Mathematical analysis of a human papillomavirus transmission model with vaccination and screening

: We formulate a mathematical model to explore the transmission dynamics of human papillomavirus (HPV). In our model, infected individuals can recover with a limited immunity that results in a lower probability of being infected again. In practice, it is necessary to revaccinate individuals within a period after the first vaccination to ensure immunity to HPV infection. Accordingly, we include vaccination and revaccination in our model. The model exhibits backward bifurcation as a result of imperfect protection after recovery and because the basic reproduction number is less than one. We conduct sensitivity analysis to identify the factors that markedly affect HPV infection rates and propose an optimal control problem that minimizes vaccination and screening cost. The optimal controls are characterized according to Pontryagin’s maximum principle and numerically solved by the symplectic pseudospectral method.


Introduction
Uterine cervical cancer is a worldwide health problem but it is especially concerning in developing countries. It is the first or second most common cancer in women [1]. It is estimated that the probability of a person being infected with human papillomavirus (HPV) in their lifetime reaches 70 to 80% [2], and the total infection rate in the global population is as high as 11.7% [3]. An estimated 233,000 deaths were attributed to HPV infection in the year 2000 [4]. There were approximately 500,000 cases and 275,000 deaths due to cervical cancer worldwide in 2002, equivalent to about a tenth of all deaths in women due to cancer [5]. The burden of cervical cancer is disproportionately high (> 80%) in the developing world [6].
HPV was discovered to be the causative agent of cervical cancer in the 1970s by the Zur Hausen group [7]. Usually, the infecting papillomavirus is eliminated from individuals; however, some individuals retain the virus. Persistent infection with oncogenic HPV is recognized as the major cause of uterine cervical cancer [8]. Cervical carcinogenesis is a complex stepwise process over a continuum of increasingly severe precancerous changes known collectively as cervical intraepithelial neoplasia (CIN) [9]. The spectrum of CIN is traditionally divided into three histopathological categories: CIN1, CIN2 and CIN3. In CIN1, cells with malignant changes are limited to the superficial layer of the cervical epithelium. Most CIN1 lesions are likely to disappear without treatment. However, a small percentage may progress to high-grade CINs (i.e., CIN2 and CIN3). The risk of progression to invasive cervical cancer increases significantly with worsening CIN grades [10][11].
Pap cytology screening for the early detection of cervical neoplasia has been successful in reducing cervical cancer incidence and mortality [12]. In unscreened populations, the risk of invasive cervical cancer occurs earlier than of most adult cancers, peaking or reaching a plateau between about 35 and 55 years of age [13]. This distribution is because cervical cancers originate mainly from HPV infections transmitted sexually in late adolescence and early adulthood [14]. HPV transmission can be reduced through the use of condoms [15]. Some studies have reported that smoking [16], multiparity [17], and long-term use of oral contraceptives [18] can double or triple the risk of precancer and cancer among women infected with carcinogenic types of HPV. There are two major kinds of anti-HPV vaccines approved for use to protect newly sexually active individuals against some of the most common HPV types and boost immunity, namely, therapeutic vaccines and prophylactic vaccines [7]. A few years after receiving a prophylactic vaccine, the individual must be revaccinated because the vaccine loses its preventive effect. Progress in the development of therapeutic vaccines for HPV has been slow [7]. In summary, there is currently no specific treatment for HPV infection [19]. There are three major treatments for cervical cancer: surgery (such as total hysterectomy and subtotal hysterectomy), radiotherapy, and chemotherapy. Among these, surgery and radiotherapy are the main treatment methods [19].
Mathematical modeling is a useful tool for assessing the potential impact of intervention strategies against HPV spread among humans [20−24]. A number of authors have reported the use of mathematical modeling to evaluate the impact of HPV vaccination. Al-arydah [20] developed a two-sex, age-structured model to describe a vaccination program for the administration of an HPV vaccine. Malik et al. [11] presented an age-structured mathematical model that incorporated sex structure and Pap screening cytology. Sharomi and Malik [21] developed a two-sex HPV vaccination model to study the effect of vaccine compliance on HPV infection and cervical cancer. Omame [22] developed a two-sex deterministic model for HPV that assessed the impact of treatment and vaccination. Elbasha [23] presented a two-sex, deterministic model for assessing the potential impact of a prophylactic HPV vaccine with several properties.
Based on the above research and understanding of HPV pathology, we develop an ordinary differential equation model with precautionary measures such as screening, which are rarely considered in previous studies, and analyze the potential effects of multiple factors on HPV transmission. The model is formulated in section 2. In section 3, the equilibria, basic reproduction number, and global stability are analyzed. We report the sensitivity analysis of the model through the partial rank correlation coefficient (PRCC) method and identify the key factors in the model in section 4. In section 5, we set the vaccination rate and screening rate as control variables and analyze an optimal control problem that minimizes vaccination and screening cost. Section 6 concludes the article. Through extensive numerical simulations with MATLAB, we obtained results to verify our conclusions.

An HPV model with vaccination and screening
The total individual population at time t is divided into 10 mutually exclusive subpopulations of susceptible individuals S(t), vaccinated individuals V(t), infectious individuals without disease symptoms E(t), infectious individuals with disease symptoms H(t), individuals with persistent HPV infection P(t), CIN1 symptomatic individuals I 1 (t), CIN2 symptomatic individuals I 2 (t), CIN3 symptomatic individuals I 3 (t), cancer-infected individuals A(t) and recovered individuals R(t). As such, the total population is It follows that the model for the transmission of HPV is given by the following system of differential equations.

Recovered individuals
Tables 1 and 2 list the associated state variables and parameters of model (2). Figure 1 shows the flow diagram of the model. We emphasize that the vaccine mentioned in model (2) is a prophylactic vaccine. In the following section, model (2) is qualitatively analyzed to derive insights into its dynamical features.
Proof. We note that along the edges of D, the time derivatives all lead the solution into the invariant domain [25] 00 . Furthermore, adding all the equations in the differential equation system of model (2) gives It follows from Eq (3) that as required.

Consider the region
.

V S E H P I I I A R R V S E H P I I I
The following steps establish the positive invariance of  (i.e., solutions in  remain in  0 t  ). It follows from Eq (3) that A standard comparison theorem can then be used to show that In particular Thus, the region  is positively invariant. Hence, it is sufficient to consider the dynamics of the flow generated by model (2) in  . In this region, the model can be considered as being epidemiologically and mathematically well-posed [27]. Thus, every solution of model (2) with initial conditions in  remains in  for all 0 t  . Therefore, the  -limit sets of model (2) are contained in  . This result is summarized below.

Lemma 3.1
The region  is positively invariant for model (2) with initial conditions in 10 R  .

Local stability of the disease-free equilibrium (DFE)
Model (2) has a DFE, which is obtained by setting the right-hand sides of the equations in the model to zero, given by , , , , , , , , , Let   1 2 3 , , , , , , , , ,

T V S E H P I I I A R 
. Using the notation from [28], the model consists of nonnegative initial conditions together with the following system of equations: and it follows that The matrices F and V  for the new infection terms and the remaining transfer terms are respectively given by Consequently, it follows from Theorem 2 of [28]. Lemma 3.2 The DFE of model (2), given by (4), is locally asymptotically stable (LAS) when R 0 < 1 and unstable if R 0 > 1.

Backward bifurcation
The epidemiological significance of forward bifurcation is that the disease will eventually disappear if the basic reproduction number is less than one. The public health significance of backward bifurcation is that the classical requirement of R 0 < 1 although necessary is no longer sufficient for effective disease control. Therefore, the presence of backward bifurcation in HPV transmission dynamics makes its effective control more difficult.

Existence of backward bifurcation
First, the possible equilibrium solutions that model (2) be the associated force of infection at a steady state. Setting the right-hand sides of model (2) to zero (steady state) gives Substituting (7) where 8 Quadratic Eq (9) can be analyzed for the possibility of multiple endemic equilibria. It is worth noting that the coefficient a is always positive, and c is positive (negative) if 0 R is less than (greater than) one. Hence, the following result is established.
it can be shown that backward bifurcation occurs for values of 10 1 RR  . This phenomenon is illustrated by simulating model (2). The parameter values are presented in Table 3. Let   The associated backward bifurcation diagram, depicted in Figure 2, shows that the model has a DFE (corresponding to Figure 3) and two endemic equilibria: One of the endemic equilibria is LAS (corresponding to Figure 4a); the other is unstable (a saddle); and the disease-free equilibrium is LAS. This clearly shows the coexistence of two stable equilibria when 0 1 R  , confirming that the model exhibits backward bifurcation for 10 1 RR  . This result is summarized below for model (2) (a more rigorous proof of the backward bifurcation phenomenon of the model, using the center manifold theory is given in Appendix B).

Global stability of the DFE
Therefore, by the Dulac−Bendixson theorem [29], there is no periodic orbit for model (2). Moreover,

Efficacy of interventions and sensitivity analysis
In this section, we performed a numerical simulation to enhance the understanding of model (2).

Efficacy of interventions
To examine the possible impact of interventions on disease infections we plot the number of infected individuals ( E ) with various vaccination rates and revaccination rates. This analysis shows that an increasing vaccination rate persistently decreases the peak value, as shown in Figure 5. Increasing the vaccination rate 2  by 1.75 times (increase from 0.4 to 0.7) or 1.43 times (increased from 0.7 to 1) will lead to a reduction in the peak value in the number of E by 20.21% or by 15.67%, respectively. In addition, the peak value of the number of people infected with 2 1   decreased by 43.82% compared with the number of people infected with 2 0   .
On the premise that the vaccine's protective effect will end after a few years, we consider the situation of vaccination and revaccination. Figure 5b indicates that increasing 2  and 4  from 0 to 0.4 will lead to a reduction in the peak value in the number of E by 34.16%. In addition, the peak value of the number of people infected with 24

Sensitivity analysis of 0 R to parameters
To identify the factors associated with a certain intervention that markedly affect the rate of new infections, we performed sensitivity analysis of the basic reproduction number.
LHS belongs to the MC class of sampling methods; it was introduced by Mckay et al. [30]. LHS allows an unbiased estimate of the average model output and has the advantage that it requires fewer samples than simple random sampling to achieve the same accuracy. For nonlinear but monotonic relationships between outputs and inputs, measures that work well are based on rank transforms such as the partial rank correlation coefficient, and standardized rank regression coefficient.
Model (2)   Detailed inspection of Table C1 (Appendix C) and Figure 6 indicates that in terms of reducing the value of 0 R , except 3  (control the disease and reduce the number of persistent infections) and d , the vaccination rate 2  is the most sensitive parameter with a leading PRCC value, followed by R decreases successively. That is, the same treatment intervention is more effective in the earlier stages. This means that more attention should be paid to patients in the early stages of infection. As asymptomatic patients are unable to diagnose themselves, regular screening for HPV should be strengthened. Smoking, overuse of contraceptive drugs, and unsafe sexual life will increase the value of 0 R , thus promoting the spread of HPV.

Optimal control in an extended model
In this section, an optimal control model for the transmission dynamics of HPV is formulated by extending model (2) to include control functions. Our goal here is to study the optimal control strategies to curtail the epidemic and minimize cost.

An extended HPV model
The optimal vaccination and screening strategy can be formulated as the following optimal control problem (P) with inequality constraints and free terminal states defined over the prescribed interval 0, f t   [31]:

Characterization of optimal control
The inequality constraints in problem (P) can be transformed into equality ones with the help of Hence, the Hamiltonian function for problem (P) is obtained as follows:      , with transversality conditions The following characterization holds Proof. The existence of an optimal control can be obtained owing to the convexity of the integrand of       12 , J u t u t with respect to       12 , u t u t [33], a priori boundedness of the state solutions, and the Lipschitz property of the state system with respect to the state variables. By Pontryagin's maximum principle [34], the optimal conditions with respect to the state, costate, and parametric variables result in a two-point boundary value problem coupled with a nonlinear complementarity problem as follows: and evaluated at the optimal control and corresponding states results in the stated adjoint system (13) with transversality (14). The optimality conditions with respect to the control variables are Mathematical Biosciences and Engineering Volume 17, Issue 5, 5449−5476.
By solving Eq (17), the optimal control can be expressed as To determine an explicit expression of the optimal control without 1  , we consider the following three cases: i. On the set  

Numerical simulations
An analytical expression of the optimal vaccination rate and screening rate was derived in Eq (15). However, an effective algorithm is still required to solve the nonlinear constrained optimal control problem numerically. Based on the generating function method, Peng et al. developed a series of symplectic methods for nonlinear optimal control problems [35−38]. Such symplectic methods have good precision and efficiency because of the structure-preserved property. Recently, Wang et al. improved the symplectic methods by incorporating the local pseudospectral discretization scheme [39−42]. Such symplectic pseudospectral methods (SPMs) have been successfully applied to solve optimal control problems in various mechanical systems [43−44]. In this paper, the SPM developed in [45] was adopted.
In the following simulation, the weights in the objective function (meaning the minimization of Table 4. Parameter values used in Figure 7. The controlled solutions together with the solutions for the uncontrolled case are presented in Figure 7. It can be seen that the control strategy is effective. Vaccinated individuals increase steadily and reach almost 400% at the terminal end. Susceptible individuals keep increasing and then stabilize during the whole period. The number of infected individuals decreases significantly when optimal control strategies are used compared to the number in the absence of control strategies. . Other parameter values are the same as those in Figure 7.

Parameter
We considered another set of weights, the simulation results are shown in Figure 8. A higher focus on the control strategies leads to a drop in the importance of the vaccination and screening strategies. As the number of asymptomatic individuals depends on the immunity of the susceptible individuals and the protection of the susceptible population, we should consider strengthening their immunity or implement regular cost-effective screening to control HPV transmission.

Conclusions
The human papillomavirus is among the most common sexually transmitted infections. Following infection, cervical carcinogenesis is a complex stepwise process characterized by slow progression. According to the known pathology, we represented the CIN stages with three corresponding components in the model. Our model accounted for the fact that preventive vaccines become ineffective over time. We derived three types of equilibria and their conditions of existence, analyzed the stability of the equilibria, and characterized the threshold condition as backward bifurcation for the stable fixed points. We also obtained the conditions for the elimination of the disease. We found that the possibility of HPV transmission to lead to endemic disease can be reduced by strengthening the protection after cure. We then simulated and compared practical mitigation strategies and performed sensitivity analysis to illustrate the key factors for the threshold condition. The results show that increasing the vaccination rate is the most effective way to reduce the basic reproduction number. The effect of optimal control was illustrated numerically, and a comparison of HPV infection was presented under different control strategies.
Thus, we have made the following conclusions Theorem A.1 Model (B.1) (or, equivalently, model (2)) undergoes a backward bifurcation at 0 1 R  if all parameters are positive.
Appendix C Table C1. PRCC values of 0 R with corresponding values of p (significant for p ≤ 0.01).
Parameter PRCC values