Qualitative analysis and chaotic behavior of respiratory syncytial virus infection in human with fractional operator

Respiratory syncytial virus (RSV) is the cause of lung infection, nose, throat, and breathing issues in a population of constant humans with super-spreading infected dynamics transmission in society. This research emphasizes on examining a sustainable fractional derivative-based approach to the dynamics of this infectious disease. We proposed a fractional order to establish a set of fractional differential equations (FDEs) for the time-fractional order RSV model. The equilibrium analysis confirmed the existence and uniqueness of our proposed model solution. Both sensitivity and qualitative analysis were employed to study the fractional order. We explored the Ulam–Hyres stability of the model through functional analysis theory. To study the influence of the fractional operator and illustrate the societal implications of RSV, we employed a two-step Lagrange polynomial represented in the generalized form of the Power–Law kernel. Also, the fractional order RSV model is demonstrated with chaotic behaviors which shows the trajectory path in a stable region of the compartments. Such a study will aid in the understanding of RSV behavior and the development of prevention strategies for those who are affected. Our numerical simulations show that fractional order dynamic modeling is an excellent and suitable mathematical modeling technique for creating and researching infectious disease models.

• Examine the stability and dynamical behavior of the SEI r I s R model.
• The Basic Reproduction number and Equilibrium points should be determined.
• Application of the Lagrange polynomial technique to obtain a numerical solution.
• The fractional order RSV model is demonstrated with chaotic behaviors.This paper is structured as follows: section one serves as an introduction, while section two presents fundamental fractional order derivatives applicable to solving the epidemic model.The third section delves into the positivity of the fractional order model, discussing endemic equilibrium, DFE, and sensitivity analysis.In "Qualitative analysis of model", the existence and uniqueness of a system of model solutions are affirmed using the fixed point concept."Ulam-Hyers stability" focuses on investigating the Ulam-Hyers stability of the RSV model."Numerical scheme" explores the impact of fractional parameters by employing numerical techniques to solve the fractionalorder system.Finally, "Simulation and discussion" and "Conclusion" discuss the results and conclusion.

Basic concepts
Before creating the model, we must first study the fundamental definitions that are essential to comprehending fractional operators.Definition 2.1 14 Assume that w(t) ∈ H 1 (0, T), T > 0, and 0 < β ≤ 1 .The following is the definition of a power- law kernel fractional derivative: Definition 2.2 14 For (1), the corresponding fractional integral operator is given by

Fractional order RSV model
Here, we propose an RSV model with memory-affected fractal fractional order.The RSV epidemic model presented in 13 is a classical derivative that has to be taken into account.The entire population is classified into five classes: susceptible S(t), exposed E(t), normally infectious I r (t) , super infectious I s (t) , and recovered R(t).The birth rate is indicated by b while the natural death rate is symbolized by µ .The overall size of the human population at a given time is denoted by N. The rate of virus transmission between people, which affects the disease's spread is represented by the symbol β .The interval from infection to the development of symptoms, (1) or the incubation time of the virus within an infected human, is shown by the symbol η .The likelihood that a new case of the virus will manifest as a normal infected human, typically characterized by typical transmission patterns indicated by p.The additional likelihood that a new case may involve an infected person who is superspreading, who has an elevated potential for transmitting the virus to a larger number of individuals indicated by (1 − p) .The rate at which individuals with normal infections recover from the disease is symbolized by r 1 .The rate at which individuals with super-spreading infections recover is indicated by r 2 .The set of non-linear fractional differential equations is the initial conditions are

Steady state analysis
Here, we will look into the RSV fractional-order system for both endemic and disease-free steady states.Setting the fractional derivative C D α t S , C D α t E , C D α t I r , C D α t I s and C D α t R of the fractional system (3) without infection to zero allows us to reach the steady-state with no infection.Disease free equilibrium points are The endemic equilibrium can be expressed as follows by setting the right-hand side of the system (3) to zero and assuming that none of the disease states is zero, where Utilizing the next-generation technique, we were able to determine the system's fundamental reproduction number, which is represented by R 0 .In 13 is given as: where � 1 = r 1 + pr 1 + pr 2 + µ + 2pµ, and � 2 = µ(µ + r 1 )(µ + r 2 )(ηµ + 1).The aforementioned R 0 serves as a threshold parameter, if R 0 is less than 1, the disease disappears, and if R 0 is more than 1, the contagion per- severes in the community.

Sensitivity analysis
We can check the sensitivity of R 0 by computing the partial derivative for the significant variables, that is: ( www.nature.com/scientificreports/R 0 is extremely responsive to changes in the parameters.In this work, the values b, β , and p are increasing, while r 1 , r 2 , µ , and η are decreasing.To avoid an illness, prevention is preferable to treatment. µ is a positive invariant set for the proposed fractional-order system (3).
Proof To prove that the system of Eqs.(3) has a non-negative solution, we have Thus, the fractional system (3) has non-negative solutions.Lastly By adding all the relations of the system (3), the total population with the fractional derivative is given as Take a Laplace transform to both sides, we get Take Laplace inverse to both sides and Theorem 7.2 in 31 , we obtained Because of this if N(0) ≤ b µ then for t > 0 , N(t) ≤ b µ .Therefore, in the context of fractional derivative, positive invariance exists for the closed set .

Remarks 3.1
The closed set is currently representing a set of conditions that are biologically significant in the context of RSV transmission modeling.It is emphasizing the relationship between population dynamics, birth Vol www.nature.com/scientificreports/and death rates, and the vulnerability of specific age groups.These insights are currently informing strategies for managing and controlling RSV transmission within a population.
Proof The system's (3) Jacobian matrix at D 0 and where It should be emphasized that the suggested model assumes positive values for the parameters.Therefore, the eigenvalue 1 < 0 .Indeed the quantity µ is strictly positive.Thus, there are no positive roots of Eq. ( 9) according to Descarte's rule of signs because there is no sign change if R 0 < 1 .In addition, if is changed to − in Eq. ( 9), then Eq. ( 9) has three signs that change if R 0 < 1, therefore there are precisely three negative roots to Eq. ( 9).Thus by the condition arg i > απ 2 , i = 1, 2, 3, 4, 5, α ∈ (0, 1] , D 0 is locally asymptotically stable.

Remarks 3.2
The value of R 0 is a critical determinant of RSV transmission dynamics.A value below 1 suggests limited transmission, while a value above 1 indicates the potential for significant and sustained spread.This insight is essential for understanding the epidemiology of RSV and for designing effective control and prevention strategies.

Qualitative analysis of model
Here, we demonstrate that the system has a unique solution.System (3) is first written as follows: The aforementioned equations can be solved by applying integral form to both sides.
We demonstrate that the Lipschitz condition and contraction are satisfied by the kernels P i = 1, 2, 3, 4, 5.
Theorem 4.1 If the following inequality holds, then the kernel P 1 fulfills both the Lipschitz condition and contraction: As a result, P 1 meets the Lipschitz condition, and P 1 is a contraction if 0 ≤ βk 1 + βk 2 + µ < 1 .Similarly, we may demonstrate that P i , i = 2, 3, 4, 5 fulfil the Lipschitz condition and P i are contractions for i = 2, 3, 4, 5, if µ are bounded functions.Take into account the following recursive forms of system (11): . Take the norm of the first equation of the above system Lipschitz condition (12) gives us Similar to this, we get Hence, we can state that  13) and ( 14), we have This means that the system is continuous and has a solution.Currently, we demonstrate how the aforementioned functions create a model solution (11).Considering that
We assume that the system has an alternative solution, such as S 1 (t), E 1 (t), I r1 (t), I s1 (t), and R 1 (t), then we have We use the norm of the aforementioned equation Then �S(t) − S 1 � = 0. So, we obtain S(t) = S 1 (t) .Similar equality may be demonstrated for E, I r , I s , R.

Ulam-Hyers stability
In this section of the paper, we will examine the stability of the system (3) considering the perspective of UH.The analysis of approximate solution stability holds significant importance.
Definition 5.1 The considered system is said to be UH stable if ∃ some constants ϒ i > 0, i ∈ N 5 and for each � i > 0 , i ∈ N 5 , for and there exist S, Ẽ, Ĩr , Ĩs , R satisfy the following Theorem 5.1 The considered system is UH stable with the above assumption.

Simulation and discussion
Here, we show how the transmission of RSV is impacted by fractional order.For a better understanding of the RSV infection phenomenon, we conducted several simulations.For numerical simulation, Table 1 contains the initial values of the compartments and the values of the model parameters.
To demonstrate how memory affects the dynamics of RSV, we consider various values of the memory index (α = 0.85, 0.90, 0.95, 1.00) in Figs. 1, 2, 3, 4 and 5. Figure 1 indicates that the susceptible population S(t) grows uniformly whenever the non-integer order α decreases.This phenomenon reflects how variations in memory index α influence immunity duration.A lower α value indicates stronger memory effects, causing individuals to retain immunity for an extended time after infection or vaccination.Figure 3 shows that, there is a sharp leap in the population of super spreading infected people in the early days when fractional order α decreases.( 27) This finding underscores the role of highly infectious individuals in initiating and driving early outbreaks.A decrease in memory index α could intensify the impact of memory-driven interactions, making highly infectious individuals more potent transmitters.This insight is crucial for anticipating and managing the initial surge of infection.Furthermore, as can be seen from Fig. 5 the recovered population R(t) decreases by decreasing memory index α .A lower memory index α value implies a shorter duration of immunity post-recovery.This aligns with the idea that reduced α values emphasize stronger memory effects, resulting in faster waning of immunity.The observation emphasizes the potential for reinfections and the importance of immunity maintenance strategies.It is clear that the memory index has a significant impact on the dynamics of RSV and lowers the number of infected people.These observations reinforce the importance of memory effects, immunity, and initial transmission dynamics in the context of RSV infection.They offer insights that can influence public health strategies, vaccination programs, and the understanding of population-level immunity.The findings contribute to a more comprehensive understanding of the biological mechanisms underlying RSV behavior and its interactions with the human immune system, ultimately aiding in more effective disease management and control.Figs. 1, 2, 3, 4 and 5 depict the effects of input parameters p on the dynamics of RSV transmission, where the impact of infection progression rate has been established.Figure 6 shows the chaotic behavior of our system (3) with different settings for the memory index α , the trajectories of the system converge to the equilibrium point.In Fig. 6a     treatment effectiveness, immune response variations, and the overall progression of the infection collectively influence this chaotic transition.We observed that the memory index α can also be used as a chaotic control parameter.The chaotic behavior of the system is significantly relied upon in numerous scientific and engineering applications.It is generally known that there is a significant propensity to imagine and represent the behavior of chaotic systems.The proposed mathematical model is made feasible and scalable by the chaotic modeling, which can then be used to study novel chaos systems.We demonstrated how α might have made a significant contribution and could be used as a useful parameter for preventative measures.

Conclusion
The most frequent cause of lower respiratory tract infections in newborns and children globally is respiratory syncytial virus (RSV).In this work, we have presented a fractional-order mathematical model for the respiratory syncytial virus (RSV) transmission in the presence of a super-spreader.Using the Caputo derivative, we provided the suggested model.The results demonstrate that the suggested model has bounded and positive solutions.The sensitivity analysis indicates that the value R0 correlates directly with the birth rate of susceptible individuals (µ) , the virus transmission rate between humans (β) , and the likelihood of a new case being a normally infected human (p).These factors are adjustable through the efficient implementation of vaccination campaigns.Using the fixed-point theorem, we investigated the existence and uniqueness of the system's solutions.Furthermore, we established UHS results for our system of viral infection RSV.We presented a two-step Lagrange polynomial numerical technique for addressing the Caputo fractional derivative to understand the dynamics of RSV.With varying input parameters, the chaotic graphs have been displayed.It has been demonstrated that the chaotic behavior of the proposed model is affected by fractional order α .Adding the memory index α is expected to improve the system and could have been employed as a controlling parameter.Every fractional order model, in our opinion, has more information than the integer orders.For example, the integer order model will only have one solution, but the fractional order in an interval will have an infinite number of solutions.The beauty of these operators is that they can find new information for orders other than one while still approaching the integer order solution for orders that are close to one.In conclusion, the utilization of data-driven approaches in Respiratory Syncytial Virus (RSV) modeling proves pivotal in understanding the complexities of disease transmission and management.
In our future work, we plan to implement optimal control analysis on this model to reduce infection rates and increase the number of healthy individuals.The biological phenomenon can be described using real data by extending our model to a new generalized fractional derivative.Additionally, we will also apply the fractional operators to stochastic models.and I r (t) . (c) Behavior of S(t) and I s (t) . (d) Behavior of S(t) and R(t). (e) Behavior of I r (t) and I s (t) . (f) Behavior of I r (t) and R(t).

Figure 6 .
Figure 6.Simulation of chaotic behavior of compartments.(a) Behavior of S(t) and E(t).(b) Behavior of S(t)and I r (t) . (c) Behavior of S(t) and I s (t) . (d) Behavior of S(t) and R(t). (e) Behavior of I r (t) and I s (t) . (f) Behavior of I r (t) and R(t).

Table 1 .
Parameters values and initial values of the compartments.