Analysis of dynamic transmission of HPV with reproduction numbers R0

In this paper, we discuss the patterns of the transmission of how the Human papillomavirus (HPV) spreads in the public. We analyze the behaviors of infectious disease in SIR, SEIR, and Multistrian models. In terms of this disease, HPV has many subtypes, some are high-risk, and some are low-risk. Unfortunately, many people in some areas are not aware of it. That will always cause many delays for the disease and aggravations. The harm caused by HPV is different for different groups of people, people will take various measures to handle the situation according to the severity. We divided them into 3 cases. That is case1-negative for people who may have caught HPV; case 2-negative people who may have caught HPV, but excluding exposures; case 3-negative for people who is sure to catch HPV. After that, by computing the disease-free equilibrium (DFE) to get the value for calculating reproduction number R 0 to estimate the stability for every case. Then the way of getting the derivatives and partitioned matrix and computing reproduction number with the measure of the Jacobian matrix were used for deciding the response for the epidemic eventually.

Theorem.Consider the disease transmission model given by with f(x) satisfying conditions.If  0 is a DFE of the model, then  0 is locally asymptotically stable if  0 < 1, but unstable if  0 > 1, where  0 is defined below [2].
Definition.  0 is the reproduction number for HPV. 0 = ( −1 ) Which  is the matrix for the positive states which transform to negatives. −1 is the inverse matrix for the rest parts [1].

Background
Human papilloma virus (HPA Pillomavirus) is a kind of genus papillomavirus A (Papilloma virus A) of the lacteoviridae family.It is a spherical DNA infectious virus that can cause the proliferation of the squamous epithelium of human skin mucosa.The symptoms are common warts, genital warts (condyloma), etc. HPV can be divided into low-risk or high-risk (hr-HPV) 2 kinds, according to the oncogenic potential [3].Currently, the Papillomavirus Family recorded more than four hundred types; Some subtypes have been discovered in different human anatomic sites.And caused proliferative lesions on the skin and mucosa, or in other places, such as the penile, anal, oropharyngeal, vulvar, and uterine cervix regions [4].And the most generally sexually transmitted human virus and perhaps the cause of a great number of cancers in both men and women, in a lot of areas of body tissues [5].Especially, among women, it remains the fourth most common type of cancer diagnosed and the fourth cause of death.Cervical cancer (CC), is regarded as one of the most threatening diseases to cause women's death and is tough to cure [6].Fortunately, because CC is caused by a virus (HPV), it seems nearly completely preventable (Eurina Yujin Cha and Hans Chun2021).In 2018, the WHO advocated control on the global scale of CC, it would welcome the eventual elimination of CC [7].For details, the aimed targets were 90% vaccination among teenage girls up to 15 years, 70% of the adult women aged 35-45 years screened for HPV, and treatment of 90% of precursor and invasive lesions [8].Despite humans having already had fine methods to prevent the disease, there are still many people dying of HPV symptoms.According to Kim, Dove, and Dang, For the people who are low educated, and have difficulties in finance, and people in Asia, Africa, and South America, the awareness of the perniciousness of HPV is of low percentage (2022) [9].That means there are still many people, especially teen girls who are not familiar with HPV and do not know how to protect themselves from it [10].
Many scientists are looking at the dynamic trends of this infectious agent in the human population, it interacts with human cells, and it can effect on other organisms to produce both benign and malignant tumors.Now, there is a discussion about a new vaccine for HPV and people hope that the new vaccination can make difference in the trends with this virus and the associated cancers it produces [11].
So far, this paper is going to demonstrate the spreading methods and patterns of HPV, give some corresponding methods to control the spread, and avoid the disease breaking out.

Method Theory
The mathematical epidemic SIR model (Susceptible-Infected-Recovered) is designed to get solutions of time depending differential equations under restrictive assumptions [12].To demonstrate the spreading ways of HPV, HPV is a kind of infectious virus, it is reasonable to discuss it in SIR models [13].Tracking how the virus invades susceptible people and makes them become conformed infected people, and suspected ones.Found that the original SIR is not suitable for the case [14].So, it is crucial to introduce the exposures for assistance (Hunter& Kelleher, 2022).Next, study how infected groups to cure the disease, and be back to susceptible people [15].To achieve our goals, the traditional SEIR model is modified in such a manner that can be mixed with the modified multi-strain model to get involved in the suspected infected people and conformed ones [16].The new thing is, some newborn children can catch HPV from their mothers [17].Meanwhile, some people may get immune by taking a vaccine [18].But sometimes, the vaccination will be invalid [19].Moreover, the infected people who caught a high-risk virus may have the chance to develop a life-threatening disease [20].So, based on this situation, The framework of the multi-strain model for studying such systems, is a caricature of the more detailed treatment model and the coupled two-strain vector-host model [21].To get access to the appropriate control methods, it is necessary to figure out a way to avoid an outbreak [22].Thus, the first step is calculating for the Disease-Free-Equilibrium (DFE) to know the situation when the disease disappeared completely [18].After that, A definition of the basic reproduction number R 0 will be introduced.It is presented for a general compartmental disease transmission model based on a system of time depending ordinary differential equations [23].It means that if is R 0 < 1, the DFE is locally asymptotically stable; while if R 0 > 1, then the equilibrium is unstable, and the disease is going to break out [24].This is an important criterion for lots of yields to test the existence and stability of endemics [25].This criterion is illustrated by treatment [26].What's more, it can be used in staged progression, multi-strain, and vector-host models and can be applied to more complex models [27].The results are significant for disease control [1].

Contribution
This paper discusses the behavior of HPV with SIR, SEIR, and mixes multi-strain models, using the follow-up extensions of the existing theorems and methods to analyze and control the spreading of HPV with a new mindset.Firstly, it describes the various possibilities of transmission to demonstrate how the virus spreads.Talking about it in an equation system, assuming 3 cases for a different situation to help people prevent it.By computing DFE and R 0 , we are going to use a novel way to get reproduction number, which is with lemmas and theorems to get matrix V and F then calculate for spectral radius and R 0 .Furthermore, with the result of R 0 , it is possible to analyze the stability of the equilibrium and predict if the epidemic will be explosive or not, also provides some solutions for health organizations and many governments to prevent the virus.Such as cutting off the source of infection, to decrease the death rate and save more lives.

Interpretation of equations and models
This model refers to the basic SIR and SEIR models and creates a new modified one.The main idea of the SIR model is that the virus first invades the public and causes some groups of people to get infected by the virus, as can be seen in figure 1.And the infected people will infect the suspected groups via touching or any other ways.After that, the infected can be cured by some medical ways to be recovered.Moreover, SEIR modifies the SIR with a new group of people who are exposed to the infected ones.It gives some new situations when we talk about dynamic behaviors [28].However, in order to demonstrate a certain virus, it is necessary to make the traditional models more specific to a certain situation.In severe cases, it can cause cancer in different areas, and that may be life-threatening.And the lowrisk virus has a lower threat to humans, but it does not mean it cannot be life-threatening [30].Thus, according to the features of HPV, the modified model for HPV is going to divide the state of the infected group into 3 parts  1 ,  2 ,   3 , which implies symptom suspected, conformed diagnosis and developed serious diseases, respectively.Then, insert the multi-strain model with ,  1 ,  2 and deduce what is going to happen afterward in the follow-up procedures.To demonstrate this in a good way, we are going to do it with a flow chart (figure 2) to show how the dynamic behaviors of HPV invade the public.And the meaning of symbols is shown in Table 1.

Notation
Table1 The natural birth rate of a susceptible person μ The natural death rate of healthy people  1 The rate that a susceptible person is exposed to HPV  2 The rate that people who are exposed to HPV but do not catch it  1 The rate of exposed people with suspicious symptoms  1 The rate of people who have symptoms but not infected with HPV  2 The rate of exposed persons with confirmed HPV infection  2 The percentage of people who were diagnosed with HPV infection but recovered because of self-immunity  1 The percentage of the susceptible population who are vaccinated  2 The percentage of vaccines which get invalid  2 The birth rate of children infected from their mothers φ The rate of confirmed HPV infection in the suspected symptomatic population ε The proportion of people with severe HPV who are cured d The mortality rate due to severe HPV infection  3 The birth rate of newborns with suspicious symptoms  1 The rate of people who cure their symptoms by medical ways in time The proportion of people who already develop suspicious symptoms and get worse because of unawareness of HPV

The dynamic time depending differential equations are shown below:
Note that, the transmission of  1 →  3 and the transmission of  1 →  2 are not symmetric, because some people may be unawareness of the virus and its symptoms so that develop a serious disease.The reason for  1 →  2 , People often goes to hospital to get HPV check after they have already had some symptoms.And another special case is that some people may take the vaccination, and get immunity, but as time goes, the effect of vaccination cannot be reserved overtime.It is invalid at some rates.

Disease Free Equilibrium
In order to analyze the dynamic behavior.It is of great significance to get the Disease-Free-Equilibrium (DFE).To achieve this, we divide the situation in 3 cases for different functions to handle with different problems.

Case 1
Sometimes, the disease may be extremely threatening, people would like to prevent it completely.In this case, taking all the 4 states , 1 ,  2 ,  3 to be negative.Take the value of 0 for them.Because the purpose is to eliminate all them away.The original equation system is: Reduce it to the corresponding case, let negative states to be 0: Solve for the equation, get the result:

Case 2
In some cases, Due to the cost for treating and preventing HPV is too high, so sometimes people will not take practical measures to the exposures because there is no definite evidence to show that they have caught HPV.So far, we only take  1 ,  2 ,  3 as the negative states, be 0.Here gives the equation below: Solve for the system, it yields that Thus, The DFE shows ( 1 ,  2 ,  3 , , , , respectively):

Case 3
In this case, it is going to regard the conformed infected people to be the negative group.Because some people may catch the low-risk virus, which can be removed by self-immunity.In order to reduce the cost, The governments do not need to take steps for them.Thus, in the following system, we do not consider , and  1 when we talk about negative states.Taking  2 and  3 be 0.

Reproduction Number 𝐑 𝟎
This paper is going to use  0 to discuss the stability of the model for HPV.If  0 > 1, the disease will break out or out of control; If  0 < 1, the disease will be controlled ideally.Details can be checked in Method Theory in chapter 1.
Here, some theorems and definitions need to be show and utilized in the following part.

Recall:
Lemma .If  0 is a DFE for the case, then the derivatives D( 0 ) and D( 0 ) are partitioned as Theorem.Consider the disease transmission model given by with f(x) satisfying conditions.If  0 is a DFE of the model, then  0 is locally asymptotically stable if  0 < 1, but unstable if  0 > 1, where  0 is defined below (Theparod&Thitiya, 2019).
Definition.  0 is reproduction number for HPV. 0 = ( −1 ) Which  is the matrix for the positive states which transform to negatives. −1 is the inverse matrix for the rest parts (Driessche& Watmough, 2001).

For case 1
For the case of which negative states are E, I 1 , I 2 , I 3 .Calculate for  0 .(E, I 1 , I 2 , I 3 , S, R ,respectively).
Recall that the DFE of case 1 is (,  1 ,  2 ,  3 , , , respectively): By the lemma we claimed at the beginning of the chapter.We got D and D: It yields that by the lemma, Calculate for the inverse matrix of V, we got: Thus, to get  0 , we use the formula we presented: Finally, we got the reproduction number for HPV in the case 1.

For case 2
For the case of which negative states are I 1 , I 2 , I 3 .Calculate for R 0 .(I 1 , I 2 , I 3 , E, S, R ,respectively).
As lemma said.Thus, we got F and V: Calculate for the inverse of the matrix of V: So far, we got the reproduction number for case 2.

For case 3
For the case of which negative states are I 2 , I 3 .Calculate for  0 ( I 2 , I 3 , I 1 , E, S, R ,respectively).
the DEF we concluded is ( 1 ,  2 ,  3 , , , , respectively): Here, we are going to calculate for  ) and V = ( ) the rest parts of the matrix does not need to take into consideration.
Finally, so far, we got the  0 which is shown below:

Analyse
In this chapter, it is going to analyse the value we got in the previous parts, and will do some attempts to control the HPV model.For example, in the view of the governments, what measures should be take can mostly achieve the idea situation we want, that means we should do something with some certain values.

Case 1
In this case, The government may treat HPV in a serious way.If there any exposures or anyone who has the HPV-like symptoms, they will be treated as 'negative people'.So that, people do not want any existence of HPV.Thus, if we do not want HPV spread very much, and avoid it breaks out.Recall  0 = v 1  2 − 2 ， the numerator v 1 should be control, do not allow to boom.It is necessary to isolate the infected people, HPV-like symptom people, and even the exposures.And as quickly as possible, determine whether a patient with suspected symptoms has HPV.

Case 2
In this case, the exposures will not be regarded as the 'negative people'.Consider that  0 =  1  1  1 [+ 2 ] (+ 2 )(++ 2 + 1 )( 2 + 1 + 2 ) , we can reduce the number of people who has the probability to get in touch with the infected people.In addition to determine whether a patient with suspected symptoms has HPV, can also advocate people who has the suspected symptoms go to see doctor as fast as possible.Moreover, make the vaccine to HPV general among common people.

Case 3
In the case, only the conformed infected people and people who develop a high risk disease will be seen as negative.refer to  0 =  2 (−− 2 + 1 )  1 (+ 1 + 2 ) .We would like to let the numerator get smaller and the denominator to be bigger.It is wise to check the suspected symptoms and accept treatment in time.

Conclusion
In this paper, we are by means of fixed SIR, SEIR and some theorems for calculating the reproduction number to determine the stability for HPV model.To achieve this goal, we demonstrated the possible transmission ways for it, and divided into 3 cases to discuss different circumstances.By Calculating the DEF (Disease-Free-Equilibrium) for auxiliary, the reproduction number for HPV is gotten.Finally with the reproduction number we can take steps to avoid the break out of the disease.This article gives some new ideas for infectious disease models.In terms of this topic, it is crucial to analyze the behavior of HPV.On one hand, it would provide us methods for the government or some health organizations to deal with the public health security issues as we analyze in the previous chapters.On the other, it can also be a proper model or demonstration for other infectious diseases.
Lemma.If  0 is a DFE for the case, then the derivatives D( 0 ) and D( 0 ) are partitioned as

Figure 1 .
Figure 1.ordinary SIR model.Known that HPV can be divided into 2 kinds, high-risk type and low-risk type.The high-risk type is mainly due to some subtypes of virus-like HPV5, 8, 14, 16,18, and 30 (2022)[29].High-risk viruses cause damage to the epidermis or mucous membranes in different parts of the body to varying degrees.In severe cases, it can cause cancer in different areas, and that may be life-threatening.And the lowrisk virus has a lower threat to humans, but it does not mean it cannot be life-threatening[30].Thus, according to the features of HPV, the modified model for HPV is going to divide the state of the infected group into 3 parts  1 ,  2 ,   3 , which implies symptom suspected, conformed diagnosis and developed serious diseases, respectively.Then, insert the multi-strain model with ,  1 ,  2 and deduce what is going to happen afterward in the follow-up procedures.To demonstrate this in a good way, we are going to do it with a flow chart (figure2) to show how the dynamic behaviors of HPV invade the public.And the meaning of symbols is shown in Table1.

Figure 2 .
Figure 2. Model of transmission of HPV.