Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2114 https://internationalpubls.com An Analytical Approach to Solve Nipah Virus Infection Model J. Sujatha1, * And N. Magesh2 1 *, 2 Post-Graduate and Research Department of Mathematics, Government Arts College for Men, Krishnagiri 635 001, Tamilnadu, India. e-mail : 1, * sujiselva8815@gmail.com., ORCID Address : http://orcid.org/0009-0006-2521-7304. e-mail: 2 nmagi_2000@yahoo.co.in, ORCID Address : http://orcid.org/0000-0002-0764-8390. Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: An epidemic is the rapid spread of disease to a large number of hosts in a given population within a short period of time. The SIR model is that it assumes homogeneous mixture of populations, however this is an over simplification of the reality of epidemic process. The Nipah virus, known for its severe outbreaks and high mortality rates, necessitates effective modeling techniques for understanding its transmission and potential control measure. In this paper, initially we discuss the stability analysis and the q − Homotopy Analysis Transform Method (q − HATM) is compared with CFDTM graphically. The q − Homotopy Analysis Transform Method offers a series solution that converges rapidly, while the CFDTM provides robust numerical approximation. Comparative analysis highlights the strengths and limitations of each method in terms of accuracy, computational efficiency, and applicability to real world scenarios. The q − Homotopy Analysis Transform Method used to solve the system of non-linear differential equations which arises due to Nipah virus infection Keywords and phrases : Stability Analysis, Nipah Virus (NiV), Epidemic Model, Non- linear Differential Equations, Mathematical Models, q - Homotopy Analysis Transform Method. 2020 AMS Classification : 37M05, 34F05, 92D30 1. Introduction Clinical manifestations of the Nipah virus infection in humans range from asymptomatic illness to severe respiratory infection and deadly encephalitis. An estimate of the case fatality rate ranges from 40 percent to 75 percent. Depending on regional capacity for epidemiological monitoring and clinical care, this rate may vary per epidemic. Humans can become infected with the Nipah virus through close contact with infected individuals or animals, particularly bats and pigs. The first major outbreak occurred in Malaysia in 1999, where the virus was linked to pigs. Later outbreaks in Bangladesh and India showed that it could also spread directly between people through bodily fluids and secretions. In 2001, it was noted that the virus might spread within a hospital environment in Siliguri, India, where medical employees or patients accounted for 75 percent of recorded cases. About half of the reported cases in Bangladesh between 2001 and 2008 were caused by transmission from person to person while treating infected patients. Individuals who are infected initially exhibit symptoms such as headaches, fever, vomiting, sore throat and myalgia (muscle pain). These symptoms can then progress to dizziness, sleepiness, altered awareness, and neurological indications that suggest severe encephalitis. It is thought that the incubation time (how long it takes for symptoms to appear after an infection) lasts mailto:sujiselva8815@gmail.com mailto:2000@yahoo.co.in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2115 https://internationalpubls.com between 4 and 14 days [6, 8, 14, 24] The Nipah Virus, a novel type of virus in the Parayxoviridae family and a member of the genus Henipavirus, has gained attention as an emerging zoonotic virus in the south-east and south Asian area. Because of this, epidemiology defines health and illness in terms of the frequencies and distributions of factors and situations in a population or in a particular subset of a population. Even though there have only been a few Nipah virus outbreaks, it is a matter of public health since it affects a broad variety of animals and leads to severe illness and even death in humans. As the effectiveness of antiviral medications is unsatisfactory, the majority of treatments are symptomatic and supportive. In light of the extremely high case fatality rate, effective and stringent control and preventative measures are required. This study applies stability analysis and the q − HATM method to Nipah virus (NiV) infections and explores potential q − HATM preventative and control measures [1, 2, 21, 25]. Finding exact solutions to differential equations with fractional orders appears to be substantially more challenging than carrying out the same tasks for equations with corresponding integer orders. Thus, much emphasis has been paid to developing effective numerical and analytical methodologies for estimating answers to this kind of issue. The Homotopy perturbation method, Laplace decomposition method, Homotopy analysis method (HAM) [21], and Adomian decomposition method are a few methods used at present. The q − HATM procedure is yet another really potent technique. This q − HATM technique merges the LTM and HAM when q ∈ [0,1/ m] Definition 1.1 [26]. The fractional R-L derivative of a function f(ι) is determined as 𝐽℘(𝑓(𝜄)) = 1 Γ( ℘ ) ∫   𝜄 0 (𝜄 − 𝜚)℘−1(𝑓(𝜚))𝑑𝜚 (1) Definition 1.2 [26] The Caputo fractional order derivative of f ∈ C is presented as follows : 𝐷𝜄 ℘(𝑓(𝜄) = { 𝑑𝑚𝑓(𝜄) 𝑑𝜄𝑚 if ℘ = 𝑚 ∈ 𝑁 1 Γ(𝑚 −℘) ∫   𝜄 0   (𝜄 − 𝜚)𝑚−℘−1(𝑓𝑚(𝜚))𝑑𝜚 if 𝑚− 1 < ℘ < 𝑚,𝑚 ∈ 𝑁 (2) Definition 1.3. [26] The LT of f(ι) with respect to fractional Caputo derivative is ℒ[𝐷𝜄 ℘(𝑓(𝜄))] = 𝑠℘𝐹(𝑠) − ∑   𝑚−1 𝑟=0 𝑠℘−𝑟−1𝑓(𝑟)(0 +), (𝑚 − 1 < ℘ ≤ 𝑚), (3) where F(s) is LT of f(ι). Here we solve fractional differential equations to illustrate how the suggested method can be applied. 𝐷 𝜄 ℘ 𝜚(ζ, ι) + R 𝜚(ζ, ι) + N 𝜚(ζ, ι) = g(ζ, ι),m − 1 < ℘ ≤ m, (4) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2116 https://internationalpubls.com ` 𝜚(ζ, 0) = g(ζ), (5) where 𝐷𝜄 ℘𝜚(𝜁, 𝜄) symbolise the Caputo derivative of 𝜚(ζ, ι), Utilizing the LT of equation (4), we obtain the following equation ℒ[𝜚(𝜁, 𝜄)] − 𝜚(𝜁,0) 𝑠 + (1−℘+ ℘ 𝑠℘ ) ℬ(℘) {ℒ[ℛ𝜚(𝜁, 𝜄)] + ℒ[𝒩𝜚(𝜁, 𝜄)] − ℒ[𝔤(𝜁, 𝜄)]} = 0 (6) Following are the steps involved in creating a homotopy for a primary function that is non zero: 𝒩[𝜗(𝜁, 𝜄; 𝑞)] = ℒ[𝜗(𝜁, 𝜄; 𝑞)] − 𝜚(𝜁, 0) 𝑠 + (1 − ℘ + ℘ 𝑠℘ ) ℬ(℘) {ℒ[ℛ𝜗(𝜁, 𝜄; 𝑞)] + ℒ[𝒩𝜗(𝜁, 𝜄; 𝑞)]} − (1 − ℘ + ℘ 𝑠℘ ) ℬ(℘) ℒ[𝔤(𝜁, 𝜄; 𝑞)] (7) where 𝑞 ∈ [0, 1 𝔪 ] and a real function of 𝜁, 𝜄 and 𝑞 is 𝜗(𝜁, 𝜄; 𝑞). For primary functions that are non zero, we establish the following homotopy: (1 − 𝔫𝑞)ℒ{𝜗(𝜁, 𝜄; 𝑞) − 𝜚(𝜁, 0)} = 𝔥𝑞𝒩{𝜗(𝜁, 𝜄; 𝑞)}, (8) where ℒ is denotes the LT, the constant parameter appears to be 𝑞 ∈ [0, 1 𝔪 ] , (𝔪 ≥ 1) and 𝔥 ≠ 0 is a supporting variable, 𝜗(𝜁, 𝜄; 𝑞) is an undetermined function, 𝜚0(𝜁, 𝜄) is an initial estimation of 𝜚(𝜁, 𝜄; 𝑞). For 𝑞 = 0 and 𝑞 = 1 𝔪 respectively, the outcomes listed below are acceptable: 𝜗(𝜁, 𝜄; 0) = 𝜚0(𝜁, 𝜄), 𝜗 (𝜁, 𝜄; 1 𝔪 ) = 𝜚(𝜁, 𝜄). (9) By expanding 𝑞 from 0 to 1 m , the result 𝜗(𝜁, 𝜄; 𝑞) converges from 𝜚0(𝜁, 𝜄) to 𝜚(𝜁, 𝜄). Taylor's theorem is used close to 𝑞 to enlarge the component of the series structure (𝜁, 𝜄; 𝑞) : 𝜗(𝜁, 𝜄; 𝑞) = 𝜚0(𝜁, 𝜄) + ∑  ∞ 𝑟=1 𝜚𝑟(𝜁, 𝜄)𝑞 𝑟, (10) where 𝜚𝑟(𝜁, 𝜄) = 1 𝑟! ∂𝑟𝜗(𝜁,𝜄;𝑞) ∂𝑞𝑟 | 𝑞=0 . (11) By using the basic linear operators 𝜚0(𝜁, 𝜄), 𝔪 and 𝔥, we could find one of the solutions for equation (4). At = 1 m , the series 𝜗(𝜁, 𝜄; 𝑞) converges: 𝜚(𝜁, 𝜄) = 𝜚0(𝜁, 𝜄) + ∑  ∞ 𝑟=1 𝜚𝑟(𝜁, 𝜄) ( 1 m ) 𝑟 . (12) By dividing r!, computing for q = 0 after differentiating the zeroth order deformation equation r times with respect to q ℒ[𝜚𝑟(𝜁, 𝜄) − 𝑘𝑟𝜚𝑟−1(𝜁, 𝜄)] = 𝔥ℜ𝑟(�⃗�𝑟−1), (13) where 𝜚𝑟 = [𝜚0(𝜁, 𝜄), 𝜚1(𝜁, 𝜄),⋯ 𝜚𝑟(𝜁, 𝜄)]. (14) Eq. (15), is reduced by using inverse LT, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2117 https://internationalpubls.com 𝜚𝑟(𝜁, 𝜄) = 𝑘𝑟𝜚𝑟−1(𝜁, 𝜄) + 𝔥ℒ −1 [ℜ𝑟 (𝜚𝑟−1)], (15) where the vectors are determined as ℜ𝑟(𝜚𝑟−1) = ℒ[𝜚𝑟−1(𝜁, 𝜄)] − (1 − 𝑘𝑟 𝔪 )( 𝜚(𝜁, 0) 𝑠 + (1 − ℘ + ℘ 𝑠℘ ) 𝒦(℘) ℒ[𝔤(𝜁, 𝜄)]) + (1 − ℘ + 𝜑 𝑠𝜑) 𝒦(℘) ℒ[ℛ𝜚𝑟−1(𝜁, 𝜄) +ℋ𝑟−1], (16) and 𝑘𝑟 = { 0 if 𝑟 ≤ 1 m if 𝑟 > 1 , ℋ𝑟 = 1 𝑟! [ ∂𝑟𝜗(𝜁, 𝜄; 𝑞) ∂𝑞𝑟 ] 𝑞=0 , and 𝜗(𝜁, 𝜄; 𝑞) = 𝜗0 + 𝑞𝜗1 + 𝑞 2𝜗2 +⋯ , we have, 𝜚𝑟(𝜁, 𝜄) = (𝑘𝑟 + 𝔥)𝜚𝑟−1(𝜁, 𝜄) − (1 − 𝑘𝑟 m )ℒ−1 (∑   𝑛−1 𝑘=0  𝑠𝜌−𝑘−1𝜚(𝑘)(𝜁, 0) + 1 𝑠𝜑 ℒ[𝔤(𝜁, 𝜄)]) +𝔥ℒ−1 1 𝑠𝜌 (ℒ[ℛ𝜚𝑟−1(𝜁, 𝜄)] +ℋ𝑟−1). (17) Therefore, by resolving the above, we may derive the iterative term 𝜚𝑟(𝜁, 𝑡). The series solution of the 𝑞 - HATM method is denoted by 𝜚(𝜁, 𝜄) = 𝜚0(𝜁, 𝜄) + ∑  ∞ 𝑟=1 𝜚𝑟(𝜁, 𝜄) ( 1 𝑚 ) 𝑟 . (18) 2. MATHEMATICAL FORMULATION The main tenet of this investigation is the notion of fractional calculus generalization. We believe that mathematical modelling is a very important component in understanding the illness and its transmission. To sustain the amounts of data produced on host pathogen communication, several types of models must be integrated. There are several models that may be used to study the transmission of a disease, however some techniques provide more precise and consistent results than others. To capture the ideal treatment for the disease, the system of nonlinear equations in this work is using the stability analysis and q − HATM technique. In the current work, we used the epidemiology model created by Sultana and N. Podder to observe the spread of a virus and to gather data for the epidemic model under consideration. The total population N (ι) is divided into three mutually disjoint compartments, namely susceptible S(ι), infected I(ι) and recovered R(ι) such that N (ι) = S(ι)+I(ι)+R(ι). To explain the fluctuation of NiV infections in the community, we take into account the framework of non-linear differential equations given below, which is a form of typical SIR disease model [24]. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2118 https://internationalpubls.com 𝑑𝒮 𝑑𝜄 = 𝜃𝒩(𝜄) − 𝛽𝒮(𝜄)ℐ(𝜄) − 𝜏𝒮(𝜄), 𝑑ℐ 𝑑𝜄 = 𝛽𝒮(𝜄)ℐ(𝜄) − (𝜓 + 𝜏 + 𝛿)ℐ(𝜄), 𝑑ℛ 𝑑𝜄 = 𝜓ℐ(𝜄) − 𝜏ℛ(𝜄), 𝑑𝒩 𝑑𝜄 = 𝜃𝒩(𝜄) − 𝛿ℐ(𝜄) − 𝜏𝒩(𝜄), (19) where the parameter β is the effective contact rate, θ is the natural birth rate and τ is the natural mortality rate, ψ is the recovery rate. δ is represent the disease induced death rate. The associated values of the parametric quantity are β = 0.75, δ = 0.01, τ = 0.002, ψ = 0.005, θ = 0.03 and the initial values are S(0) = 0.90, I(0) = 0.05, R(0) = 0.05. Using the above assumptions, the model of the fractional-order dynamical system represented mathematically (19) as follows [24]. 3. STABILITY ANALYSIS In this section, we discuss the local stability of endemic equilibrium and disease-free equilibrium of system (1) by using Hurwitz criterion. Since the first three equations of system (1) are independent of N(t), without loss of generality, we omit this one and then the system (1) is reduced to the following : 𝑑𝒮 𝑑𝜄 = 𝜃𝒩(𝜄) − 𝛽𝒮(𝜄)ℐ(𝜄) − 𝜏𝒮(𝜄), 𝑑ℐ 𝑑𝜄 = 𝛽𝒮(𝜄)ℐ(𝜄) − (𝜓 + 𝜏 + 𝛿)ℐ(𝜄), 𝑑ℛ 𝑑𝜄 = 𝜓ℐ(𝜄) − 𝜏ℛ(𝜄). (20) 3.1 Equilibrium Points. For the disease-free equilibrium points point E0 𝐸0 = ( 𝜃 𝜏 , 0,0). The Jacobian of the system (2) is given by, 𝐽 = [ −𝛽ℐ − 𝜏 𝛽𝒮 0 𝛽ℐ 𝛽𝑆 − 𝛿 − 𝜓 − 𝜏 0 0 𝜓 −𝜏 ] while the Jacobian of free equilibrium point 𝐸0 is 𝐽(𝐸0) = [ −𝜏 −𝛽𝜃 𝜏 0 0 𝛽𝜃−𝛿−𝜓𝜏−𝜏2 𝜏 0 0 𝜓 −𝜏] . For the reproduction number to be largest eigenvalues or spectral radius of characteristic equation |F V −1 − λI| = 0, for the model in (3), the associated matrices F and V for the new infectious terms Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2119 https://internationalpubls.com and the remaining transition terms evaluated at the disease-free equilibrium are respectively given by 𝐹 = [ 𝛽𝜃 𝜏 0 0 0 ] , 𝑉 = [ 𝜓 + 𝛿 + 𝜏 0 −𝜓 𝜏 ]. From which the next generation matrix can be calculated as F V−1=[ βθ τ(ψ + δ + τ ) 0 0 0 ] , where 𝑅0 = βθ τ(ψ + δ + τ ) is the required reproduction number. Theorem 3.1. For R0 > 1, there exist a positive disease equilibrium point E* . Proof. We set the left side of the system (20) is equal to zero. Then we have E* 𝜃 − 𝛽𝒮ℐ − 𝜏𝒮 = 0 (21) 𝛽𝒮ℐ − (𝜓 + 𝛿 + 𝜏)ℐ = 0 (22) 𝜓ℐ − 𝜏𝑅 = 0. (23) From (22) we have 𝛽𝒮ℐ = (𝜓 + 𝛿 + 𝜏)ℐ 𝛽𝒮 = (𝜓 + 𝛿 + 𝜏) 𝒮 = (𝜓 + 𝛿 + 𝜏) 𝛽 . It follows from (21) 𝜃 − 𝛽𝒮ℐ − 𝜏𝒮 = 0 𝜃 = (𝛽ℐ + 𝜏)𝒮 𝛽𝐼 + 𝜏 = 𝜃 𝒮 𝛽𝐼 = 𝜃 𝒮 − 𝜏 𝛽𝐼 = 𝛽𝜃 𝜓+𝛿+𝜏 − 𝜏 𝛽𝐼 = 𝛽𝜃−𝜏𝜓−𝜏𝛿−𝜏2 𝜓+𝛿+𝜏 𝐼 = 𝛽𝜃−𝜏𝜓−𝜏𝛿−𝜏2 𝛽(𝜓+𝛿+𝜏) 𝐼 = −𝜏(𝜓+𝛿+𝜏)+ 𝑠𝜃+𝜏) 𝛽(𝜓+𝛿+𝜏) 𝐼 𝐼 = 𝜏(𝑅0−𝛿+𝜏)𝜏(𝑅0−1) 𝛽(𝜓+𝛿+𝜏) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2120 https://internationalpubls.com From (23) we have 𝜓ℐ − 𝜏ℛ = 0 𝜓ℐ = 𝜏ℛ 𝜏ℛ = 𝜏(𝛽𝜃 − 𝜏𝜓 − 𝜏𝛿 − 𝜏2) 𝛽(𝜓 + 𝛿 + 𝜏) ℛ = 𝜏(𝛽𝜃 − 𝜏(𝜓 + 𝛿 + 𝜏) 𝛽𝜏(𝜓 + 𝛿 + 𝜏) ℛ = 𝜓𝜏(𝜓 + 𝛿 + 𝜏) − ( 𝛽𝜃 𝜓 + 𝛿 + 𝜏 − 1 ) 𝛽𝜏(𝜓 + 𝛿 + 𝜏) ℛ = 𝜓 𝛽 (𝑅0 − 1) 𝒮 = 𝜓+𝛿+𝜏 𝛽 I = τ β (R0 − 1) R = ψ β (R0 − 1) from which it can clearly be seen that, the equilibrium exists only if R0 > 1. 4. 𝒒 - HATM SOLUTION FOR THE PROJECTED MODEL We will discuss the model presented in this section, by mean of q − HATM solution approach. The system of equations of fractional order (19) are now defined as follows 𝐷𝑡 𝜌 𝒮(𝜄) = 𝜃𝒩(𝜄) − 𝛽𝒮(𝜄)ℐ(𝜄) − 𝜏𝒮(𝜄), 𝐷𝐿 𝜌 ℐ(𝜄) = 𝛽𝒮(𝜄)ℐ(𝜄) − (𝜓 + 𝜏 + 𝛿)ℐ(𝜄), 𝐷𝐿 𝜌 ℛ(𝜄) = 𝜓ℐ(𝜄) − 𝜏ℛ(𝜄), 𝐷𝐿 𝜌 𝒩(𝜄) = 𝜃𝒩(𝜄) − 𝛿ℐ(𝜄) − 𝜏𝒩(𝜄) (24) with initial conditions 𝒮(0) = 𝒮0, ℐ(0) = ℐ0, ℛ(0) = ℛ0,𝒩(0) = 𝒩0. (25) Applying the LT to the system (19) we obtain equation. ℒ{𝒮(𝜄)} − 1 𝑠 𝒮0 − 1 ℬ(𝜑) (1 − 𝜑 + 𝜑 𝑠𝜑 )ℒ{𝜃𝒩(𝜄) − 𝛽𝒮(𝜄)ℐ(𝜄) − 𝜏𝒮(𝜄)} = 0, ℒ{ℐ(𝜄)} − 1 𝑠 ℐ0 − 1 ℬ(𝜑) (1 − 𝜑 + 𝜑 𝑠𝜑 )ℒ{𝛽𝒮(𝜄)ℐ(𝜄) − (𝜓 + 𝜏 + 𝛿)ℐ(𝜄)} = 0, ℒ{ℛ(𝜄)} − 1 𝑠 ℛ0 − 1 ℬ(𝜑) (1 − 𝜑 + 𝜑 𝑠𝜑 )ℒ{𝜓ℐ(𝜄) − 𝜏ℛ(𝜄)} = 0, ℒ{𝒩(𝜄)} − 1 𝑠 𝒩0 − 1 ℬ(𝜑) (1 − 𝜑 + 𝜑 𝑠𝜑 )ℒ{𝜃𝒩(𝜄) − 𝛿ℐ(𝜄) − 𝜏𝒩(𝜄)} = 0. (26) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2121 https://internationalpubls.com Next taking non-linear operator to (26) we get , 𝒩1[𝜗1, 𝜗2, 𝜗3, 𝜗4] = ℒ{𝜗1(𝜄; 𝑞)} − 1 𝑠 𝒮0 − 1 ℬ(𝜑) (1 − ∅ + 𝜑 𝑠𝜑 ) × ℒ{𝜃𝜗4(𝜄; 𝑞) − 𝛽𝜗1(𝜄; 𝑞)𝜗2(𝜄; 𝑞) − 𝜏𝜗1(𝜄; 𝑞)}, 𝒩2[𝜗1, 𝜗2, 𝜗3, 𝜗4] = ℒ{𝜗2(𝜄; 𝑞)} − 1 𝑠 ℐ0 − 1 ℬ(∅) (1 − ℘ + ℘ 𝑠𝑝 ) × ℒ{𝛽𝜗1(𝜄; 𝑞)𝜗2(𝜄; 𝑞) − (𝜓 + 𝜏 + 𝛿)𝜗2(𝜄; 𝑞)}, 𝒩3[𝜗1, 𝜗2, 𝜗3, 𝜗4] = ℒ{𝜗3(𝜄; 𝑞)} − 1 𝑠 ℛ0 − 1 ℬ(℘) (1 − ℘ + ℘ 𝑠℘ ) × ℒ{𝜓𝜗2(𝜄; 𝑞) − 𝜏𝜗3(𝜄; 𝑞)} 𝒩4[𝜗1, 𝜗2, 𝜗3, 𝜗4] = ℒ{𝜗4(𝜄; 𝑞)} − 1 𝑠 𝒩0 − 1 ℬ(℘) (1 − ℘ + ℘ 𝑠℘ ) × ℒ{𝜃𝜗4(𝜄; 𝑞) − 𝛿𝜗2(𝜄; 𝑞) − 𝜏𝜗4(𝜄; 𝑞)}. (27) For ℋ(𝑥, 𝜄) = 1 the 𝑟th order deformation equation is presented as ℒ[𝒮𝑟(𝜄) − 𝐾𝑟𝒮𝑟−1(𝜄)] = 𝔥ℜ1,𝑟[𝒮𝑟−1, ℐ⃗𝑟−1, ℛ⃗⃗𝑟−1, �⃗⃗⃗�𝑟−1], ℒ[ℐ𝑟(𝜄) − 𝐾𝑟ℐ𝑟−1(𝜄)] = 𝔥ℜ3,𝑟[𝒮𝑟−1, ℐ⃗𝑟−1, ℛ⃗⃗𝑟−1, �⃗⃗⃗�𝑟−1], ℒ[𝑅𝑟(𝜄) − 𝐾𝑟𝑅𝑟−1(𝜄)] = 𝔥ℜ1,𝑟[𝒮𝑟−1, ℐ⃗𝑟−1, ℛ⃗⃗𝑟−1, �⃗⃗⃗�𝑟−1], ℒ[𝑁𝑟(𝜄) − 𝐾𝑟𝑁𝑟−1(𝜄)] = 𝔥ℜ3,𝑟[𝒮𝑟−1, ℐ⃗𝑟−1, ℛ⃗⃗𝑟−1, �⃗⃗⃗�𝑟−1], (28) ℒ[ℛ𝑟(𝜄) − 𝐾𝑟ℛ𝑟−1(𝜄)] = 𝔥ℜ4,𝑟[𝒮𝑟−1, ℐ⃗𝑟−1, ℛ⃗⃗𝑟−1, �⃗⃗⃗�𝑟−1] ℒ[𝒩𝑟(𝜄) − 𝐾𝑟𝒩𝑟−1(𝜄)] = 𝔥ℜ5,𝑟[𝒮𝑟−1, ℐ⃗𝑟−1, ℛ⃗⃗𝑟−1, �⃗⃗⃗�𝑟−1] where, ℜ1,𝑟[𝒮𝑟−1, ℐ⃗𝑟−1, ℛ⃗⃗𝑟−1, �⃗⃗⃗�𝑟−1] = ℒ{𝒮𝑟−1(𝜄)} − (1 − 𝐾𝑟 n ) 𝒮0 𝑠 − 1 ℬ(℘) (1 − ℘ + ℘ 𝑠℘ ) × ℒ {𝜃𝒩𝑟−1(𝜄) − 𝛽∑   𝑟−1 𝑖=0   𝒮𝑖(𝜄)ℐ𝑟−1−𝑖(𝜄) − 𝜏𝒮𝑟−1(𝜄)} , ℜ2,𝑟[𝒮𝑟−1, ℐ⃗𝑟−1, ℛ⃗⃗𝑟−1, �⃗⃗⃗�𝑟−1] = ℒ{ℐ𝑟−1(𝜄)} − (1 − 𝐾𝑟 n ) ℐ0 𝑠 − 1 ℬ(℘) (1 − ℘ + 𝜑 𝑠𝜑 ) × ℒ {𝛽∑   𝑟−1 𝑖=0  𝒮𝑖(𝜄)ℐ𝑟−1−𝑖(𝜄) − (𝜓 + 𝜏 + 𝛿)) ℐ𝑟−1(𝜄)} , ℜ3,𝑟[𝒮𝑟−1, ℐ⃗𝑟−1, ℛ⃗⃗𝑟−1, �⃗⃗⃗�𝑟−1] = ℒ{ℛ𝑟−1(𝜄)} − (1 − 𝐾𝑟 n ) ℛ0 𝑠 − 1 ℬ(℘) (1 − ℘ + ℘ 𝑠𝜑 ) × ℒ{𝜓ℐ𝑟−1(𝜄) − 𝜏ℛ𝑟−1(𝜄)}, ℜ4,𝑟[𝒮𝑟−1, ℐ⃗𝑟−1, ℛ⃗⃗𝑟−1, �⃗⃗⃗�𝑟−1] = ℒ{𝒩𝑟−1(𝜄)} − (1 − 𝐾𝑟 n ) 𝒩0 𝑠 − 1 ℬ(℘) (1 − ℘ + ℘ 𝑠℘) × ℒ{𝜃𝒩𝑟−1(𝜄) − 𝛿ℐ𝑟−1(𝜄) − 𝜏𝒩𝑟−1(𝜄)}. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2122 https://internationalpubls.com Taking inverse LT on both side of (28) we have 𝒮𝑟(𝜄) = 𝐾𝑟𝒮𝑟−1(𝜄) + 𝔥ℒ −1{ℜ1,𝑟[𝒮𝑟−1, ℐ⃗𝑟−1, ℛ⃗⃗𝑟−1, �⃗⃗⃗�𝑟−1]}, ℐ𝑟(𝜄) = 𝐾𝑟ℐ𝑟−1(𝜄) + 𝔥ℒ −1{ℜ3,𝑟[𝒮𝑟−1, ℐ⃗𝑟−1, ℛ⃗⃗𝑟−1, �⃗⃗⃗�𝑟−1]}, ℛ𝑟(𝜄) = 𝐾𝑟ℛ𝑟−1(𝜄) + 𝔥ℒ −1{ℜ4,𝑟[𝒮𝑟−1, ℐ⃗𝑟−1, ℛ⃗⃗𝑟−1, �⃗⃗⃗�𝑟−1]}, 𝒩𝑟(𝜄) = 𝐾𝑟𝒩𝑟−1(𝜄) + 𝔥ℒ −1{ℜ5,𝑟[𝒮𝑟−1, ℐ⃗𝑟−1, ℛ⃗⃗𝑟−1, �⃗⃗⃗�𝑟−1]}. (29) On solving the system (29), we get 𝒮0(𝜄) = 0.90 ℐ0(𝜄) = 0.05, ℛ0(𝜄) = 0.05, 𝒩0(𝜄) = 1.00, 𝒮1(𝜄) = 0.005550𝔥 ℬ(℘) {1 − ℘ + ℘𝑡℘ Γ(℘ + 1) }, ℐ1(𝜄) = −0.032900𝔥 ℬ(℘) {1 − ℘ + ℘𝑡℘ Γ(℘ + 1) }, ℛ1(𝜄) = −0.00015𝔥 ℬ(℘) {1 − ℘ + ℘𝑡℘ Γ(℘ + 1) }, 𝒩1(𝜄) = −0.02750𝔥 ℬ(℘) {1 − ℘ + ℘𝑡℘ Γ(℘ + 1) }, 𝒮2(𝜄) = 0.005550𝔥(𝔫 + 𝔥) ℬ(℘) {1 − ℘ + ℘𝑡℘ Γ(℘ + 1) } − 0.0211632750𝔥2 [ℬ(℘)]2 (1 − 2℘ + ℘2 + 2℘(1 − ℘)𝑡℘ Γ(℘ + 1) + ℘2𝑡2℘ Γ(2℘ + 1) ), Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2123 https://internationalpubls.com and so on simplifying the above system of equations and the values given can be acquired. As described by the solutions of the q − HATM, we obtain the series as follows: 𝒮(𝜄) = 𝒮0(𝜄) +∑   ∞ 𝑟=1  𝒮𝑚(𝜄) ( 1 𝔪 ) 𝑟 , ℐ(𝜄) = ℐ0(𝜄) +∑   ∞ 𝑟=1   ℐ𝑚(𝜄) ( 1 𝔪 ) 𝑟 , ℛ(𝜄) = ℛ0(𝜄) +∑   ∞ 𝑟=1  ℛ𝑚(𝜄) ( 1 𝔪 ) 𝑟 , (31) 5. RESULT AND DISCUSSION This study establishes the local stability of both the endemic and disease-free equilibriums of system (19) using the Hurwitz criterion. we made use of q − HATM technique (q − HATM) to provide a suitable answer to a set of non-linear equations and solve an epidemic Nipah virus illness model. The initial requirements regarding the model that is offered in this paper are S(0) = 𝑆0 = 0.90, I(0) = 𝐼0 = 0.05, R(0) = 𝑅0 = 0.05. A series of solutions have been examined in order to understand the behaviour of the model. We establish and identify a technique for S(ι), I(ι) and R(ι) using variations in fractional orders (℘) with regard to time (ι). It is evident from the diagram that the planned version matches the order and provides more comfort. As a consequence, success is predicted when the fractional operator is applied to the future version. The diagram shows how the sequence has a substantial impact on the anticipated model and provides additional flexibility. Here, we evaluated the tabular data and diagrams produced by q − HATM. Table 1, 2 and 3 show the answer to S(ι), I(ι) and R(ι) correspondingly for variations in fractional orders (℘) with regard to time (ι) acquired by q − HATM. From the charts, it is clear that the system’s time and order significantly influence the values of S(ι), I(ι) and R(ι). Additionally, we record embedding behavior using the value of ℘ = 0.6, 0.7, 0.8, 0.9 and for classical order ℘ = 1 and the order of the fractional differential coefficient at h = −1 and m = 1. As can be seen, the proposed system must imitate behaviors’ with regard to parameters and fractional order. The proposed fractional operator also gives other exciting outcomes to explore and project the future of the proposed model. The recent study may aid in understanding the lethal virus because epidemic models heavily rely on genetic characteristics. TABLE 1. Table of the susceptible class of S(ι) for various ℘ values ι ℘ = 0.6 ℘ = 0.7 ℘ = 0.8 ℘ = 0.9 ℘ = 1 0 0.8943938760 0.8964303052 0.8980434690 0.8992333672 0.9000000000 1 0.8723831960 0.8740240779 0.8765310374 0.8798538867 0.8838683625 2 0.8556267369 0.8515618372 0.8483633345 0.8465448548 0.8465734500 3 0.8393687934 0.8272321253 0.8141067684 0.800756083 0.7881152625 4 0.8232160227 0.8011902661 0.7744866197 0.7433693061 0.7084938000 5 0.8070412178 0.7735978359 0.7299972459 0.6749925801 0.6077090625 6 0.7907912215 0.7445877208 0.6810028025 0.5960768778 0.48576105500 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2124 https://internationalpubls.com TABLE 2. Table of the infective class of I(ι) for various ℘values Ι ℘ = 0.6 ℘ = 0.7 ℘ = 0.8 ℘ = 0.9 ℘ = 1 0 0.06659041200 0.06179960675 0.05743760300 0.05350440075 0.05000000000 1 0.1072058919 0.1055129490 0.1026609786 0.09866297055 0.09362003750 2 0.1336278505 0.1413923592 0.1485607804 0.1545145311 0.1586801500 3 0.1577451162 0.1772442353 0.1988958201 0.2218808745 0.2451803375 4 0.1807804428 0.2137417469 0.2536356816 0.3002924200 0.3531206000 5 0.2031954688 0.2510705611 0.3125881524 0.3893329883 0.4825009375 6 0.2252195365 0.2892854813 0.3755563451 0.4886561530 0.6333213500 TABLE 3. Table of the recovered class of R(ι) for various ℘ values ι ℘ = 0.6 ℘ = 0.7 ℘ = 0.8 ℘ = 0.9 ℘ = 1 0 0.05008627200 0.05005977800 0.05003656800 0.05001664200 0.05000000000 1 0.05046117025 0.05049310148 0.05052099088 0.05054365211 0.05056050000 2 0.05069644883 0.05082942615 0.05097484619 0.05112829150 0.051285200 3 0.05090780422 0.05115653905 0.05145245440 0.05179309994 0.05217410000 4 0.05110747387 0.05148349032 0.05195773351 0.05253751992 0.053227200 5 0.05130015308 0.05181334016 0.05249103017 0.05335975052 0.05444450000 6 0.05148820996 0.05214739658 0.05305185035 0.05425793308 0.055826000 Figure 1: Plot of q − HATM and CFDTM solution for S(ι) with respect to ι at h = −1, ; m = 1, for varying ℘. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2125 https://internationalpubls.com Figure 2: Plot of q − HATM and CFDTM solution for I(ι) with respect to ι at h = −1, ; m = 1, for varying ℘. Figure 3: Plot of q − HATM and CFDTM solution for R(ι) with respect to ι at h = −1, ; m = 1, for varying ℘. 6 CONCLUSION The complex structure of fractional calculus is studied using Hurwitz criterion, the local stability of the disease-free equilibrium and endemic equilibrium of system is proved and utilizing the q − Homotopy Analysis Transform Method in the current inquiry of Nipah viral sickness. The results are provided in a series form that converges quickly. We have stabilized a more accurate and realistic model by taking into account the Caputo fractional derivatives. The present technique also saves time and does not call for any deviation while solving nonlinear models. It is concluded that findings for real, imagined, and other phenomenal variables may be derive using the proposed nonlinear model. Data Availability: No data were used in this paper. Ethical Approval: This article does not contain any studies with human participants or animals performed by any of the authors. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2126 https://internationalpubls.com Conflicts of Interest: The authors confirm no competing interests. Funding Statement: The research did not receive any funding. Author's Contributions: The authors read and approved the final manuscript. REFERENCES [1] M. Abdullah, A. Ahmad, N. Raza, M. Farman and M. O. Ahmad, Approximate solution and analysis of smoking epidemic model with Caputo fractional derivatives, Int. J. Appl. Comput. Math., 4 (5) (2018), 1-16, doi:10.1007/s40819-018-0543-5. [2] E. Addai, D. F. M. Torres, Z. Abdul-Hamid, M. Nwaife Mesue and J. K. K. Asamoah, Modelling the dynamics of online food delivery services on the spread of food-borne disease. Model. Earth Syst. Environ., 10(2) (2024), 4993-5008, doi : 10.1007/s40808-024-02046-8. [3] E. Alzahrani and A. Zeb, Stability analysis and prevention strategies of tobacco smoking model, Bound. Value Probl., 3 (2020), 1–13, doi:10.1186/s13661-019-01315-1. [4] A. Atangana and J. F. G´omez-Aguilar, A new derivative with normal distribution kernel, Theo. Meth. Appl. Phys. A, 476 (2017), 1-–14, doi:10.1016/j.physa.2017.02.016. 14. [5] A. Atangana and J. F. G´omez-Aguilar, Hyperchaotic behavior obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws, Chaos Solitons Fractals, 102 (2017), 285–294, doi:10.1016/j.chaos.2017.03.022. [6] M. Biswas, Model and control strategy of the deadly Nipah virus (NiV) infections in Bangladesh, Res. Rev. Biosci., 6 (12) (2012), 370–377. [7] M. H. A. Biswas, Optimal control of Nipah virus (NiV) infections: A Bangladesh scenario, J. Pure Appl. Math. Adv. Appl., 12 (1) (2014). 77-104. [8] M. S. Chadha, J. A. Comer, L. Lowe, P. A. Rota, P. E. Rollin, W. J. Bellini, T. G. Ksiazek and A. Mishra, Nipah virus associated encephalitis outbreak, Siliguri, India, Emerg Infect Dis., 12 (2) (2006), 235-240, doi:10.3201/eid1202.051247. [9] D. K. Chanchal, S. Alok, M. Sabharwal, R. K. Bijauliya, and S. Rashi, Nipah: silently rising infection, Int. J. Pharm. Sci. Res., 9 (8) (2018), 3128–3135, doi:10.13040/IJPSR.0975- 8232.9(8).3128-35. [10] V. K. Chattu, R. Kumar, S. Kumary, F. Kajal, and J. K. David, Nipah virus epidemic in Southern India and emphasizing one health approach to ensure global health security, J. Fam Med Prim Care, 85 (2) (2018), 275–283, doi:10.4103/jfmpc.jfmpc.137.18. [11] H. T. Chong, M. J. Hossain and C. T. Tan, Differences in epidemiological and clinical features of Nipah virus encephalitis between the Malaysian and Bangladesh outbreaks, Neurology Asia, 13 (2008), 23-26. [12] G. Chowell, N. W. Hengartner, C. Castillo-Chavez, P. W. Fenimore and J. M. Hyman, The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda, J. Theor. Biol., 229 (2004), 119-126, doi:10.1016/j.jtbi.2004.03.006. [13] K. B. Chua, W. J. Bellini, P. A. Rota, B. H. Harcourt, A.Tamin, S. K. Lam, T. G. ksiazek, P. E. Rollin and B. W. Mahy, Nipah Virus: A recently emergent deadly paramyxovirus, J Sci., 288 (5470) (2000), 1432-1435, doi:10.1126/science.288.5470.1432. [14] H. Field, P. Young, J. M. Yob, J. Mills, L. Hall and J. MacKenzie, The natural history of Hendra and Nipah viruses, Microbes and Infection, 3 (4) (2001), 307-314, doi:10.1016/S1286- Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2127 https://internationalpubls.com 4579(01)01384-3. [15] H. F. Huo, R. Chen and X.Y. Wang, Modelling and stability of HIV/AIDS epidemic model with treatment, Appl. Math. Model., 40 (13) (2016), 6550–6559, doi:10.1016/j.apm.2016.01.054. [16] K. M. Owolabi and A. Atangana, Mathematical analysis and computational experiments for an epidemic system with nonlocal and nonsingular derivative, Chaos Solitons and Fractals, 126 (2019), 41–49, doi:10.1016/j.chaos.2019.06.001. [17] V. Padmavathi, A. Prakash, K. Alagesan, N. Magesh, Analysis and numerical simulation of novel coronavirus (COVID-19) model with Mittag-Leffler kernel. Math. Method Appl. Sci., 44 (2) (2021), 1863-1877, doi:10.1002/mma.6886. [18] V. Padmavathi, N. Magesh , K. Alagesan , M. I. Khan, S. Elattar, M. Alwetaishi, and A. M. Galal, Numerical modeling and symmetry analysis of a pine wilt disease model using the Mittag-Leffler kernel, Symmetry, 14 (2022), 1–15, doi:10.3390/sym14051067. [19] S. Pathak, A. Maiti, G. Samanta and Rich dynamics of an SIR epidemic model, Nonlinear Anal. Model. Control, 15 (1) (2010), 71-–81, doi:10.15388/NA.2010.15.1.14365. [20] A. Prakash, M. Goyal, H. M. Baskonus and S. Gupta, A reliable hybrid numerical method for a time dependent vibration model of arbitrary order, AIMS Math., 5 (2) (2020), 979-1000, doi:10.3934/math.2020068. [21] A. Prakash and H. Kaur, Numerical solution for fractional model of Fokker-Planck equation by using q-HATM, Chaos Solitons Fractals 105 (2017), 99-110, doi:10.1016/j.chaos.2017.10.003. [22] A. Prakash and H. Kaur, Analysis and numerical simulation of fractional order Chan-Allen model with Atangana-Baleanu derivatives, Chaos Solitons Fractals, 124 (2019), 134–142, doi:10.1016/j.chaos.2019.05.005 . [23] J. Singh, D. Kumar and D. Baleanu, New aspects of fractional Biswas-Milovic model with Mittag-Leffler law, Math. Model. Nat. Phenom., 14 (3) (2019), 1–23, doi:10.1051/mmnp/2018068. [24] J. Sultana and C. N. Podder, Mathematical analysis of Nipah virus infections using optimal control theory, J. App. Maths. Phy., 4 (2016), 1099-1111, doi:10.4236/jamp.2016.46114. [25] S. Thakur, V. Singh, A. Kumar and S. K. Srivastava, A time-fractional order HIV/AIDS epidemic model woth q-HATM. Int. J. Appl. Comput. Math., 10(26) (2024), doi: 10.1007/s40819-023-01664-7. [26] P. Veeresha, D. G. Prakasha, N. Magesh, A. John Christopher, D. U. Sarwe, Solution for fractional potential KdV and Benjamin equations using the novel technique, J. Ocean Eng Sci., 6 (2021), 265–275, doi:10.1016/j.joes.2021.01.003.