vaccination. The progression of the measles sickness is represented in Fig. 3.

Figure 3: Flow chart for measles infection model

3.1 Qualitative Aspects of Measles Epidemic Model

Here, we will use the structural evaluation of deterministic dynamics to analyze the evolution of stochastic frameworks in this part. Additionally, the (23) solution pertains to the (24) mean. To get equilibria, we first investigate a measles epidemic model (23), and then we examine adequate requirements wherein the equilibria are locally stable.

Theorem 3.1. If there be a viable domain Ω=(S(0),V1(0),V2(0),I(0),R(0)) of the systems (23) and (24), then Ω∗={(S(0),V1(0),V2(0),I(0),R(0))∈R+5:0≤N≤Ξ+Ωφ} is bounded.

Proof. Considering, the overall population in the system under discussion is provided by

N(ξ)=S(ξ)+V1(ξ)+V2(ξ)+I(ξ)+R(ξ).

After differentiation with respect to ξ and utilizing (23), yield

N˙(ξ)=Ξ+Ω−φN−ηI≤Ξ+Ω−φN.

Simple computations yield

ln⁡(Ξ+Ω−φN)≤−(ξ+C1),where C1 is a constant.

It follows that

Ξ+Ω−φN≥exp⁡(−φξ)exp⁡(−φC1).

Assume that N0=exp⁡(−φC1) and limξ↦∞exp⁡(−φξ)=0. Therefore (Ξ+Ω−φN)≥0 gives

N≤Ξ+Ωφ,for all ξ≥0.

Hence, Ω∗={(S(0),V1(0),V2(0),I(0),R(0))∈R+5:0≤N≤Ξ+Ωφ} is positive invariant for the frameworks (23) and (24), so that (24) introduce stochastic perturbations to (23).

3.2 Disease-Free Equilibrium Point (DFEP)

In order to find the EP at which the outbreak is eliminated from the community is found in this section. Allowing the right hand sides of (23) to zero and I=R=0, yields

{Ξ+ϑV1∗−(χ1+φ)S0=0,Ω+χ1S0−(ϑ+χ3+φ)V1∗=0,χ3V1∗−(χ2+φ)V2∗=0.(25)

After simplification, we have

(S0,V1∗,V2∗,0,0)=(Ξ(ϑ+χ3+φ)+ϑΩ(χ1+φ)(ϑ+χ3+φ)−χ1ϑ,χ1Ξ+Ω(χ1+φ)(χ1+φ)(ϑ+χ3+φ)−χ1ϑ,Ξχ1χ3+Ωχ3(χ1+φ)(χ1+φ)(χ2+φ)(ϑ+χ3+φ)−χ1ϑ),

i.e., is the phase where no disease exists in the environment.

3.3 Stochastic Fundamental Reproductive Number

Here, the fundamental reproductive number for (24) can be evaluated by employing Itôs formula for twice differentiable mapping on [0,T] to f1(I)=ln⁡(I). Utilizing the fact of Taylor series expansion so that we have

df1(ξ,I(ξ))=∂f1dξdξ+∂f1dIdI+12∂2f1∂I2(dI)2+12∂2f1∂ξ2(dξ)2+∂2f1∂ξ∂IdξdI.

Now, by means of system (24), we have

dI=[ωSI−(φ+η+δ)I]dξ+ρ4IdB4.

It follows that

df1(ξ,I(ξ))=([ωS−(φ+η+δ)]dξ+ρ4dB4)−12ρ42(dB42).

Higher order differentials (dξ,dB) approach quickly zero; (dξ)2↦0 and dξdB(ξ)↦0. The stochastic component dB2(ξ) is supplied as dB2(ξ)=dξ pertaining to Brownian motion principles. where we implement the underlying features to compute (dI(ξ))2, we have

df1(ξ,I(ξ))=([ωS−(φ+η+δ)−12ρ42]dξ+ρ4dB4).

Taking into account the next-generation matrix, we computed E0=(S,V1∗,V2∗,0,0), where S0=Ξ(ϑ+χ3+φ)+ϑΩ(χ1+φ)(ϑ+χ3+φ)−χ1ϑ.

Hence, the stochastic fundamental reproductive number is

R0s=2ωS0−ρ422(φ+η+χ1).(26)

3.4 Local Stability of DFEP

For the (24), we deliver an analogous stochastic eradication of transmission. In the long term, if R0<1, the population of the contaminated group will approach zero. If there is an R0<1, then the DFE of the system (24) is locally asymptotically stable.

Theorem 3.2. If there be R0<1, then I(ξ) will approaches to zero almost certainly tremendously stable of system (24) is locally asymptotically stable, i.e., limξ↦∞supln⁡(ξ)ξ<0.

Proof. By means of (23), we have

dln⁡(I)={ωS−(φ+η+δ)−12ρ42}dξ+ρ4dB4(ξ).

Performing integration from 0 to ξ, we have

ln⁡(I)−ln⁡(I0)={ωS−(φ+η+δ)−12ρ42}ξ+∫0ξρ4dB4(ξ).

It follows that

ln⁡(I)−ln⁡(I0)ξ≤{ωS−(φ+η+δ)−12ρ42}+𝒬(ξ)ξ.

Assume that 𝒬(ξ)=∫0ξρ4dB4(s1). As 𝒬(ξ) represents the martingale [34] having a quadratic variation described by

⟨𝒬(ξ),𝒬(ξ)⟩ξ=∫0ξρ42ds1=ρ42ξ.

In view of strong law, we have

limξ↦∞sup⟨𝒬(ξ),𝒬(ξ)⟩ξξ=ρ42<∞,

arrives at

limξ↦∞sup𝒬(ξ)ξ=0.

This leads to limξ↦∞supln⁡(ξ)ξ=ωS−φ−η−δ−12ρ42.

If R0<1, then ωS−φ−η−δ−12ρ42<0.

At DFE, we have

limξ↦∞supln⁡(ξ)ξ=ωS0−φ−η−δ−12ρ42,=(φ+η+δ)(ω0S0φ+η+δ−ρ422(φ+η+δ)−1),=(φ+η+δ)(R0−1),≤(R0−1)<0.

This concludes that R0<1.

3.5 Endemic Equilibrium Point (EEP)

Here, the equilibrium point at which the infection survives in the population is determined in this part. By putting all of the systems equations equal to zero, the EEP of model (23) can be determined as

{Ξ+ϑV1(ξ)−ωS(ξ)I(ξ)−χ1S(ξ)−φS(ξ)=0,Ω+χ1S(ξ)−ϑV1(ξ)−χ3V1(ξ)−φV1(ξ)=0,χ3V1(ξ)−χ2V2(ξ)−φV2(ξ)=0,ωS(ξ)I(ξ)−δI(ξ)−ηI(ξ)−φI(ξ)=0,δI(ξ)+χ2V2(ξ)−φR(ξ)=0.(27)

After simplifying, then system (23) has the EEP as follows:

(S∗,V1∗,V2∗,I∗,R∗)=(δ+η+φω,χ1(δ+η+φ)+ωΩω(χ3+ϑ+φ),χ3χ1(δ+η+φ)+χ3ωΩω(χ2+φ)(χ3+ϑ+φ),Ξ+Ω−φNη,δ(Ξ+Ω−φN)η+χ3χ1(δ+η+φ)+χ3ωΩω(χ2+φ)(χ3+ϑ+φ)).

3.6 Sensitivity Result

This section determines the importance of each component in the spread and prevention of measles infection through sensitivity evaluation. When a criterion improves, the investigation can help measure and compare the variation in a factor. This data is critical for studying the disease’s emergence and spread. With regard to a factor ℳ, the sensitivity criterion is determined by

ℳxj=∂R0∂xj∗xjR0,j=1,2,...,9,ℳω=∂R0∂ω∗ωR0=2ω(Ξ(ϑ+χ3+φ)+ϑΩ2ω(Ξ(ϑ+χ3+φ)+ϑΩ)−ρ42(χ3χ1+φ(φ+ϑ+χ3+χ1))>0,ℳΞ=∂R0∂Ξ∗ΞR0=2Ξω(ϑ+χ3+φ+χ3)2ω(Ξ(ϑ+χ3+φ)+ϑΩ)−ρ42(χ3χ1+φ(φ+ϑ+χ3+χ1))>0,ℳΩ=∂R0∂Ω∗ΩR0=ωϑΩ(φ+η+δ)(φ(φ+ϑ+χ3+χ1))(2ω(Ξ(ϑ+χ3+φ)+ϑΩ)−ρ42(χ3χ1+φ(φ+ϑ+χ3+χ1)))>0,ℳϑ=∂R0∂ϑ∗ϑR0=4ϑ(φ+η+δ)(φ(φ+ϑ+χ3+χ1)(ωχ3Ξ+ωφΞ+ωΩΞ)−ωρ42(φ(φ+χ3+χ1)+χ3χ1−χ1))(φ+η+δ)(φ(φ+ϑ+χ3+χ1))(2ω(Ξ(ϑ+χ3+φ)+ϑΩ)−ρ42(χ3χ1+φ(φ+ϑ+χ3+χ1)))−4ϑ(φ+η+δ)(Ξ(ϑ+χ3+φ)+ϑΩ)(φ(φ+χ3+χ1+χ3)−χ1)(φ+η+δ)(φ(φ+ϑ+χ3+χ1))(2ω(Ξ(ϑ+χ3+φ)+ϑΩ)−ρ42(χ3χ1+φ(φ+ϑ+χ3+χ1)))>0,ℳφ=∂R0∂φ∗φR0=12(φ+η+δ)(φ(φ+ϑ+χ3+χ1))(2ω(Ξ(ϑ+χ3+φ)+ϑΩ)+2ωϑΩ−ρ42(χ3χ1+φ)(χ1+φ)−ϑχ1)){4ωΞφ+2ρ42φ(2φ+ϑ+χ3+χ1)(φ+η+δ)(φ(φ+ϑ+χ3+χ1)+χ3χ1)+6φ3+2φ2(ϑ+χ3+χ1+η+δ)+(2φδ+2φδ)(ϑ+χ3+χ1+χ1χ3)(2ω(Ξ(ϑ+χ3+φ)+ϑΩ)−ρ42(φ(φ+ϑ+χ3+χ1)+χ3χ1))<0,ℳη=∂R0∂η∗ηR0=−η(φ+η+δ)<0,ℳδ=∂R0∂δ∗ϕR0=−δ(φ+η+δ)<0,ℳρ4=∂R0∂ρ4∗ρ4R0=2ρ42(φ(φ+ϑ+χ3+χ1+χ3χ1))2ω(Ξ(ϑ+χ3+φ)+ϑΩ)−ρ42(φ(φ+ϑ+χ3+χ1)+χ3χ1)<0.(28)

The sensitivity values of the components of the measles illness framework are shown in the aforementioned evaluation. The following is a summary of sensitivity index comprehension.

The fundamental reproducing number’s responsiveness levels to the essential factors are addressed and presented in the previous analysis. As a result, the factors ω,Ξ,Ω,ϑ and χ1 have high sensitivity values, indicating that increasing their levels has a significant effect on the spread of infection in the population. However, because the factors φ,η,δ and ρ4 have deleterious sensitivity values, diminishing their level will result in the infection outbreak spreading more.

4  A Fractional Stochastic Model of Measles Epidemic

Next, we investigate a generic measles stochastic model in which the standard temporal derivative is transformed into the global derivative in this part. It is specified as the GD of a differentiable mapping f1 with respect to an increasing non-negative continuous mapping 𝒢 as follows:

D𝒢f1(ξ)=limt↦ξf1(t)−f1(ξ)𝒢(t)−𝒢(ξ).

In fact, if 𝒢 is differentiable then

D𝒢f1(ξ)=f1 ′(ξ)𝒢 ′(ξ).

Considering the system (24), we can simply evaluate the disease’s elimination and prevalence. For this, we assume the system (24) with respect to the global derivative as follows:

{D𝒢S(ξ)=[Ξ+ϑV1(ξ)−ωS(ξ)I(ξ)−χ1S(ξ)−φS(ξ)]−σS(ξ)I(ξ),D𝒢V1(ξ)=[Ω+χ1S(ξ)−ϑV1(ξ)−χ3V1(ξ)−φV1(ξ)],D𝒢V2(ξ)=[χ3V1(ξ)−χ2V2(ξ)−φV2(ξ)],D𝒢I(ξ)=[ωS(ξ)I(ξ)−δI(ξ)−ηI(ξ)−φI(ξ)]+σS(ξ)I(ξ),D𝒢R(ξ)=[δI(ξ)+χ2V2(ξ)−φR(ξ)].

Since 𝒢 is differentiable, we have

{dS(ξ)=[Ξ+ϑV1(ξ)−ωS(ξ)I(ξ)−χ1S(ξ)−φS(ξ)]𝒢 ′(ξ)dξ−ρ1S(ξ)dB1(ξ),dV1(ξ)=[Ω+χ1S(ξ)−ϑV1(ξ)−χ3V1(ξ)−φV1(ξ)]𝒢 ′(ξ)dξ+ρ2V1(ξ)𝒢 ′(ξ)dB2(ξ),dV2(ξ)=[χ3V1(ξ)−χ2V2(ξ)−φV2(ξ)]𝒢 ′(ξ)dξ+ρ3V2(ξ)𝒢 ′(ξ)dB3(ξ),dI(ξ)=[ωS(ξ)I(ξ)−δI(ξ)−ηI(ξ)−φI(ξ)]𝒢 ′(ξ)dξ+ρ4I(ξ)𝒢 ′(ξ)dB4(ξ),dR(ξ)=[δI(ξ)+χ2V2(ξ)−φR(ξ)]𝒢 ′(ξ)dξ+ρ5R(ξ)𝒢 ′(ξ)dB5(ξ).(29)

It is important to mention that the framework is simple to determine if the ambient noise (ρj),j=1,2,...,5.

Taking the integral on both sides, we have

{S(ξ)=S(0)+∫0ξ(Ξ+ϑV1(τ)−ωS(τ)I(τ)−χ1S(τ)−φS(τ))𝒢 ′(τ)dτ−∫0ξσS(ξ)I(τ)𝒢 ′(τ)dτ,V1(ξ)=V1(0)+∫0ξ(Ω+χ1S(τ)−ϑV1(τ)−χ3V1(τ)−φV1(τ))𝒢 ′(τ)dτ+∫0ξρ2V1(τ)𝒢 ′(τ)dB2(τ),V2(ξ)=V2(0)+∫0ξ(χ3V1(τ)−χ2V2(τ)−φV2(τ))𝒢 ′(τ)dτ+∫0ξρ3V2(τ)𝒢 ′(τ)dB3(τ),I(ξ)=I(0)+∫0ξ(ωS(τ)I(τ)−δI(τ)−ηI(τ)−φI(τ))𝒢 ′(τ)dτ+∫0ξσS(ξ)I(τ)𝒢 ′(τ)dτ,R(ξ)=R(0)+∫0ξ(δI(τ)+χ2V2(τ)−φR(τ))𝒢 ′(τ)dτ+∫0ξρ5R(τ)𝒢 ′(τ)dB5(τ).(30)