Deterministic-stochastic analysis of fractional differential equations malnutrition model with random perturbations and crossover effects

To boost the handful of nutrient-dense individuals in the societal structure, adequate health care documentation and comprehension are permitted. This will strengthen and optimize the well-being of the community, particularly the girls and women of the community that are welcoming the new generation. In this article, we extensively explored a deterministic-stochastic malnutrition model involving nonlinear perturbation via piecewise fractional operators techniques. This novel concept leads us to analyze and predict the process from the beginning to the end of the well-being growth, as it offers the possibility to observe many behaviors from cross over to stochastic processes. Moreover, the piecewise differential operators, which can be constructed with operators such as classical, Caputo, Caputo-Fabrizio, Atangana-Baleanu and stochastic derivative. The threshold parameter is developed and the role of malnutrition in society is examined. Through a rigorous analysis, we first demonstrated that the stochastic model’s solution is positive and global. Then, using appropriate stochastic Lyapunov candidates, we examined whether the stochastic system acknowledges a unique ergodic stationary distribution. The objective of this investigation is to design a nutritional deficiency in pregnant women using a piecewise fractional differential equation scheme. We examined multiple options and outlined numerical methods of coping with problems. To exemplify the effectiveness of the suggested concept, graphical conclusions, including chaotic and random perturbation patterns, are supplied. Consequently, fractional calculus’ innovative aspects provide more powerful and flexible layouts, enabling us to more effectively adapt to the system dynamics tendencies of real-world representations. This has opened new doors to readers in different disciplines and enabled them to capture different behaviors at different time intervals.


Model and preliminaries
This portion describes a mathematical model of nutritional deficiencies in pregnant women.Figure 1 depicts the evolution of this concept.Table 1 contains all requirements and their understandings.
Malnourished pregnant women S f lead up to malnourished boys Mb and girls Mg .Boys and girls are undernourished as a result of vulnerable females at propagation rates b and g , respectively.Even before lowweight newborns are not granted immediate healthcare treatment, they develop into underdeveloped kids at rates of γ b and γ g for both genders, respectively.The fundamental fatality rate is signified by ϑ , and the hand- ful of underweight people is symbolized by Ū .Furthermore, N h determines the the entire community, where N h = S f + Mb + Mg + Ū .The respective complex differential equations framework illustrates malnutrition at various varying phases of an entire evolution 40 : where ε indicates the progression of disease to a newly born community and B signifies the new female recruit- ment rate.In (2.1), the other specifications δ g , χ b , χ g depict the restoration proportion of malnutrition girls and the proportions of underweight newborns progressing to Mb and Mg , respectively.Because framework (2.1) is concerned with human population demographics, all specifications and system specifications are assumed to (2.1)The proportion of boys who progress from malnourishment to underweight 0.01 cell ml −1 day −1 person −1 γ g The proportion of girls who progress from malnourishment to underweight 0.1 day www.nature.com/scientificreports/be non-negative, respectively 21,40 .The mathematical formulation provided by (2.1) has been investigated earlier in 40 to explore the propagation of food insecurity and underweight participants in a community.This framework, even so, excludes the consequences of memory, which are present in several natural systems.When analyzing the existence of stochastic processes using the Has'minskii concept 41 , the challenges experienced include how to assemble a Lyapunov function and determining an appropriate subset such that the dispersion operator is negative beyond the subset.Encouraged by the monitoring and evaluation process, we contemplate the stochastic theory of underweight four-species cooperative frameworks in this article as follows: where W k (ζ ), k = 1, ..., 4 denotes the standard one-dimensional Brownian motion described on a complete filtered probability space { ℧, P, {P ζ } ζ ≥0 , P} having a σ-filtration {P ζ } ζ ≥0 .Also, σ k , k = 1, ..., 4 is the white noise intensity.The framework is hypothesized all segmentation based this inquiry (2.1) is acknowledged as a complete probability space ( ℧, P, {P ζ } ζ >0 , P) having a right continuous filtration {P ζ } ζ >0 and an {P 0 } constituted all the elements with criterion zero.
The stochastic DE in d-dimensions is presented below: where u : Introducing the functional where then Itô's method can be described as: Here, we furnish the associated overview here to assist viewers who are familiar with FC (see; [25][26][27] ).Definition 2.1 ( 25 ) The Caputo fractional derivative of order for a continuous function G is defined by Our second notion is a fractional derivative without singular kernel introduced by Caputo and Fabrizio 26 .

Qualitative aspects of proposed model
Shah et al. 40 researched the global features of the nonlinear malnutrition model, explaining how boys and girls relocate from one cohort to another as described in (2.1).For the sake of simplicity, we denote For framework (2.1), there is always a feasible region as follows: Thus, the malnutrition steady state E 0 = B+ε ϑ , 0, 0, 0 .Now, using the next-generation matrix approach 42 , compute the basic reproduction number R 0 .The next- generation matrix can be described as FV −1 , where F and V are both Jacobian matrices of individuals in an experimental setting is presented as follows: Therefore, the basic reproduction number R 0 is the spectral radius of matrix FV −1 which is presented by .
Vol.:(0123456789) Existence of ergodic stationary distribution.When an ailment arrives in a community and begins to grow rapidly, health authorities are notably interested in its protracted behaviour, which can be efficaciously dealt with mathematically by incorporating stability techniques.In terms of deterministic modelling techniques, it is possible to demonstrate that in specific settings, the accompanying framework has an endemic equilibrium that is globally asymptotically stable.However, there is no endemic equilibrium in stochastic structures such as model (2.1), making it difficult to predict when the disorder will persist in communities.Depending on Has'minskii's 41 concept, we aim to demonstrate in this portion that framework (2.1) has an ESD, indicating that the ailment will endure.If we assume that σ 1 = σ 2 = σ 3 = σ 4 = 0, we can conveniently procure a deterministic preview of scheme (2.1); nevertheless, the stochastic model is remarkably distinct from its corresponding deterministic one.It is also understood that there is no endemic disorder state in the stochastic framework.As a result, the linear stability explanation cannot be used to investigate the disease's perseverance.As a result, we focused on the envisaged system's stationary distribution (2.2), which assumes that the concern will persist.Assume that the function X(ζ ) is a regular time-homogeneous Markov process R n 1 + with the mathematical version The diffusion matrix is as shown in: Suppose there is a Markov technique X(ζ ) admits a unique stationary distribution π(.) if there is a bounded region Ū ∈ R d having a regular boundary such that its closure Ū ∈ R d has the subsequent criterion: (M 1 ) The smallest eigenvalue of the diffusion matrix A(ζ ) is very close to zero in the open region Ū and some of its neighbours. (M 2 ) For κ ∈ R d Ū, the mean time it takes for a path emanating from κ to approach the set Ū is finite, and sup κ∈κ Eφ κ < ∞ for each compact subset.Moreover, if f (.) is an integrable mapping with regard to the measure π(.), then Now let's classify some other threshold significance for future needs: Proof First, we should indeed illustrate the design specifications M 1 of Lemma 3.2 to validate the Theorem, we assert a positive C 2 -mapping H 1 : R 4 The positive components must be determined later in this case.These specifications must be found out later on.
To communicate directly with (3.11), we first should apply Itô's approach to the design process (2.2) as (3.9) (3.11) where Consequently, we have Furthermore, one can achieve that here, ψ 4 > 0 is a fixed value that will be discovered later.It is critical to illustrate that The following procedure will demonstrate that H 2 ( X) has a least value H 2 ( X(0)).
The partial derivative of H 2 ( X) regarding to X is as follows ), (3.15) www.nature.com/scientificreports/It is straightforward to show that H 2 has a distinctive stagnation point, which seems ascertained by the afore- mentioned computation: Also, the Hessian matrix of H 2 X at X(0) is presented by the following The preceding link demonstrates unequivocally that B is a non-negative definite matrix.Thus, H 2 ( X) has mini- mum value ( X(0)).Finally, Lemma 3.2 concludes and the continuity of H 2 ( X) that it has a distinct lowest value of about ( X(0)) in the interior of R 4 + .Further, we define a positive C 2 : R 4 The application of Itô's strategy and the structure (2.2) will give us or finally we can express where The representation of a collection is supplied by where ε k , k = 1, 2 , are fixed which are extremely small and will have to be revealed afterward.The domain R 4 + \ Y is separated into ten zones, which are as follows: (3.16) )  Finally, all of the previous contexts demonstrate that a non-negative B exists, so where a path starting with ū led directly to the collection Y, Performing integration on (3.24) over 0 to φ (n 1 ) (ζ ), applying expectation and Dynkins process, we conclude that (3.24) Thus, the M ∞ of Lemma 3.2 is fulfilled.The proposed stochastic structure has a unique ESD as an outcome of Lemma 3.2.

Numerical simulation
In what follows, we will contemplate the numerical modelling using the power-law kernel, the exponential decay kernel and generalized Mittag-Leffler kernel, respectively.
Power-law kernel.Here, we will examine at the nonlinear dynamics of poor nutrition systems (2.1) and (2.2) that incorporate malnutrition and underweight, using conventional, index-law and subsequently stochastic treatments.If we consider T to be the final time of dissemination, then the computational structure will be con- structed during the initial process utilizing the classical derivative implementation, followed by the power-law kernel in the other approach and eventually the random perturbations in the later stages.The computational framework that accounts for this occurrence is then given as follows: (3.25) (3.27) Vol

Exponential decay kernel.
In this segment, we will take a glance at the simulation framework of a poor nutrition framework that involves malnutrition and underweight congregation members, as well as conventional, exponential decay and random perturbations.If we define T as the ultimate dissemination duration, then the computational formation will be established during the initial phase that uses the integer-order derivative implementation, then comes the exponentially decaying kernel in the other phase, and finally the Gaussian noise in the future period.In this reference, the scientific model we are using to exemplify this incidence is as follows: Here, we employ the method reported in 36 for the situation of Caputo-Fabrizio derivative to calculate and investigate the piecewise configuration (4.7)-(4.9).We begin the methodology by doing the following: (4.12) where ℑ 1 , ℑ 2 and ℑ 3 are defined in (4.4)-(4.6).

Results and discussion
The simulation results of the framework (2.2) for all four sets of data reveal that malnutrition and body immunization have a massive effect on undernourished pregnant females S f , conceive famished boys Mb , girls Mg , and underweight individuals Ū via the crossover effects.Various dietary prestige and distinct immune defence stages have various sorts of effects, as shown by the graphs of differential equations for malnourished individuals utilizing the numerical scheme proposed by Atangana and Araz 36 .To overcome the malnutrition problem, an initial value and random intensities are required.It determines the variation in attributes over time depending on that initial value.To test the modifications in all three scenarios, we employ one initial value and steadily observe how initial values affect the modification.Also, R s 0 = 1.245 > 1, where R s 0 is described in Section 3. We can verify that system (2.2) will persist for a long time using the findings of Theorem 3.3 and a distribution of π(.).The numerical simulations below confirm this.Let us now examine the consequences for each individual.
Figures 2a and b depict the modifications in undernourished pregnant females' cases for normal nutrient intake, Figure 3a and b represents the view of birth to malnourished boys, Figure 4a and b denotes birth of malnourished girls and Figure 5a and b represents the under weight individuals with immune function in the sets of parameters under various random intensities σ 1 = 0.08, σ 2 = 0.09, σ 3 = 0.1, 4 = 0.12 and initial condi- tions S f (0) = 30, Mb = 2, Mg = 4 and Ū = 1, respectively via the piecewise fractional differential equations techniques.For the first set of values, we explore that a starting value of 30 results in linear decay, whereas values 2, 4, and 1 result in logarithmic growth.It shows logarithmic and wave growth for all random intensities in the second, third, and fourth sets of values when the Caputo fractional derivative is convoluted with the deterministic-stochastic case.The significance of immune function and nourishment is evident from the research, and it is interesting to note that maintaining strong immunity and appropriate nourishment in the bloodstream will substantially decrease hypersensitivity, decrease the risk of infestation, and improve the mental health process.
Figure 6a and b depict the modifications in undernourished pregnant females' cases for normal nutrient intake, Fig. 7a and b represents the view of birth to malnourished boys, Fig. 8a and b denotes birth of  The significance of immune function and nourishment is evident from the research, and it is interesting to note that maintaining strong immunity and appropriate nourishment in the bloodstream will substantially decrease hypersensitivity, decrease the risk of infestation, and improve the mental health process.It is indeed clear from simulation analysis that the consequences of dietary patterns and immune function tend to vary with changes in other attributes connected to the model's conceptualization.This system will assist those responsible for attempting to make decisions to ameliorate losses incurred by complexities in pregnancy.Figure 10a and b depict the modifications in undernourished pregnant females' cases for normal nutrient intake, Fig. 11a and b represents the view of birth to malnourished boys, Fig. 12a and b denotes birth of malnourished girls and Fig. 13a and b represents the under weight individuals utilization and rational immune function were ascertained in a variety of variables with various different random intensities, σ 1 = 0.08, σ 2 = 0.09, σ 3 = 0.1, σ 4 = 0.12 and initial conditions S f (0) = 30, Mb = 2, Mg = 4 and Ū = 1, respectively via the piecewise fractional differential equations approaches.For the first set of values, we notice that a starting value of 30 results in linear decay, whereas values 2, 4, and 1 result in logarithmic growth.When the Atangana-Baleanu fractional derivative is combined with the deterministic-stochastic case, it exhibits logarithmic and wave expansion for all random intensities in the second, third, and fourth value systems.Individual's undernutrition has been assessed to quantify their resistance to destabilization during pregnancy.The immune system is impacted by an effective diet and nutritional requirements.As a result, the only long-term strategy for surviving in the current environment is to boost the immune system, develop diet and exercise plans.This article examines the relevance of nourishment in boosting resistance and provides some skilful and truthful nutritional recommendations for coping with the intricacies of pregnancy.
The quality of the graphs is very high, with a numerical scheme with respect to the fractional-order = 1 in Figs.14, 15, 16,  respectively.This fact shows that an integer-order model can approximate, within a given random perturbation, data generated by a fractional-order one with very high precision without the need for excessively high orders of derivation or computational resources.Figures 17, 18, represents the histogram plots for the proposed system (2.2).In reality, controlling poor nutrition will not affect the disruption of health issues or the spread of various infections.Simultaneously, when other considerations hinder development of resistance to a newborn child population, such as constant treatment and a healthy life campaign, the number of deaths decreases along with the number of malnourished and underweight.This is essentially consistent with the system (2.2) research findings in this paper.Finally, these findings indicate that fractional-order techniques are instinctively superior to classical ones when dealing with phenomena such as memory effects and non-local behaviour in general.

Conclusion
Numerical modelling is useful for analysing societal problems and following up with cost-effective remedies.
Fractional calculus and stochastic perturbation, among existing schemes, have a phenomenal capacity for recording, eventually afflicted by noise sources and memory effects, which have been revealed to include almost all biomedical functions.This research represents a deterministic-stochastic framework that employs crossover consequences to predict the intricacies of undernutrition in pregnant women.Initially, we use an inventive interconnection of Lyapunov candidates to determine the existence and uniqueness of the global non-negative outcome corresponding to the unit likelihood of occurrence.The necessary prerequisites for the stationary distribution of poor nutrition are therefore calculated.Whereas the generalized Mittag-Leffler kernel, exponential decay and index law have been shown to be competent at portraying numerous crossover tendencies, we assert that their abilities to achieve this might be strictly limited to the true extent of the environment.In the intervention of undernourishment as well as other insatiable hungers and dietary patterns influencing ailments, the concentration of Gaussian white noise is pivotal.The strategy requires stochastic perturbations (noise) and biological methods to enhance understanding of the scientific studies, which have critical repercussions for antibacterial drugs and genetic engineering.Several other intriguing discussions need to be researched further, such as the fractional nutrition model with Lévy noise and Poisson noise 44,45 , which can generalize Brownian motion and include several important jump and impulsive random processes often found in neural and financial engineering models.

Figure 2 .
Figure 2. Two-dimensional view and phase portrait of dynamic pattern of malnutrition system (4.1)-(4.3)for undernourished pregnant women S f using Caputo fractional derivative of order = 0.95 with lowest random perturbations.

Figure 3 .
Figure 3. Two-dimensional view and phase portrait of dynamic pattern of malnutrition system (4.1)-(4.3)for birth to malnourished boys Mb using Caputo fractional derivative of order = 0.95 with lowest random perturbations.

Figure 4 .
Figure 4. Two-dimensional view and phase portrait of dynamic pattern of malnutrition system (4.1)-(4.3)for birth to malnourished girls Mg using Caputo fractional derivative of order = 0.95 with lowest random perturbations.

Figure 5 .Figure 6 .
Figure 10a and b depict the modifications in undernourished pregnant females' cases for normal nutrient intake, Fig. 11a and b represents the view of birth to malnourished boys, Fig. 12a and b denotes birth of malnourished girls and Fig. 13a and b represents the under weight individuals utilization and rational immune function were ascertained in a variety of variables with various different random intensities, σ 1 = 0.08, σ 2 = 0.09, σ 3 = 0.1, σ 4 = 0.12 and initial conditions S f (0) = 30, Mb = 2, Mg = 4 and Ū = 1, respectively via the piecewise fractional differential equations approaches.For the first set of values, we notice that a starting value of 30 results in linear decay, whereas values 2, 4, and 1 result in logarithmic growth.When the Atangana-Baleanu fractional derivative is combined with the deterministic-stochastic case, it exhibits logarithmic and wave expansion for all random intensities in the second, third, and fourth value systems.Individual's undernutrition has been assessed to quantify their resistance to destabilization during pregnancy.The immune system is impacted by an effective diet and nutritional requirements.As a result, the only long-term strategy for surviving in the current environment is to boost the immune system, develop diet and exercise plans.This article examines the relevance of nourishment in boosting resistance and provides some skilful and truthful nutritional recommendations for coping with the intricacies of pregnancy.The quality of the graphs is very high, with a numerical scheme with respect to the fractional-order = 1 in Figs.14,15, 16, close to that of the identified nutritional Caputo-derivative fractional model (4.1)-(4.2),Caputo-Fabrizio fractional derivative model (4.7)-(4.8)and Atangana-Baleanu fractional derivative model (4.12)-(4.14),

Figure 7 .Figure 8 .
Figure 7. Two-dimensional view and phase portrait of dynamic pattern of malnutrition system (4.7)-(4.9)for birth to malnourished boys Mb using Caputo-Fabrizio fractional derivative of order = 0.95 with lowest random perturbations.

Figure 17 .
Figure 17.Frequency plots of malnutrition model (2.2) for malnourished pregnant women S f and birth to malnourished boys Mb having probability density function of normal distribution N(mean, variance)..

Figure 18 .
Figure 18.Frequency plots of malnutrition model (2.2) for birth to malnourished girls Mg and underweight individuals Ū having probability density function of normal distribution N(mean, variance)..

Table 1 .
Explanation of system's feature.
is GAS in .