THE MATHEMATICAL NIPAH VIRUS MODEL WITH ATANGANA-BALEANU DERIVATIVE

. The applicability of the Atangana-Baleanu derivative in modeling and assessing the dynamics of the Nipah virus is investigated in this paper. The Atangana-Baleanu derivative, a fractional derivative operator, is used in the mathematical model of the Nipah virus to add memory effects and non-local behaviour. To do this, we ﬁrst use ﬁxed point theory to establish the existence and uniqueness of the solutions for the fractional order model. Using various fractional order values, we got a number of numerical simulations emphasizing the signiﬁcance of the aforementioned derivative. The ﬁndings of solving the Nipah virus (NiV) model using the Atangana-Baleanu derivative provide a better understanding of the dynamics and behaviors of the studied systems


INTRODUCTION
The Nipah virus is a zoonotic pathogen that is a member of the Henipavirus genus and family Paramyxoviridae.It was first identified during an outbreak of encephalitis (inflammation of the brain) in Malaysia and Singapore in 1998-1999.The name "Nipah" is derived from a village in Malaysia where the outbreak occurred ( [31], [27]).
Nipah virus is primarily transmitted to people via direct contact with infected pigs, most notably fruit bats (specifically, some species of flying foxes) or their infected bodily fluids.
Nipah virus disease can cause moderate to serious symptoms.They commonly include fever, headache, muscle soreness, disorientation, and respiratory problems ( [16], [8]).In severe cases, it can progress to encephalitis, characterized by seizures, altered mental status, and coma.Nipah virus disease has a significant fatality rate, ranging from 40% to 75%.
Nipah epidemics have largely occurred in South and Southeast Asia, most notably in Bangladesh and India [10].Such outbreaks are frequently caused by a mix of transmission from person to person and contact with diseased pigs or their products, such as tainted fruits or raw date sap from palm trees ( [23], [32].[23], [12]).
Pteropus fruit bats are thought to be the Nipah virus's natural reservoir hosts.These bats do not show symptoms of sickness, but the virus is often found in their urine, spit, and stool.
In certain outbreaks, intermediate hosts such as pigs have been implicated in multiplying and spreading the virus to people ( [15], [15], [20]).Because of its ability to cause severe sickness and epidemics, the Nipah virus remains an important threat to public safety.Ongoing research and surveillance efforts are focused on understanding the virus, its transmission dynamics, and developing effective preventive and therapeutic strategies.
The Atangana-Baleanu derivative is a fractional derivative operator that extends the concept of differentiation to include fractional orders.It was introduced by Dumitru Baleanu and Jean Roger Atangana in 2016 as a generalization of the classical derivative.The Atangana-Baleanu derivative has found applications in various fields, including physics, engineering, and biology.
It is particularly useful for modeling phenomena such as viscoelasticity, anomalous diffusion, fractal behavior, and population dynamics, where memory effects and non-local behaviors play a crucial role.In fractional differential equations, researchers use the Atangana-Baleanu derivative to accurately characterize and evaluate systems having memory effects.It is a mathematical instrument for investigating the stability, its current state, uniqueness, and attributes of answers in these systems.The use of the Atangana-Baleanu derivative contributes to a more comprehensive understanding of complex dynamics and facilitates the development of more accurate models for real-world phenomena.
Several writers played a role in the formation of fractional mathematics beginning in 1695, after L'Hospital asked Leibiz what if the order of a derivative is n = 1/2.The modeling of biological processes, engineering, physics, finance, and many other fields have shown increased interest in fractional operators [9].Despite the fact that the classic fractional Riemann Liouville and Caputo derivatives have various advantages for describing reality as more reliable, the singularity that results from the strength of their kernels presents several significant processing challenges [18], and [17].To overcome these concerns, Caputo and Fabrizio presented the Caputo-Fabrizio (CF) derivative, a novel non-singular fractional derivative that uses an increasing kernel.[17].Atangana and Baleanu have introduced Atangana-Baleanu (AB) derivatives with Mittag-Leffler kernel function, which are influenced by the concept of CF derivative [2].
Apart from being a derivative, these operators have been viewed as a filter regulator [3].Atangana and Alkahtani performed a comprehensive examination of the existence and uniqueness of Keller-Segel model solutions including the CF derivative [4].Baggs and Freedman models with exponential kernels were examined by Atangana and Koca [2].Singh et al. investigated the epidemiological model for computer viruses with CF derivatives using Banach fixed point theory [13].Yavuz et al [25] solved fractional partial differential equations with an AB derivative.The traditional model of a contaminated lake system was changed using the notion of fractional differentiation [28].Uc ¸ar [34] used CF and AB variants to investigate a smoking model as it relates to determination and education-related activities.
Therefore, motivated by the applicability of Atangana-Baleanu derivatives, we intend to further explore a Nipah Virus model in the perspective of fractional concept in relation to the effects of the vaccine and condoms.Following is how the remainder of the document is organized: On the following page, we give some introductory information about fractional order derivatives.
The method described in Section 2 for the Nipah virus.In Section 3, it is established that our fractional NiV model's solutions exist and are distinct using fixed point theory.A few number findings are presented in Section 4 along with a brief commentary on them.The conclusions are found in section 5 of the research.
1.1.Basic Definitions.In this part, we provide some essential definitions of fractional derivative.
Definition 1.1.The Sobolev space of order one(1) in (k, l) is defined as [19]: derivative is Caputo type of order α of p is given by [?]: (1.2) derivative in Riemann-Liouville type of order α of p is given by [34]: . The fractional integral is defined by [34]: [34]: We assume the following: Natural sickness recovery can take place because of powerful antibodies [7].Casual touching of the dead bodies will expose the individuals to the virus [5].There is an interaction between the farmer and the infectious pigs [6].Since they are continuously watched, medical personnel safeguard themselves against the virus, and infection can happen in a therapy class, isolated individuals do not aid in the spread of NiV [7].The general public has easy access to and can afford condoms, an infected isolation facility, and vaccinations [6].After some time, people who have recovered become susceptible to infection once more [35].
where the force of infections are

EXISTENCE AND UNIQUENESS OF NIV MODEL SOLUTION
The solution of nonlinear equations is a complicated matter in differential calculus [?].The fractional order model under examination is nonlinear, accurate solutions to these kind of problems may be hard to find.Therefor, we analyze the existence and uniqueness of Niv model solution with the use of fixed point theory. Consider with the norm defined as We reorganize the model (2.1)-(2.17) in the simple method shown below.17 are the kernels Applying fractional integral [11] to the equation (2.1)-(2.17),we have then the kernels W i , i = 1, 2, ..., 17 fulfill the Lipschitz condition and contraction such that Proof.Given two functions S and S * , we have Hence, the Lipschitz condition satisfied for W 1 and 0 ≤ χ 1 + χ 2 + µ < 1 implies W 1 is also contraction.
Similarly, it can be demonstrated that the other kernels satisfy the Lipschitz condition and contraction.However, observe that S(t) + S u (t) + S v (t) + S uc (t) + S un (t) + S vc (t) + S vn (t) + E(t) +C(t) + I(t) Without loss of generality, ξ2 = (τ Therefore the kernels W i , i = 1, 2, ..., 17 fulfill the Lipschitz condition and 0 ≤ ξi < 1 implies We define the system (3.18)-(3.34) in the following recursive form: with the initial conditions: Next, we look at the difference in the successive terms as follows: With the same procedure from (3.88)-(3.96),we reduced the remaining expressions to the form: If we can determine t 0 that satisfies the equation This establishes the Existence and continuity of the aforementioned solutions.In order to demonstrate that the functions are solutions of equation (2.1)-(2.17),let's say that when we continue recursively at t 0 , we have Observe that N ≤ Λ µ and N p ≤ Λ p µ p .Therefore without loss of generality, we have Since all the kernel satisfied Lipschitz condition, applying norm on the both side we have b 5 0.000324 Inferred from [22] q 1 0.75 Inferred from [22] q 2 0.4617 Inferred from [22] q 3 0.531 Inferred from [22] q 4 0.513 Inferred from [22] q 5 0.000648 Inferred from [22] ε 0.03 Estimated  Due to the introduction of the vaccine, the susceptible population was marginally altered as increased but remained steady while responding to the virus.On the other hand, because to the lack of vaccine administration, the susceptible unvaccinated population dramatically shrank and eventually disappeared.Hence, the immune system was compromised.The number of susceptible people who have received vaccinations, on the other hand, rises over time as the value of fractional order α rises, whereas the number of susceptible people who have not received vaccinations falls with time.This suggests that the fractional order derivatives of the dynamical variables are more useful for estimating the proportion of susceptible, susceptible vaccinated, and susceptible unvaccinated people.become exposed and sick.First growing and then declining but never reaching zero was the number of infected and exposed pigs.The exposed and infected pigs grow slowly and eventually diminish slowly while the susceptible pigs decrease slowly when the fractional order is tiny.

CONCLUSION
A derivative of AB made up of a Mittag-Leffler kernel has been described by Atangana and Baleanu.We introduce the fractional with vaccination and condoms linked with the model for the first time by the idea of Atangana and Baleanu derivatives in order to see further applications of these fractional derivatives and better investigate Nipah virus dynamics.As with the fixed point approach, our goal is to provide the prerequisites for the models' existence and uniqueness as solutions.To comprehend the efficacy of the fractional order α as well as vaccine and condoms, numerical calculations for these fractional models have been carried out.These simulations show that, depending on the various fractional orders, increasing vaccination and condom use result in a reduction in the Nipah virus's ability to propagate.The description of the Nipah virus's mechanics in light of vaccines and condoms is where we believe the current research will be most helpful.
Finally, the utilization of the Atangana-Baleanu derivative in solving differential equations with memory effects and non-local behaviors offers significant advantages in understanding and modeling complex systems.By incorporating this fractional derivative operator, we can capture the long-term memory effects and non-local behaviors exhibited by various real-world phenomena more accurately.Through the application of the Atangana-Baleanu derivative, we have gained deeper insights into the dynamics and behaviors of systems that cannot be adequately described by classical differential equations.The inclusion of memory effects in the modeling process has allowed us to achieve a more comprehensive and accurate representation of the underlying dynamics.
For a continuous function p on [k, l].The inequality given below holds on [k, l] Taking norm of both sides of equations (3.54)-(3.86),applying triangular inequality and Lipschitz condition, we have

(
A) Susceptible vaccinated: condoms S vc and without condoms S vn (B) Susceptible unvaccinated: condoms S uc and without condoms S un

TABLE 2
4 Illness-related death rate in infectious people undergoing treatment δ d Illness-related death rate in infectious pigs µ d The rate at which deceased bodies are disposed of (burial/cremation) µ p Pig mortality rate µ Natural death rate ABC 0 , S u , S vc , S vn , S uc , S un , E,C, I, I it , I t , R, D, S p , E p , I p satisfied Lipschitz condition and they 2, 3, ..., 17, the fractional model provided as (2.1)-(2.17)has a unique solution.
Taking the limit as n → ∞, we have θin → 0, i = 1, 2, ..., 17 which implies that the fractional model (2.1)-(2.17)has solution Next, we show that the solution is unique: suppose there is another solution to the model say S * , S v * , S u * , S uc * , S un * , S vc * , S vn * , E * ,C * , I * , I it * , I t * , R * , D * , S p * , E p * , I p * , then we have