Novel Numerical Estimates of the Pneumonia and Meningitis EpidemicModel via theNonsingularKernelwithOptimalAnalysis

Department of Mathematics, Government College University, Faisalabad 38000, Pakistan Department of Mathematics, COMSATS University, Islamabad, Pakistan Department of Mathematics, National Mathematical Centre Abuja, Abuja 900211, Nigeria Department of Mathematics Education, University of Education, Winneba, Kumasi Campus, Ghana Department of Mathematics and Computer Sciences, Faculty of Science Menoufia University, Shebin, Elkom 32511, Egypt Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia


Introduction
Fractional calculus has gained popularity over the years for representing a plethora of new challenges in fields such as computational virology, quantum theory, technology, and numerous others, wherein fractional-order (FO) operators are either singular (Caputo derivative and Riemann-Liouville (RL) fractional derivatives) or nonsingular (Atangana-Baleanu and Caputo-Fabrizio derivatives) [1][2][3][4].
However, the variation between integer-order and FO derivatives is that the integer-order derivative depicts the functionality of a complex nonlinear network for the entirety of the period, whereas the FO derivative operator represents a characteristic of a logistic scheme for the enormous moment. Furthermore, the integer-order derivative reflects a dynamic state's spatial information, while the FO derivative formulation of a complex process encompasses the project process domain [5][6][7][8]. On the other hand, in modelling specific cases, implementing derivation operators via noninteger values is critical for articulating generational requirements and the reliability of memories as a key component of various systems [9,10]. erefore, the advent of multiple meanings of a fractional derivative is fascinating and creates an incentive to identify the intricacies of natural surroundings in the context that certain challenges in existence pursue the index law for the RL fractional operator, some also implement the Mittag-Leffler (ML) rules for the Atangana-Baleanu fractional derivative operator, and many others try to emulate the exponentially decaying law for the Caputo-Fabrizio fractional operator or an amalgamation of such regulations [11,12].
Furthermore, the fact that many nonlinear mechanisms in connection to complex processes are discovered to be nonlocal with protracted recollection in time, and that inherently fractional derivation operators can characterize such kinds of mechanisms more appropriately than integer derivatives, has resulted in a rapid boost in the description of various nonlinear dynamic structures utilizing fractionalorder derivatives [13]. In other respects, fractional-order operators are the ideal way to characterize or reveal crucial characteristics of several complex processes.
Pneumonia, which is classified as an infectious agent, is responsible for the loss of individual lives worldwide due to the inhalation of potential pathogens, primarily Streptococcus pyogenes [14]. Other disorders susceptible to infection include meningitis, respiratory ailments, and nasal congestion, among others (see Figure 1).
Infections can occur in adults, including infants to adults, and pneumonia turns severe whenever the immune system is weakened, as it is in vulnerable populations, as well as when it is co-infected with several other infections, such as meningitis [15]. Meningitis is a contagious bacterial ailment of the membrane that surrounds the cerebrum and spinal cord [16] (see Figures 2(a)and 2(b)).
Meningitis is caused by viral illness in 80 percent of cases, which is caused by pneumococcal pneumonia, streptococci, and Neisseria meningitidis. A majority of the research was carried out to determine the governing strategies of contagious ailments in a population (see [17][18][19]). In order to analyze the evolution of contagious infections in a population, various researchers have designed simulations to examine the evolution of various contagious infections. e authors [20,21] also mentioned the co-dynamics of pneumonia and waterborne illness infections [22], as well as economic evaluation and optimal monitoring. e findings of this investigation demonstrated that among the suggested techniques, preventing infectious disease and treating pneumonia seem to be the most cost-effective. Many researchers have examined co-infection of communicable ailments with HIV and pneumonia co-infection [23], while Akinyi et al. [24] looked into bacterial meningitis and plasmodium co-infection.
To the best of our knowledge, since its invention in 2016 by Atangana and Baleanu [2], the innovative fractional derivative operator has been significantly employed in a multitude of disciplines of scientific innovation [25,26]. For a short period of time, modelling using the nonsingular fractional derivatives culminates in a stochastic, deterministic, and physical process [27][28][29]. Atangana [30] presented the extension of rate of change concept from local to nonlocal operators with applications. Baleanu et al. [31] constructed a fractional framework for a malignant cell's ability to use information and assessed how chemotherapeutics influenced the system. As a consequence of the observations, the optimum modulation technique was found to be effective. Sene [32] studied the formulation of the governing equations of a fractional diffusion equation in the setting of the fractional operator with a Rabotnov fractional exponential kernel. Zhao [33] presented the fuzzy-based strategy to suppress the novel coronavirus (2019-NCOV) massive outbreak.
To approach the ABC fractional derivative, Owolabi [34] adopted a two-step family of Adams-Bashforth procedures to perform evaluation and modelling techniques on a problem. Demirci et al. [35] took into account an SEIR fractional model and its experimental validation. In [36], the SEIRA mathematical model is examined using the ABC fractional derivative operator with the ML kernel. us, we can conclude from the above-mentioned research that fractional derivatives have several implications in numerical techniques and in the study of realworld processes. e newly formed Atangana-Baleanu fractional operator, notably, has gained appreciation and acceptance as a result of its variety of uses in ecological, chemical, and biomedical sciences, as well as a variety of other complex studies.
In this research, we investigate a pneumonia and meningitis (P) mathematical formulation with a dominating interaction incidence, which is inspired by the aforesaid considerations. e fractional derivative formulation of the system is generated, and its analytical simulation is estimated by applying the newly reported Atangana-Baleanu fractional derivative and Toufik-Atangana numerical solutions [37]. e P epidemiological theory utilizing the Atangana-Baleanu fractional derivative has not previously been examined, to the best of the researchers' expertise. e researchers further claim that in the relevant research, robust regulation characterization of mathematical formulae in the context of Atangana-Baleanu fractional operators is infrequent in the earlier research. As a response, we examine the sensitivity characterization of the P model in this article.

Depiction of the Model
In this part, the ABC fractional derivative form of the P epidemic mathematical systems is introduced. Let us just continue with a review of the ML kernels' notions and their concerning consequences.
Lemma 3 (see [40]). Let there be a function h 1     investigation has been developed utilizing the sequence chart below (Figure 3). e numerical approach with numerical approximation employed in this work is represented by the governing formulae, which are predicated on the workflow.
Here, we examine a community that is diversified in this scenario. In this framework, we explore the probabilistic seven-dimensional global species. In this framework, we explore the probabilistic seven dimensional global species such as sensitive group (S), pneumonia virulent (I p ), meningitis epidemic(I m ), pneumonia and meningitis co-infectious(I pm ), pneumonia healed(R p ), meningitis regained(R m ), and pneumonia-meningitis co-infectious retrieved are the numerous subgroups. (R pm ). Susceptible are recruited with rate of Π through birth or immigration, and their number increases from individuals that come from subclasses of pneumonia recovered, meningitis recovered, and coinfectious recovered by losing their temporary immunity with rate of ϕ 1 , ϕ 2 , and ϕ 3 , respectively. In the entire susceptible population, individuals can get pneumonia with contact rate of q 1 from a pneumonia-only infected or coinfected person with force of infection of pneumonia g 1 � q 1 (I p + I pm )/Nand join I p compartment. In a similar way, individuals can get meningitis by a contact rate of q 2 from a meningitis-only infected or coinfected person with force of infection of meningitis g 2 � q 2 (I m + I pm )/N and join I m compartment. Pneumonia-only infected individuals also can get an additional meningitis infection with a force of infection g 2 and join coinfected compartment (I pm ). Participants emerge from meningitis merely affected by a segment when attacked by pneumonia using g 1 a high intensity of transmission to boost the postoperative subsystem. Patients who have only been afflicted with pneumonia may recuperate at a frequency of ρ 1 approximately and can represent the pneumonia only cured category (Ip). As a result, meningitis-only people who are contaminated recover at a rate of ρ 2 and are assigned to the meningitis-only healed group (I pm ). People in the immunecompromised class also retrieve at a speed ρ, but they either heal just from pneumonia ailment and enter pneumonia only retrieved storage area, with the possibility of ρ(1 − ϵ), or revive just from meningitis ailment and participate in meningitis

Complexity
only retrieved zone with the plausibility of ρh 1 (1 − ϵ), or restore from all these maladies and participate in co-infected recapture zone with a likely hood of ρ(1 − ϵ)(1 − h 1 ). Furthermore, ϱ represents the natural death rate. Furthermore, η 1 is the relative risk induced entirely by pneumonia, and η 2 is the number of fatalities due to meningitis.

Model Configuration.
In this part, we investigate the framework's descriptive characteristics. We segmented the comprehensive framework into various DEs as presented in (5), which are estimates for pneumonia and meningitis (P), to make the process easier. e combination of DEs describes formal system (6) that incorporates the hypotheses, the saturate interaction frequency, and the flowchart ( Figure 3) and analyzes model (6) by ABC fractional derivative.
where the kernels are set as follows: subject to the nonnegative initial values S(0) � S 0 , I p (0) � I p0 , I m (0) � I m0 , us, in the absence of disease, the differential equation of the total population size is dN/dζ � π − ϱN − η 1 (I p + I pm ) − η 2 (I m + I pm ). Table 1 summarizes the characteristics that were considered in the analysis (7).

Consequences on the Existence-Uniqueness.
Here, employing the Banach fixed point f p theorem for contraction mapping, the existence-uniqueness of the result for the ABC fractional model suggested in (7) is demonstrated. Before actually moving on, it is important to remember the two additional theorems. For further details, see [41] and the references cited therein.
We continue as follows to demonstrate the method's existence-uniqueness. We acquire framework (7) while we implement the AB fractional integral.
e contraction and the Lipschitz hypothesis are the foundations of our following theorem.

Theorem 1. For the following kernels
Proof. In order to satisfy Lipschitz assumptions, we have where It is worth noting that Θ 1 (ζ, S 1 ) holds the Lipschitz assumption containing Lipschitz constant Analogously, we can analyze the existence consequences of L ℓ , ℓ � 2, 3, . . . , 7 and the contraction technique for Complexity At ζ � ζ n , n � 1, 2, . . ., introduce the subsequent recursive version of (9): In (12), the variations of successive components are written in the following form: Implementing the norm on the aforementioned system (13), we have In addition, the first identity in (14) can be simplified to the representations: Ultimately, we have Repeating the same procedure, the subsequent terms of (14) can be simplified to the following: 8 Complexity □ Theorem 2. e fractional P model presented in (7) has a solution if U 0 holds the variant Proof. By means of (16) and (19), we have eorem 1 verifies the validity of the solution (the existence of a f p ), and which shows that the mappings S(ζ), I p (ζ), I m (ζ), I pm (ζ), R p (ζ), R m (ζ), and R pm (ζ) are solution of the model (7).

□
Here, the Banach f p theorem asserts the validity of the result of system (7) by eorems 1 and 2. eorem 3 verifies the solution's uniqueness.

Theorem 3.
Let there be a unique solution of the fractional P model (7) given that Proof. Suppose that S 1 , I p 1 , I m 1 , I pm 1 , R p 1 , R m 1 , and R pm 1 are another solution to fractional P model (7). en, Employing the norm to the aforesaid equation, we have Since (1 − (1 − β/M(β) + ζ β /Γ(β)M(β))L 1 ) > 0, we acquire ‖S − S 1 ‖ � 0. Consequently, we have S � S 1 . Repeating the same process, we can prove that and R pm � R pm 1 . is completes the proof.

Qualitative Analysis
Several key aspects of the developed framework, including boundedness, the existence of equilibria, and fundamental reproductive quantity, will be outlined and discussed.

Positively Invariant and Boundedness.
In order to find the invariant domain, we surmise the overall population N � S + I p + I m + I pm + R p + R m + R pm .
Let the domain of the fractional P model (7) that is epidemiologically sustainable determined by In order to prove that the collection [ is positively invariant, apply Lemma 1, and we have It is clear from (29) that every outcome of (7) is positive and will be in R 7 + . erefore, the set [ presented in (28) is positively invariant for fractional P model (7).
Additionally, to prove the boundedness of the solutions of the fractional P framework, we proceed by aggregating all of the system's components, offering Using the Laplace transform, we have

Complexity
where λ � (β − 1)ϱ/M(β). In view of inverse Laplace transform, we have where E β,c presents the ML function. Based on the assumption that the ML function exhibits asymptotic nature, It is not really hard to perceive that N(ζ)↦π/ϱ, and this concludes that system (7) is biologically sustainable in the domain.
Proof. By means of P model, we can write and It is clear by expression (43) that χ � (πϱ, 0) is the global asymptotic state.
is is reinforced by the result, which is as follows: When ζ↦∞, the solution follows S↦π/ϱ. is implies the global convergence of (43) en, Y 2 (χ 1 , χ 2 ) can be expressed as which illustrates that Y 2 (χ 1 , χ 2 ) < 0, and this suggests that the secondary assumption (ii) is not provided, and V � (χ, 0) may not be globally asymptotically stable when R 0 < 1.

Sensitivity Analysis.
rough sensitivity analysis, the most important components for the development and regulation of transmission in the population are identified. We employ the strategies provided in [43] to accomplish this. With consideration to a factor, assume x 1 , the responsiveness factor of R 0 , is determined by ψ x R 0 1 � zR 0 /zx 1 x 1 /R 0 . As R 0 � max R 0p , R 0m , the sensitivity evaluation of R 0p and R 0m is conducted individually as follows: Complexity 13 (47) e analysis shows that characteristics with highly accurate index values, especially q 1 and q 2 , have a huge promise for spreading bacterial viruses, meningitis, and their co-infections in the congregation even though they boost their corresponding propagation quantity, which seems to be the mean value of supplementary illnesses. If the estimated quantities containing a negative sensitivity value are improved while the quantities of the other characteristics remain stable, the factors having a negative sensitivity level contribute significantly to reducing the spread of P in the population.

Numerical Approaches of PM Model.
Here, we leverage the Toufik-Atangana [37] approach to generate a systematic formula for scheme (7) in this part.
In consideration of the first component of (7), we find Observing (9), one can calculate for (48) in the problem described in the following: Utilizing Lagrange's interpolating polynomial technique on [ζ ℘ , ζ ℘+1 ]to∇ 1 (y, S(y)) � π/(g 1 + g 2 + ϱ) − (ϕ 1 R p (y) + ϕ 2 R m (y) + ϕ 3 R pm (y))/(g 1 + g 2 + ϱ) − S(y), then we have where Z � ζ ℘ − ζ ℘− 1 . Inserting (49) into (50), we have 14 Complexity where and Also, utilizing ζ κ � κZ into (52) and (53) represents and Eventually, we can write (56) in the form of (54) and (55) as follows: ∇ 1 ζ n , S ζ n , I p ζ n , I m ζ n , I pm ζ n , R p ζ n , R m ζ n , R pm ζ n , Additionally, the formulations for the remaining model factors are as follows: ∇ 3 ζ n , S ζ n , I p ζ n , I m ζ n , I pm ζ n , R p ζ n , R m ζ n , R pm ζ n 16 Complexity 3.8. Results and Discussion. In this section, simulation studies for the resulting structure (P co-infection model) are carried out, taking into consideration the ABC derivative fractional operator having ML kernel. We employed MATLAB 2022 to assess the influence of several factors in the proliferation as well as to prevent P co-infection. For modelling purposes, the model parameters in Table 1 are considered. In Figure 4, by maintaining the interaction rate stable, q 1 � 0.9, we evaluated the influence of ρ 1 in reducing the amount of pneumonia exclusively infected people. Figures 5  and 6 show that as the quantity of ρ 1 increases, the proportion of pneumonia exclusively susceptible people decreases. Figure 5, 6, and 7 indicate that the proportion of instances in categories I p , I m , I pm , R p , R m , R pm decreased dramatically when contrasted to Figures 4-7, which were replicated lacking the control approach. For various orders of the fractional derivative β, the trajectories have varying asymptotic behaviour. As a result, authorities and regulators should focus on maximizing the levels of the survival rate, either by addressing sick populations or by increasing specific susceptibility to the pneumonia virus.
In Figure 8, we can observe that ρ 2 is essential to minimize the meningitis development. e proportion of  18 Complexity contagious community owing to meningitis decreases as the level of ρ 2 increases from 0.1 to 0.9, but the interaction rate stays unchanged at q 2 � 0.06. Figures 9, 10, and 11 indicate that the proportion of instances in categories I p , I m , I pm , R p , R m , R pm decreased dramatically when contrasted to Figures 8-11, which were replicated inducing the control approach. erefore, the outbreak spreads gradually as the fractional order diminishes from 1, and the majority of patients at the apex drops significantly (Figures 8-11). Consequently, normal individuals or the administration must pay special consideration to healing the afflicted individuals in their locality when combating the meningitis infection.
In Figure 12, the meningitis connection incidence q 2 and the success percentage of the co-infectious community ρ are both assumed unchanged. Figure 13 demonstrates that as the interaction frequency of pneumonia improves, the co-infectious numbers boost, implying that the proliferation of   pneumonia and meningitis co-infection will expand as well. According to Figure 14, it is critical to reduce the incidence and prevalence of pneumonia in order to prevent co-infection. As a result, organizations should aim to minimize the interaction risk of pneumonia by quarantining sick individuals or implementing an effective mitigation technique to limit the spread of co-infection in the population. e influence of the survival incidence of P on the coinfectious community was investigated. According to the scenario characterization in Section 2, co-infectious populations generally heal from pneumonia solely or resume the corresponding healed section, alleviating the symptoms or additional processes. As a result, Figure 15 indicates that boosting the co-infectious majority's survival intensity has a significant impact on eliminating both infections in the region.

Conclusion
is research examines seven-dimensional pneumonia and meningitis. e Atangana-Baleanu fractional derivative is applied to describe the integer-order framework, and the Banach contraction hypothesis is employed to assess the presence of systems in the fractional formulation of the numerical method. e Atangana-Baleanu fractional operator featuring generational characteristics is responsible for this beneficial outcome. e findings of this article argue that formal frameworks utilizing the Atangana-Baleanu fractional operator can effectively disclose the underlying or realistic features of real-world situations.
is hypothesis can be supported by continuing research into the effects of various fractional operators, including such fractal-fractional derivatives, and reporting the performance of the Atangana-Baleanu fractional operator outcome on the relatively similar system or additional relevant epidemiological concepts. e paucity of a comparison of the present findings for the SEAIR paradigm employing the Atangana-Baleanu fractional operator versus findings acquired for the equivalent system assuming alternative fractional operators could potentially represent a deficit for ongoing studies.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.