Novel insights for a nonlinear deterministic-stochastic class of fractional-order Lassa fever model with varying kernels

Lassa fever is a hemorrhagic virus infection that is usually spread by rodents. It is a fatal infection that is prevalent in certain West African countries. We created an analytical deterministic-stochastic framework for the epidemics of Lassa fever employing a collection of ordinary differential equations with nonlinear solutions to identify the influence of propagation processes on infected development in individuals and rodents, which include channels that are commonly overlooked, such as ecological emergent and aerosol pathways. The findings shed light on the role of both immediate and subsequent infectiousness via the power law, exponential decay and generalized Mittag-Leffler kernels. The scenario involves the presence of a steady state and an endemic equilibrium regardless of the fundamental reproduction number, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Re _{0}<1$$\end{document}ℜ0<1, making Lassa fever influence challenging and dependent on the severity of the initial sub-populations. Meanwhile, we demonstrate that the stochastic structure has an exclusive global positive solution via a positive starting point. The stochastic Lyapunov candidate approach is subsequently employed to determine sufficient requirements for the existence and uniqueness of an ergodic stationary distribution of non-negative stochastic simulation approaches. We acquire the particular configuration of the random perturbation associated with the model’s equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Re _{0}^{s}<1$$\end{document}ℜ0s<1 according to identical environments as the presence of a stationary distribution. Ultimately, modeling techniques are used to verify the mathematical conclusions. Our fractional and stochastic findings exhibit that when all modes of transmission are included, the impact of Lassa fever disease increases. The majority of single dissemination pathways are less detrimental with fractional findings; however, when combined with additional spread pathways, they boost the Lassa fever stress.

Lassa fever , formerly known as Lassa hemorrhagic fever, is a deadly infectious species with serious consequences for the public's health 1 .The Lassa fever is primarily circulated by rodents (a multi-mammate rat) and is prevalent in West African countries 1 .Lassa fever is an extremely infectious condition characterized by an elevated temperature (38 C • ) and the degeneration of internal organs (including the spleen) 2 .The condition has been named after Lassa, a municipality in Borno State, located in Nigeria's northeastern region, where the initially identified Lassa fever case was discovered in 1969 [2][3][4] .
Lassa fever ways of dissemination encompass rodent-to-rodent, rodent-to-human, human-to-human, humanto-rodent and human-to-environment [5][6][7][8][9][10][11][12][13][14] .In accordance with the World Health Organization 14 , approximately 80% of Lassa fever-affected individuals exhibit no clinical signs (i.e., are asymptomatic), and one in every five contaminated individuals has been determined to be in a severe inflammation scenario [13][14][15] .Lassa virus infection is the underlying cause of Lassa fever, and it has an elevated death rate, particularly among expectant mothers and individuals with pre-existing medical histories 4 .According to an investigation carried out by Richmond et al. 16 , Lassa virus could potentially be employed to deploy missiles featuring infectious bacteria or physical arsenals.The infection has impressive efficacy in its spread.One instance of Lassa fever becoming infected in an entire community may initiate a pandemic 17 .As a result of the socioeconomic and biological consequences of Lassa virus, more research on the virus is required to gain a better understanding of transmission mechanisms and control.
Undoubtedly, the Lassa virus's host is a multi-mammate rodent (commonly referred to as Mastomys natalensis) that develops repeatedly and spreads extensively throughout the West Africa.Rodents from infected environments are seven times more likely to become infected compared with animals from controlled environments 16 .Yearly, roughly thirty thousand intriguing Lassa virus ailments occur, with 5500-15,000 casualties 3,5,13 .Despite this, there is currently no approved vaccine for Lassa fever.Nonetheless, it can be successfully alleviated by the antiviral drug, which is widely accessible and highly efficient if administered shortly after the start of the course of infection (i.e., throughout the six weeks of illness onset) 14 .Furthermore, as reported in 10 , Lassa fever implementation may necessitate medication for viruses, substance substitution, and bloodstream transplants.As a result, successful treatment for Lassa fever getting sick is unable to ensure permanent resistance to recurrence 10 .Certain elements may contribute to the prevalence (for example, human-to-human sickness, rodent-to-rodent illness, and ecological damage), whereas individuals (for example, therapy, ecological decontamination, and medical education campaigns) can lower the illness stress in a particular region.
Because of its significant incidence and the possibility of dissemination, the World Health Organization decided to place Lassa fever on its model, identifying critical illnesses that require greater involvement via healthcare administrators and scientists to enable greater focus on mitigation and regulation strategies 18 .To the extent of our understanding, few research investigations have been conducted with the objective of shedding more insight into the prevalence and medical manifestations of Lassa fever.As a result, additional scientific backing and epidemiological inquiries on the evolution of the propagation of Lassa fever are required, particularly with regard to the effect of elements influencing the environment.The Lassa fever time series case data were obtained from the open website of the Nigeria Centre for Disease Control 19 for the period of November 28, 2022, to April 13, 2023.All case data are laboratory confirmed by the Nigeria Centre for Disease Control situation report 19 .Figure 1 presents the number of Lassa fever laboratory cases confirmed weekly by states in Nigeria.
While searching for evidence, we discovered that multiple scholars have proposed finding algorithms that are capable of being utilized for obtaining fractional differential operators.The primary explanation for why this happens is that in practical application, obstacles prove manifestations of procedures that are analogous to the behaviours displayed by certain mathematical formulas.The discoveries of Hadamard, Caputo, Riez and Hilfer contribute to a fractional calculus that contains an index-law kernel.Because of Caputo's subsequent improvements, which enabled the use of classical initial values, the resulting form has been used in a variety of scientific fields 20 .Prabhakar 21 contemplated an alternative kernel via three settings as an outcome of the powerlaw and the generalized Mittag-Leffler function.Numerous investigators have felt drawn to this adaptation, and investigations on both concepts and their implementation were carried out [22][23][24][25][26][27][28][29][30][31][32] .Actually, both of the algorithms have distinct principles; e.g., the index-law kernel merely aids in the replication of procedures that demonstrate power-law actions, whereas the combination of the index-law and the generalized three-parameter kernel assists in the replication of procedures that indicate power-law behaviour.Mittag-Leffler encounters a sphere with potential as well 21,33 .Because the environment is convoluted, Caputo and Fabrizio 34 proposed an innovative kernel: an unusual exponential kernel alongside Delta Dirac features.A differential operator that is well-noted currently since it has the capability to reproduce procedures after diminishing memory.In fact, the notion of the fractional derivative that works with a non-singular kernel was developed by this kernel, ushering in an entirely novel era in fractional calculus 35 .Several of the investigators observations of the kernel's non-fractionality prompted the development of an additional kernel, the generalized Mittag-Leffler work, that had one setting.Atangana and Baleanu 35 suggested this formulation, which signifies yet another expansion breakthrough in the field of fractional calculus.The fractional derivative techniques are being successfully implemented in a variety of research disciplines of research.Author 36 presents the fundamental concepts of fractional differentiation, existence-uniqueness concepts and computational approaches to solving fractional differential equation.Nevertheless, whereas the crossover features of the Mittag-Leffler and the exponential kernel are widely identified as powerful mathematical approaches for illustrating practical problems, it is critical to recall that solely the core problems observing the crossover features of each of these approaches can be simulated according to multiple limitations, as in major difficulties, these two components are likely ineffective in confirming precisely at which

Model configuration
The cumulative human community, denoted as N h (t) , is classified into five categories, including those who are vulnerable to the pathogen, or X h (t) , individuals who carry the pathogen but aren't contagious, P h (t) , those who are contagious yet do not exhibit symptoms, Q ha (t) , those who are contagious but have symptoms, Q hs (t) and individuals who have healed from Lassa fever, or R h (t) are presented as: The overall rodent community, denoted as N r (t) , is categorized as follows: rodents vulnerable to the pathogen, denoted as X r (t) rodents contaminated well with Lassa virus infection and not contagious, denoted as P r (t) and contaminated rodents, denoted as N r (t) to: We take into consideration the aforementioned limited propagation routes: human-to-human, rodent-to-human and rodent-to-rodent.We also take into account informal pathogens like E-H interaction, A-H interaction and E-R interaction.We employ G s to represent the accumulation of the Lassa fever pathogen on ecological interfaces and G a to represent the accumulation of the viral infection in the atmosphere and as such, to account for unin- tended propagation mechanisms, where G s , G a , provides the highest pathogen maximum load on interfaces and equipment and in the atmosphere is presented by v with G a ≤ v .
We presume that 1 represents the steady rate of vulnerable living organisms recruiting new members.Throughout an infectious disease outbreak, the vulnerable people advance to the exposure group P h as described Here, ρ 1 is the reconfiguration value which suggests interaction with Q hs is less contagious than interacting to Q r .γ h is the enhanced surface rate between highly vulnerable individuals and afflicted rodents, vulnerable beings and contagious beings, vulnerable beings, the viral disease in the atmosphere, and the pathogen in the atmosphere.In this manner, the adjustment specifications ρ 2 , ρ 3 and ρ 4 also consider the degree of reinfec- tion of interaction with Q ha , G s and G a , respectively.Indications from the guarantees of inclusivity stated as The unprotected individuals advance to the contagious cohort at a speed of α 1 , where µα 1 is the ratio of affected populations who are latent and (1 − µ)α 1 is the fraction who develop symptoms.All categories of indi- viduals instinctually pass away at the speed ϑ 1 .Individuals who are infectiously indicative can pass away from the ailment at a speed of δ , whereas there are no incidences of infectiously subclinical people passing away from the infestation.Individuals who are infectiously displaying symptoms or not come back at rates of φ 1 and φ 2 , respectively.Throughout a power of infestation, the vulnerable rodents are attracted to the unprotected class P r at a steady rate 2 and relocate there for , where γ r is the proportion of efficacious interac- tion between rodents that are vulnerable to getting sick and afflicted rodents, as well as between vulnerable rodents and potentially polluted in the surroundings.The modifying variable, φ 1 , demonstrates that interaction with G s becomes less contagious than interaction with Q r .All rodents inherently pass away at a speed of ϑ 2 , and unprotected rodents transition to the contagious category at a rate of α 2 .Because they are consumed by human beings as meals, rodents can indeed perish at a rate of υ .Since afflicted rodents can persist to absorb the pathogen for the rest of their lives, pathogens do not cause rodents to drop dead.By urinating, excreting feces, haemorrhage, and secreting mucus, afflicted rodents, contagious indicative beings, and contagious symptom less beings, respectively, release the Lassa fever pathogen into the surroundings at rates of β 1 , β 2 and β 3 , respectively.We also make the assumption that a component of the pathogen accumulation advances into the atmosphere via air flow and anthropogenic at a rate of ξ 2 , whereas the remaining pathogen accumulation degrades on contaminated interfaces and in the atmosphere at a rate of ξ 2 and ξ 3 , respectively.The first order nonlinear ordinary differential equations that represent the Lassa fever framework in Fig. 2 are as follows: Table 1 displays the elements and representations for the parameters and their variables.However, in the framework (3), it is assumed that people currently reside in a steady environment.Even so, the perturbation in the surroundings will indeed influence certain aspects of the outbreak model's process variables.Having subsequently discovered that the stochastic framework can further adequately represent biological mechanisms and viral infections, there has been a significant rise in enthusiasm for taking random perturbation into account in virology configurations 42,43 .The framework can currently be perturbed stochastically in a variety of manners.Assuming that random perturbations constitute a single sort of white noise that is proportional to every component, respectively, comprises one of the most crucial steps.Considering the foregoing, it really is supposed in the proposed investigation that the white noise is individually proportional to the compartments s and G * a , respectively.Regarding that, the dynamical framework presumes the respective structure, linking the deterministic framework (3): (3) Where B m (t), m = 1, ..., 10 are mutually independent standard Brownian motions described on a com- plete probability space (�, F, {F t } t≥0 , P) with a {F t } t≥0 filtration entertaining the regular requirements 50 , and σ m , m = 1, ..., 10 represents the intensity of white noises B m , (m = 1, ..., 10) , respectively).
For the sake of inconvenience, we use the following symbols: For any x, ȳ ∈ ℜ, then x ∨ ȳ = max{x, ȳ} and x ∧ ȳ = min{x, ȳ}.The stochastic differential equation in d-dimensions is presented below: valued Wiener process, and v 0 is an ℜ d -valued random variable presented as .
Therefore, C2,1 . The differential for- mulation L for the stochastic differential Eq. ( 6) is given as then Itô's method can be described as: Here, we furnish the associated overview here to assist viewers who are familiar with FC (see; 20,34,35 ).
The Caputo fractional derivative involves the power-law function.The Caputo fractional-order derivative allows usual initial conditions when playing with the integral transform, for instance the Laplace transform 51,52 .
The Caputo-Fabrizio operator which has attracted many research scholars due to the fact that it has a nonsingular kernel.Also the Caputo-Fabrizio operator is most appropriate for modeling some class of real-world problem which follows the exponential decay law 53 .With the passage of time, developing a mathematical model using the Caputo-Fabrizio fractional-order derivative became a remarkable field of research.In recent times, several mathematicians were busy in development and simulation of Caputo-Fabrizio fractional differential equations 54 .
The fractional derivative operator of the Atangana-Baleanu of Caputo type is defined as: (5) Vol.:(0123456789) where ABC(χ) = 1 − χ + χ Ŵ(χ) indicates the normalization mapping.The kernel used in Atangana-Baleanu fractional differentiation appears naturally in several physical problems as generalized exponential decay and as a power-law asymptotic for a very large time 55,56 .The choice of this derivative is motivated by the fact that the interaction is not local, but global, and also, the trend observed in the field does not follow the power-law.The generalized Mittag-Leffler function completely induced the effect of memory, which is very important in the nonlinear Baggs-Freedman model 57 .
As fractional-order models describe the non-local behavior of biological systems and posses hereditary property, moreover, it provides information about its past and present state for the future, therefore, we represent the dynamical system (3) of Lassa virus in the framework of fractional-order Caputo's derivative to conceptualize the transmission of Lassa fever in a more accurate way.Thus, the system consist of fractional derivatives is presented by The structure of this essay can be described as follows: In Section "Model configuration", we demonstrate that the deterministic framework (3) has a forward and backward at ℜ 0 = 1.In Section "Stochastic analysis", we use a stochastic Lyapunov candidate technique to develop the necessary requirements for an ergodic stationary distribution of effective solutions to the stochastic system (5) to arise and be distinct.Also, the unique global positive solution for every positive initial conditions is provided in detail.We accurately communicate the piecewise fractional differential equations with varying kernels of the stochastic system (5) in Section "Numerical simulations" under the same assumptions as stated in 37 , reflecting the strong extinction and persistence of the illness.In Section "Results and discussion", simulation results are provided to certify our diagnostic results gained in Sections "Stochastic analysis and Numerical simulations".This manuscript is concluded with a concise summary.

Deterministic behaviour
Here, we demonstrate the mathematical and biophysical significance of our framework.Additionally, we will calculate the fundamental reproduction number and evaluate the steady state's consistency.

Theorem 2.1
The closed set ˜ := X h , P h , Q ha , Q hs , R h , X r , P r , Q r , G s , G a is a positive invariant set for the pro- posed fractional-order system (7).
Proof To prove that the system of Eq. ( 7) has a non-negative solution, the system of Eq. ( 7) implies Thus, the fractional system (7) has non-negative solutions.In the end, from the first four equations of the fractional system (7), we obtain Vol:.( 1234567890 www.nature.com/scientificreports/Solving the above inequality, we obtain so by the asymptotic behavior of Mittag-Leffer function 33 , we obtain Taking the same steps for the sixth, seventh and eighth of system (8), we get N r = � 2 ν+ϑ 2 .Analogously, we can deal the ninth and tenth compartment of (8), which yields . Hence, the closed set ˜ is a positive invariant region for the fractional-order Lassa fever model (7).
• We demonstrate that the solutions continue to stay positive and bounded in the suggested region, , under the assumption that all specifications are positive for time t .We shall examine the framework for Lassa fever spreads in the domain, which is as follows: • The biological meaningful equilibria of fractional system (7) are disease-free equilibrium and endemic equi- librium, depending on infected classes in both the populations.To obtain the infection-free equilibrium, we set the fractional derivative c 0 D

zero of the fractional system (7) without infection, and get
• To use the next generation matrix strategy 58 , the dominant eigenvalue of the matrix FG −1 corresponds to the fundamental reproduction number ℜ 0 of system (3).Therefore, we have After, making use of the Jacobian of F and G reviewed at E 0 , we obtain the next generation matrix at disease-free equilibrium is where , 0, 0, 0, 0 .The fundamental reproductive number ℜ 0 can be formulated as which is employed to establish whether the ailment manifests itself or not.
Next, we will illustrate the persistence of infection in the fractional-order system.It describes the level of endemicity of infection in the system.Biologically speaking, the infection persists in the system if the level of infected fraction stays at a higher level for t large enough.
(ii) When ℜ 0 > 1, then E 0 is unstable and the fractional-order system ( 7) is uniformly persistent.Thus, there is a unique globally asymptotically stable endemic equilibrium in the interior of , where and , , , µ 2 w 6 w 8 with b 1 > 0, then the model (3) will endure a forward bifurca- µ 2 w 2 w 6 with b 1 > 0, then the model (3) will endure a backward bifurcation at ℜ 0 = 1.
Proof Suppose y m = (y 1 , y 2 , y 3 , y 4 , y 5 , y 6 , y 7 , y 8 , y 9 , y 10 ) T = X h , P h , Q ha , Q hs , R h , X r , P r , Q r , G s T .Then, frame- work (3) can be composed as ẏm = g 1 (x 1 ) as shown in: where y 6 +y 7 +y 8 .By adjusting ℜ 0 = 1 , we select γ h as the bifurcation deviates.Let γ r ∝ γ h , which suggests that γ r = τ γ h for such t > 0 .Following that we obtain from the value of ℜ 0 where Now, the Jacobian matrix of model ( 3) is provided as assessed at the DFE E 0 in view of the bifurcation criterion γ * h presented as where The zero eigenvalue is connected with an appropriate eigenvector w = (w 1 , w 2 , w 3 , w 4 , w 5 , w 6 , w 7 , w 8 , w 9 , w 10 ) T .
The occurrence of bifurcation in the fractional-order Lassa fever transmission model (7) has important epidemiological implications.It implies that the conventional criterion of ℜ 0 < 1 is no longer sufficient for disease eradication, although it is still necessary.In this case, disease eradication would be determined by the initial sizes of the sub population in the model (i.e., state variables).Therefore, the practicality of controlling Lassa fever when ℜ 0 < 1 may depend on the starting sizes of the sub population.

Stochastic analysis
Before providing insights into the system dynamics of an Lassa fever model ( 5), we must guarantee that the solution is both global and non-negative.The existence and uniqueness of the global non-negative solution of system (5) with a certain non-negative initial value are guaranteed by the following formula.Theorem 3.1 Assume that there is an initial value �(0) = X h (0), P h (0), Q ha (0), Q hs (0), R h (0), X r (0), P r (0), + , there is a non-negative solution �(t) = X h (t), P h (t), Q ha (t), Q hs (t), R h (t), X r (t), P r (t), Q r (t), G s (t), G a (t) of the stochastic system (5) for t ≥ 0 and the solution will stay in ℜ 10 + almost surely (a.s).
Proof Because the parameters in the mathematical formulas are locally Lipschitz continuous for the specified preliminary community composition �(0) ∈ ℜ 10 + , there exists a distinctive local solution �(t) ∈ ℜ 10 + when t ∈ [0, τ ǫ ) (for information, see 50 ).To demonstrate that this finding is global in nature, we must demonstrate that τ ǫ = ∞ a.s.Suppose T ≥ 0 be large enough that �(0) all fall inside that interval 1 T 0 , T 0 .Determine the stopping time for every integer T ≥ T 0 .Introducing the stopping time In this investigation, we designated inf ∅ = ∞ so when ∅ signifies the empty set.By interpretation, τ T increases as T → ∞ .Select τ ∞ = lim T� →∞ τ T having τ ∞ ∈ [0, τ ǫ ] a.s.By asserting τ ∞ = ∞ a.s., we can show that τ ǫ = ∞ and �(t) a.s for all t ≥ 0 .To put it another way, we have to demonstrate that τ ǫ = ∞ a.s.If the assertion is false, a couple of parameters T > 0 and ε ∈ (0, 1) exist such that Since N h (t) = X h (t), P h (t), Q ha (t), Q hs (t), R h (t), then for t ≤ τ T , as is evident, By attempting to solve (23), we obtain Analogously, we assume N r (t) = X r (t), P r (t), Q r (t), G s (t), G a (t), then for t ≤ τ T , we have Again, solving (25), we have Furthermore, we introduce a C2 mapping h : ℜ 10 www.nature.com/scientificreports/Obviously, the function h is positive, as demonstrated by the reality that y 1 − ln(ey 1 ) ≥ 0 for all y 1 ≥ 0 .Assume that T ≥ T 0 and T > 0 be arbitrary, and the Itô methodology applied to (27) generates In view of (28), Lh : ℜ 10 + � → ℜ + the continuity formula defines as Here, K is a positive fixed number that is free of �(t) and t .Accordingly, Performing integration over 0 to τ T ∧ T, we have ( 28) www.nature.com/scientificreports/Inserting � T = τ T ≤ T for T ≥ T 1 and utilizing (32), P(� T ) ≥ ε.Additionally, it is important to keep in mind that for every ω ∈ � T , ∃ at least one �(τ T , ω) that are identical to T or 1/T, and therefore is not less than T − 1 − logT or 1 T − 1 + log T. As a result, As a result of ( 22) and ( 31), it describes that where 1 �(ω) denotes the indicator mapping of .Choosing T → ∞ shows the contradiction ∞ = T( Ũ + M 2 ) + h( �(0)) < ∞, which implies that τ ∞ = ∞ a.s and this is the immediate consequence.

Extinction and ergodic stationary distribution of Lassa fever model
We are interested in establishing adequate prerequisites for the extinction and existence-uniqueness of an ergodic stationary distribution of non-negative solutions to the dynamical model ( 5) in this segment.We begin by discussing a few explanations about stationary distribution (see; Khasminskii 59 ).Allow for the sake of simplicity Our next result is the strong law of large numbers, which is mainly due to 60 .

Theorem 3.4 If ℜ
p 0 < 1, then the disease Q ha , Q hs and Q r will wipe out exponentially with unit probability, that is., and Also, Proof Performing the integration on both sides of the proposed model ( 5) yields the following formulas We utilize the conception φ(t) in (39) for simplicity, and with several algebraic estimation, we emerge at the accompanying where the value of φ(t) is described as Clearly, we have Similarly, we integrate both sides of the last three cohorts of the developed framework (5), yielding the formula given (36)  �Q ha (t)� = 0, a.s.Similarly, by employing the Itô technique to the fourth cohort of model ( 5), employing the limits [0, t] , and then dividing by t , we have By integrating (43) over [0, t] and dividing it by t leads to (40)   X r (t) − X r (0) �Q r (t)� = 0, a.s.As a result, we noticed that illness extermination is determined by the setting of the parameter ℜ p 0 , i.e., for ℜ p 0 < 1 , the illness will eventually disappear.
Regardless of the omission of an EEP in the random perturbation model ( 5), we aim to explore the existence of an ergodic stationary distribution, which could prove disease perseverance more clearly.First, we shall discuss some of the results of Has'minskii's notion.Additional data is available at 59 .
Lemma 3.5 ( 50 ) Assume a bounded domain U ⊂ χ d 1 with regular boundary Ŵ such that (Z 1 ) Suppose a positive number M such that then the Markov technique Y(t) has a unique ergodic stationary distribution π(.), and satisfies ∀ y ∈ χ d 1 , where (.) is an integrable function in relation to the measure π.
In addition, based on Has'minskii's theory 59 , we will demonstrate essentials that guarantees the presence of an ergodic stationary distribution.www.nature.com/scientificreports/Proof The argument is separated into two phases: the initial one is to demonstrate that the uniform elliptic scenario is fulfilled, and the subsequent step is to generate a positive Lyapunov function that meets the criteria (Z 2 ) of Lemma 3.5.

Phase I:
The diffusion matrix of model ( 5) is presented as Finally, the criteria (Z 1 ) in Lemma 3.5 satisfies.
Phase II: Assume that Using the fact of model ( 5), we have utilizing the fact of ln Analogously, we have we have , Rh = 1 , Pr = 1 , Ḡs = 1
As a result, the requirement (Z 2 ) in Lemma 3.5 as well possesses.According to Lemma 3.5, mechanism (5) has a unique stationary distribution π(.) and the ergodicity applies.This completes the proof.

Numerical simulations
This section devotes itself to implementing the piecewise derivatives whenever the interrelated derivatives are the deterministic and fractional differential operators, taking into account local/nonlocal and singular/non-singular kernels.Thus, the order of the derivative χ lies in (0,1].

Fractional derivative with power kernel
In this segment, we shall examine the analysis of Lassa fever models (3) and ( 5)to ascertain how distinctive pathogen advancement mechanisms, which include those that are typically disregarded, like particulate and environmental interfacial pathways, affect both individuals and rodents using classical, index-law, and subsequently stochastic methods.The computational mechanism will be established in the initial process utilizing the classical derivative enactment, followed by the power law kernel in the second level, and eventually X r ∈ ǫ 3 , 1/ǫ 3 , P r ∈ ǫ 4 , 1/ǫ 4 , Q r ∈ ǫ 4 , 1/ǫ 4 , G s ∈ ǫ 5 , 1/ǫ 5 , G a ∈ ǫ 5 , 1/ǫ 5 , Vol:.( 1234567890

Fractional derivative with exponential decay kernel
Now, we shall demonstrate the dynamical analysis of Lassa fever models (3) and ( 5)to ascertain how distinctive pathogen advancement mechanisms, which include those that are typically disregarded, like particulate and environmental interfacial pathways, affect both individuals and rodents using classical, exponential decay law and subsequently stochastic methods.The computational mechanism will be established in the initial process utilizing the classical derivative enactment, followed by the exponential decay kernel in the other level, and thus the stochastic surroundings in the final stages, if we describe T as the ultimate propagation duration, such that the final attempt.The explanation for this hypothesis is then given using the corresponding formulaic framework: Now, we use the method outlined in 37 to analyze the piecewise configuration (78-80) in the context of Caputo-Fabrizio derivative.We immediately begin the methodology by doing the following:  2. Figure 3 illustrates the bifurcations with respect to susceptible humans and rodents, respectively.Following that, we intend to concentrate our efforts by examining the following two scenarios.We intend to concentrate on the three resulting components: (i) If the criterion ℜ 0 > 1 entails, there is a unique ergodic stationary distribution.
(ii) The effect of environmental noise on mechanisms (5) illness extermination.

Experimental examples
In what follows, we illustrate two examples in order to support the mathematical outcomes provided in previous sections.
Example 5.1 Assume that the random perturbations (σ 1 , σ 2 , ..., σ 10 ) = (0.001, 0.001, ..., 0.001), we determined ℜ 0 = 3.1301 > 1 and ℜ s 0 = 13.001> 1, which indicates that the illness will persist over time in a framework that is deterministic (3).Furthermore, we can deduce from Theorems 3.1 and 3.6 that model (5) confesses a global non-negative stationary outcomes on ℜ 10 + , as shown in the Fig. 5(a,b,c,d,e,f,g,h,i and j).According to the biological significance of Fig. 5(i,j), the most efficient interaction given between X h and Q r causes the greatest harm in terms of pathological advancement when power law kernels have been employed.This is accompanied by a low transmission rate within X h and fewer transmissible infectious asymptomatic humans when fractional-order is assumed to be χ = 0.95.By interacting with polluted atmospheres, dirty air and spreading indicative individuals, one can diminish the virus infection.We find that each means of dissemination contributes to the spread of Lassa fever, but certain are more important than others as well.It is clear that there is a huge disparity in the extent of dissemination of the pathways in N r as well as infection individuals.This demonstrates that while certain routes prove more lethal than others, every process contributes to a distinctive approach.shows that model (5) has a stable endemic equilibrium ℜ 10 + .On the contrary, we can derive the condition systemic from Theorem (3.4) will become disappearing with a unit probability.Figure 6(a,b,c,d,e,f,g,h,i, and j) indicate the appropriate numerical modeling of the solution ˜ ∈ ℜ 10 + to model (5) with the low random intensities and piecewise fractional differential equations scheme.
Taking into account the fractional calculus and biophysical approach, Fig. 6(i,j) shows that combining two dissemination processes improves the prevalence of diseases compared to an individual process when an exponential decay type fractional derivative has been applied with a fractional-order χ = 0.95 .We additionally discover that specific blends are more lethal than other people.Any interaction involving the efficient interaction rate among X h and Q r results in a spike of transmission, which is subsequently accompanied by any pairing via the successful interaction rate between X h and Q ha and then additional processes.

Conclusion
In this manuscript, we construct and verify a deterministic-stochastic scheme for analyzing the Lassa fever infection, including several modes of transmission, to address their effect on contamination growth in a community.
To begin, we illustrate that the framework ( 5) has a single global positive findings for any particular initial conditions.Following that, we employ the stochastic Lyapunov candidate technique to identify sufficient requirements for determining the presence and distinctive characteristics of an ergodic stationary distribution, known as a distribution of chances exhibiting certain inflexible features.In view of the distinct features of ℜ s 0 , we succeeded in demonstrating the way including multiple dissemination processes influences illness incidence.We employed stochastic tools to determine the extinction.We obtained mathematical formulas from our investigation that demonstrate the situations that dictate whether the illness can endure or can be regulated in the framework, as well as the manner in which the response of setting modifications results in system operation improvements.According to our system modeling, every single propagation route has an effect on the advancement of Lassa fever.But certain pathways of transmission are contributing substantially greater amounts than others as well.
Our findings indicate that measures in these fields ought to be avoided and overlooked when developing health strategies.Additional research can be conducted: (i) an amalgam of various methods for propagation that takes into account the unpredictable nature of disease, (ii) Effectively sanitary conditions, approaches to intervention, and multifaceted prevention initiatives that incorporate such several dissemination processes can aid in the reduction of illness incidence in the overall health sector, (iii) Employing the technique of cost-effective assessment, optimize the expense of multiple intervention strategies to ensure people in regions struggling with impoverishment challenges can be adequately aided.
Being able to get to actual information and involving Lévy noise, Poisson noise and telegraph noise may additionally enhance the model's anticipatory capability.Vertical propagation of Lassa fever in rodents may additionally be included in future research.Other approaches, such as system scaling, may be employed to assist in the evaluation when the settings are without dimensions and convey proportions of tangible repercussions instead of capacities of specific implications.

Figure 4 .
Figure 4. Lassa fever fitting outcomes considering the data obtained from Nigeria Centre for Disease Control 19 .(a) Cumulative cases (b) Weekly cases.

Table 1 .
Explanation of system's feature.
Vol.:(0123456789) Scientific Reports | (2023) 13:15320 | https://doi.org/10.1038/s41598-023-42106-0 ) April 13, 2023, Nigeria.The Ordinary Least Square solution was utilized for reducing the error terms with the help of (86), and the related relative error is used in the goodness of fit, Here, the notion I i is the reported cumulative infected cases and Îi is the cumulative infected cases obtained from simulating the model.The simulated values of cumulative infection are calculated by summing up the individuals, which moves from the infected compartment to the quarantined class each day.The infected population predicted by the proposed system (3).It is clear from the figure that the deterministic curves are in good agreement with the real data.All the parameters are estimated except ζ 1 = 0.167 , which is assumed.Estimated values of parameters are shown in Table Vol.:(0123456789) Scientific Reports | (2023) 13:15320 | https://doi.org/10.1038/s41598-023-42106-0www.nature.com/scientificreports/