Finite Difference Method for Infection Model of HPV with Cervical Cancer under Caputo Operator

In this paper, a fractional model in the Caputo sense is used to characterize the dynamics of HPV with cervical cancer. Generalized mean value theorem has been used to examine whether the infection model has a unique positive solution. Te model has two equilibrium points: the disease-free point and the endemic point. Te examination of the system’s local and global stability is provided in terms of the basic reproductive number ( R p ° ) . Te global stability analysis has been carried out using an appropriate Lyapunov function and the LaSalle invariant principle. Te results demonstrate that in the infection model, if R p ° < 1, then the solution converges to the disease-free equilibrium, which is both locally and globally asymptotically stable. Whilst R p ° > 1, the endemic equilibrium is considered to exist. Simulations are implemented via a fnite diference method with Gr¨unwald-Letnikov discretization approach for Caputo derivative operator to defne how changes in parameters impact the dynamic behavior of the system using Matlab.


Introduction
Te most prevalent sexually transmitted infectious agent in both sexes worldwide is the human papillomavirus (HPV), which gets its name from warts (papillomas) [1].HPVs are small DNA viruses that attack the cutaneous or mucosal epithelium [2].Tere are more than 170 diferent varieties of HPV that have been found and categorized, and more than 40 of these viruses are the most widespread sexually transmitted ailment in the world [3].One of the main causes of anal cancer, cervical cancer (CC), and other cancers is HPV.Each year, more than 400000 new cases are reported worldwide, and more than 300 women's lives were lost to HPV-related causes in just 2018 [4].Based on the severity of clinical manifestations, genital HPV types are divided into high-risk as subtypes (16,18,31,33,35,39,45,51,52,56,58,59, 68, 73, 82) associated with premalignant and malignant cervical, penile, vulvar, vaginal, anal, head, and neck cancers; and low-risk such as subtypes (6,11,42,44,51,53,83) that cause warts or benign, highly proliferative lesions on the genitals [5].
It has been determined that a high-risk HPV DNA sequence, notably HPV 16 and 18, which are present in around 70% of invasive cancers, is present in approximately 99.8% of CC [6].Te World Health Organization (WHO) developed a global strategy to hasten the elimination of CC by 2020 and urged for its eradication as a public health problem in 2018 [7].It classes as the fourth most frequent malignancy in women, with an anticipated 604,000 new cases and 342,000 fatalities worldwide in 2020 [8].
Infectious disease modeling has drawn a lot of interest recently in an efort to fnd cures for the diseases and calamities that have beset humanity [9].A system of equations with state variables and parameters is used in the mathematical model to show the interconnectedness of theories and observations and to investigate approximations and the efects of the parameters as well as anticipate the behavior of the problem over a particular period of time [10] and provide policymakers in public health with information on how to carry out efcient infection control interventions [11].Te fundamental quantities like as birth rates, transmission rates, recovery rates, and mortality rates are expressed by parameters, which are constants incorporated into the equations [12].Fractional-order diferential equation models are utilized as an alternative approach since integer-order differential equations can't adequately describe experimental and feld measurement data.More degrees of freedom and the inclusion of memory efects are advantages of fractionalorder diferential equation systems over conventional differential equation systems.To put it another way, they ofer a useful tool for expressing memory and hereditary aspects that were not inclosed in the classical integer-order system [13].A variety of fractional operator types are introduced to obtain a deeper understanding of the behavior of the models.Tese operators have various advantages and disadvantages over each other, such as Riemann-Liouville, Caputo, Caputo-Fabrizio, Hadamard, Katugampola, Atangana-Baleanu, and many more [14].Furthermore, unlike conventional integer-order derivatives, fractional-order derivatives such as the Caputo-Fabrizio derivative; have a nonsingular kernel property.Tis property makes it very evident that when modeling realistically, the model's future state depends on both its current and past states [15].Moreover, the fractional-order explains some characteristics of the dynamical system for the entire time and provides a complete description of the system that covers the entire process space, whereas the classical integer-order derivative addresses certain dynamic characteristics at a specifc time [16].Te realization that the majority of challenging dynamical systems are found to be non-singular is recently credited with the enormous increase in the treatment of dynamical systems with fractional-order derivatives.Additionally, they have a lengthy memory, which allows them to provide overall systems efectively [17].Te Mittag-Lefer function and the exponential decay functions, respectively, are used as the kernels of non-singular derivatives like the Atangana-Beleanu and Caputo-Fabrizio fractional derivatives, whilst the power function is used as the kernel of singular derivatives like Caputo [18].It's hardly surprising that a lot of models have been researched using fractional-order derivatives given their enormous and wealthy properties.As an illustration, Paul et al. [19] created and studied the SEIR model using fractional-order Caputo derivatives, considering two time-lags: the time required to heal sick individuals and the temporary immune period.Jahanshahi et al. [20] investigated a fractional-order SIRD model in Caputo's sense with time-dependent memory indexes to include the multi-fractional properties of COVID-19.Caputo fractional derivatives were used by Naik et al. [21] to analyze a fractional-order model for the HIV epidemic's propagation with optimal control.Chu et al. [22] examined the Holling type II form, a vector-host infectious illness compartmental model with nonlinear saturated incidence and therapy functions.Firstly, the model is formulated mathematically as a nonlinear classical integer-order deferential system.To more accurately represent the dynamics of the disease, the subsequently extended the model to the fractional order by utilizing the well-known Caputo-Fabrizio operator with an exponential decay kernel.Atangana-Baleanu Caputo operator has been used in [9] to examine the diabetes mellitus fractional-order model.Qureshi [14] suggested using Caputo fractional-order operator to create a novel epidemiological system for the measles pandemic.In [23], a model of the dynamics of HIV and malaria transmission with optimal control was created using a Caputo fractional derivative.Owolabi and Edson [24] used the Caputo operator and fxed point theory to model and investigate tuberculosis (TB).Utilizing the Caputo derivative, the dynamical analysis of a fractionalorder time-delay glucose-insulin model was carried out in [25].Te researcher in [26] studied a variable-order fractional mathematical model driven by Lévy noise describing the model of the Omicron virus using the concept of Caputo derivative.For additional information on several diferent forms of fractional derivatives, see reference [27] as well as the references cited therein.
Many researchers created mathematical models to represent the ailment's dynamics, which helped them to propose ailment control strategies and characterize the dynamics of co-infection with other infectious ailments.Te dynamics of HPV and CC (HPV-CC) with preventative strategies including screening, vaccination, and revaccination were described using an ordinary diferential equation model developed in [28].Chakraborty et al. [29] developed a mathematical model to analyze how vaccination afects the dynamics of HPV infection prevention and looked into the viability of a vaccination strategy, illustrating analytically and numerically how vaccination can ensure a predictable preventive policy against disease transmission among sexually active individuals who are more likely to develop CC from high-risk and low-high-risk papillomavirus strains, whilst the same mathematical model in the fractional-order with Atangana-Baleanu derivatives was studied, and numerical solutions were obtained using the Adams-Bashforth-Moulton method by [30].Te most crucial epidemiological aspects of HPV infection and related cancers were incorporated into a two-sex deterministic mathematical model that was developed by the researchers in [31].Te model included catch-up vaccination for adults and school-based vaccine administration for teenagers to evaluate the population-level efects of HPV immunization programs.Te center manifold theorem, normal forms theory, and the next-generation operator were all used to thoroughly study the model's dynamics.For several conceivable scenarios, they created an optimal control problem to identify the best HPV vaccination deployment method.Tey proved that there are optimum control issue solutions and used Pontryagin's Maximum Principle to describe the prerequisites for optimal control solutions.Akimenko and Fajar studied the stability of the age-structured model of CC cells and HPV dynamics [32].In [33], the author has researched how antiviral treatments afect the spread of cervical cancer.Te model includes two pharmacological therapies.Te function of the frst one is to prevent new infections, while the second's function is to prevent viral replication.Te numerical outcomes have demonstrated the role of the provided medication in regulating HPV infection and cancer cell growth rate.In [34], the mathematical model for the dynamics of HPV transmission in-host in the presence of an immune response represented by Cytotoxic T-lymphocytes cells has been studied; the model presented taking into account the efects of latent HPV infections, and the dynamics of the model were successfully analyzed.A new mathematical model of CC based on the age-structured of cells at the tissue level has been put out by researchers in [35].Goshu and Alebachew [36] created a mathematical model for the spread of CC transmission disease in the presence of vaccination and therapy.Omame et al. [37] created and presented a coinfection classical integer-order model for syphilis and HPV with cost-efectiveness analysis and optimal control.While the fractional-order in the Caputo-Fabrizio sense of the coinfection model for HPV and Syphilis is investigated using the nonsingular kernel derivative [38].HPV and Chlamydia trachomatis coinfection mathematical model with cost-efectiveness optimal control analysis has been formulated and examined.It has been investigated how HPV screening, Chlamydia trachomatis therapy, and both diseases' preventive measures afect the management of their coinfections, the consequent prevention of malignancies, and pelvic infammatory disease [39], and Nwajeri et al. [40] researched the fractional-order with Caputo derivatives of the codynamics model for HPV and Chlamydia trachomatis, and created the numerical simulations utilizing the fractional predictor-corrector method.In [41] they developed an optimal control strategy for the HPV-Herpes Simplex Virus type 2 codynamics model that minimizes the cost of implementing controls while also minimizing the number of infectious individuals over the intervention interval.Te Runge-Kutta forwardbackward sweep numerical approximation method was used to implement the optimal control systems.
In light of these accomplishments, we are inspired to investigate the HPV-CC model in this study with the Caputo fractional-order operator, which is best appropriate for modeling biological and physical facts.Te motivation to continue our investigation with the Caputo derivative which is a modifcation of the Riemann-Liouville defnition is that this type of fractional operator has advantages concerning other present derivatives; the Caputo derivative of a constant function yields zero, which is usual in mathematics.Te Caputo operator frst solves an ordinary diferential equation, then it takes a fractional integral to get the desired fractional derivative order.More crucially, Caputo's fractional diferential equation allows the use of local initial conditions to be included in the derivation of the model, and fnally, the inclusion of the memory efect on the Caputo derivative.
Te purpose of this study is to investigate the dynamics of the fractional-order of the HPV-CC model, using the Caputo derivative.Te organization of the paper is as follows: Some essential defnitions and fractional calculus properties are given in Section 2. Te fractional HPV-CC model is introduced in Section 3, along with proof of the solutions' existence and uniqueness.In Section 4, it is shown that the suggested model is locally stable at the disease-free and endemic equilibrium points.Te global stability of the suggested model at the disease-free and endemic equilibrium points is examined in Section 5.In Section 6, we construct a fnite diference scheme for the suggested model and show that it maintains the boundedness and positivity of the solutions to the investigated model; here, we will discuss the asymptotic stability of this scheme.To demonstrate the applicability and efectiveness of the fnite diference method, some numerical simulations for the suggested model are shown in Section 7, and the fnal Section is devoted to the conclusion.

Preliminary Concepts
Some basic results pertaining to fractional calculus are presented in this section.Defnition 1. Laplace transform of the function g(x) is defned by the following improper integral, ( Defnition 2. MittagLefer function is defned as Defnition 3. Mittag-Lefer function of two parameters (generalization of Mittag-Lefer function) is given by Lemma 4. For c ∈ R and ω, ] > 0, we acquire Defnition 5. Riemann-Liouville fractional integral of order β > 0, a ≥ 0 of a function g: R ≥0 ⟶ R is defned by Defnition 6. Caputo fractional derivative of order β > 0, a ≥ 0 for a function g(x) where g ∈ C m ([a, ∞), R) and m � [β] is given by Defnition 7. Caputo fractional derivative on the half axis When 0 < β < 1, (7) takes the the following form

HPV-CC Model Formulation
Tis work examines a modifed version of the classical integer-order HPV-CC model that was previously examined in [42].
where, B p represents the rate of HPV infection transmission, q p represents efective contact rate, r 3 , and r 4 represent the relative infectiousness of A p and I p respectively with (r 3 < r 4 ).
With probability q where 0 < q < 1, individuals in class E p develop to individuals in class I p .individuals in class E p develop to individuals in class A p with probability (1 − q).After experiencing an HPV symptom, individuals in class A p proceed to individuals in class C with a rate w 3 .Individuals in class I p compartments may have CC with the rate of development α 3 .Based on the formulations and presumptions mentioned above, the HPV-CC model takes the following form (associated biological parameters of the model are shown in Table 1) [42]: With the following non-negative initial conditions, Te reason for examining the fractional-order case is the noteworthy uniqueness of these fractional-order systems with hereditary qualities and non-local features (memory) that have not been observed with the integer-order diferential operators commonly found in biology.Additionally, mistakes resulting from overlooked parameters can be minimized by modeling real-life processes with fractionalorder diferential equations [21].As a result, our suggested fractional-order model for HPV-CC using the Caputo derivatives is: Subject to the initial conditions (11), where β(∈ 0, 1], N p (t) � S(t) + E p (t) + A p (t) + I p (t) + C(t), and (S(t), E p (t), A p (t), I p (t), C(t)) ∈ R 5 ≥0 .If β � 1 then Model (12) reduces to Model (10).
Biological parameters in Model (12) have been modifed to make sure that the left and right-hand sides of the model have the same dimension (t − β ) as follows (the schematic diagram of Model ( 13) is shown in Figure 1): here, λ β p � (B p q p (r

Existence and Uniqueness of Non-Negative Solutions.
Consider the following initial value problem of autonomous nonlinear Caputo fractional-order derivative system combined with the initial condition where g 0 ∈ R n and the function f(g(t)): R n ⟶ R n , n ≥ 1 is called vector fled.
Proof.Let us reformulate Model (13) in the form of Caputo fractional derivative system of order β(∈ 0, 1], as follows where f: Hence, the second condition of Teorem 11 is proven.Next, we need to prove that this solution is nonnegative.From (13), we have We conclude that the solution of Model (13) will remain in R 5 ≥0 based on Lemmas 9 and 10.Finally, we demonstrate that the solution is bounded.Adding Model's (13), results in Taking Laplace transform in (19) into account, we get Consequently, we have It follows that as t ⟶ ∞, 0 ≤ N p ≤ (π β /μ β ), Tis completes the proof.

Local Stability
In this section, the local stability analysis of equilibrium points will be covered.

Theorem 13. Te equilibrium points of Model (13) are locally asymptotically stable if the following Matignon condition is satisfed
where λ(J) stands for the class of all eigenvalues (λ) of the Jacobian matrix (J) of Model (13).

Local Stability Analysis of Disease-Free Equilibrium Point.
Model's (13) equilibrium points are derived by setting Model's (13) right side to zero.Te disease-free equilibrium point is given by T °p � (S °, E p°, A p°, I p°, C °) � ((π β /μ β ), 0, 0, 0, 0).Te threshold value of Model (13) which is known as a basic reproduction number (R p °) can be computed using the next-generation matrix (G) indicated in Van den Driessche and Watmough [46].Te transmission matrix (F p ) and transition matrix (V p ) of Model ( 13) are obtained respectively as follows (13).6 Discrete Dynamics in Nature and Society Ten, we compute the derivative of matrices F p and V p with respect to infected classes at T °p, and we get Jacobian of F p and V p , respectively so that Te aforementioned matrices are utilized to determine R p °for Model (13) using the spectral radius (ρ).Tat is, , and is given by Theorem 14. T °p is locally asymptotically stable if R p °< 1 and unstable if R p °> 1.
Ten the characteristic equation of the linearized Model (24) is where It is clear that λ 1,2 � − μ β are negative.Te three remaining eigenvalues are the roots of the characteristic equation Te necessary and sufcient conditions on three-order Routh-Hurwitz determinants such that the zeros of (26) have negative real parts or (20) to be satisfed are ð 1 , ð 3 > 0 and ð 1 ð 2 − ð 3 > 0.
It is easy to show that and All stability conditions have been met, then, T °p is locally asymptotically stable if R p °< 1 and unstable if R p °> 1. □ Discrete Dynamics in Nature and Society

Local Stability Analysis of Endemic Equilibrium Point.
Te endemic equilibrium point of Model ( 13) denoted by Theorem 15.T ⋆ p is locally asymptotically stable if R p °> 1 and unstable if R p °< 1.
Proof.Te linearization matrix of Model (13) around T ⋆ p , is Matrix (30) eigenvalues yield the following ffth-degree polynomial equation where ), and T 0 � _ a _ d_ e((R p °− 1)/R p °). Te following eigenvalue is easily obtained − μ β .Te remaining eigenvalues are the roots of the characteristic equation Te necessary and sufcient conditions on fourth-order Routh-Hurwitz determinants for the zeros of (32) to have negative real parts or (20) to be fulflled are T 0 , T 1 , T 3 > 0 and It is simple to establish that T 3 > 0, and Finally, we show that where H � B p q p η β ((1 − q)r β 3 + qr β 4 ).T ⋆ p has just been demonstrated to be locally asymptotically stable if R p °> 1 and unstable if R p °< 1.

Global Stability
In this section, we examine the global stability for model (13) by building appropriate Lyapunov functions.In order to show global stability, we take the function 0 ≤ Υ(z) � z − 1 − ln(z), ∀z > 0 into consideration.Lemma 1 (see [47]).Assume g(t) ∈ R ≥0 be a continuous function.Ten, for any time t ≥ 0 Theorem 17. T °p is globally asymptotically stable whenever R p °< 1.
Proof.Suppose the following Lyapunov function: along the solution of model ( 13) as Terefore, R p °< 1 guarantees for all S(t), E p (t), A p (t), We conclude that T °p is globally asymptotically stable provided R p °< 1 as a result of LaSalle's invariance principle [48].
Proof.Let us construct the following Lyapunov function: Obviously, V ⋆ p (t) > 0, ∀S(t), E p (t), A p (t), I p (t), C(t) > 0 and along the trajectories of model ( 13), yields 10 Discrete Dynamics in Nature and Society At T ⋆ p , we possess p , and _ eI ⋆ p � qη β E ⋆ p , where λ ⋆β p � (B p q p (r Discrete Dynamics in Nature and Society which according to the arithmetic mean-geometric mean inequality, is less than or equal to zero.Tus, R p °> 1 ensures for all S(t), E p (t), A p (t), According to LaSalle's invariance principle [48], T ⋆ p is globally asymptotically stable so long as R p °> 1.

Construction of Finite Difference Scheme
To acquire numerical solutions for HPV-CC fractionalorder model ( 13), we create the fnite diference scheme in this section.To estimate the fractional derivative, one can utilize Grünwald-Letnikov approach.Te suggested fnite diference scheme maintains the positivity and stability of the equilibrium points of the solution corresponding to the continuous epidemiological models of fractional order.
Let t ∈ [0, T], a fnite interval.Te usual notation t k � kδ t , k � 0, 1, . . ., N and u(t k ) ≡ u k are used, where δ t � (T/N) and u k denotes the numerical solution of the analytical solution of u(t) at the grid points t k .
Grünwald-Letnikov discretization approach based on Caputo derivative is defned as where Lemma 19 (see [49]).Suppose that β ∈ (0, 1), therefore the coefcients d (β) j and ϑ β j fulfl for j ≥ 1 the properties Te fnite diference scheme of Model ( 13) using Grünwald-Letnikov discretization approach ( 41) is as follows here λ β p k � (B p q p (r 12 Discrete Dynamics in Nature and Society Because each of these equations is linear in S k+1 , E p k+1 , A p k+1 , I p k+1 , and C k+1 , hence through some calculations the following explicit expressions can be obtained

Non-Negativity and Boundedness of Finite Diference
Scheme.Tis subsection examines a few characteristics of proposed Scheme (44).Keeping in mind that system (13) have unique non-negative solutions and also all the parameters are positive.
Applying the induction principle and utilizing Lemma (19), it follows that for k � 0 For k � 1, we have 14 Discrete Dynamics in Nature and Society For k � 2, we have Now, we assume that for k � 3, . . ., N − 1, is Tus for k � N, we get Discrete Dynamics in Nature and Society 2. Stability of Finite Diference Scheme.Tis subsection investigates the stability of Scheme (44).

Numerical Simulations
Tere are no general techniques for solving systems of fractional diferential equations analytically, similar to the classical theory of diferential equations.Te fractional case is much more challenging to treat even approximately [50].Tere are various techniques based on a continuous expansion formula for the fractional derivative that, in some circumstances, can be used to approximately solve the original fractional-order model [51].
In this section, we'll simulate the solution to HPV-CC fractional-order Model (13) using suggested Scheme (44).Te presented model's parameter values are either extrapolated from earlier model studies or based on assumptions.Table 2 lists the values of these parameters for the two scenarios R p °< 1 and R p °> 1.
Proposed Scheme ( 44) is implemented with a time step size δ t � 0.02, and fgures of the numerical solutions are presented using varied initial conditions that satisfy Teorem 22 and various values of q p and derivative order β(∈ 0, 1]. Te convergence of all solutions toward the equilibrium point T °p is depicted in Figures 2-4 across a range of diferent values for β(∈ 0, 1] and q p .Tis result was attained when R p °< 1. Te infected classes converge to zero over time while the susceptible compartment (S(t)) rises initially and then converges to S °� (π β /μ β ).T °p is thereby shown to be globally asymptotically stable.When R p °> 1, all solutions converge toward the equilibrium point T ⋆ p which is globally asymptotically stable.Tis result is shown in Figures 5-7 for a variety of β(∈ 0, 1] and q p values indicating that upon the introduction of HPV, the susceptible compartment (S) over time (t) gradually reduces and stays constant and can't be treated.While the exposed compartment (E p ), the asymptomatic compartment (A p ), the infected compartment (I p ), and the compartment afected by cervical cancer (C) gradually increase with time (t).
Drawing from the preceding discussion and the numerical results presented in Figures 2-7, we deduce that β may be infuenced by an individual's prior experience with or understanding of the illness.Consequently, the numerical results validate that diferential equations with fractionalorder derivatives explain biological systems more accurately than classical integer-order models.
Depending on the values of the parameters mentioned in Table 2, Tables 3 and 4 show how q p has an impact on decreasing and increasing R p °, respectively.Te prevalence of HPV-CC is infuenced by the level of interpersonal contact within a community (q p ).
By analyzing these variations, we may predict more about how fractional orders afect the model's dynamics and how crucial the parameter q p is in infuencing population dynamics.Moreover, Figures 2-7 present results that provide a basis for additional investigation and refnement of the model, which could result in the development of more potent disease control and prevention techniques.13) when R p °> 1 applying scheme (44) at a time step size δ t � 0.02 with fve diferent initial conditions that satisfy Teorem 23 as well as fve diferent values of β(∈ 0, 1], and q p � 6.8.All other parameters are as in Table 2 when 13) when R p °� 8.6474 > 1 applying scheme (44) at a time step size δ t � 0.02 with initial condition (S(0), E p (0), A p (0), I p (0), C(0)) � (0.5425, 0.2790, 0.1600, 0.0147, 0.0038) and β � 0.96, q p � 6.8.All other parameters are as in Table 2 when R p °> 1.

Conclusion and Future Directions
In this article, we proposed a fractional-order HPV with cervical cancer (HPV-CC) model with Caputo derivative of order β(∈ 0, 1].Tis dynamical model was more compatible for describing biological phenomena with memory than the integer-order model.Some mathematical results related to the model are presented.Te local stability of the model for R p °< 1 and R p °> 1 was given.Te model had two equilibrium points: the disease-free point (T °p) and the endemic point (T ⋆ p ). Te examination of the system's local and global stability was provided in terms of the basic reproductive number (R p °).Using LaSalle invariant principle and the appropriate Lyapunov function, the analysis of global stability had been conducted.Te results demonstrated that in the infection model, if R p °< 1, then the solution converged to T °p, which was both locally and globally asymptotically stable.Whilst R p °> 1, T ⋆ p was considered to exist.To determine how changes in parameters afect the dynamic behavior of the proposed system, simulations were constructed using a fnite diference scheme with Grünwald-Letnikov discretization approach for Caputo derivative   (10) and fractional-order model ( 13) with β � 0.99 and 0.89 applying scheme (44) at a time step size δ t � 0.02 with initial condition (S(0), E p (0), A p (0), I p (0), C(0)) � (0.5425, 0.2790, 0.1600, 0.0147, 0.0038).q p � 7.5 and all other parameters are as in Table 2 when   operator.Te outcomes acquired demonstrated that the fnite diference method is a precise and efectual technique to obtain the numerical solution of the suggested nonlinear fractional-order HPV-CC model.In future work, the presented model after being modifed, will be integrated with HIV-AIDS; employing nonlocal and nonsingular kernel operators such such Caputo-Fabrizio.Te generalized Adams-Bashforth Moulton method will be applied for the numerical simulations.

Figure 3 Figure 3 :
Figure 2: Dynamics of model (13) when R p °< 1 applying scheme (44) at a time step size δ t � 0.02 with fve diferent initial conditions that satisfy Teorem 23 as well as fve diferent values of β(∈ 0, 1], and q p � 0.68.All other parameters are as in Table2 whenR p °< 1.(a) Dynamics of susceptible individuals.(b) Dynamics of individuals exposed to HPV.(c) Dynamics of individuals asymptomatic with HPV.(d) Dynamics of individuals infected with HPV.(e) Dynamics of individuals having CC.

Figure 5 :
Figure5: Dynamics of model(13) when R p °> 1 applying scheme (44) at a time step size δ t � 0.02 with fve diferent initial conditions that satisfy Teorem 23 as well as fve diferent values of β(∈ 0, 1], and q p � 6.8.All other parameters are as in Table2 whenR p °> 1.(a) Dynamics of susceptible individuals.(b) Dynamics of individuals exposed to HPV.(c) Dynamics of individuals asymptomatic with HPV.(d) Dynamics of individuals infected with HPV.(e) Dynamics of individuals having CC.
Figure5: Dynamics of model(13) when R p °> 1 applying scheme (44) at a time step size δ t � 0.02 with fve diferent initial conditions that satisfy Teorem 23 as well as fve diferent values of β(∈ 0, 1], and q p � 6.8.All other parameters are as in Table2 whenR p °> 1.(a) Dynamics of susceptible individuals.(b) Dynamics of individuals exposed to HPV.(c) Dynamics of individuals asymptomatic with HPV.(d) Dynamics of individuals infected with HPV.(e) Dynamics of individuals having CC.

Table 2 :
Parameter values for the numerical simulations.

Table 3 :
Impact of changing q p on R p °in the case of disease-free.

Table 4 :
Impact of changing q p on R p °in the case of epidemic.