# Novel Numerical Estimates of the Pneumonia and Meningitis Epidemic Model via the Nonsingular Kernel with Optimal Analysis

Authors: Saima Rashid; Bushra Kanwal; Abdulaziz Garba Ahmad; Ebenezer Bonyah; S.K. Elagan
Journal: Complexity
 (2022)

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## Abstract

In this article, we investigated a deterministic model of pneumonia-meningitis coinfection. Employing the Atangana–Baleanu fractional derivative operator in the Caputo framework, we analyze a seven-component approach based on ordinary differential equations (DEs). Furthermore, the invariant domain, disease-free as well as endemic equilibria, and the validity of the model’s potential results are all investigated. According to controller design evaluation and modelling, the modulation technique devised is effective in diminishing the proportion of incidences in various compartments. A fundamental reproducing value is generated by exploiting the next generation matrix to assess the properties of the equilibrium. The system’s reliability is further evaluated. Sensitivity analysis is used to classify the impact of each component on the spread or prevention of illness. Using simulation studies, the impacts of providing therapy have been determined. Additionally, modelling the appropriate configuration demonstrated that lowering the fractional order from 1 necessitates a rapid initiation of the specified control technique at the largest intensity achievable and retaining it for the bulk of the pandemic’s duration.


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## Body

## 1. 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) [1–4].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 [5–8]. 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].Therefore, 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 fractional-order 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, primarilyStreptococcus pyogenes [14]. Other disorders susceptible to infection include meningitis, respiratory ailments, and nasal congestion, among others (see Figure 1).Figure 1
Basic indications of infectious pneumonia.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)).Figure 2
(a) Bacterial meningitis. (b) Anatomy of meningitis.
(a)(b)Meningitis is caused by viral illness in 80 percent of cases, which is caused by pneumococcal pneumonia, streptococci, andNeisseria meningitidis. A majority of the research was carried out to determine the governing strategies of contagious ailments in a population (see [17–19]). 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. The authors [20, 21] also mentioned the co-dynamics of pneumonia and waterborne illness infections [22], as well as economic evaluation and optimal monitoring. The 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 [27–29]. 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. Thus, we can conclude from the above-mentioned research that fractional derivatives have several implications in numerical techniques and in the study of real-world processes. The 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 meningitisℙ mathematical formulation with a dominating interaction incidence, which is inspired by the aforesaid considerations. The 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]. The ℙ epidemiological theory utilizing the Atangana–Baleanu fractional derivative has not previously been examined, to the best of the researchers’ expertise. The 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.


## 2. Depiction of the Model

In this part, the ABC fractional derivative form of theℙ epidemic mathematical systems is introduced. Let us just continue with a review of the ML kernels’ notions and their concerning consequences.Definition 1 (see [2]).
Forβ∈0,1, c<d and ℱ1∈H1c,d; then, the ABC derivative of fractional-order for ℱ1 is presented as(1)DABCβℱ1ζ=Mβ1−β∫cζdℱ1dζEβββ−1ζ−>βd>,where Mβis a normalization mapping that satisfying M0=M1=1and the Mittag–Leffler function is denoted by Eβz1 defined as (2)Eβz1=∑γ=0∞z1γΓβγ+1,β,z1∈ℂ,ℜβ>0.Definition 2 (see [2]).
Forβ∈0,1β∈0,1, c<d and ℱ1∈H1c,d; then, the AB fractional integral of ℱ1 is presented as(3)IcABζβℱ1ζ=1−βMβℱ1ζ+βΓβMβ∫cζℱ1ϱζ−ϱβ−1dϱ.Lemma 1 (see [38]) (Newton–Leibniz identity).
Forℱ1∈ℂ1c,d, the ABC fractional derivative and integral for ℱ1 hold:(4)IζβcABDζβcABℱ1ζ=ℱ1ζ−ℱ1c.Lemma 2 (see [39]).
Forc<d,ℱ1,ℱ2∈H1c,d, then the ABC fractional derivative holds for the subsequent variant:(5)DζβcABℱ1ζ−DζβcABℱ2ζ≤Hℱ1ζ−ℱ2ζ.Our next result is the generalized mean-value theorem, which is mainly due to [40].Lemma 3 (see [40]).
Let there be a functionh1ϱ∈ℂc,d and also suppose Dζβ0ABhϱ∈ℂc,q2,β∈0,1. Then, h1ϱ=h1c+1/Γβ0ABCDζβh1θϱ−cβ,θ∈0,ϱ.Followed by Lemma3, for β∈0,1, if h1ϱ∈0,d,Dζβ0ABh1ϱ∈0,d, and 0ABCDζβh1ϱ≥0 for all ϱ∈0,d, then the mapping h1ϱ is increasing. Otherwise, h1ϱ is said to be decreasing for all ϱ∈0,d.We shall now continue on to the system’s development. The numerical framework for this investigation has been developed utilizing the sequence chart below (Figure3).Figure 3
Flowchart forPM co-infection model.The numerical approach with numerical approximation employed in this work is represented by the governing formulae, which are predicated on the workflow.(6)S.ζ=π+ϕ1Rp+ϕ2Rm+ϕ3Rpm−g1+g2+ϱS,Ip.ζ=g1S−g2+ϱ+ρ1+η1Ip,Im.ζ=g2S−g1+ϱ+ρ2+η2Im,Ipm.ζ=g2Ip+g1Im−ρ+ϱ+η1+η2Ipm,Rp.ζ=ρ1Ip+ϵρIpm−ϕ1+ϱRp,Rm.ζ=ρ2Im+ϵρh11−ϵIpm−ϕ2+ϱRm,Rpm.ζ=ρ1−h11−ϵIpm−ϕ3+ϱRpm.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 groupS, pneumonia virulent Ip, meningitis epidemicIm, pneumonia and meningitis co-infectiousIpm, pneumonia healedRp, meningitis regainedRm, and pneumonia-meningitis co-infectious retrieved are the numerous subgroups. Rpm. 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 ϕ1,ϕ2, and ϕ3, respectively. In the entire susceptible population, individuals can get pneumonia with contact rate of q1 from a pneumonia-only infected or coinfected person with force of infection of pneumonia g1=q1Ip+Ipm/Nand join Ip compartment. In a similar way, individuals can get meningitis by a contact rate of q2 from a meningitis-only infected or coinfected person with force of infection of meningitis g2=q2Im+Ipm/N and join Im compartment. Pneumonia-only infected individuals also can get an additional meningitis infection with a force of infection g2 and join coinfected compartment Ipm. Participants emerge from meningitis merely affected by a segment when attacked by pneumonia using g1 a high intensity of transmission to boost the postoperative subsystem. Patients who have only been afflicted with pneumonia may recuperate at a frequency of ρ1 approximately and can represent the pneumonia only cured category Ip. As a result, meningitis-only people who are contaminated recover at a rate of ρ2 and are assigned to the meningitis-only healed group Ipm. People in the immune-compromised 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 ρ1−ϵ, or revive just from meningitis ailment and participate in meningitis only retrieved zone with the plausibility of ρh11−ϵ, or restore from all these maladies and participate in co-infected recapture zone with a likely hood of ρ1−ϵ1−h1. Furthermore, ϱ represents the natural death rate. Furthermore, η1 is the relative risk induced entirely by pneumonia, and η2 is the number of fatalities due to meningitis.


### 2.1. 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.The 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.(7)Dζβ0ABSζ=Θ1ζ,S,Dζβ0ABIpζ=Θ2ζ,Ip,Dζβ0ABImζ=Θ3t3,Im,Dζβ0ABIpmζ=Θ4ζ,Ipm,Dζβ0ABRpζ=Θ5ζ,Rp,Dζβ0ABRmζ=Θ6ζ,Rm,Dζβ0ABRpmζ=Θ7ζ,Ipm,where the kernels are set as follows:(8)Θ1ζ,S=π+ϕ1Rp+ϕ2Rm+ϕ3Rpm−g1+g2+ϱS,Θ2ζ,Ip=g1S−g2+ϱ+ρ1+η1Ip,Θ3ζ,Im=g2S−g1+ϱ+ρ2+η2Im,Θ4ζ,Ipm=g2Ip+g1Im−ρ+ϱ+η1+η2Ipm,Θ5ζ,Rp=ρ1Ip+ϵρIpm−ϕ1+ϱRp,Θ6ζ,Rm=ρ2Im+ϵρh11−ϵIpm−ϕ2+ϱRm,Θ7ζ,Rpm=ρ1−h11−ϵIpm−ϕ3+ϱRpm,subject to the nonnegative initial values S0=S0,Ip0=Ip0,Im0=Im0,Ipm0=Ipm0,Rp0=Rp0,Rm0=Rm0,Rpm0=Rpm0.Thus, in the absence of disease, the differential equation of the total population size isdN/dζ=π−ϱN−η1Ip+Ipm−η2Im+Ipm. Table 1 summarizes the characteristics that were considered in the analysis (7).Table 1
Explanation of attributed values assumed in the model.
SymbolsValueReferencesϕ10.003–0.1[20]ϕ20.00904–0.99[16]ϕ30.01Supposedq10.007–0.6[22]q20.9[16]η10.006–0.5Estimatedη20.002–0.2Estimatedρ0.1[20]h10.5–1.0[22]ϵ0.5–1.0[22]ϱ0.01Supposedρ10.9Supposedρ20.8Supposed


### 2.2. Consequences on the Existence-Uniqueness

Here, employing the Banach fixed pointfp 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.(9)%Sζ−S0=1−βMβΘ1ζ,S+βMβΓβ∫0ζΘ1ς,Sζ−ςβ−1dς,Ipζ−Ip0=1−βMβΘ2ζ,Ip+βMβΓβ∫0ζΘ2ς,Ipζ−ςβ−1dς,Imζ−Im0=1−βMβΘ3ζ,Im+βMβΓβ∫0ζΘ3ς,Imζ−ςβ−1dς,Ipmζ−Ipm0=1−βMβΘ4ζ,Ipm+βMβΓβ∫0ζΘ4ς,Ipmζ−ςβ−1dς,Rpζ−Rp0=1−βMβΘ5ζ,Rp+βMβΓβ∫0ζΘ5ς,Rpζ−ςβ−1dς,Rmζ−Rm0=1−βMβΘ6ζ,Rp+βMβΓβ∫0ζΘ6ς,Rpζ−ςβ−1dς,Rpmζ−Rpm0=1−βMβΘ7ζ,Rpm+βMβΓβ∫0ζΘ7ς,Rpmζ−ςβ−1dς.Surmise that the collectionB=ΛJ×ΛJ×ΛJ×ΛJ×ΛJ×ΛJ×ΛJ, where ΛJ=C0,T¯ indicates real-valued continuous mappings for the Banach space on J=0,T¯ considering the norm presented as S,Ip,Im,Ipm,Rp,Rm,Rpm=S+Ip+Im+Ipm+Rp+Rm+Rpm, where S=supζ∈JSζ, Ip=supζ∈JIpζ, Im=supζ∈JImζ, Ipm=supζ∈JIpmζ, Rp=supζ∈JRpζ, Rm=supζ∈JRmζ, and Rpm=supζ∈JRpmζ.The contraction and the Lipschitz hypothesis are the foundations of our following theorem.Theorem 1.
For the following kernelsΘℓ,ℓ=1,2,…,7 in (7), there exists Lℓ>0,ℓ=1,2,…7, such that(10)Θ1ζ,S−Θ1ζ,S1≤L1Sζ−S1ζ,Θ2ζ,Ip−Θ2ζ,Ip1≤L2Ipζ−Ip1ζ,Θ3ζ,Im−Θ3ζ,Im1≤L3Imζ−Im1ζ,Θ4ζ,Ipm−Θ4ζ,Ipm≤L4Ipmζ−Ipmζ,Θ5ζ,Rp−Θ5ζ,Rp1≤L5Rpζ−Rp1ζ,Θ6ζ,Rm−Θ6ζ,Rm1≤L6Rmζ−Rm1ζ,Θ7ζ,Rpm−Θ7ζ,Rpm≤L7Rpmζ−Rpmζ,are contractions for Lℓ∈0,1,ℓ=1,2,…,7.Proof.
In order to satisfy Lipschitz assumptions, we have(11)Θ1ζ,S−Θ1ζ,S1=π+ϕ1Rp+ϕ2Rm+ϕ3Rpm−g1+g2+ϱS−π+ϕ1Rp+ϕ2Rm+ϕ3Rpm−g1+g2+ϱS1‖=−g1+g2+ϱS−S1≤g1+g2+ϱS−S1≤L1S−S1,where L1=q1K2+K4/N+q2K3+K4/N+ϱ, S=supζ∈JSζ=K1, Ip=supζ∈JIpζ=K2, Im=supζ∈JImζ=K3, Ipm=supζ∈JIpmζ=K4, Rp=supζ∈JRpζ=K5, Rm=supζ∈JRmζ=K6, and Rpm=supζ∈JRpmζ=K7.
It is worth noting thatΘ1ζ,S1 holds the Lipschitz assumption containing Lipschitz constant L1=q1K2+K4/N+q2K3+K4/N+ϱ. Furthermore, if L1∈0,1, then Θ1ζ,S1 is a contraction.
Analogously, we can analyze the existence consequences ofLℓ,ℓ=2,3,…,7 and the contraction technique for Θ2ζ,Ip,Θ3ζ,Im,Θ4ζ,Ipm,Θ5ζ,Rp,Θ6ζ,Rm, and Θ7ζ,Rpm for Lℓ∈0,1,ℓ=2,3,…,7.
Atζ=ζn,n=1,2,…, introduce the subsequent recursive version of (9):(12)Snζ=1−βMβΘ1ζ,Sn−1+βMβΓβ∫0ζΘ1ς,Sn−1ζ−ςβ−1dς,Ipnζ=1−βMβΘ2ζ,Ipn−1+βMβΓβ∫0ζΘ2ς,Ipn−1ζ−ςβ−1dς,Imnζ=1−βMβΘ3ζ,Imn−1+βMβΓβ∫0ζΘ3ς,Imn−1ζ−ςβ−1dς,Ipmnζ=1−βMβΘ4ζ,Ipmn−1+βMβΓβ∫0ζΘ4ς,Ipmn−1ζ−ςβ−1dς,Rpnζ=1−βMβΘ5ζ,Rpn−1+βMβΓβ∫0ζΘ5ς,Rpn−1ζ−ςβ−1dς,Rmnζ=1−βMβΘ6ζ,Rmn−1+βMβΓβ∫0ζΘ6ς,Rmn−1ζ−ςβ−1dς,Rpmnζ=1−βMβΘ7ζ,Rpmn−1+βMβΓβ∫0ζΘ7ς,Rpmn−1ζ−ςβ−1dς,subject to ICs S0=S0,Ip0=Ip0,Im0=Im0,Ipm0=Ipm0,Rp0=Rp0,Rm0=Rm0,Rpm0=Rpm0.
In (12), the variations of successive components are written in the following form:(13)Ξ1nζ=Snζ−Sn−1ζ=1−βMβΘ1ζ,Sn−1−Θ1ζ,Sn−2+βΓβMβ∫0ζΘ1ς,Sn−1−Θ1ς,Sn−2ζ−ςβ−1dς,Ξ2nζ=Ipnζ−Ipn−1ζ=1−βMβΘ2ζ,Ipn−1−Θ2ζ,Ipn−2+βΓβMβ∫0ζΘ2ς,Ipn−1−Θ2ς,Ipn−2ζ−ςβ−1dς,Ξ3nζ=Imnζ−Imn−1ζ=1−βMβΘ3ζ,Imn−1−Θ3ζ,Imn−2+βΓβMβ∫0ζΘ3ς,Imn−1−Θ3ς,Imn−2ζ−ςβ−1dς,Ξ4nζ=Ipmnζ−Ipmn−1ζ=1−βMβΘ4ζ,Ipmn−1−Θ4ζ,Ipmn−2+βΓβMβ∫0ζΘ4ς,Ipmn−1−Θ4ς,Ipmn−2ζ−ςβ−1dς,Ξ5nζ=Rpnζ−Rpn−1ζ=1−βMβΘ5ζ,Rpn−1−Θ5ζ,Rpn−2+βΓβMβ∫0ζΘ5ς,Rpn−1−Θ5ς,Rpn−2ζ−ςβ−1dς,Ξ6nζ=Rmnζ−Rmn−1ζ=1−βMβΘ2ζ,Rmn−1−Θ6ζ,Rmn−2+βΓβMβ∫0ζΘ6ς,Rmn−1−Θ6ς,Rmn−2ζ−ςβ−1dς,Ξ7nζ=Rpmnζ−Rpmn−1ζ=1−βMβΘ7ζ,Rpmn−1−Θ7ζ,Rpmn−2+βΓβMβ∫0ζΘ7ς,Rpmn−1−Θ7ς,Rpmn−2ζ−ςβ−1dς.
Implementing the norm on the aforementioned system (13), we have(14)Ξ1nζ=Snζ−Sn−1ζ=1−βMβΘ1ζ,Sn−1−Θ1ζ,Sn−2+βΓβMβ∫0ζΘ1ς,Sn−1−Θ1ς,Sn−2ζ−ςβ−1dς,Ξ2nζ=Ipnζ−Ipn−1ζ=1−βMβΘ2ζ,Ipn−1−Θ2ζ,Ipn−2+βΓβMβ∫0ζΘ2ς,Ipn−1−Θ2ς,Ipn−2ζ−ςβ−1dς,Ξ3nζ=Imnζ−Imn−1ζ=1−βMβΘ3ζ,Imn−1−Θ3ζ,Imn−2+βΓβMβ∫0ζΘ3ς,Imn−1−Θ3ς,Imn−2ζ−ςβ−1dς,Ξ4nζ=Ipmnζ−Ipmn−1ζ=1−βMβΘ4ζ,Ipmn−1−Θ4ζ,Ipmn−2+βΓβMβ∫0ζΘ4ς,Ipmn−1−Θ4ς,Ipmn−2ζ−ςβ−1dς,Ξ5nζ=Rpnζ−Rpn−1ζ=1−βMβΘ5ζ,Rpn−1−Θ5ζ,Rpn−2+βΓβMβ∫0ζΘ5ς,Rpn−1−Θ5ς,Rpn−2ζ−ςβ−1dς,Ξ6nζ=Rmnζ−Rmn−1ζ=1−βMβΘ2ζ,Rmn−1−Θ6ζ,Rmn−2+βΓβMβ∫0ζΘ6ς,Rmn−1−Θ6ς,Rmn−2ζ−ςβ−1dς,Ξ7nζ=Rpmnζ−Rpmn−1ζ=1−βMβΘ7ζ,Rpmn−1−Θ7ζ,Rpmn−2+βΓβMβ∫0ζΘ7ς,Rpmn−1−Θ7ς,Rpmn−2ζ−ςβ−1dς.
In addition, the first identity in (14) can be simplified to the representations:(15)Ξ1nζ=Snζ−Sn−1ζ=1−βMβΘ1ζ,Sn−1−Θ1ζ,Sn−2+βΓβMβ∫0ζΘ1ς,Sn−1−Θ1ς,Sn−2ζ−ςβ−1dς≤L11−βMβSn−1−Sn−2+βL1ΓβMβ∫0ζSn−1−Sn−2ζ−ςβ−1dς≤L1Ξ1n−1ζ1−βMβ+ζβMβ.
Ultimately, we have(16)Ξ1nζ≤L1Ξ1n−1ζ1−βMβ+ζβMβ.
Repeating the same procedure, the subsequent terms of (14) can be simplified to the following:(17)Ξ2nζ≤L2Ξ2n−1ζ1−βMβ+ζβMβ,Ξ3nζ≤L3Ξ3n−1ζ1−βMβ+ζβMβ,Ξ4nζ≤L4Ξ4n−1ζ1−βMβ+ζβMβ,Ξ5nζ≤L5Ξ5n−1ζ1−βMβ+ζβMβ,Ξ6nζ≤L6Ξ6n−1ζ1−βMβ+ζβMβ,Ξ7nζ≤L7Ξ7n−1ζ1−βMβ+ζβMβ.Theorem 2.
The fractionalℙ model presented in (7) has a solution if U0 holds the variant(18)1−βMβ+U0βMβΓβLℓ<1,ℓ=1,2,…,7.Proof.
By means of (16) and (19), we have(19)Ξ1nζ≤S0L11−βMβ+ζβMβn,Ξ2nζ≤Ip0L21−βMβ+ζβMβn,Ξ3nζ≤Im0L31−βMβ+ζβMβn,Ξ4nζ≤Ipm0L41−βMβ+ζβMβn,Ξ5nζ≤Rp0L51−βMβ+ζβMβn,Ξ6nζ≤Rm0L61−βMβ+ζβMβn,Ξ7nζ≤Rpm0L71−βMβ+ζβMβn.
Theorem 1 verifies the validity of the solution (the existence of afp), and which shows that the mappings Sζ, Ipζ, Imζ, Ipmζ, Rpζ, Rmζ, and Rpmζ are solution of the model (7).
Let us begin by defining which requirements are satisfied:(20)Sζ−S0=Sn−B1n˜ζ,Ipζ−Ip0=Ipn−B2n˜ζ,Imζ−Im0=Imn−B3n˜ζ,Ipmζ−Ipm0=Ipmn−B4n˜ζ,Rpζ−Rp0=Rpn−B5n˜ζ,Rmζ−Rm0=Rmn−B6n˜ζ,Rpmζ−Rpm0=Rpmn−B7n˜ζ.
Utilizing the fact of (20), we find(21)B1nζ≤1−βMβΘ1ζ,Sn−Θ1ζ,Sn−1+βΓβMβ∫0ζΘ1ς,Sn−Θ1ς,Sn−1ζ−ςβ−1dς≤L11−βMβSn−Sn−1+βnL1ΓβMβSn−Sn−1.
Iteratively performing the technique yields(22)B1nζ≤L1n1−βMβ+L1nζβΓβMβn+1Sn−Sn−1n.
ζ=U0β gives(23)B1nζ≤L1n1−βMβ+ζβΓβMβn+1Sn−Sn−1n.
Since, the expression can be formulated as(24)B1nζ↦0.
Take limit to (23) as n↦∞. Clearly, observe that B1nζ↦0 for 1−β/Mβ+ζβ/ΓβMβL1<1.
Identically, we can illustrateBℓnζ↦0,forℓ=2,3,…,7; then, 1−β/Mβ+ζβ/ΓβMβLℓ<1,ℓ=1,2,…,7.
This completes the proof.Here, the Banachfp theorem asserts the validity of the result of system (7) by Theorems 1 and 2. Theorem 3 verifies the solution’s uniqueness.Theorem 3.
Let there be a unique solution of the fractionalℙ model (7) given that(25)1−βMβ+ζβΓβMβLℓ<1,ℓ=1,2,…,7.Proof.
Suppose thatS1,Ip1,Im1,Ipm1,Rp1,Rm1, and Rpm1 are another solution to fractional ℙ model (7). Then,(26)Sζ−S1ζ=1−βMβΘ1ζ,S−Θ1ζ,S1+βMβΓβ∫0ζΘ1ς,S−Θ1ς,S1ζ−ςβ−1dς.
Employing the norm to the aforesaid equation, we have(27)Sζ−S1ζ≤1−βMβS−S1L1+ζβMβΓβS−S1.
Since1−1−β/Mβ+ζβ/ΓβMβL1>0, we acquire S−S1=0. Consequently, we have S=S1. Repeating the same process, we can prove that Ip=Ip1,Im=Im1,Ipm=Ipm1,Rp=Rp1,Rm=Rm1, and Rpm=Rpm1. This completes the proof.


## 2.1. 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.The 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.(7)Dζβ0ABSζ=Θ1ζ,S,Dζβ0ABIpζ=Θ2ζ,Ip,Dζβ0ABImζ=Θ3t3,Im,Dζβ0ABIpmζ=Θ4ζ,Ipm,Dζβ0ABRpζ=Θ5ζ,Rp,Dζβ0ABRmζ=Θ6ζ,Rm,Dζβ0ABRpmζ=Θ7ζ,Ipm,where the kernels are set as follows:(8)Θ1ζ,S=π+ϕ1Rp+ϕ2Rm+ϕ3Rpm−g1+g2+ϱS,Θ2ζ,Ip=g1S−g2+ϱ+ρ1+η1Ip,Θ3ζ,Im=g2S−g1+ϱ+ρ2+η2Im,Θ4ζ,Ipm=g2Ip+g1Im−ρ+ϱ+η1+η2Ipm,Θ5ζ,Rp=ρ1Ip+ϵρIpm−ϕ1+ϱRp,Θ6ζ,Rm=ρ2Im+ϵρh11−ϵIpm−ϕ2+ϱRm,Θ7ζ,Rpm=ρ1−h11−ϵIpm−ϕ3+ϱRpm,subject to the nonnegative initial values S0=S0,Ip0=Ip0,Im0=Im0,Ipm0=Ipm0,Rp0=Rp0,Rm0=Rm0,Rpm0=Rpm0.Thus, in the absence of disease, the differential equation of the total population size isdN/dζ=π−ϱN−η1Ip+Ipm−η2Im+Ipm. Table 1 summarizes the characteristics that were considered in the analysis (7).Table 1
Explanation of attributed values assumed in the model.
SymbolsValueReferencesϕ10.003–0.1[20]ϕ20.00904–0.99[16]ϕ30.01Supposedq10.007–0.6[22]q20.9[16]η10.006–0.5Estimatedη20.002–0.2Estimatedρ0.1[20]h10.5–1.0[22]ϵ0.5–1.0[22]ϱ0.01Supposedρ10.9Supposedρ20.8Supposed


## 2.2. Consequences on the Existence-Uniqueness

Here, employing the Banach fixed pointfp 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.(9)%Sζ−S0=1−βMβΘ1ζ,S+βMβΓβ∫0ζΘ1ς,Sζ−ςβ−1dς,Ipζ−Ip0=1−βMβΘ2ζ,Ip+βMβΓβ∫0ζΘ2ς,Ipζ−ςβ−1dς,Imζ−Im0=1−βMβΘ3ζ,Im+βMβΓβ∫0ζΘ3ς,Imζ−ςβ−1dς,Ipmζ−Ipm0=1−βMβΘ4ζ,Ipm+βMβΓβ∫0ζΘ4ς,Ipmζ−ςβ−1dς,Rpζ−Rp0=1−βMβΘ5ζ,Rp+βMβΓβ∫0ζΘ5ς,Rpζ−ςβ−1dς,Rmζ−Rm0=1−βMβΘ6ζ,Rp+βMβΓβ∫0ζΘ6ς,Rpζ−ςβ−1dς,Rpmζ−Rpm0=1−βMβΘ7ζ,Rpm+βMβΓβ∫0ζΘ7ς,Rpmζ−ςβ−1dς.Surmise that the collectionB=ΛJ×ΛJ×ΛJ×ΛJ×ΛJ×ΛJ×ΛJ, where ΛJ=C0,T¯ indicates real-valued continuous mappings for the Banach space on J=0,T¯ considering the norm presented as S,Ip,Im,Ipm,Rp,Rm,Rpm=S+Ip+Im+Ipm+Rp+Rm+Rpm, where S=supζ∈JSζ, Ip=supζ∈JIpζ, Im=supζ∈JImζ, Ipm=supζ∈JIpmζ, Rp=supζ∈JRpζ, Rm=supζ∈JRmζ, and Rpm=supζ∈JRpmζ.The contraction and the Lipschitz hypothesis are the foundations of our following theorem.Theorem 1.
For the following kernelsΘℓ,ℓ=1,2,…,7 in (7), there exists Lℓ>0,ℓ=1,2,…7, such that(10)Θ1ζ,S−Θ1ζ,S1≤L1Sζ−S1ζ,Θ2ζ,Ip−Θ2ζ,Ip1≤L2Ipζ−Ip1ζ,Θ3ζ,Im−Θ3ζ,Im1≤L3Imζ−Im1ζ,Θ4ζ,Ipm−Θ4ζ,Ipm≤L4Ipmζ−Ipmζ,Θ5ζ,Rp−Θ5ζ,Rp1≤L5Rpζ−Rp1ζ,Θ6ζ,Rm−Θ6ζ,Rm1≤L6Rmζ−Rm1ζ,Θ7ζ,Rpm−Θ7ζ,Rpm≤L7Rpmζ−Rpmζ,are contractions for Lℓ∈0,1,ℓ=1,2,…,7.Proof.
In order to satisfy Lipschitz assumptions, we have(11)Θ1ζ,S−Θ1ζ,S1=π+ϕ1Rp+ϕ2Rm+ϕ3Rpm−g1+g2+ϱS−π+ϕ1Rp+ϕ2Rm+ϕ3Rpm−g1+g2+ϱS1‖=−g1+g2+ϱS−S1≤g1+g2+ϱS−S1≤L1S−S1,where L1=q1K2+K4/N+q2K3+K4/N+ϱ, S=supζ∈JSζ=K1, Ip=supζ∈JIpζ=K2, Im=supζ∈JImζ=K3, Ipm=supζ∈JIpmζ=K4, Rp=supζ∈JRpζ=K5, Rm=supζ∈JRmζ=K6, and Rpm=supζ∈JRpmζ=K7.
It is worth noting thatΘ1ζ,S1 holds the Lipschitz assumption containing Lipschitz constant L1=q1K2+K4/N+q2K3+K4/N+ϱ. Furthermore, if L1∈0,1, then Θ1ζ,S1 is a contraction.
Analogously, we can analyze the existence consequences ofLℓ,ℓ=2,3,…,7 and the contraction technique for Θ2ζ,Ip,Θ3ζ,Im,Θ4ζ,Ipm,Θ5ζ,Rp,Θ6ζ,Rm, and Θ7ζ,Rpm for Lℓ∈0,1,ℓ=2,3,…,7.
Atζ=ζn,n=1,2,…, introduce the subsequent recursive version of (9):(12)Snζ=1−βMβΘ1ζ,Sn−1+βMβΓβ∫0ζΘ1ς,Sn−1ζ−ςβ−1dς,Ipnζ=1−βMβΘ2ζ,Ipn−1+βMβΓβ∫0ζΘ2ς,Ipn−1ζ−ςβ−1dς,Imnζ=1−βMβΘ3ζ,Imn−1+βMβΓβ∫0ζΘ3ς,Imn−1ζ−ςβ−1dς,Ipmnζ=1−βMβΘ4ζ,Ipmn−1+βMβΓβ∫0ζΘ4ς,Ipmn−1ζ−ςβ−1dς,Rpnζ=1−βMβΘ5ζ,Rpn−1+βMβΓβ∫0ζΘ5ς,Rpn−1ζ−ςβ−1dς,Rmnζ=1−βMβΘ6ζ,Rmn−1+βMβΓβ∫0ζΘ6ς,Rmn−1ζ−ςβ−1dς,Rpmnζ=1−βMβΘ7ζ,Rpmn−1+βMβΓβ∫0ζΘ7ς,Rpmn−1ζ−ςβ−1dς,subject to ICs S0=S0,Ip0=Ip0,Im0=Im0,Ipm0=Ipm0,Rp0=Rp0,Rm0=Rm0,Rpm0=Rpm0.
In (12), the variations of successive components are written in the following form:(13)Ξ1nζ=Snζ−Sn−1ζ=1−βMβΘ1ζ,Sn−1−Θ1ζ,Sn−2+βΓβMβ∫0ζΘ1ς,Sn−1−Θ1ς,Sn−2ζ−ςβ−1dς,Ξ2nζ=Ipnζ−Ipn−1ζ=1−βMβΘ2ζ,Ipn−1−Θ2ζ,Ipn−2+βΓβMβ∫0ζΘ2ς,Ipn−1−Θ2ς,Ipn−2ζ−ςβ−1dς,Ξ3nζ=Imnζ−Imn−1ζ=1−βMβΘ3ζ,Imn−1−Θ3ζ,Imn−2+βΓβMβ∫0ζΘ3ς,Imn−1−Θ3ς,Imn−2ζ−ςβ−1dς,Ξ4nζ=Ipmnζ−Ipmn−1ζ=1−βMβΘ4ζ,Ipmn−1−Θ4ζ,Ipmn−2+βΓβMβ∫0ζΘ4ς,Ipmn−1−Θ4ς,Ipmn−2ζ−ςβ−1dς,Ξ5nζ=Rpnζ−Rpn−1ζ=1−βMβΘ5ζ,Rpn−1−Θ5ζ,Rpn−2+βΓβMβ∫0ζΘ5ς,Rpn−1−Θ5ς,Rpn−2ζ−ςβ−1dς,Ξ6nζ=Rmnζ−Rmn−1ζ=1−βMβΘ2ζ,Rmn−1−Θ6ζ,Rmn−2+βΓβMβ∫0ζΘ6ς,Rmn−1−Θ6ς,Rmn−2ζ−ςβ−1dς,Ξ7nζ=Rpmnζ−Rpmn−1ζ=1−βMβΘ7ζ,Rpmn−1−Θ7ζ,Rpmn−2+βΓβMβ∫0ζΘ7ς,Rpmn−1−Θ7ς,Rpmn−2ζ−ςβ−1dς.
Implementing the norm on the aforementioned system (13), we have(14)Ξ1nζ=Snζ−Sn−1ζ=1−βMβΘ1ζ,Sn−1−Θ1ζ,Sn−2+βΓβMβ∫0ζΘ1ς,Sn−1−Θ1ς,Sn−2ζ−ςβ−1dς,Ξ2nζ=Ipnζ−Ipn−1ζ=1−βMβΘ2ζ,Ipn−1−Θ2ζ,Ipn−2+βΓβMβ∫0ζΘ2ς,Ipn−1−Θ2ς,Ipn−2ζ−ςβ−1dς,Ξ3nζ=Imnζ−Imn−1ζ=1−βMβΘ3ζ,Imn−1−Θ3ζ,Imn−2+βΓβMβ∫0ζΘ3ς,Imn−1−Θ3ς,Imn−2ζ−ςβ−1dς,Ξ4nζ=Ipmnζ−Ipmn−1ζ=1−βMβΘ4ζ,Ipmn−1−Θ4ζ,Ipmn−2+βΓβMβ∫0ζΘ4ς,Ipmn−1−Θ4ς,Ipmn−2ζ−ςβ−1dς,Ξ5nζ=Rpnζ−Rpn−1ζ=1−βMβΘ5ζ,Rpn−1−Θ5ζ,Rpn−2+βΓβMβ∫0ζΘ5ς,Rpn−1−Θ5ς,Rpn−2ζ−ςβ−1dς,Ξ6nζ=Rmnζ−Rmn−1ζ=1−βMβΘ2ζ,Rmn−1−Θ6ζ,Rmn−2+βΓβMβ∫0ζΘ6ς,Rmn−1−Θ6ς,Rmn−2ζ−ςβ−1dς,Ξ7nζ=Rpmnζ−Rpmn−1ζ=1−βMβΘ7ζ,Rpmn−1−Θ7ζ,Rpmn−2+βΓβMβ∫0ζΘ7ς,Rpmn−1−Θ7ς,Rpmn−2ζ−ςβ−1dς.
In addition, the first identity in (14) can be simplified to the representations:(15)Ξ1nζ=Snζ−Sn−1ζ=1−βMβΘ1ζ,Sn−1−Θ1ζ,Sn−2+βΓβMβ∫0ζΘ1ς,Sn−1−Θ1ς,Sn−2ζ−ςβ−1dς≤L11−βMβSn−1−Sn−2+βL1ΓβMβ∫0ζSn−1−Sn−2ζ−ςβ−1dς≤L1Ξ1n−1ζ1−βMβ+ζβMβ.
Ultimately, we have(16)Ξ1nζ≤L1Ξ1n−1ζ1−βMβ+ζβMβ.
Repeating the same procedure, the subsequent terms of (14) can be simplified to the following:(17)Ξ2nζ≤L2Ξ2n−1ζ1−βMβ+ζβMβ,Ξ3nζ≤L3Ξ3n−1ζ1−βMβ+ζβMβ,Ξ4nζ≤L4Ξ4n−1ζ1−βMβ+ζβMβ,Ξ5nζ≤L5Ξ5n−1ζ1−βMβ+ζβMβ,Ξ6nζ≤L6Ξ6n−1ζ1−βMβ+ζβMβ,Ξ7nζ≤L7Ξ7n−1ζ1−βMβ+ζβMβ.Theorem 2.
The fractionalℙ model presented in (7) has a solution if U0 holds the variant(18)1−βMβ+U0βMβΓβLℓ<1,ℓ=1,2,…,7.Proof.
By means of (16) and (19), we have(19)Ξ1nζ≤S0L11−βMβ+ζβMβn,Ξ2nζ≤Ip0L21−βMβ+ζβMβn,Ξ3nζ≤Im0L31−βMβ+ζβMβn,Ξ4nζ≤Ipm0L41−βMβ+ζβMβn,Ξ5nζ≤Rp0L51−βMβ+ζβMβn,Ξ6nζ≤Rm0L61−βMβ+ζβMβn,Ξ7nζ≤Rpm0L71−βMβ+ζβMβn.
Theorem 1 verifies the validity of the solution (the existence of afp), and which shows that the mappings Sζ, Ipζ, Imζ, Ipmζ, Rpζ, Rmζ, and Rpmζ are solution of the model (7).
Let us begin by defining which requirements are satisfied:(20)Sζ−S0=Sn−B1n˜ζ,Ipζ−Ip0=Ipn−B2n˜ζ,Imζ−Im0=Imn−B3n˜ζ,Ipmζ−Ipm0=Ipmn−B4n˜ζ,Rpζ−Rp0=Rpn−B5n˜ζ,Rmζ−Rm0=Rmn−B6n˜ζ,Rpmζ−Rpm0=Rpmn−B7n˜ζ.
Utilizing the fact of (20), we find(21)B1nζ≤1−βMβΘ1ζ,Sn−Θ1ζ,Sn−1+βΓβMβ∫0ζΘ1ς,Sn−Θ1ς,Sn−1ζ−ςβ−1dς≤L11−βMβSn−Sn−1+βnL1ΓβMβSn−Sn−1.
Iteratively performing the technique yields(22)B1nζ≤L1n1−βMβ+L1nζβΓβMβn+1Sn−Sn−1n.
ζ=U0β gives(23)B1nζ≤L1n1−βMβ+ζβΓβMβn+1Sn−Sn−1n.
Since, the expression can be formulated as(24)B1nζ↦0.
Take limit to (23) as n↦∞. Clearly, observe that B1nζ↦0 for 1−β/Mβ+ζβ/ΓβMβL1<1.
Identically, we can illustrateBℓnζ↦0,forℓ=2,3,…,7; then, 1−β/Mβ+ζβ/ΓβMβLℓ<1,ℓ=1,2,…,7.
This completes the proof.Here, the Banachfp theorem asserts the validity of the result of system (7) by Theorems 1 and 2. Theorem 3 verifies the solution’s uniqueness.Theorem 3.
Let there be a unique solution of the fractionalℙ model (7) given that(25)1−βMβ+ζβΓβMβLℓ<1,ℓ=1,2,…,7.Proof.
Suppose thatS1,Ip1,Im1,Ipm1,Rp1,Rm1, and Rpm1 are another solution to fractional ℙ model (7). Then,(26)Sζ−S1ζ=1−βMβΘ1ζ,S−Θ1ζ,S1+βMβΓβ∫0ζΘ1ς,S−Θ1ς,S1ζ−ςβ−1dς.
Employing the norm to the aforesaid equation, we have(27)Sζ−S1ζ≤1−βMβS−S1L1+ζβMβΓβS−S1.
Since1−1−β/Mβ+ζβ/ΓβMβL1>0, we acquire S−S1=0. Consequently, we have S=S1. Repeating the same process, we can prove that Ip=Ip1,Im=Im1,Ipm=Ipm1,Rp=Rp1,Rm=Rm1, and Rpm=Rpm1. This completes the proof.


## 3. Qualitative Analysis

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


### 3.1. Positively Invariant and Boundedness

In order to find the invariant domain, we surmise the overall populationN=S+Ip+Im+Ipm+Rp+Rm+Rpm.Let the domain of the fractionalℙ model (7) that is epidemiologically sustainable determined by(28)℧≕S,Ip,Im,Ipm,Rp,Rm,Rpm∈ℝ+7:0≤S+Ip+Im+Ipm+Rp+Rm+Rpm≤N≤πϱ.In order to prove that the collection℧ is positively invariant, apply Lemma 1, and we have(29)Dζβ0ABCSζ|S=0=π≥0,Dζβ0ABCIpζ|Ip=0=g1S≥0,Dζβ0ABCImζ|Im=0=g2S≥0,Dζβ0ABCIpmζ|Ipm=0=g2Ip+g1Im≥0,Dζβ0ABCRpζ|Rp=0=ρ1Ip+ϵρIpm≥0,Dζβ0ABCRmζ|Rm=0=ρ2Im+ϵρh11−ϵIpm≥0,Dζβ0ABCRpmζ|Rpm=0=ρ1−h11−ϵIpm≥0.It is clear from (29) that every outcome of (7) is positive and will be in ℝ+7. Therefore, the set ℧ presented in (28) is positively invariant for fractional ℙ model (7).Additionally, to prove the boundedness of the solutions of the fractionalℙ framework, we proceed by aggregating all of the system’s components, offering(30)Dζβ0ABCNζ=π−ϱN−η1Ip+Ipm−η2Im+Ipm≤π−ϱN.Using the Laplace transform, we have(31)ℒDζβ0ABCNζ+−ϱN≤ℒπℒN1−λs1β−λβ1−β−ωβ−1N0≤1−βMβωβ+β1−βπω≤1−λβ1−λ1−βωβ1−β1−λMβ1+β1−βωβπω+N01−λω,where λ=β−1ϱ/Mβ. In view of inverse Laplace transform, we have(32)Nζ=πϱ−πϱ1−λddζ∫0ζEβλβζ−x1β1−λ1−βdx1+11−λEβλβζβ1−λ1−βN0,where Eβ,γ presents the ML function. Based on the assumption that the ML function exhibits asymptotic nature,(33)Eβ,γz˜≈∑J=1wz˜−λΓγ−βJ+Oz˜−w−1,z˜↦∞,βπ2<argz˜≤π.It is not really hard to perceive thatNζ↦π/ϱ, and this concludes that system (7) is biologically sustainable in the domain.


### 3.2. Disease-Free EquilibriumDℱℰ

SetIp=0,Im=0, and Ipm=0 in (6); then, Dℱℰ reduces to E˜0=π/ϱ,0,0,0,0,0,0.


### 3.3. Fundamental Reproductive Number

Assume that the infected component of system (6) is(34)Ip.ζ=g1S−g2+ϱ+ρ1+η1Ip,Im.ζ=g2S−g1+ϱ+ρ2+η2Im,IPm.ζ=g2Ip+g1Im−ρ+ϱ+η1+η2Ipm.As in [42], it is constructed employing the next generation matrix. For this, we construct the matrix ℱ and V as follows:(35)ℱ=q1πϱ0q1πϱ0q2πϱq2πϱ000andV=ρ1+η1+ϱ000ρ2+η2+ϱ000ρ+η1+η2+ϱ,ℱV−1=q1πϱρ1+η1+ϱ0q1πϱρ1+η1+η1+ϱ0q2πϱρ2+η2+ϱq2πϱρ+η1+η2+ϱ000.The eigenvalues of the system areλ1=0,λ2=q1π/ϱρ1+η1+ϱ=ℜ0p,λ3=q2π/ϱρ2+η2+ϱ=ℜ0m. Clearly, we see that ℜ0=maxℜ0p,ℜ0m is the fundamental reproductive number.


### 3.4. Local Stability ofDℱℰ

Theorem 4.
Let there be aDℰℱ point which is locally asymptotically stable if ℜ0<1. Also, if ℜ0>1, then it is unstable.Proof.
ForDℱℰ, the Jacobian matrix of ℙ model (6) is presented as(36)JE˜0=R3×3Q3×4ℛ4×3S4×4,where(37)R3×3=−ϱ,−q1πϱ,0,0,−q1πϱ−ρ1+η1+ϱ,0,0,0,q2πϱ−ρ2+η2+ϱ,,Q3×4=−q2πϱϕ1ϕ2ϕ3−q2πϱ0000000−ρ+η1+η2+ϱ000,ℛ4×3=0000ϕ1000ϕ2000,S4×4=ρe−ϱ+ϕ100ρh11−ϵ0−ϱ+ϕ20ρ1−h11−ϵ00−ϱ+ϕ3.
By the virtue of (36), one can acquire the subsequent characteristic polynomial as(38)Dλ=−ϱ+λϕ1+ϱ+λϕ2+ϱ+λρ+η1+η2+ϱ+λϕ3+ϱ+λ−q1πϱ+ρ1+η1+ϱ+λ×−q2πϱ+ρ2+η2+ϱ+λ.
After simplification, we get(39)λ1=−,λ2=−ϕ1+,λ3=−ϕ2+,λ4=−ρ+η1+η2+,λ5=−+ϕ3,λ6=−ρ+η1+η1+ϱ,λ7=−ρ+η1+η1+ϱ.
Sinceλ6,λ7<0, that is, ρ1+η1+ϱ<q1π/ϱ and ρ2+η2+ϱ<q2π/ϱ, this yields that ℜ0p<1. Analogously, ρ2+η2+ϱ<q2π/ϱ shows that ℜ0m<1. This concludes that ℜ0=maxℜ0p,ℜ0m<1 which shows that ℱℰ is locally asymptotically stable.


### 3.5. Global Stability ofDℱℰ

Here, we employ the mechanism described in [22] to examine the global stability of Dℱℰ. To begin, rewrite ℙ model (6) as(40)χ1.ζ=Y1χ1,χ2,χ2.ζ=Y2χ1,χ2,subject to Y2χ1,0=0. It is worth mentioning that χ1=S,Rp,Rm,Rpm represents the uncontaminated population while χ2=Ip,Im,Ipm indicate the contaminated population, respectively. Furthermore, the ℱℰ of the ℙ model is presented by V=χ¯,0. This demonstrates that V=χ¯,0 is globally asymptotically stable equilibrium for ℙ framework given that ℜ0<1 (by Theorem 1) and the subsequent assumptions hold:(i)
Forχ1.ζ=Y1χ1,0,χ¯ is globally asymptotically stable.(ii)
Y2χ1,χ2=Aχ2−Y2˜χ1,χ2,Y2˜χ1,χ2≥0 for χ1,χ2∈℧.The preceding hypothesis applies ifℙ model (6) satisfies the aforementioned requirements.Theorem 5.
Ifi and ii hold, then V=χ¯,0 is globally asymptotically stable equilibrium given that ℜ0<1.Proof.
By means ofℙ model, we can write(41)Y1χ1,χ2=π+ϕ1Rp+ϕ2Rm+ϕ3Rpm−g1+g2+ϱSρ1Ip+ϵρIpm−ϕ1+ϱRpρ2Im+ϵρh11−ϵIpm−ϕ2+ϱRmρ1−h11−ϵIpm−ϕ3+ϱRpm,and(42)Y2χ1,χ2=g1S−g2+ϱ+ρ1+η1Ip,g2S−g1+ϱ+ρ2+η2Im,g2Ip+g1Im−ρ+ϱ+η1+η2Ipm..Suppose the system is diminished to(43)χ1.ζ|χ2=0=π−ϱS000.It is clear by expression (43) that χ¯=πϱ,0 is the global asymptotic state.This is reinforced by the result, which is as follows:(44)S=πϱ+S0−πϱexp−ϱζ.Whenζ↦∞, the solution follows S↦π/ϱ. This implies the global convergence of (43) in ℧.Consider(45)A=q1−ρ1−η1−ϱ0q10−ρ2+η2+ϱ000−ρ+η1+η1+ϱ.Then,Y2χ1,χ2 can be expressed as(46)Y2˜χ1,χ2=q1Im+Ipm1−SN+g2Ipg1Im−g1Im+g2Ip,which illustrates that Y2˜χ1,χ2<0, and this suggests that the secondary assumption (ii) is not provided, and V=χ¯,0 may not be globally asymptotically stable when ℜ0<1.


### 3.6. Sensitivity Analysis

Through 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 x1, the responsiveness factor of ℜ0, is determined by ψx1ℜ0=∂ℜ0/∂x1x1/ℜ0. As ℜ0=maxℜ0p,ℜ0m, the sensitivity evaluation of ℜ0p and ℜ0m is conducted individually as follows:(47)ψq1ℜ0=∂ℜ0∂q1q1ℜ0=πq1ϱρ1+β1+ϱq1πϱρ1+β1+ϱ=1>0,ψη1ℜ0=∂ℜ0∂η1η1ℜ0=−η1ρ1+β1+ϱ<0,ψρ1ℜ0=∂ℜ0∂ρ1ρ1ℜ0=−ρ1ρ1+β1+ϱ<0,ψϱℜ0=∂ℜ0∂ϱ1ϱ1ℜ0=−ρ1+η1+2ϱρ1+β1+ϱ<0,ψq2ℜ0=∂ℜ0∂q2q2ℜ0=πq2ϱρ2+β2+ϱq1πϱρ2+β2+ϱ=1>0,ψη2ℜ0=∂ℜ0∂η2η2ℜ0=−η2ρ2+β2+ϱ<0,ψρ2ℜ0=∂ℜ0∂ρ2ρ2ℜ0=−ρ2ρ2+β2+ϱ<0,ψϱℜ0=∂ℜ0∂ϱϱℜ0=−ρ2+η2+2ϱρ2+β2+ϱ<0.The analysis shows that characteristics with highly accurate index values, especiallyq1 and q2, 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.


### 3.7. Numerical Approaches ofPM 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(48)Dζβ0ABCSζ=∇1ζ,Sζ,S0=S0.Observing (9), one can calculate for (48) in the problem described in the following:(49)Sζ=S0+1−βMβ∇1ζ,Sζ+βMβΓβ∫0ζ∇1ς,Sςζ−ςβ−1dς.Utilizing Lagrange’s interpolating polynomial technique onζ℘,ζ℘+1to∇1y,Sy=π/g1+g2+ϱ−ϕ1Rpy+ϕ2Rmy+ϕ3Rpmy/g1+g2+ϱ−Sy, then we have(50)S℘≈1ℏy−ζ℘−1∇1ζ℘,Sζ℘,Ipζ℘,Imζ℘,Ipmζ℘,Rpζ℘,Rmζ℘,Rpmζ℘−y−ζ℘∇1ζ℘−1,Sζ℘−1,Ipζ℘−1,Imζ℘−1,Ipmζ℘−1,Rpζ℘−1,Rmζ℘−1,Rpmζ℘−1,where ℏ=ζ℘−ζ℘−1.Inserting (49) into (50), we have(51)Sζn+1=S0+1−βMβ∇1ζ℘,Sζ℘,Ipζ℘,Imζ℘,Ipmζ℘,Rpζ℘,Rmζ℘,Rpmζ℘+βMβΓβ∑κ=1n∇1ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,Rpmζκℏ×∫ζκζκ+1y−ζκ−1ζn+1−yβ−1dy−∇1ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×∫ζκζκ+1y−ζκ−1ζn+1−yβ−1dy.=S0+1−βMβ∇1ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn+βMβΓβ∑κ=1n∇1ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,Rpmζκℏℑκ−1−∇1ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏℑκ,where(52)ℑκ−1=∫ζκζκ+1y−ζκ−1ζn+1−yβ−1dy=−1βζκ+1−ζκ−1ζn+1−ζκ+1β−ζκ−ζκ−1ζn+1−ζκβ−1ββ+1ζn+1−ζκ+1β+1ζn+1−ζκ+1β−ζn+1−ζκβ+1,and(53)ℑκ=∫ζκζκ+1y−ζκ−1ζn+1−yβ−1dy=−1βζκ+1−ζκ−1ζn+1−ζκ+1β−1ββ+1ζn+1−ζκ+1β+1−ζn+1−ζκβ+1.Also, utilizingζκ=κℏ into (52) and (53) represents(54)ℑκ−1=ℏβ+1ββ+1n+1−κβn−κ+2+β−n−κβn−κ+2+2β,and(55)ℑκ=ℏβ+1ββ+1n+1−κβ+1−n−κβn−κ+1+β.Eventually, we can write (56) in the form of (54) and (55) as follows:(56)Sζn+1=Sζ0+1−βMβ∇1ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn,+βMβΓβ∑κ=1n∇1ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,RpmζκΓβ+2×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β−∇1ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β.Additionally, the formulations for the remaining model factors are as follows:(57)Ipζn+1=Ipζ0+1−βMβ∇2ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn+βMβΓβ∑κ=1n∇2ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,RpmζκΓβ+2×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β−∇2ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β,Imζn+1=Imζ0+1−βMβ∇3ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn+βMβΓβ∑κ=1n∇3ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,RpmζκΓβ+2×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β−∇3ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β,,Ipmζn+1=Ipmζ0+1−βMβ∇4ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn+βMβΓβ∑κ=1n∇4ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,RpmζκΓβ+2×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β−∇4ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β,,(58)Rpζn+1=Rpζ0+1−βMβ∇5ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn+βMβΓβ∑κ=1n∇5ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,RpmζκΓβ+2×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β−∇5ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β,,Rmζn+1=Rmζ0+1−βMβ∇6ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn+βMβΓβ∑κ=1n∇6ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,RpmζκΓβ+2×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β−∇6ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β,,Rpmζn+1=Rpmζ0+1−βMβ∇7ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn+βMβΓβ∑κ=1n∇7ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,RpmζκΓβ+2×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β−∇7ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β..


### 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 Figure4, by maintaining the interaction rate stable, q1=0.9, we evaluated the influence of ρ1 in reducing the amount of pneumonia exclusively infected people. Figures 5 and 6 show that as the quantity of ρ1 increases, the proportion of pneumonia exclusively susceptible people decreases. Figure 5, 6, and 7 indicate that the proportion of instances in categories Ip,Im,Ipm,Rp,Rm,Rpm 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.Figure 4
Two-dimensional illustration of the susceptible classS and pneumonia infectious Ip when q1=0.9 with multiple fractional orders β∈0,1.
(a)(b)Figure 5
Two-dimensional illustration of the meningitis infectiousIm and ℙM co-infection Ipm when q1=0.9 with multiple fractional orders β∈0,1.
(a)(b)Figure 6
Two-dimensional illustration of the pneumonia recoveredRp and meningitis recovered Rm when q1=0.9 with multiple fractional orders β∈0,1.
(a)(b)Figure 7
Two-dimensional illustration of theℙM co-infection recovered Rpm when q1=0.9 with multiple fractional orders β∈0,1.In Figure8, we can observe that ρ2 is essential to minimize the meningitis development. The proportion of contagious community owing to meningitis decreases as the level of ρ2 increases from 0.1 to 0.9, but the interaction rate stays unchanged at q2=0.06. Figures 9, 10, and 11 indicate that the proportion of instances in categories Ip,Im,Ipm,Rp,Rm,Rpm decreased dramatically when contrasted to Figures 8–11, which were replicated inducing the control approach. Therefore, 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.Figure 8
Two-dimensional illustration of the susceptible classS and pneumonia infectious Ip when q2=0.06, ρ2∈0.1,0.9 with multiple fractional orders β∈0,1.
(a)(b)Figure 9
Two-dimensional illustration of the meningitis infectiousIm and ℙM co-infectious Ipm when q2=0.06, ρ2∈0.1,0.9 with multiple fractional orders β∈0,1.
(a)(b)Figure 10
Two-dimensional illustration of the pneumonia recoveredRp and meningitis recovered Rm when q2=0.06, ρ2∈0.1,0.9 with multiple fractional orders β∈0,1.
(a)(b)Figure 11
Two-dimensional illustration of theℝpm co-infectious recovered Rpm when q2=0.06, ρ2∈0.1,0.9 with multiple fractional orders β∈0,1.In Figure12, the meningitis connection incidence q2 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.Figure 12
Two-dimensional illustration of the susceptible classS and pneumonia infectious Ip with varying contact rate q2 and recovery rate co-infection ρ for multiple fractional orders β∈0,1.
(a)(b)Figure 13
Two-dimensional illustration of the meningitis infectionIm and ℙM infectious Ipm with varying contact rate q2 and recovery rate co-infection ρ for multiple fractional orders β∈0,1.
(a)(b)Figure 14
Two-dimensional illustration of the pneumonia recoveredRp and meningitis recovered Rm with varying contact rate q2 and recovery rate co-infection ρ for multiple fractional orders β∈0,1.
(a)(b)The influence of the survival incidence ofℙ on the co-infectious 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.Figure 15
Two-dimensional illustration of theℙM recovered Rpm with varying contact rate q2 and recovery rate co-infection ρ for multiple fractional orders β∈0,1.


## 3.1. Positively Invariant and Boundedness

In order to find the invariant domain, we surmise the overall populationN=S+Ip+Im+Ipm+Rp+Rm+Rpm.Let the domain of the fractionalℙ model (7) that is epidemiologically sustainable determined by(28)℧≕S,Ip,Im,Ipm,Rp,Rm,Rpm∈ℝ+7:0≤S+Ip+Im+Ipm+Rp+Rm+Rpm≤N≤πϱ.In order to prove that the collection℧ is positively invariant, apply Lemma 1, and we have(29)Dζβ0ABCSζ|S=0=π≥0,Dζβ0ABCIpζ|Ip=0=g1S≥0,Dζβ0ABCImζ|Im=0=g2S≥0,Dζβ0ABCIpmζ|Ipm=0=g2Ip+g1Im≥0,Dζβ0ABCRpζ|Rp=0=ρ1Ip+ϵρIpm≥0,Dζβ0ABCRmζ|Rm=0=ρ2Im+ϵρh11−ϵIpm≥0,Dζβ0ABCRpmζ|Rpm=0=ρ1−h11−ϵIpm≥0.It is clear from (29) that every outcome of (7) is positive and will be in ℝ+7. Therefore, the set ℧ presented in (28) is positively invariant for fractional ℙ model (7).Additionally, to prove the boundedness of the solutions of the fractionalℙ framework, we proceed by aggregating all of the system’s components, offering(30)Dζβ0ABCNζ=π−ϱN−η1Ip+Ipm−η2Im+Ipm≤π−ϱN.Using the Laplace transform, we have(31)ℒDζβ0ABCNζ+−ϱN≤ℒπℒN1−λs1β−λβ1−β−ωβ−1N0≤1−βMβωβ+β1−βπω≤1−λβ1−λ1−βωβ1−β1−λMβ1+β1−βωβπω+N01−λω,where λ=β−1ϱ/Mβ. In view of inverse Laplace transform, we have(32)Nζ=πϱ−πϱ1−λddζ∫0ζEβλβζ−x1β1−λ1−βdx1+11−λEβλβζβ1−λ1−βN0,where Eβ,γ presents the ML function. Based on the assumption that the ML function exhibits asymptotic nature,(33)Eβ,γz˜≈∑J=1wz˜−λΓγ−βJ+Oz˜−w−1,z˜↦∞,βπ2<argz˜≤π.It is not really hard to perceive thatNζ↦π/ϱ, and this concludes that system (7) is biologically sustainable in the domain.


## 3.2. Disease-Free EquilibriumDℱℰ

SetIp=0,Im=0, and Ipm=0 in (6); then, Dℱℰ reduces to E˜0=π/ϱ,0,0,0,0,0,0.


## 3.3. Fundamental Reproductive Number

Assume that the infected component of system (6) is(34)Ip.ζ=g1S−g2+ϱ+ρ1+η1Ip,Im.ζ=g2S−g1+ϱ+ρ2+η2Im,IPm.ζ=g2Ip+g1Im−ρ+ϱ+η1+η2Ipm.As in [42], it is constructed employing the next generation matrix. For this, we construct the matrix ℱ and V as follows:(35)ℱ=q1πϱ0q1πϱ0q2πϱq2πϱ000andV=ρ1+η1+ϱ000ρ2+η2+ϱ000ρ+η1+η2+ϱ,ℱV−1=q1πϱρ1+η1+ϱ0q1πϱρ1+η1+η1+ϱ0q2πϱρ2+η2+ϱq2πϱρ+η1+η2+ϱ000.The eigenvalues of the system areλ1=0,λ2=q1π/ϱρ1+η1+ϱ=ℜ0p,λ3=q2π/ϱρ2+η2+ϱ=ℜ0m. Clearly, we see that ℜ0=maxℜ0p,ℜ0m is the fundamental reproductive number.


## 3.4. Local Stability ofDℱℰ

Theorem 4.
Let there be aDℰℱ point which is locally asymptotically stable if ℜ0<1. Also, if ℜ0>1, then it is unstable.Proof.
ForDℱℰ, the Jacobian matrix of ℙ model (6) is presented as(36)JE˜0=R3×3Q3×4ℛ4×3S4×4,where(37)R3×3=−ϱ,−q1πϱ,0,0,−q1πϱ−ρ1+η1+ϱ,0,0,0,q2πϱ−ρ2+η2+ϱ,,Q3×4=−q2πϱϕ1ϕ2ϕ3−q2πϱ0000000−ρ+η1+η2+ϱ000,ℛ4×3=0000ϕ1000ϕ2000,S4×4=ρe−ϱ+ϕ100ρh11−ϵ0−ϱ+ϕ20ρ1−h11−ϵ00−ϱ+ϕ3.
By the virtue of (36), one can acquire the subsequent characteristic polynomial as(38)Dλ=−ϱ+λϕ1+ϱ+λϕ2+ϱ+λρ+η1+η2+ϱ+λϕ3+ϱ+λ−q1πϱ+ρ1+η1+ϱ+λ×−q2πϱ+ρ2+η2+ϱ+λ.
After simplification, we get(39)λ1=−,λ2=−ϕ1+,λ3=−ϕ2+,λ4=−ρ+η1+η2+,λ5=−+ϕ3,λ6=−ρ+η1+η1+ϱ,λ7=−ρ+η1+η1+ϱ.
Sinceλ6,λ7<0, that is, ρ1+η1+ϱ<q1π/ϱ and ρ2+η2+ϱ<q2π/ϱ, this yields that ℜ0p<1. Analogously, ρ2+η2+ϱ<q2π/ϱ shows that ℜ0m<1. This concludes that ℜ0=maxℜ0p,ℜ0m<1 which shows that ℱℰ is locally asymptotically stable.


## 3.5. Global Stability ofDℱℰ

Here, we employ the mechanism described in [22] to examine the global stability of Dℱℰ. To begin, rewrite ℙ model (6) as(40)χ1.ζ=Y1χ1,χ2,χ2.ζ=Y2χ1,χ2,subject to Y2χ1,0=0. It is worth mentioning that χ1=S,Rp,Rm,Rpm represents the uncontaminated population while χ2=Ip,Im,Ipm indicate the contaminated population, respectively. Furthermore, the ℱℰ of the ℙ model is presented by V=χ¯,0. This demonstrates that V=χ¯,0 is globally asymptotically stable equilibrium for ℙ framework given that ℜ0<1 (by Theorem 1) and the subsequent assumptions hold:(i)
Forχ1.ζ=Y1χ1,0,χ¯ is globally asymptotically stable.(ii)
Y2χ1,χ2=Aχ2−Y2˜χ1,χ2,Y2˜χ1,χ2≥0 for χ1,χ2∈℧.The preceding hypothesis applies ifℙ model (6) satisfies the aforementioned requirements.Theorem 5.
Ifi and ii hold, then V=χ¯,0 is globally asymptotically stable equilibrium given that ℜ0<1.Proof.
By means ofℙ model, we can write(41)Y1χ1,χ2=π+ϕ1Rp+ϕ2Rm+ϕ3Rpm−g1+g2+ϱSρ1Ip+ϵρIpm−ϕ1+ϱRpρ2Im+ϵρh11−ϵIpm−ϕ2+ϱRmρ1−h11−ϵIpm−ϕ3+ϱRpm,and(42)Y2χ1,χ2=g1S−g2+ϱ+ρ1+η1Ip,g2S−g1+ϱ+ρ2+η2Im,g2Ip+g1Im−ρ+ϱ+η1+η2Ipm..Suppose the system is diminished to(43)χ1.ζ|χ2=0=π−ϱS000.It is clear by expression (43) that χ¯=πϱ,0 is the global asymptotic state.This is reinforced by the result, which is as follows:(44)S=πϱ+S0−πϱexp−ϱζ.Whenζ↦∞, the solution follows S↦π/ϱ. This implies the global convergence of (43) in ℧.Consider(45)A=q1−ρ1−η1−ϱ0q10−ρ2+η2+ϱ000−ρ+η1+η1+ϱ.Then,Y2χ1,χ2 can be expressed as(46)Y2˜χ1,χ2=q1Im+Ipm1−SN+g2Ipg1Im−g1Im+g2Ip,which illustrates that Y2˜χ1,χ2<0, and this suggests that the secondary assumption (ii) is not provided, and V=χ¯,0 may not be globally asymptotically stable when ℜ0<1.


## 3.6. Sensitivity Analysis

Through 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 x1, the responsiveness factor of ℜ0, is determined by ψx1ℜ0=∂ℜ0/∂x1x1/ℜ0. As ℜ0=maxℜ0p,ℜ0m, the sensitivity evaluation of ℜ0p and ℜ0m is conducted individually as follows:(47)ψq1ℜ0=∂ℜ0∂q1q1ℜ0=πq1ϱρ1+β1+ϱq1πϱρ1+β1+ϱ=1>0,ψη1ℜ0=∂ℜ0∂η1η1ℜ0=−η1ρ1+β1+ϱ<0,ψρ1ℜ0=∂ℜ0∂ρ1ρ1ℜ0=−ρ1ρ1+β1+ϱ<0,ψϱℜ0=∂ℜ0∂ϱ1ϱ1ℜ0=−ρ1+η1+2ϱρ1+β1+ϱ<0,ψq2ℜ0=∂ℜ0∂q2q2ℜ0=πq2ϱρ2+β2+ϱq1πϱρ2+β2+ϱ=1>0,ψη2ℜ0=∂ℜ0∂η2η2ℜ0=−η2ρ2+β2+ϱ<0,ψρ2ℜ0=∂ℜ0∂ρ2ρ2ℜ0=−ρ2ρ2+β2+ϱ<0,ψϱℜ0=∂ℜ0∂ϱϱℜ0=−ρ2+η2+2ϱρ2+β2+ϱ<0.The analysis shows that characteristics with highly accurate index values, especiallyq1 and q2, 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.


## 3.7. Numerical Approaches ofPM 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(48)Dζβ0ABCSζ=∇1ζ,Sζ,S0=S0.Observing (9), one can calculate for (48) in the problem described in the following:(49)Sζ=S0+1−βMβ∇1ζ,Sζ+βMβΓβ∫0ζ∇1ς,Sςζ−ςβ−1dς.Utilizing Lagrange’s interpolating polynomial technique onζ℘,ζ℘+1to∇1y,Sy=π/g1+g2+ϱ−ϕ1Rpy+ϕ2Rmy+ϕ3Rpmy/g1+g2+ϱ−Sy, then we have(50)S℘≈1ℏy−ζ℘−1∇1ζ℘,Sζ℘,Ipζ℘,Imζ℘,Ipmζ℘,Rpζ℘,Rmζ℘,Rpmζ℘−y−ζ℘∇1ζ℘−1,Sζ℘−1,Ipζ℘−1,Imζ℘−1,Ipmζ℘−1,Rpζ℘−1,Rmζ℘−1,Rpmζ℘−1,where ℏ=ζ℘−ζ℘−1.Inserting (49) into (50), we have(51)Sζn+1=S0+1−βMβ∇1ζ℘,Sζ℘,Ipζ℘,Imζ℘,Ipmζ℘,Rpζ℘,Rmζ℘,Rpmζ℘+βMβΓβ∑κ=1n∇1ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,Rpmζκℏ×∫ζκζκ+1y−ζκ−1ζn+1−yβ−1dy−∇1ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×∫ζκζκ+1y−ζκ−1ζn+1−yβ−1dy.=S0+1−βMβ∇1ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn+βMβΓβ∑κ=1n∇1ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,Rpmζκℏℑκ−1−∇1ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏℑκ,where(52)ℑκ−1=∫ζκζκ+1y−ζκ−1ζn+1−yβ−1dy=−1βζκ+1−ζκ−1ζn+1−ζκ+1β−ζκ−ζκ−1ζn+1−ζκβ−1ββ+1ζn+1−ζκ+1β+1ζn+1−ζκ+1β−ζn+1−ζκβ+1,and(53)ℑκ=∫ζκζκ+1y−ζκ−1ζn+1−yβ−1dy=−1βζκ+1−ζκ−1ζn+1−ζκ+1β−1ββ+1ζn+1−ζκ+1β+1−ζn+1−ζκβ+1.Also, utilizingζκ=κℏ into (52) and (53) represents(54)ℑκ−1=ℏβ+1ββ+1n+1−κβn−κ+2+β−n−κβn−κ+2+2β,and(55)ℑκ=ℏβ+1ββ+1n+1−κβ+1−n−κβn−κ+1+β.Eventually, we can write (56) in the form of (54) and (55) as follows:(56)Sζn+1=Sζ0+1−βMβ∇1ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn,+βMβΓβ∑κ=1n∇1ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,RpmζκΓβ+2×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β−∇1ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β.Additionally, the formulations for the remaining model factors are as follows:(57)Ipζn+1=Ipζ0+1−βMβ∇2ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn+βMβΓβ∑κ=1n∇2ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,RpmζκΓβ+2×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β−∇2ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β,Imζn+1=Imζ0+1−βMβ∇3ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn+βMβΓβ∑κ=1n∇3ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,RpmζκΓβ+2×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β−∇3ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β,,Ipmζn+1=Ipmζ0+1−βMβ∇4ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn+βMβΓβ∑κ=1n∇4ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,RpmζκΓβ+2×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β−∇4ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β,,(58)Rpζn+1=Rpζ0+1−βMβ∇5ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn+βMβΓβ∑κ=1n∇5ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,RpmζκΓβ+2×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β−∇5ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β,,Rmζn+1=Rmζ0+1−βMβ∇6ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn+βMβΓβ∑κ=1n∇6ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,RpmζκΓβ+2×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β−∇6ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β,,Rpmζn+1=Rpmζ0+1−βMβ∇7ζn,Sζn,Ipζn,Imζn,Ipmζn,Rpζn,Rmζn,Rpmζn+βMβΓβ∑κ=1n∇7ζκ,Sζκ,Ipζκ,Imζκ,Ipmζκ,Rpζκ,Rmζκ,RpmζκΓβ+2×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β−∇7ζκ−1,Sζκ−1,Ipζκ−1,Imζκ−1,Ipmζκ−1,Rpζκ−1,Rmζκ−1,Rpmζκ−1ℏ×ℏβn+1−κβn−κ+2+β−n−κβn−κ+2+2β..


## 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 Figure4, by maintaining the interaction rate stable, q1=0.9, we evaluated the influence of ρ1 in reducing the amount of pneumonia exclusively infected people. Figures 5 and 6 show that as the quantity of ρ1 increases, the proportion of pneumonia exclusively susceptible people decreases. Figure 5, 6, and 7 indicate that the proportion of instances in categories Ip,Im,Ipm,Rp,Rm,Rpm 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.Figure 4
Two-dimensional illustration of the susceptible classS and pneumonia infectious Ip when q1=0.9 with multiple fractional orders β∈0,1.
(a)(b)Figure 5
Two-dimensional illustration of the meningitis infectiousIm and ℙM co-infection Ipm when q1=0.9 with multiple fractional orders β∈0,1.
(a)(b)Figure 6
Two-dimensional illustration of the pneumonia recoveredRp and meningitis recovered Rm when q1=0.9 with multiple fractional orders β∈0,1.
(a)(b)Figure 7
Two-dimensional illustration of theℙM co-infection recovered Rpm when q1=0.9 with multiple fractional orders β∈0,1.In Figure8, we can observe that ρ2 is essential to minimize the meningitis development. The proportion of contagious community owing to meningitis decreases as the level of ρ2 increases from 0.1 to 0.9, but the interaction rate stays unchanged at q2=0.06. Figures 9, 10, and 11 indicate that the proportion of instances in categories Ip,Im,Ipm,Rp,Rm,Rpm decreased dramatically when contrasted to Figures 8–11, which were replicated inducing the control approach. Therefore, 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.Figure 8
Two-dimensional illustration of the susceptible classS and pneumonia infectious Ip when q2=0.06, ρ2∈0.1,0.9 with multiple fractional orders β∈0,1.
(a)(b)Figure 9
Two-dimensional illustration of the meningitis infectiousIm and ℙM co-infectious Ipm when q2=0.06, ρ2∈0.1,0.9 with multiple fractional orders β∈0,1.
(a)(b)Figure 10
Two-dimensional illustration of the pneumonia recoveredRp and meningitis recovered Rm when q2=0.06, ρ2∈0.1,0.9 with multiple fractional orders β∈0,1.
(a)(b)Figure 11
Two-dimensional illustration of theℝpm co-infectious recovered Rpm when q2=0.06, ρ2∈0.1,0.9 with multiple fractional orders β∈0,1.In Figure12, the meningitis connection incidence q2 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.Figure 12
Two-dimensional illustration of the susceptible classS and pneumonia infectious Ip with varying contact rate q2 and recovery rate co-infection ρ for multiple fractional orders β∈0,1.
(a)(b)Figure 13
Two-dimensional illustration of the meningitis infectionIm and ℙM infectious Ipm with varying contact rate q2 and recovery rate co-infection ρ for multiple fractional orders β∈0,1.
(a)(b)Figure 14
Two-dimensional illustration of the pneumonia recoveredRp and meningitis recovered Rm with varying contact rate q2 and recovery rate co-infection ρ for multiple fractional orders β∈0,1.
(a)(b)The influence of the survival incidence ofℙ on the co-infectious 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.Figure 15
Two-dimensional illustration of theℙM recovered Rpm with varying contact rate q2 and recovery rate co-infection ρ for multiple fractional orders β∈0,1.


## 4. Conclusion

This research examines seven-dimensional pneumonia and meningitis. The 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. The Atangana–Baleanu fractional operator featuring generational characteristics is responsible for this beneficial outcome. The 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. This 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. The 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.



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*Source: 4717663-2022-07-31.xml*