Fractional view analysis of sexual transmitted human papilloma virus infection for public health

The infection of human papilloma virus (HPV) poses a global public health challenge, particularly in regions with limited access to health care and preventive measures, contributing to health disparities and increased disease burden. In this research work, we present a new model to explore the transmission dynamics of HPV infection, incorporating the impact of vaccination through the Atangana–Baleanu derivative. We establish the positivity and uniqueness of the solution for the proposed model HPV infection. The threshold parameter is determined through the next-generation matrix method, symbolized by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_0$$\end{document}R0. Moreover, we investigate the local asymptotic stability of the infection-free steady-state of the system. The existence of the solutions of the recommended model is determined through fixed-point theory. A numerical scheme is presented to visualize the dynamical behavior of the system with variation of input factors. We have shown the impact of input parameters on the dynamics of the system through numerical simulations. The findings of our investigation delineated the principal parameters exerting significant influence for the control and prevention of HPV infection.


Theory of fractional-calculus
In this section, we will unveil pivotal theory of the Atangana-Baleanu operator alongside the classical Caputo derivative, as elucidated in reference 33 .Additionally, we will delve into the Atangana-Baleanu operator, as outlined in reference 33 .These basic concepts and findings will be utilized in the analysis of the model.Definition 2.1 33 .Consider a function k such that k : [p, q] → R , then the Caputo fractional derivative of order υ on k can be stated pas where r ∈ Z and υ ∈ (r − 1, r).Definition 2.2 Suppose a function k such that k ∈ H 1 (p, q) , q > p , and υ ∈ [0, 1], then AB fractional operator in Caputo structure represented by ABC is defined as Definition 2.3 Integral of AB derivative is represented by ABC p I υ t k(t) and defined as Since the fractional-order υ → 0 implies that the initial function can be attained.
Theorem 2.1 33 .Consider a function k such as k ∈ C[p, q] , then the following holds Moreover, the Lipschitz holds for the ABC derivative as Theorem 2.2 33 .The following system of fractional differential equation

Fractional order model formulation
In this part, we present a mathematical model of HPV transmission.In formulation of the model, we represent the total population by N (t) .According to their illness state, the model splits the whole population into six sub-classes: Susceptible S(t) , Vaccinated V(t) , Asymptomatic A(t) , Infected I(t) , Recovered R(t) , and Cervical cancer C(t) .Here, the following presumptions are used to build a mathematical model of the human papilloma virus.A portion p are vaccinated on the onset of an outbreak and a fraction α of the susceptible are vaccinated.
After vaccination a portion ϕ of vaccinated individuals moves to susceptible class after losing the effectiveness of the vaccination while the recovered individuals lose the immunity with a rate ω and become susceptible.The susceptible population is at risk of infection, whether it is from asymptomatic or symptomatic individuals, and this risk is quantified by a force of infection denoted as Here, β is calculated as the product of κ (the contact rate) and τ(the probability that a contact leads to infection), and γ represents the transmission coef- ficient for asymptomatic individuals.In cases where γ > 1 , asymptomatic individuals are more likely to infect susceptible individuals than symptomatic ones.When γ = 1 , both asymptomatic and symptomatic individuals have an equal chance of infecting the susceptible population.However, if γ < 1 , symptomatic individuals have a greater likelihood of infecting susceptible individuals compared to asymptomatic ones.
The HPV vaccine is considered to provide only temporary immunity, and individuals who have been vaccinated may still have a chance of being infectious or asymptomatic, although this likelihood is relatively low.The force of infection for the vaccinated population is denoted as v = ε , where ε falls within the range of 0 to 1. ε represents the proportion of the serotype not covered by the vaccine.Individuals acquiring new infections due to the force of infection face dual potential outcomes.They can either undergo asymptomatic infection, characterized by a probability denoted as ρ , and subsequently join the asymptomatic class.Alternatively, there is a probability of 1 − ρ that they progress to the infected class.Within the asymptomatic class, individuals encounter two distinct trajectories.They may either manifest disease symptoms or opt for screening, triggering a transition into the infected class at a rate represented as θ .Alternatively, they may naturally recover, acquiring immunity at a rate denoted as φ .Individuals within the infected class undergo transitions based on treatment.At a rate of η , some individuals move to the recovered compartment through effective treatment, where a pro- portion q successfully joins the recovered class.Others, constituting the remaining (1 − q) proportion, opt for an alternative treatment path, joining the asymptomatic class.Unfortunately, in cases where the treatment fails, individuals may progress to develop cervical cancer at a rate δ , leading to a shift into the cervical cancer com- partment.Individuals who are afflicted with cervical cancer may face mortality due to the infection, occurring at a rate denoted as ξ .Within all compartments, µ represents the natural mortality rate of individuals.The flow chart of the transmission dynamics of HPV with the above assumptions is illustrated in Fig. 1.Then, the model of HPV with the above assumptions in the form of mathematical expression is as follows: The initial conditions for the system of model ( 1) are all non-negative and represented as follows: The above model (1) in fractional form can written as: where 0 < υ ≤ 1.The adoption of fractional derivatives in epidemic modeling enhances the models' ability to reflect the complexity of real-world scenarios, making them more effective tools for predicting and managing the spread of infectious diseases.The Atangana-Baleanu derivative is known for its ability to model non-local and non-singular behaviors, which may be crucial for accurately describing certain physical processes.The Atangana-Baleanu derivative provides a flexible mathematical framework that can be adapted to describe systems with memory and long-range dependencies.Its versatility makes it suitable for a wide range of applications.
Theorem 3.1 The solutions the system (3) of the disease are nonnegative and bounded for nonnegative initial vales of state variables of the system.
The solutions of our fractional system (3) of the disease is evidently constrained and remains nonnegative for nonnegative initial values of state variables.Consequently, the system is biologically valid.Further analysis of the model will be presented in the upcoming investigation of the system.

Analysis of the model
In this section of the, we will investigate our model of HPV for disease-free steady-state, reproduction number and local asymptotic stability.Let the disease-free steady-state is denoted by E 0 and can be determined by taking the steady-state of system (3) without infection, then, we have Here, we assume that the basic reproduction number is indicated by R 0 which can be calculated through different technique.We take the following step to determined R 0 of our model: Taking the Jacobian of the above, we have F and V as given below www.nature.com/scientificreports/Here, assume that the greatest eigenvalue of and the basic reproduction number through next-generation matrix method is the greatest eigenvalue of FV −1 , thus, we have Theorem 4.1 If R 0 < 1 , then the steady-state E 0 is locally asymptotically stable and is unstable in other cases.
Proof For the required stability result, we take the the Jacobian matrix at E 0 as For the required result, we will show that all the eigenvalues of J (E 0 ) are negative.For which, we take the char- acteristic equation det[J (E 0 ) − χ I] = 0 as: From the above, the first and second eigenvalue are −µ and −f which are negative while the other eigenvalues can be determined from here, we have the third and fourth eigenvalue are −b and −e which are negative.The remaining eigenvalues can be calculated from Here, if det(J (E 1 )) < 0 and trc(J (E 1 )) > 0 for R 0 < 1 , then the disease-free steady-state of our model of HPV is locally asymptotically stable.

Fractional-order model solution
In this section of the manuscript, we will utilize fixed-point theory to confirm the uniqueness and existence of solutions of our model of the disease.The described system for HPV with the Atangana-Baleno derivative is provided as follows Vol:.( 1234567890) In this context, we have the state variables represented by w(t) = (S, V, A, I, R, C) , and J is a continuous func- tion.To clarify, the vector function J can be more clearly expressed as follows: with appropriate initial conditions specified as w 0 (t) = (S(0), V(0), A(0), I(0), R(0), C(0)) , and furthermore, the function J satisfies the Lipschitz condition as outlined below: Subsequently, we will explore the uniqueness and existence of system (4) in the following outcome.

Theorem 5.1 A unique solution for the suggested system (4) of HPV is present if the following condition is met
Proof To establish the intended result, we apply the AB fractional integral (2.3) to the system (5) which provide the following Consider the interval (0, T ) represented as I, and the operator � : P(I, R 6 ) → P(I, R 6 ) is defined as follows: Then, Eq. ( 8) can be written as the supremum norm over the set I is represented by .I , and defined as One can observe with patience that P(I, R 6 ) forms a Banach space equipped with the norm .I .Moreover, it is evident that both w(t) and K(t, ̟ ) are members of P(I, R 6 ) and P(I 2 , R) , respectively, in a way that Using the definition of as outlined in (10), we obtain the following result Furthermore, by making use of the Lipschitz condition ( 6) and the outcome from ( 12), the following is derived (4) ABC 0  7) holds, then is a contraction.This, in turn, implies that the HPV system (4) possesses a unique solution.

Fractional dynamics via Newton polynomial
In this section of the paper, our focus is on the numerical solution of our system (4) of the infection.To do this, we consider the below stated Atangana-Baleanu derivative system transform the previously stated equation into the subsequent form according to 34 : the above at t r+1 = (r + 1)�t can be stated as this can be further transformed into: In the subsequent phase, we employ the Newton polynomial method to estimate f(t, g(t)) as follows Utilizing the above stated polynomial in (20), we get that Moreover, we get ABC 0 www.nature.com/scientificreports/ the following result is achieved after simplification the integrals above can be evaluated using the following method After simplification, we get that We will employ the aforementioned approach to depict the time series of the proposed infection model.Time series analysis holds significant importance in comprehending, monitoring, and managing diseases.It furnishes valuable insights into the dynamics of the disease, aids in the early detection of outbreaks, and enables the assessment of intervention effectiveness.This, in turn, contributes to more informed and targeted public health initiatives.The numerical values of system parameters and state variables will be assumed for computational purposes.Various simulations will be conducted to illustrate how these parameters impact the infection system.In the initial simulation, illustrated in Figs. 2 and 3, we scrutinized the impact of the fractional parameter υ on the dynamics of HPV.In Fig. 2, we consider the values of υ to be 1.00, 0.95, 0.90, and 0.85, while in Fig. 3, the value of υ is varied as 0.80, 0.70, 0.60, and 0.50.This systematic exploration of diverse values for the input parameter υ allows us to thoroughly investigate the characteristic solution pathways of the system.The outcomes of these simulations unequivocally highlight the substantial influence exerted by the fractional parameter on the dynamics of the infection.Notably, υ emerges as a promising tool for effectively managing the spread of the infection within the community.Therefore, we strongly advocate for a more in-depth exploration and analysis of this fractional parameter by policymakers to enhance their understanding of its potential in mitigating the impact of the infection on public health.This comprehensive investigation can contribute valuable insights for developing targeted strategies in the control and prevention of the infection.Figure 4 depicts the impact of the input parameter β on the dynamics of HPV infection.In this simulation, we considered β values of 0.20, 0.40, (24) Vol.:(0123456789) www.nature.com/scientificreports/0.60, and 0.80.Our observations highlight the crucial role of this parameter, indicating a direct association with an increased risk of the infection.In Figs. 5 and 6, we have illustrated the biological implications of varying input parameters ρ and θ on the dynamics of HPV.In Fig. 5, we explored the effects of different values of ρ (0.45, 0.55, 0.65, and 0.75), while maintaining θ at values of 0.2, 0.3, 0.4, and 0.5 in Fig. 6.Our investigation specifically focuses on discerning how changes in these parameters influence the behaviors of asymptomatic and infected individuals within the HPV system.In the conclusive simulation, depicted in Fig. 7, we investigated the impact of the input parameter η on the solution pathways of HPV infection.For this analysis, we considered values of η as 0.25, 0.30, 0.35, and 0.40.The observation centered on understanding how variations in η contribute to the dynamics of the asymp- tomatic and infected classes within the model.These insights hold significant relevance for informing public health strategies, intervention measures, and the formulation of effective control policies aimed at managing and mitigating the repercussions of infectious diseases on populations.Understanding the intricate relationships between input parameters and the dynamics of HPV infection is essential for the development of targeted and efficient approaches to tackle such public health challenges.

Conclusion
The infection HPV had posed a global public health challenge, especially in regions with limited access to healthcare and preventive measures, contributing to health disparities and an increased disease burden.In our research, we structured a mathematical model for the transmission dynamics of HPV infection with the effect of vaccination, asymptomatic carrier and cervical cancer.We have shown that the solution of the recommended model are positive and bounded for positive initial values of state variables.We utilized the next-generation matrix method for the calculation of the basic reproduction number R 0 .In addition to this, we proved that the infection-free steady-state of the system are locally asymptotically stable for R 0 < 1 and unstable in other cases.The existence of the solution has been investigated with the help of fixed-point theory.We introduced a numerical scheme to elucidate the dynamic behavior of the system, aiming to demonstrate the influence of the system's input parameters.The most critical factors of the proposed system has been visualized and are recommended to the policy makers for the control and management of the infection.In the future research work, we will examine the impact of pulse vaccination on the dynamics of HPV infection.Additionally, we intend to incorporate the dynamics of HPV infection within a stochastic framework and conduct a comparative analysis of their respective outcomes.

Ethical approval
There is no ethical issue in this work.All the authors actively participated in this research and approved it for publication.
Illustration of the flow chart of the dynamics of the infection of human papilloma virus.