CN111834015A - Optimal vaccination strategy and screening strategy making method for HPV and cervical cancer caused by HPV - Google Patents

Abstract

An optimal vaccination strategy and screening strategy making method for HPV and cervical cancer caused by HPV, belonging to the field of public health and medical treatment. The method comprises the following steps: firstly, according to the epidemic dynamics of HPV, a chamber model is established by using a differential equation for describing the transmission rule of HPV. Secondly, estimating model parameters according to actual data, counting the number of population in each chamber, and taking the vaccination rate and the screening rate as control variables to establish a nonlinear optimal control problem for minimizing the number of HPV infectors, the vaccine cost and the screening cost. And finally, solving the constructed optimal control problem by adopting a numerical method to obtain an optimal HPV long-term vaccination strategy and screening strategy. The invention can realize the comprehensive consideration of the prevention and treatment effect and the economic cost in the inoculation strategy and the screening strategy, and has important scientific significance for the scientific prevention and treatment work of cervical cancer; and the adopted method has strong operability and feasibility and is convenient for practical application.

Description

Optimal vaccination strategy and screening strategy making method for HPV and cervical cancer caused by HPV

Technical Field

The invention belongs to the field of public health and medical treatment, and relates to an optimal vaccination strategy and screening strategy formulation method for HPV and cervical cancer caused by HPV.

Background

HPV is a common sexually transmitted virus and has a plurality of types, the main types are HPV-1, -2, -6, -11, -16, -18, -31, -33 and-35, and the like, wherein part of types of HPV-16 and HPV-18 have carcinogenicity. Pathological studies have demonstrated that long-term infection with HPV is the leading cause of cervical cancer in women. By vaccination and screening of susceptible people and the like, the HPV infection number can be effectively reduced and the incidence of cervical cancer can be reduced. Therefore, how to determine an effective and economical vaccination strategy and screening strategy has important scientific significance for the scientific prevention and treatment work of HPV and cervical cancer caused by HPV.

Disclosure of Invention

In order to solve the technical problems, the invention provides an optimal vaccination strategy and a screening strategy formulation method for HPV and cervical cancer caused by the HPV based on HPV epidemiology dynamics and an optimal control method. The method can realize the comprehensive consideration of the prevention and treatment effect and the economic cost in the vaccination strategy and the screening strategy, and has good practicability.

In order to achieve the purpose, the invention adopts the technical scheme that:

firstly, according to the epidemic disease dynamics of HPV, a chamber model is established by using a differential equation so as to describe the transmission rule of HPV and the process of morbidity after infecting HPV. Secondly, estimating model parameters according to actual data, counting the number of population in each chamber, taking the vaccination rate and the screening rate as control variables, and establishing a nonlinear optimal control problem for minimizing the number of HPV infected people, the vaccination cost and the screening cost. And finally, solving the constructed optimal control problem by using a numerical method to obtain the optimal HPV long-term vaccination strategy and screening strategy. The method comprises the following steps:

step 1: building chamber model for describing HPV propagation law

All populations were divided into those susceptible to HPV (V compartment), susceptible to HPV (S compartment), asymptomatic infected HPV (E compartment), symptomatic infected HPV (H compartment), long-term infected HPV (P compartment), patients with long-term infection leading to precancerous lesions (I compartment)₁Bin), patients with precancerous lesions in stage II (I)₂Bin), patients in the third stage of precancerous lesions (I)₃Chamber), cervical cancer patients caused by HPV (a chamber) and HPV rehabilitators (R chamber); the total number of N is recorded as V + S + E + H + P + I₁+I₂+I₃+ A + R; thus, the following bin model can be established:

wherein the infection rate is alpha₁(t) can be expressed as:

the meaning of each parameter in the formula (1) and the formula (2) is as follows: lambda is the number of human mouth growth in the system, d is the natural mortality, d₁In the case of the mortality caused by cervical cancer,₁in order to determine the failure rate of the vaccine,₂and₄the ratio of the first and second vaccinations for a susceptible person,₃correction factor for the probability of infection of the convalescent person relative to the susceptible person, c_pAnd c_qFor the effectiveness of the screening and the proportion of asymptomatic patients who receive the screening, c_nProbability of obtaining a new sexual partner for a male or female, c_kProbability of male or female being transmitted by the virus in contact, c_cAnd c_aFor condom utility and rate of use, alpha for taking contraceptivesEffects of substances on HPV infection, beta is the effect of smoking on HPV infection, alpha₂Probability of conversion of an asymptomatic patient to HPV symptomatology, α₃Probability of persistent infection for patients with HPV symptoms, alpha₄The probability of a premalignant lesion at the site of infection of a patient persistently infected with HPV to enter a patient of stage one, alpha₅The probability of a patient in the first stage of a precancerous lesion to develop into a patient in the second stage, α₆The probability of a patient in the second stage of a precancerous lesion to develop into a patient in the third stage, α₇The probability of a patient in the third stage of precancerous lesions developing cervical cancer, σ₁Probability of recovery for an asymptomatic patient of HPV, σ₂Probability of recovery of a patient with HPV symptoms to an asymptomatic patient, σ₃Probability of recovery of patients with persistent HPV infection to symptomatic patients, σ₄Probability of recovery of patients with HPV precancerous lesions in the first stage of recovery into patients with persistent HPV infection, σ₅Probability of recovery of patients with HPV precancerous lesions in stage II to patients in stage I, σ₆Probability of recovery of patients with HPV precancerous lesions in the third stage to patients in the second stage, gamma₁，γ₂，γ₃，γ₄，γ₅And gamma₆The self-healing rate sigma of each drug pair₁，σ₁，σ₁，σ₁，σ₁And σ₁Positive effect of (2), gamma₇The cure rate of cervical cancer patients, theta₁，θ₂，θ₃，θ₄And theta₅Correction factors for the infection rate in contact with an HPV-symptomatic patient, a patient with persistent infection with HPV, a patient with a precancerous lesion at the first stage, a patient with a precancerous lesion at the second stage, a patient with a precancerous lesion at the third stage, and the like, respectively;

step 2: establishing a nonlinear optimal control problem for minimizing the number of HPV infectors, the vaccine cost and the screening cost, and estimating parameters in a chamber model according to historical data;

considering vaccination of susceptible populations and screening of asymptomatic patients for HPV, parameters₂And c_qDue to the probabilistic nature, it satisfies:

0≤₂≤1，1≤c_q≤2, (3)

the time for starting and stopping the implementation of the inoculation strategy and the screening strategy is t₁And t₂(ii) a Statistics of t₁The number of people in each chamber is respectively

Comprehensively considering the number of HPV infectors in the strategy implementation stage, the cost of vaccine and the economic cost of screening cost, the following optimal control problems are established:

s.t.

V′＝₂(t)S+₄S-(d+₁)V,

S′＝Λ+₁V-(₂(t)+₄+α₁+d)S,

E′＝α₁S+₃α₁R+σ₂H-(α₂+c_pc_q(t)γ₁σ₁+d)E,

H′＝α₂E+σ₃P-(σ₂+γ₂σ₂+α₃+d)H,

I′₃＝α₆I₂-(σ₆+γ₆σ₆+α₇+d)I₃,

A′＝α₇I₃-(γ₇+d+d₁)A,

R′＝c_pc_q(t)γ₁σ₁E+γ₂σ₂H+γ₃σ₃P+γ₄σ₄I₁+γ₅σ₅I₂+γ₆σ₆I₃+γ₇A-(₃α₁+d)R,

V(t₁)＝V₀,S(t₁)＝S₀,E(t₁)＝E₀,H(t₁)＝H₀,P(t₁)＝P₀,

I₁(t₁)＝I_1,0,I₂(t₁)＝I_2,0,I₃(t₁)＝I_3,0,A(t₁)＝A₀,R(t₁)＝R₀,

0≤₂(t)≤1,1≤c_q(t)≤2,

wherein, the parameter C₁，C₂，C₃，C₄，C₅Represents the weight of the number of patients with HPV symptoms, patients with persistent HPV infection, patients with precancerous lesions in the first stage, patients with precancerous lesions in the second stage and patients with precancerous lesions in the third stage in the long-term implementation strategy; b is₁And B₂Weights representing vaccine costs and screening costs in the implemented strategy;

and step 3: solving the optimal control problem, and guiding the establishment of HPV long-term vaccination strategy and screening strategy

Solving the optimal control problem in formula (5); obtained₂(t) and c_q(t) represents the optimal vaccination and screening rates in the sense of optimal control; and V (t), S (t), E (t), H (t), P (t), I₁(t),I₂(t),I₃(t), A (t), R (t) respectively represent the change history of the number of vaccinees, HPV susceptible persons, HPV asymptomatic persons, HPV infected persons, HPV long-term infected persons, patients with pre-cancerous lesions caused by HPV in the first stage, patients with pre-cancerous lesions in the second stage, patients with pre-cancerous lesions in the third stage, patients with cervical cancer caused by HPV and patients with HPV rehabilitation under a long-term strategy.

Compared with the prior art, the invention has the beneficial effects that: the invention describes the HPV propagation rule and the process of morbidity after HPV infection through a chamber model, guides the formulation of an optimal HPV long-term vaccination strategy and a screening strategy through an optimal control method, and has important theoretical value and practical value for scientific prevention and treatment of cervical cancer. The method disclosed by the invention has strong operability and feasibility and is convenient for practical application.

Drawings

FIG. 1 is a flow chart of the calculation of the present invention.

FIG. 2 is a graph of the history of the number of vaccinees in an example of the invention.

FIG. 3 is a graph illustrating the history of the number of human subjects that are susceptible to a change in population in an embodiment of the present invention.

FIG. 4 is a graph showing the course of change in the population of asymptomatic infected persons caused by HPV infection in the examples of the present invention.

FIG. 5 is a graph showing the course of the population change of an infected person with an illness due to HPV infection in the example of the present invention.

FIG. 6 is a graph showing the progression of population changes in patients chronically infected with HPV in an example of the invention.

FIG. 7 is a graph of the progression of the population of patients with pre-cancerous lesions resulting from HPV infection in an embodiment of the present invention.

FIG. 8 is a graph of the progression of the population of patients with pre-cancerous lesions in the second phase of the present invention as a result of HPV infection.

FIG. 9 is a graph of the progression of the population of patients in the third stage of premalignant lesions caused by HPV infection in an embodiment of the present invention.

FIG. 10 is a history of changes in the population number of cervical cancer patients caused by HPV in the example of the present invention.

Fig. 11 is a graph showing the history of the number of rehabilitees in the example of the present invention.

FIG. 12 is a graph showing the variation of the vaccination rate and screening rate in the examples of the present invention.

Detailed Description

The present invention is further illustrated by the following specific examples.

All populations were divided into those susceptible to HPV (V compartment), susceptible to HPV (S compartment), asymptomatic infected HPV (E compartment), symptomatic infected HPV (H compartment), long-term infected HPV (P compartment), patients with long-term infection leading to precancerous lesions (I compartment)₁Bin), patients with precancerous lesions in stage II (I)₂Bin), patients in the third stage of precancerous lesions (I)₃Chamber), cervical cancer patients caused by HPV (a chamber) and HPV rehabilitators (R chamber); the total number of N is recorded as V + S + E + H + P + I₁+I₂+I₃+ A + R; thus, the following bin model can be established:

wherein the unit of the time variable t is a list of the number of population of each chamber in yearThe position is human, the number of human mouth growth in the system is Λ ═ 288802, the natural mortality rate is d ═ 0.0162, and d is₁0.01 is the mortality rate due to cervical cancer,₁0.1 is the failure rate of the vaccine,₂and₄0.27 is the ratio of the first and second vaccinations of a susceptible person,₃0.3 is a correction coefficient of the infection probability of the patient with respect to the susceptible patient, c_p0.9 and c_qFor the effectiveness of the screening and the proportion of asymptomatic patients who receive the screening, c_n1.1 probability of acquiring a new sexual partner for male or female, c_k0.6181 is the probability of a male or female being infected by a virus during contact, c_c0.99 and c_a0.2 is the effect of condom and its usage rate, α ═ 1.4 is the effect of taking contraceptive medicine on HPV infection, β ═ 1.3 is the effect of smoking on HPV infection, α ═ 1.3 is the effect of smoking on HPV infection₂0.5 is the probability of conversion of an asymptomatic patient with HPV to an HPV symptomatic, α₃0.005 is the probability of persistent infection in patients with HPV symptoms, α₄0.1 is the probability of a premalignant lesion at the site of infection in patients with persistent HPV, α₅0.02 is the probability of a patient with precancerous lesions in the first stage progressing to a patient in the second stage, α₆0.04 is the probability of a precancerous lesion for the second stage patient to progress to the third stage patient, α₇0.08 is the probability of a patient in the third stage of precancerous lesions developing cervical cancer, σ₁0.99 is the probability of recovery of an HPV asymptomatic patient, σ₂9e-4 is the probability of recovery of a patient with HPV symptoms to an asymptomatic patient, σ₃0.5 is the probability of recovery of patients with persistent HPV infection to symptomatic patients, σ₄1.9e-7 is the probability of recovery of patients with HPV precancerous lesions in stage one to patients with persistent infection of HPV, σ₅1.9e-7 is the probability of recovery of patients with HPV precancerous lesions in stage ii to stage i, σ₆1.9e-7 is the probability that patients in stage three will recover to stage two in HPV precancerous lesions, γ₁＝1.5，γ₂＝1.5，γ₃＝1.2，γ₄＝1.1，γ₅1.05 and γ₆1.03 is the self-healing rate of the drug pair₁，σ₂，σ₃，σ₄，σ₅And σ₆Positive effect of (2), gamma₇1.01 is the cure rate for cervical cancer patients, theta₁＝1，θ₂＝0.8，θ₃＝0.7，θ₄0.6 and θ₅0.5 is a correction coefficient for the infection rate when a patient is exposed to HPV symptoms, a patient with persistent infection with HPV, a patient with precancerous lesions in the first stage, a patient with precancerous lesions in the second stage, a patient with precancerous lesions in the third stage, or the like;

step 2: establishing a non-linear optimal control problem to minimize HPV infectors population and vaccine costs and screening costs

Considering vaccination of susceptible populations and screening of asymptomatic patients for HPV, parameters₂And c_qDue to the probabilistic nature, it satisfies

0≤₂≤1，1≤c_q≤2,

The time for starting and stopping the implementation of the vaccination strategy and the screening strategy is t ₁0 and t ₂50. Statistics of t₁The number of people in each chamber is respectively

V(t₁)＝3.2607e6,S(t₁)＝3.2212e5,E(t₁)＝4.2204e4,H(t₁)＝1.1162e7,P(t₁)＝4e4,

I₁(t₁)＝1.15e5,I₂(t₁)＝4.15e4,I₃(t₁)＝1.68e4,A(t₁)＝1.29e3,R(t₁)＝6.3e3,

Comprehensively considering the number of HPV infectors in the strategy implementation stage, the vaccine cost and the economic cost of the screening cost, the following optimal control problem is established

s.t.

V′＝₂(t)S+₄S-(d+₁)V,

S′＝Λ+₁V-(₂(t)+₄+α₁+d)S,

E′＝α₁S+₃α₁R+σ₂H-(α₂+c_pc_q(t)γ₁σ₁+d)E,

H′＝α₂E+σ₃P-(σ₂+γ₂σ₂+α₃+d)H,

P′＝α₃H+σ₄I₁-(α₄+γ₃σ₃+σ₃+d)P,

I′₁＝α₄P+σ₅I₂-(α₅+γ₄σ₄+σ₄+d)I₁,

I′₂＝α₅I₁+σ₆I₃-(σ₅+γ₅σ₅+α₆+d)I₂,

I′₃＝α₆I₂-(σ₆+γ₆σ₆+α₇+d)I₃,

A′＝α₇I₃-(γ₇+d+d₁)A,

R′＝c_pc_q(t)γ₁σ₁E+γ₂σ₂H+γ₃σ₃P+γ₄σ₄I₁+γ₅σ₅I₂+γ₆σ₆I₃+γ₇A-(₃α₁+d)R,

V(t₁)＝V₀,S(t₁)＝S₀,E(t₁)＝E₀,H(t₁)＝H₀,P(t₁)＝P₀,

I₁(t₁)＝I_1,0,I₂(t₁)＝I_2,0,I₃(t₁)＝I_3,0,A(t₁)＝A₀,R(t₁)＝R₀,

0≤₂(t)≤1,1≤c_q(t)≤2,

Wherein, the parameter C₁＝4.5e-2，C₂＝1e-7，C₃＝1e-4，C₄＝1e-5，C₅2e-4 indicates the weight of the number of patients with HPV symptoms, patients with persistent HPV infection, patients with pre-cancerous lesions in the first stage, patients with pre-cancerous lesions in the second stage, patients with pre-cancerous lesions in the third stage of the long-term implementation strategy; b is₁8.3e8 and B₂4e8 represents the weight of vaccine cost and screening cost in the implemented strategy;

and step 3: solving the optimal control problem, and guiding the establishment of HPV long-term vaccination strategy and screening strategy

The calculated change history of the population number of the class-10 bins according to the above steps is shown in fig. 2 to 11. Under the condition of exerting control, the number of all types of HPV infectors is obviously reduced, and the number of cervical cancer patients caused by HPV is also obviously reduced. In fig. 12, the course of the vaccination rate and screening rate is plotted.

The above-mentioned embodiments only express the embodiments of the present invention, but not should be understood as the limitation of the scope of the invention patent, it should be noted that, for those skilled in the art, many variations and modifications can be made without departing from the concept of the present invention, and these all fall into the protection scope of the present invention.

Claims (1)

1. An optimal vaccination strategy and screening strategy formulation method for HPV and cervical cancer caused by the HPV is characterized in that firstly, a chamber model is established by utilizing a differential equation according to the epidemic disease dynamics of the HPV, and is used for describing the propagation rule of the HPV and the process of morbidity after the HPV infection; secondly, estimating model parameters according to actual data, counting the number of population in each chamber, taking the vaccination rate and the screening rate as control variables, and establishing a nonlinear optimal control problem for minimizing the number of HPV infected people, the vaccination cost and the screening cost; finally, solving the constructed optimal control problem by using a numerical method to obtain an optimal HPV long-term vaccination strategy and screening strategy; the method comprises the following steps:

step 1: building chamber model for describing HPV propagation law

All populations were classified as HPV susceptible to vaccinationV chamber, HPV susceptible person as S chamber, HPV asymptomatic infected person as E chamber, HPV symptomatic infected person as H chamber, HPV long-term infected person as P chamber, and patient with long-term infection causing precancerous lesion in the first stage as I chamber₁Second-stage patients with precancerous lesions in the Bingo area are scored as I₂Patients in the third stage of precancerous lesions in the Bingo area are scored as I₃The chamber, cervical cancer patients resulting from HPV are scored as chamber a and HPV rehabilitators as chamber R; and N is the total population number, and is V + S + E + H + P + I₁+I₂+I₃+ A + R; thus, the following bin model is established:

wherein the infection rate is alpha₁(t) is expressed as:

the meaning of each parameter in the formula (1) and the formula (2) is as follows: lambda is the number of human mouth growth in the system, d is the natural mortality, d₁In the case of the mortality caused by cervical cancer,₁in order to determine the failure rate of the vaccine,₂and₄the ratio of the first and second vaccinations for a susceptible person,₃correction factor for the probability of infection of the convalescent person relative to the susceptible person, c_pAnd c_qFor the effectiveness of the screening and the proportion of asymptomatic patients who receive the screening, c_nProbability of obtaining a new sexual partner for a male or female, c_kProbability of male or female being transmitted by the virus in contact, c_cAnd c_aAlpha is the effect of taking contraceptives on HPV infection, beta is the effect of smoking on HPV infection, alpha is the effect of smoking on HPV infection and the rate of use of condoms₂Probability of conversion of an asymptomatic patient to HPV symptomatology, α₃For persistent infection of patients with HPV symptomsProbability, α₄The probability of a premalignant lesion at the site of infection of a patient persistently infected with HPV to enter a patient of stage one, alpha₅The probability of a patient in the first stage of a precancerous lesion to develop into a patient in the second stage, α₆The probability of a patient in the second stage of a precancerous lesion to develop into a patient in the third stage, α₇The probability of a patient in the third stage of precancerous lesions developing cervical cancer, σ₁Probability of recovery for an asymptomatic patient of HPV, σ₂Probability of recovery of a patient with HPV symptoms to an asymptomatic patient, σ₃Probability of recovery of patients with persistent HPV infection to symptomatic patients, σ₄Probability of recovery of patients with HPV precancerous lesions in the first stage of recovery into patients with persistent HPV infection, σ₅Probability of recovery of patients with HPV precancerous lesions in stage II to patients in stage I, σ₆Probability of recovery of patients with HPV precancerous lesions in the third stage to patients in the second stage, gamma₁，γ₂，γ₃，γ₄，γ₅And gamma₆The self-healing rate sigma of each drug pair₁，σ₁，σ₁，σ₁，σ₁And σ₁Positive effect of (2), gamma₇The cure rate of cervical cancer patients, theta₁，θ₂，θ₃，θ₄And theta₅Correction factors for the infection rate in contact with an HPV-symptomatic patient, a patient with persistent infection with HPV, a patient with a precancerous lesion at the first stage, a patient with a precancerous lesion at the second stage, a patient with a precancerous lesion at the third stage, and the like, respectively;

step 2: establishing a nonlinear optimal control problem for minimizing the number of HPV infectors, the vaccine cost and the screening cost, and estimating parameters in a chamber model according to historical data;

considering vaccination of susceptible populations and screening of asymptomatic patients for HPV, parameters₂And c_qDue to the probabilistic nature, it satisfies:

0≤₂≤1，1≤c_q≤2, (3)

the time for starting and stopping the implementation of the inoculation strategy and the screening strategy is t₁And t₂(ii) a Statistics of t₁The number of people in each chamber at the moment is respectively as follows:

comprehensively considering the number of HPV infectors in the strategy implementation stage, the cost of vaccine and the economic cost of screening cost, the following optimal control problems are established:

wherein, the parameter C₁，C₂，C₃，C₄，C₅Represents the weight of the number of patients with HPV symptoms, patients with persistent HPV infection, patients with precancerous lesions in the first stage, patients with precancerous lesions in the second stage and patients with precancerous lesions in the third stage in the long-term implementation strategy; b is₁And B₂Weights representing vaccine costs and screening costs in the implemented strategy;

and step 3: solving the optimal control problem, and guiding the establishment of HPV long-term vaccination strategy and screening strategy, which comprises the following steps:

solving the optimal control problem in formula (5); obtained₂(t) and c_q(t) represents the optimal vaccination and screening rates in the sense of optimal control; and V (t), S (t), E (t), H (t), P (t), I₁(t),I₂(t),I₃(t), A (t), R (t) respectively represent the change history of the number of vaccinees, HPV susceptible persons, HPV asymptomatic persons, HPV infected persons, HPV long-term infected persons, patients with pre-cancerous lesions caused by HPV in the first stage, patients with pre-cancerous lesions in the second stage, patients with pre-cancerous lesions in the third stage, patients with cervical cancer caused by HPV and patients with HPV rehabilitation under a long-term strategy.

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