CN113113147A - Method for distributing hepatitis B vaccines with priority based on crowd heterogeneity - Google Patents

Method for distributing hepatitis B vaccines with priority based on crowd heterogeneity Download PDF

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CN113113147A
CN113113147A CN202110466111.1A CN202110466111A CN113113147A CN 113113147 A CN113113147 A CN 113113147A CN 202110466111 A CN202110466111 A CN 202110466111A CN 113113147 A CN113113147 A CN 113113147A
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张贺婷
纪文强
邱剑彬
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Harbin Institute of Technology
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Abstract

The invention discloses a method for distributing hepatitis B vaccines with priorities based on population heterogeneity, which comprises the following steps: dividing the crowd into a plurality of groups; establishing a differential equation SVIR model of hepatitis B virus transmission based on heterogeneity under the condition of incomplete immunization of an vaccinee; calculating a disease-free balance point and an endemic balance point of the differential equation SVIR model; determining a hepatitis B virus transmission threshold; determining the condition of hepatitis B virus transmission; determining the condition for extinguishing the hepatitis B virus based on the hepatitis B virus propagation threshold value; judging the spreading condition of hepatitis B viruses in different groups, and when the immune resources are limited, preferentially considering the groups in which the viruses are spread, and making a vaccine distribution strategy in the groups; the invention has the advantages that: wide application range, diversified evaluation indexes, effective control of hepatitis B virus propagation, and suitability for effectively controlling epidemic spread or economically and reasonably distributing immune resources in a short time or in a limited immune resource.

Description

Method for distributing hepatitis B vaccines with priority based on crowd heterogeneity
Technical Field
The invention relates to the crossing field of a differential equation power system, biological mathematics and public health services, in particular to a hepatitis B vaccine distribution method with priority based on crowd heterogeneity.
Background
Viral hepatitis B (hepatitis B for short) is an infectious disease caused by hepatitis B virus, mainly including liver disease and causing damage to multiple organs. If effective measures are not taken in time for intervention, the disease condition can be converted into cirrhosis or liver cancer along with the continuous development of the disease condition. For historical reasons, our country has been deficient in many aspects of hepatitis B virus control. The hepatitis B vaccine is inoculated in a large range at the end of the last century, so that the incidence rate of hepatitis B in China is always at a high level, and the hepatitis B vaccine becomes one of the most widely spread infectious diseases. At present, hepatitis B virus carriers in China exceed one hundred million people, and liver functions of about four patients are damaged in different degrees. Therefore, it is necessary to take corresponding prevention and cure measures to effectively prevent hepatitis B virus.
The application range of the vaccination in the prevention and control of infectious diseases is very wide and plays an irreplaceable role, and the vaccination not only protects susceptible people, but also protects the whole society. Therefore, the most economical and effective method for preventing hepatitis B virus infection is vaccination. The hepatitis B vaccine is a hepatitis B virus inactivated vaccine obtained by a gene recombination technology, cannot be replicated in vivo, and can be injected for multiple times to stimulate an organism to generate enough antibodies so as to protect a person to be inoculated. The immunization program of hepatitis B vaccine in China is 0-1-6, and 3 needles are needed to be inoculated in the whole process. After the first injection of hepatitis B vaccine is inoculated, only 30% of people generate hepatitis B surface antibodies, and the antibody effect is very unstable; after inoculation of the second needle, 90% of the humans produced antibodies; the positive rate of the antibody after the third injection is inoculated can reach more than 96 percent, and the antibody effect is continuously maintained at a higher level. Because the vaccination cannot be used for life, whether to be used for vaccination again in the future is determined according to the measurement of the titer of the hepatitis B surface antibody, and most scholars in China recommend that the vaccination needs to be enhanced within 3-5 years after the immunization. Therefore, considering that the risk of disease still exists after vaccination, how to establish a long-term stable prevention and treatment measure according to the actual demand of the public on the vaccination service so as to protect the result of special intervention and maintain a long-term high vaccination coverage rate is a problem to be solved urgently in the future. In addition, in order to better exert the preventive effect of the vaccine on hepatitis B and improve the health of the whole people, it is also vital to find a proper measure to ensure that the vaccination can be stably popularized and continued for a long time.
On the other hand, different groups of people have different risk degrees for the transmission and infection of hepatitis B virus due to the influence of virus factors, host factors, environmental factors and the like. For example: the main reason for the high incidence of hepatitis B in China is familial transmission, mainly mother-infant vertical transmission. Since the immune system of the infant body is not fully developed, the infant has a higher infection rate compared with the adult. Once the infant has infection, the infant is extremely easy to become a hepatitis B virus carrier along with the development of the disease. The annual average incidence rate of hepatitis B of the middle-aged and old people is higher than that of other age groups, mainly because the middle-aged and old people are weak and sick to cause low immune function and are easy to attack after being infected with hepatitis B virus. Meanwhile, the elderly are easy to find by screening because the hospitalization rate of other various chronic diseases is much higher than that of other people. Secondly, for people with low immune function, such as kidney transplantation, tumor, leukemia, AIDS, hemodialysis or other people with viral infection with liver history, the hepatitis B patients are easy to develop, and the recovery effect after healing is poor. In addition, men are more likely to develop diseases because they have a relatively large number of smoking and drinking, heavy liver burden, wide social contact area, and large exposure infectivity, compared to women. It is worth mentioning that the most professions of hepatitis B are farmers, housekeeping staff and workers, which may be related to the low education level, poor health knowledge, bad living habits and lack of awareness of prevention for these people. In conclusion, the tissue development of targeted hepatitis B vaccine population preventive vaccination is of great practical significance.
Chinese patent application No. 202011144668.5 discloses a method for quantitatively screening and evaluating the risk of a vaccination weak area, which comprises the following steps: step 1: dividing the vaccination children of suitable age into different age groups; step 2: determining the types of corresponding doses of the vaccination of different age groups; and step 3: collecting data, and counting the inoculation rate of the corresponding dose of the vaccine in the district of the previous year; and 4, step 4: quantitatively evaluating a risk value based on the inoculation rate of the corresponding dose of the vaccine; calculating a comprehensive risk value in the district to realize quantitative evaluation of the safety risk; and 5: dividing weak types from different jurisdictions according to the risk values; step 6: and then, classifying and guiding the work of the current year according to different weak types and checking. The invention is more objective, high in comparability and strong in repeatability, has a guiding effect on immunization work, and can realize effective monitoring by regional guidance and reduce the workload. However, the patent application also has the following disadvantages:
1) the method is based on the situation of vaccination and prevention of children of the right age, the situation of sufficient immune resources is considered, but the method is not suitable for various complex practical situations that when the vaccinated population becomes complex and is not limited to children, and when the vaccine resources are in short supply and the full coverage of the population cannot be realized, the vaccination only can reduce the infection risk but can not realize lifetime immunity, and the like.
2) The risk level is evaluated according to the inoculation rate, and the evaluation index is single and has no comprehensiveness. Second, economic benefits are not measured in terms of cost of investment and expected effectiveness, and social losses and medical burdens are directly or indirectly increased.
Chinese patent application No. 202011072093.0 discloses a vaccination monitoring method and system, the method comprising: obtaining vaccination registration information; the vaccination registration information includes registration information of the reserved recipient; calling corresponding target recipients in sequence according to the vaccination registration information, and automatically popping up corresponding vaccines when preset conditions are met; and after the fact that the vaccine meets the inoculation requirement and is matched with the registration information of the target seed receiver is determined, uploading the electronic supervision code of the vaccine injected into the target seed receiver to a server. The invention relates the information of the vaccine in the whole period, thereby effectively and reliably storing the vaccine while effectively monitoring. However, the patent application also has the following disadvantages:
1) the monitoring method used in the present invention is based on the registered information of the recipient, but since the information of the population is not considered due to the limitation of economic conditions, lack of preventive consciousness, missed diagnosis and the like, the spread of the virus cannot be effectively controlled by means of vaccination.
2) The invention provides a reservation-inoculation mode, which is equivalent to the situation that only the vaccine distribution condition under the condition of sufficient immune resources is considered, and is not suitable for the situations that the immune resources are limited or the epidemic spread needs to be effectively controlled in a short time, and the like.
In conclusion, the distribution method of the hepatitis B vaccine in the prior art has limited application range, single evaluation index, no consideration of heterogeneity of population, no effective control of hepatitis B virus propagation, and is not suitable for situations that the immune resource is limited or epidemic spread needs to be effectively controlled in a short time.
Disclosure of Invention
The invention aims to solve the technical problems that the prior hepatitis B vaccine distribution method has limited application range, single evaluation index, can not effectively control the spread of hepatitis B virus, and is not suitable for the situations that the immune resource is limited, or the epidemic situation is effectively controlled to spread in a short time or the immune resource is economically and reasonably distributed, and the like.
The invention solves the technical problems through the following technical means: a method for prioritized hepatitis b vaccine distribution based on population heterogeneity, the method comprising:
the method comprises the following steps: dividing the population into a plurality of groups according to the required measuring indexes according to the characteristics of the population;
step two: establishing a differential equation SVIR model of hepatitis B virus transmission based on heterogeneity under the condition of incomplete immunization of an vaccinee;
step three: calculating a disease-free balance point and an endemic balance point of the differential equation SVIR model;
step four: determining a hepatitis B virus propagation threshold value according to the disease-free balance point;
step five: determining the condition of spreading hepatitis B virus according to the disease-free balance point, the endemic balance point and the hepatitis B virus spreading threshold;
step six: determining the condition for extinguishing the hepatitis B virus based on the hepatitis B virus propagation threshold value;
step seven: according to the condition of hepatitis B virus propagation and the condition of hepatitis B virus extinction, the propagation conditions of hepatitis B viruses in different groups are judged, when the immune resources are limited, the groups in which the viruses have been propagated are preferentially considered, and vaccine distribution strategies are formulated in the groups.
The invention considers the actual background of incomplete immunity of hepatitis B vaccine, divides the population into a plurality of groups according to the required measurement indexes according to the characteristics of the population, has wide application range, predicts the conditions of hepatitis B virus propagation and the conditions of hepatitis B virus extinction and formulates the vaccine distribution strategy based on the conditions, evaluates the risk level by the inoculation rate instead of only, has diversified evaluation indexes and effectively controls the hepatitis B virus propagation, preferentially considers the groups in which the virus has propagated when the immune resources are limited, formulates the vaccine distribution strategy in the groups, can more economically delay the propagation of the hepatitis B virus, and is suitable for effectively controlling the propagation of epidemic situation or economically and reasonably allocating the immune resources and the like in a short time or in a limited immune resource.
Further, the first step comprises: in each group i (i is more than or equal to 0 and less than or equal to N), the crowd is divided into four chambers which are respectively as follows: number of susceptible population Si(t) number of vaccinated groups Vi(t) number of affected persons Ii(t), number of convalescent population Ri(t)。
Further, the second step comprises:
Figure BDA0003044046580000061
constructing a differential equation SVIR model based on hepatitis B virus transmission with heterogeneity under the condition of incomplete immunization of an vaccinee, and simplifying the differential equation SVIR model into the following form
Figure BDA0003044046580000062
Wherein, ΛiIndicates the inflow of neonate, beta, in group iijRepresenting susceptible population in group iAnd the disease transmission rate between the affected population in cluster j,
Figure BDA0003044046580000063
respectively representing the natural mortality of susceptible persons, vaccinators, infected persons and convalescent persons in the group i,
Figure BDA0003044046580000064
Figure BDA0003044046580000065
representing the ratio of successful vaccinations of susceptible persons in group i,. etaiIndicates the coverage of the susceptible person in group i, σiMultiplier, 1-sigma, representing the capacity of the vaccinee to contract disease in group iiIs an index used to measure the effectiveness of a vaccine, viIndicates the disease mortality, gamma, within cohort iiThe recovery rate of patients in cohort i is shown.
Further, the third step includes:
solving the following system of equations
Figure BDA0003044046580000071
When in use
Figure BDA0003044046580000072
i belongs to the N condition, the disease-free balance point of the differential equation SVIR model is solved as
Figure BDA0003044046580000073
Order to
Figure BDA0003044046580000074
Then there is no balance point of disease
Figure BDA0003044046580000075
When in use
Figure BDA0003044046580000076
When i belongs to N, solving the endemic balance point of the differential equation SVIR model
Figure BDA0003044046580000077
Wherein the content of the first and second substances,
Figure BDA0003044046580000078
representing the number of susceptible populations at the point of endemic balance,
Figure BDA0003044046580000079
representing the number of affected people at the point of endemic balance,
Figure BDA00030440465800000710
representing the number of vaccinated populations at the point of endemic balance,
Figure BDA00030440465800000711
representing a 3N-dimensional positive real space.
Further, the fourth step includes:
constructing a next generation reproduction matrix of
Figure BDA00030440465800000712
Wherein the content of the first and second substances,
Figure BDA00030440465800000713
threshold of transmission of hepatitis B virus
Figure BDA00030440465800000714
Is composed of
Figure BDA00030440465800000715
Where ρ (·) represents a spectrum radius, and σ (·) is a matrix eigenvalue.
Further, the fifth step includes:
according to the disease-free balance point, the endemic balance point and the hepatitis B virus propagation threshold value, the condition for obtaining the hepatitis B virus propagation is the hepatitis B virus propagation threshold value by utilizing a reverse-positive method and constructing a Lyapunov function to prove that the condition for obtaining the hepatitis B virus propagation is the hepatitis B virus propagation threshold value
Figure BDA0003044046580000081
Greater than 1.
Further, the fifth step further includes:
definition of
X＝{(S，I，V)：Si≥0，Ii≥0，Vi≥0，i∈N}，
Figure BDA0003044046580000082
Wherein the content of the first and second substances,
Figure BDA0003044046580000083
denotes a boundary, let Φ (t): x → X is the solution pattern of formula (2), i.e.,. phi. (t) (A)0) A (t), where Φ (t): x → X denotes a mapping of X to X, A0Representing the initial value of equation (2), A (t) represents the solution of equation (2), X is point dissipative with respect to Φ (t) since equation (2) is positive invariant and eventually bounded, as can be seen from the second equation of equation (2)
Figure BDA0003044046580000084
If Ii(0) > 0, then
Figure BDA0003044046580000085
For Φ (t), X0Is positive and constant, so
Figure BDA00030440465800000815
Is relatively closed in X, it is apparent that equation (2) has a globally asymptotically stable equilibrium point
Figure BDA0003044046580000086
Thus the disease-free balance point E0In that
Figure BDA0003044046580000087
Is the global attractor of phi (t),
order to
Figure BDA0003044046580000088
I (t) represents a vector (I)1(t)，I2(t)，…IN(t))T，
Then
Figure BDA0003044046580000089
Suppose passing A0∈X0Satisfies the solution A (t) for the normal number ε2Is provided with
Figure BDA00030440465800000810
Wherein lim inf represents the lower limit,
with the counter-syndrome method, assuming the conclusion is false, there is a certain time T1> 0 such that
Figure BDA00030440465800000811
For all T ≧ T1Is formed by
Figure BDA00030440465800000812
There is a sufficiently small constant epsilon3So that
Figure BDA00030440465800000813
Wherein
Figure BDA00030440465800000814
()N×NRepresenting a matrix of dimensions N x N,
if it is not
Figure BDA0003044046580000091
Wherein, WsIndicates the disease-free balance point E0The flow pattern of the water-soluble polymer (A),
Figure BDA0003044046580000092
representing an empty set, n represents an intersection symbol, then equation (2) has an initial value at X0Solution (S) of (1)i(t)，Ii(t)，Vi(t)), i ∈ N is such that when t → ∞ there are
Figure BDA0003044046580000093
i e N holds, then there is a certain time T2> 0 such that T ≧ T2When there is
Figure BDA0003044046580000094
And Ii(t)＜ε2I ∈ N holds, as can be seen from the second equation of equation (2)
Figure BDA0003044046580000095
Because of (beta)ij)1≤i，j≤NIs irreducible, so
Figure BDA0003044046580000096
Is also irreducible, let (α)1，α2，…，αN) Is that
Figure BDA0003044046580000097
Corresponding to the radius of the spectrum
Figure BDA0003044046580000098
Positive left eigenvector of, satisfy
Figure BDA0003044046580000099
The Lyapunov function is defined as follows:
Figure BDA00030440465800000910
wherein alpha isiIs represented by (alpha)1，α2，…，αN) The i term in (1) calculates the derivative of L (T), obtained by equation (4), for all T ≧ T2Is provided with
Figure BDA00030440465800000911
It is true that T ≧ T is all2L (T) is not less than L (T)2) > 0, which is in accordance with
Figure BDA00030440465800000912
Conflict, therefore
Figure BDA0003044046580000101
From the above-mentioned demonstration, the disease-free balance point E0In X is an isolated invariant set, and
Figure BDA0003044046580000102
it is clear that,
Figure BDA0003044046580000103
the solution orbits of any of (1) converge to the disease-free equilibrium point E0And no disease balance point E0In that
Figure BDA0003044046580000104
Is acyclic, so equation (2) relates to
Figure BDA0003044046580000105
Is consistently sustained and has at least one point of endemic balance E*At this time I*>>0；
Therefore if hepatitis B virus transmission threshold
Figure BDA0003044046580000106
If the number of the hepatitis B virus particles is more than 1, the hepatitis B virus particles exist and spread in the group I all the time, and the infected patients are finally stabilized in a certain balance state, namely I*。
Furthermore, the condition for inactivating the hepatitis B virus in the sixth step is the hepatitis B virus transmission threshold value
Figure BDA0003044046580000107
Less than 1.
Still further, the sixth step includes:
let S (t) represent the vector (S)1(t)，S2(t)，…SN(t))TDue to (beta)ij)1≤i，j≤NIs irreducible, then M is also irreducible, so there is a constant ωi> 0, i ∈ N such that
(ω1，ω2，…，ωN)ρ(M)＝(ω1，ω2，…，ωN)M
Order to
Figure BDA0003044046580000108
Then there is
Figure BDA0003044046580000109
If ρ (M) < 1, then V'DFE0 if and only if i (t) is 0;
if ρ (M) ═ 1, then there is V'DFEIf 0 is true, it indicates that
Figure BDA0003044046580000118
If it is not
Figure BDA0003044046580000111
Then
Figure BDA0003044046580000112
Then equation (6) has only one trivial solution i (t) ═ 0, then V'DFE0 if and only if i (t) is 0 or ρ (M) ≦ 1,
Figure BDA0003044046580000113
of the set is the only invariant compact subset V'DFE0 is a single point set { E }0And according to the Lasell invariant principle, if rho (M) is less than or equal to1Then no disease balance point E0Is stable in the process of approaching to the target,
therefore if hepatitis B virus transmission threshold
Figure BDA0003044046580000114
Less than 1, the hepatitis B virus cannot be spread on a large scale, and the number of infected persons will decrease and eventually tend to 0, i.e., in this case, the hepatitis B virus will eventually disappear.
Still further, the seventh step includes:
by the formula
Figure BDA0003044046580000115
Obtaining N groups at tfThe total amount of immune control resources s required in a day,
by the formula
Figure BDA0003044046580000116
Obtaining daily allocatable immune control resources s*Wherein the number of susceptible persons of group 1S 1, the number of susceptible persons of group 2S 2, and the number of susceptible persons of group 3S 3 are known, then
Inoculum coverage η in cohort 33Is shown as
Figure BDA0003044046580000117
Substituting into equation (2) to solve the end time tfNumber of patients with lower infection I1(tf)、I2(tf)、I3(tf) I.e. the number of patients who finally get infected with the disease, and adjusting eta1，η2If the number of patients is the minimum, the regulation is finished and the final eta is obtained1，η2Calculating N groups at tfThe total amount of immune control resources required within a day and ultimately the allocable immune control resources for each day.
The invention has the advantages that: the invention considers the actual background of incomplete immunity of hepatitis B vaccine, divides the population into a plurality of groups according to the required measurement indexes according to the characteristics of the population, has wide application range, predicts the conditions of hepatitis B virus propagation and the conditions of hepatitis B virus extinction and formulates the vaccine distribution strategy based on the conditions, evaluates the risk level by the inoculation rate instead of only, has diversified evaluation indexes and effectively controls the hepatitis B virus propagation, preferentially considers the groups in which the virus has propagated when the immune resources are limited, formulates the vaccine distribution strategy in the groups, can more economically delay the propagation of the hepatitis B virus, and is suitable for effectively controlling the propagation of epidemic situation or economically and reasonably allocating the immune resources and the like in a short time or in a limited immune resource.
Drawings
FIG. 1 is a flow chart of a method for assigning a prioritized hepatitis B vaccine based on population heterogeneity according to an embodiment of the present invention;
fig. 2 is a diagram of numerical simulations of different distribution strategies for a prioritized hepatitis b vaccine distribution method based on population heterogeneity according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the embodiments of the present invention, and it is obvious that the described embodiments are some embodiments of the present invention, but not all embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
As shown in fig. 1, a method for prioritized hepatitis b vaccine distribution based on population heterogeneity, the method comprising:
step 1: dividing the population into a plurality of groups according to the required measuring indexes according to the characteristics of the population; the specific process is as follows: in each group i (i is more than or equal to 0 and less than or equal to N), the crowd is divided into four chambers which are respectively as follows: number of susceptible population Si(t) number of vaccinated groups Vi(t) number of affected persons Ii(t), number of convalescent population Ri(t)。
Step 2: establishing a differential equation SVIR model of hepatitis B virus transmission based on heterogeneity under the condition of incomplete immunization of an vaccinee; the specific process is as follows:
Figure BDA0003044046580000131
constructing a differential equation SVIR model based on hepatitis B virus transmission with heterogeneity under the condition of incomplete immunization of an vaccinee, and simplifying the differential equation SVIR model into the following form
Figure BDA0003044046580000132
Wherein, ΛiIndicates the inflow of neonate, beta, in group iijRepresenting the disease transmission rate between the susceptible population in cohort i and the affected population in cohort j,
Figure BDA0003044046580000133
respectively representing the natural mortality of susceptible persons, vaccinators, infected persons and convalescent persons in the group i,
Figure BDA0003044046580000134
representing the ratio of successful vaccinations of susceptible persons in group i,. etaiIndicates the coverage of the susceptible person in group i, σiMultiplier, 1-sigma, representing the capacity of the vaccinee to contract disease in group iiIs an index used to measure the effectiveness of a vaccine, viIs shown in group iOf the cause of disease mortality, gammaiThe recovery rate of patients in cohort i is shown.
Step 3: calculating a disease-free balance point and an endemic balance point of the differential equation SVIR model; the specific process is as follows:
solving the following system of equations
Figure BDA0003044046580000141
When in use
Figure BDA0003044046580000142
i belongs to the N condition, the disease-free balance point of the differential equation SVIR model is solved as
Figure BDA0003044046580000143
Order to
Figure BDA0003044046580000144
Then there is no balance point of disease
Figure BDA0003044046580000145
When in use
Figure BDA0003044046580000146
When i belongs to N, solving the endemic balance point of the differential equation SVIR model
Figure BDA0003044046580000147
Wherein the content of the first and second substances,
Figure BDA0003044046580000148
representing the number of susceptible populations at the point of endemic balance,
Figure BDA0003044046580000149
representing the number of affected people at the point of endemic balance,
Figure BDA00030440465800001410
representing the number of vaccinated populations at the point of endemic balance,
Figure BDA00030440465800001411
representing a 3N-dimensional positive real space.
Step 4: determining a hepatitis B virus propagation threshold value according to the disease-free balance point; the specific process is as follows:
constructing a next generation reproduction matrix of
Figure BDA00030440465800001412
Wherein the content of the first and second substances,
Figure BDA00030440465800001413
threshold of transmission of hepatitis B virus
Figure BDA00030440465800001414
Is composed of
Figure BDA00030440465800001415
Where ρ (·) represents a spectrum radius, and σ (·) is a matrix eigenvalue.
Step 5: determining the condition of spreading hepatitis B virus according to the disease-free balance point, the endemic balance point and the hepatitis B virus spreading threshold; the specific process is as follows:
according to the disease-free balance point, the endemic balance point and the hepatitis B virus propagation threshold value, the condition for obtaining the hepatitis B virus propagation is the hepatitis B virus propagation threshold value by utilizing a reverse-positive method and constructing a Lyapunov function to prove that the condition for obtaining the hepatitis B virus propagation is the hepatitis B virus propagation threshold value
Figure BDA0003044046580000151
Greater than 1. First, define
X＝{(S，I，V)：Si≥0，Ii≥0，Vi≥0，i∈N}，
Figure BDA0003044046580000152
Wherein the content of the first and second substances,
Figure BDA0003044046580000153
denotes a boundary, let Φ (t): x → X is the solution pattern of formula (2), i.e.,. phi. (t) (A)0) A (t), where Φ (t): x → X denotes a mapping of X to X, A0Representing the initial value of equation (2), A (t) represents the solution of equation (2), X is point dissipative with respect to Φ (t) since equation (2) is positive invariant and eventually bounded, as can be seen from the second equation of equation (2)
Figure BDA0003044046580000154
If Ii(0) > 0, then
Figure BDA0003044046580000155
For Φ (t), X0Is positive and constant, so
Figure BDA0003044046580000156
Is relatively closed in X, it is apparent that equation (2) has a globally asymptotically stable equilibrium point
Figure BDA0003044046580000157
Thus the disease-free balance point E0In that
Figure BDA0003044046580000158
Is the global attractor of phi (t),
order to
Figure BDA0003044046580000159
I (t) represents a vector (I)1(t)，I2(t)，…IN(t))T，
Then
Figure BDA00030440465800001510
Lower certificate
Figure BDA00030440465800001511
This is true. The reciprocal method, if the conclusion is not true, there is t0Not less than 0 such that I (t)0) Is greater than 0. Divide {1, 2, …, n } into two groups Z1And Z2So that
Ii(t0)＝0，
Figure BDA00030440465800001512
And is
Figure BDA00030440465800001513
When in use
Figure BDA00030440465800001514
It is easy to verify for epsilon0Greater than 0, sufficiently small with Ii(t) > 0 for t0＜t＜t0+ε0，
Figure BDA0003044046580000161
This is true. Suppose that
Figure BDA0003044046580000162
Is composed of I (t)0) Greater than 0 to know
Figure BDA0003044046580000163
Then for
Figure BDA0003044046580000164
Figure BDA0003044046580000165
Because of (beta)ij)1≤i，j≤nIs irreducible, so j exists1，j2，…，jnAnd j is1＝j，jnI is such that
Figure BDA0003044046580000166
Then there are
Figure BDA0003044046580000167
This is true. Thus, there is ∈0> 0 such that Ii(t0) 0 for t0＜t＜t0+ε0This is true. If epsilon 00 is small enough to be of interest for
Figure BDA0003044046580000168
Has Ii(t)＞0，t0＜t＜t0+ε0Is established, then has Ii(t0)＞0，i∈N，t0＜t＜t0+ε0This is true. Using I in the system (2)iEquation to obtain Ii(t0) > 0 or Ii(t0) Is equal to 0 and
Figure BDA0003044046580000169
it holds that for both cases, at ε1When more than 0 hour, all have Ii(t)＞0，t0＜t＜t0+ε1，
Figure BDA00030440465800001610
This is true. Then when t is0＜t＜t0+min{ε0，ε1At this time, there are
Figure BDA00030440465800001611
Is established by
Figure BDA00030440465800001612
The assumptions of (c) contradict. Therefore, when i (t) is 0,
Figure BDA00030440465800001613
as described above with reference to
Figure BDA00030440465800001621
The conclusion of (2) is true.
Lower syndrome, passing through A0∈X0Solution A (t) of (A) satisfies for positiveConstant epsilon2Is provided with
Figure BDA00030440465800001614
Wherein lim inf represents the lower limit,
with the counter-syndrome method, assuming the conclusion is false, there is a certain time T1> 0 such that
Figure BDA00030440465800001615
For all T ≧ T1Is formed by
Figure BDA00030440465800001616
There is a sufficiently small constant epsilon3So that
Figure BDA00030440465800001617
Wherein
Figure BDA00030440465800001618
()N×NRepresenting a matrix of dimensions N x N,
if it is not
Figure BDA00030440465800001619
Wherein, WsIndicates the disease-free balance point E0The flow pattern of the water-soluble polymer (A),
Figure BDA00030440465800001620
representing an empty set, n represents an intersection symbol, then equation (2) has an initial value at X0Solution (S) of (1)i(t)，Ii(t)，Vi(t)), i ∈ N is such that when t → ∞ there are
Figure BDA0003044046580000171
i e N holds, then there is a certain time T2> 0 such that T ≧ T2When there is
Figure BDA0003044046580000172
And Ii(t)＜ε2I ∈ N holds, as can be seen from the second equation of equation (2)
Figure BDA0003044046580000173
Because of (beta)ij)1≤i，j≤NIs irreducible, so
Figure BDA0003044046580000174
Is also irreducible, let (α)1，α2，…，αN) Is that
Figure BDA0003044046580000175
Corresponding to the radius of the spectrum
Figure BDA0003044046580000176
Positive left eigenvector of, satisfy
Figure BDA0003044046580000177
The Lyapunov function is defined as follows:
Figure BDA0003044046580000178
wherein alpha isiIs represented by (alpha)1，α2，…，αN) The i term in (1) calculates the derivative of L (T), obtained by equation (4), for all T ≧ T2Is provided with
Figure BDA0003044046580000179
It is true that T ≧ T is all2L (T) ≧ L (T2) > 0, which is true
Figure BDA00030440465800001710
Conflict, therefore
Figure BDA00030440465800001711
From the above-mentioned demonstration, the disease-free balance point E0In X is an isolated invariant set, and
Figure BDA0003044046580000181
it is clear that,
Figure BDA0003044046580000182
the solution orbits of any of (1) converge to the disease-free equilibrium point E0And no disease balance point E0In that
Figure BDA0003044046580000183
Is acyclic, so equation (2) relates to
Figure BDA0003044046580000184
Is consistently sustained and has at least one point of endemic balance E*At this time I*>>0；
Therefore if hepatitis B virus transmission threshold
Figure BDA0003044046580000185
If the number of the hepatitis B virus particles is more than 1, the hepatitis B virus particles exist and spread in the group I all the time, and the infected patients are finally stabilized in a certain balance state, namely I*。
Step 6: determining the condition for extinguishing the hepatitis B virus based on the hepatitis B virus propagation threshold value; the specific process is as follows:
let S (t) represent the vector (S)1(t)，S2(t)，…SN(t))TDue to (beta)ij)1≤i，j≤NIs irreducible, then M is also irreducible, so there is a constant ωi> 0, i ∈ N such that
(ω1，ω2，…，ωN)ρ(M)＝(ω1，ω2，…，ωN)M
Order to
Figure BDA0003044046580000186
Then there is
Figure BDA0003044046580000187
If ρ (M) < 1, then V'DFE0 if and only if i (t) is 0;
if ρ (M) ═ 1, then there is V'DFEIf 0 is true, it indicates that
Figure BDA0003044046580000188
If it is not
Figure BDA0003044046580000189
Then
Figure BDA0003044046580000191
Then equation (6) has only one trivial solution i (t) ═ 0, then V'DFE0 if and only if i (t) is 0 or ρ (M) ≦ 1,
Figure BDA0003044046580000192
of the set is the only invariant compact subset V'DFE0 is a single point set { E }0By the Lasell invariant principle, if rho (M) is less than or equal to 1, then the disease-free equilibrium point E0Is stable in the process of approaching to the target,
therefore if hepatitis B virus transmission threshold
Figure BDA0003044046580000193
Less than 1, so the hepatitis B virus can not be spread on a large scale, and the number of infected patients can be reduced and finally tends to 0, namely under the condition, the hepatitis B virus finally disappears, so the condition that the hepatitis B virus is extinct is the hepatitis B virus spreading threshold value
Figure BDA0003044046580000194
Less than 1.
Step 7: according to the condition of hepatitis B virus propagation and the condition of hepatitis B virus extinction, the propagation conditions of hepatitis B viruses in different groups are judged, when the immune resources are limited, the groups in which the viruses have been propagated are preferentially considered, and vaccine distribution strategies are formulated in the groups.
Appropriate vaccine distribution strategies are developed according to cost-benefit maximization principles. In the process, the conditions for spreading and extinguishing the hepatitis B virus in the group i are obtained, so that the propagation conditions of the hepatitis B virus in different groups are judged, and the immune resources are reasonably distributed. When the immune resources are limited, the groups where the virus has spread need to be considered preferentially, and an appropriate vaccine distribution strategy is established in the groups, and the groups where the virus does not spread need to be considered temporarily. When the immune resources are sufficient, the virus is properly distributed by considering the actual situation of virus propagation in each group. In the invention, the social benefit is calculated by calculating the number of final infected persons which can be reduced by each unit of immune control resources, so that the advantages and disadvantages of the distribution strategy are measured, and the strategy control effect is better when the number of final infected persons is less. Considering that the population has heterogeneity and has different infection capacity to hepatitis B virus, the vaccination coverage rate of susceptible people in different groups is different according to the characteristics, and a plurality of vaccine distribution strategies are found. And analyzing social benefits brought by different distribution strategies by researching and comparing prevention costs under different distribution strategies, and determining a proper distribution strategy.
Setting the inoculation coverage η of susceptible subjects in each cohorti∈[0，1]Constant, each susceptible person requires one unit of immune control resource per day, so the daily immune control resource required for each group can be expressed as ηi·S i1. Thus N groups are at tfThe total amount of immune control resources required in a day s can be expressed as
Figure BDA0003044046580000201
To clearly describe the allocation strategy, consider the case when the group size is N-3. The grade of infectivity of cohorts 1 to 3 was assumed to decrease from strong to weak; the daily neonatal inflow in groups 1 to 3 was Λ1＝270、Λ2＝879、Λ3661; required days tf15; the disease transmission rates between susceptible individuals in the group and affected individuals inside and outside the group are shown as
β11＝4.64×10-6、β12＝2.23×10-6、β13＝1.51×10-6、
β21＝5.74×10-7、β22＝5.74×10-7、β23＝3.93×10-7、
β31＝1.26×10-7、β32＝3.36×10-8、β33＝2.10×10-8；
Disease mortality in cohorts vi(i-1, 2, 3) in the range of [0, 0.2 ]](ii) a The natural mortality rates of susceptible, vaccinated, infected and rehabilitated individuals in the cohort
Figure BDA0003044046580000202
0.00029; ratio of successful vaccinations of susceptible persons in cohorts
Figure BDA0003044046580000203
The variation range is [0, 1%](ii) a Recovery rate of patients in cohorts γi(i ═ 1, 2, 3) 0.024318; effective disease transmission between vaccinees in the cohort and those with disease both within and outside the cohort
Figure BDA0003044046580000204
Is composed of
Figure BDA0003044046580000205
Figure BDA0003044046580000206
Figure BDA0003044046580000207
The initial values of the susceptible are S1(0)＝119500、S2(0)＝274900、S3(0)＝859330；
The initial values of the patients are I1(0)＝60、I2(0)＝35、I3(0)＝150；
Initial value of vaccinee is V1(0)＝140、V2(0)＝150、V3(0)＝100；
The initial value of the rehabilitative is R1(t)＝0、R2(t)＝0、R3(t)＝0；
Taking the total amount s of immune resources as s respectively1＝12、s2＝17.2434、s3＝25、s4＝35、s5＝45、s6＝55；η1，η2Is [0, 1 ]]Is constant.
The number of final patients to be studied with respect to the constant coverage η1，η2A change in (c). Because immune control resources are limited, when the total amount of immune resources s is taken and the number of days t is givenfLater, at this point in time, daily allocatable immune control resources s*The approximation is:
Figure BDA0003044046580000211
wherein the number of susceptible population in cohort 1S 1, the number of susceptible population in cohort 2S 2, and the number of susceptible population in cohort 3S 3 are known, the inoculation coverage η in cohort 3 is η3Is shown as
Figure BDA0003044046580000212
Substituting into equation (2) to solve the end time tfNumber of patients with lower infection I1(tf)、I2(tf)、I3(tf) The number of the patients who finally get ill is the following reasonη1，η2Is at [0, 1 ]]The total amount of immune resources can be regulated and controlled according to actual conditions, so that the multi-group distribution strategy is obtained by repeated calculation according to the method. In order to observe and compare the superiority of the different strategies, a numerical simulation is given here in fig. 2.
In order to simulate the situation that the total quantity of the immune control resources is abundant and in short supply according to the actual situation, the value of s is also selected. As can be seen from FIG. 2, the greater η is given when the total amount of immunoprophylaxis resources is less than s < 17.24341Is the best choice. This means that a distribution strategy for multiple vaccination to high risk groups would be very beneficial in controlling disease transmission and reducing the number of final patients. When the immune prevention and treatment resources are moderate and s is less than 45, eta1，η2The value of (a) is very specific, and can be neither too large nor too small, and an optimal value exists. The optimal value can reduce the number of final patients to the maximum extent, however, the theory is not suitable for the situation that the resources are relatively sufficient. If the immune control resource is sufficient s > 45, e.g. 55, eta1，η2The larger the maximum possible, the better the prevention and control effect, and the maximum value eta1＝0.75，η2When 1, the final infection is minimized. This result is in accordance with common sense of life, when η1，η2As large as possible, this means that vaccination of as many people as possible best suppresses the spread of hepatitis b virus, which is the most effective vaccine distribution strategy.
From the above discussion, the vaccine allocation strategy formulated according to the cost-benefit maximization principle depends mainly on the size of the total amount s of the immune control resources. When the resource shortage of immune prevention and treatment resources exists, the vaccine distribution strategy is to preferentially inoculate people with high infection risk level, such as middle-aged and old people, people with low immune function, infants and the like; when the immune prevention and control resources are moderate, the optimal distribution strategy exists, the number of final infected persons can be reduced to the maximum extent, but the corresponding coverage rate eta needs to be determined by calculation according to actual conditions1，η2，η3The size of (d); when the immune prevention and treatment resources are sufficient, the immune prevention and treatment method is suitable for all peopleIt is the most effective distribution strategy to achieve full population coverage with as much vaccination as possible. Therefore, in this case, it is necessary to improve the coverage of hepatitis B vaccine by active preventive measures in order to eradicate the spread of hepatitis B virus.
Through the technical scheme, the actual background of incomplete immunity of the hepatitis B vaccine is considered, the crowd is divided into a plurality of groups according to the required measurement indexes according to the characteristics of the population, the application range is wide, the conditions for spreading the hepatitis B virus and the conditions for extinction of the hepatitis B virus are predicted, the vaccine distribution strategy is formulated based on the conditions, the risk level is not evaluated only by the inoculation rate, the evaluation indexes are diversified, the spreading of the hepatitis B virus is effectively controlled, when the immune resources are limited, the groups in which the virus is spread are preferentially considered, the vaccine distribution strategy is formulated in the groups, the spreading of the hepatitis B virus can be more economically delayed, and the method is suitable for effectively controlling the spreading of the epidemic situation or economically and reasonably distributing the immune resources in a short time or in a limited immune resource.
The above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.

Claims (10)

1. A method for assigning a prioritized hepatitis b vaccine based on population heterogeneity, the method comprising:
the method comprises the following steps: dividing the population into a plurality of groups according to the required measuring indexes according to the characteristics of the population;
step two: establishing a differential equation SVIR model of hepatitis B virus transmission based on heterogeneity under the condition of incomplete immunization of an vaccinee;
step three: calculating a disease-free balance point and an endemic balance point of the differential equation SVIR model;
step four: determining a hepatitis B virus propagation threshold value according to the disease-free balance point;
step five: determining the condition of spreading hepatitis B virus according to the disease-free balance point, the endemic balance point and the hepatitis B virus spreading threshold;
step six: determining the condition for extinguishing the hepatitis B virus based on the hepatitis B virus propagation threshold value;
step seven: according to the condition of hepatitis B virus propagation and the condition of hepatitis B virus extinction, the propagation conditions of hepatitis B viruses in different groups are judged, when the immune resources are limited, the groups in which the viruses have been propagated are preferentially considered, and vaccine distribution strategies are formulated in the groups.
2. The method for distributing hepatitis B vaccine with priority based on human population heterogeneity according to claim 1, wherein said step one comprises: in each group i (i is more than or equal to 0 and less than or equal to N), the crowd is divided into four chambers which are respectively as follows: number of susceptible population Si(t) number of vaccinated groups Vi(t) number of affected persons Ii(t), number of convalescent population Ri(t)。
3. The method for distributing hepatitis B vaccine with priority based on human population heterogeneity according to claim 2, wherein said step two comprises:
by the formula
Figure FDA0003044046570000021
Constructing a differential equation SVIR model based on hepatitis B virus transmission with heterogeneity under the condition of incomplete immunization of an vaccinee, and simplifying the differential equation SVIR model into the following form
Figure FDA0003044046570000022
Wherein, ΛiIndicates the inflow of neonate, beta, in group iijRepresenting the disease transmission rate between the susceptible population in cohort i and the affected population in cohort j,
Figure FDA0003044046570000023
respectively representing the natural mortality of susceptible persons, vaccinators, infected persons and convalescent persons in the group i,
Figure FDA0003044046570000024
Figure FDA0003044046570000025
representing the ratio of successful vaccinations of susceptible persons in group i,. etaiIndicates the coverage of the susceptible person in group i, σiMultiplier, 1-sigma, representing the capacity of the vaccinee to contract disease in group iiIs an index used to measure the effectiveness of a vaccine, viIndicates the disease mortality, gamma, within cohort iiThe recovery rate of patients in cohort i is shown.
4. The method for distributing hepatitis B vaccine with priority based on human population heterogeneity according to claim 3, wherein said step three comprises:
solving the following system of equations
Figure FDA0003044046570000031
When in use
Figure FDA0003044046570000032
In the case of (1), the disease-free balance point of the differential equation SVIR model is solved as
Figure FDA0003044046570000033
Order to
Figure FDA0003044046570000034
Then there is no balance point of disease
Figure FDA0003044046570000035
When in use
Figure FDA0003044046570000036
Then, the endemic balance point of the differential equation SVIR model is solved
Figure FDA0003044046570000037
Wherein S isi *Representing the number of susceptible populations at the point of endemic equilibrium, Ii *Representing the number of affected persons at the point of endemic balance, Vi *Representing the number of vaccinated populations at the point of endemic balance,
Figure FDA0003044046570000038
representing a 3N-dimensional positive real space.
5. The method for distributing hepatitis B vaccine with priority based on human heterogeneity according to claim 4, wherein said step four comprises:
constructing a next generation reproduction matrix of
Figure FDA0003044046570000039
Wherein the content of the first and second substances,
Figure FDA00030440465700000310
threshold of transmission of hepatitis B virus
Figure FDA00030440465700000311
Is composed of
Figure FDA00030440465700000312
Where ρ (·) represents a spectrum radius, and σ (·) is a matrix eigenvalue.
6. The method for distributing hepatitis B vaccine with priority based on human population heterogeneity according to claim 5, wherein said step five comprises:
according to the disease-free balance point, the endemic balance point and the hepatitis B virus propagation threshold value, the condition for obtaining the hepatitis B virus propagation is the hepatitis B virus propagation threshold value by utilizing a reverse-positive method and constructing a Lyapunov function to prove that the condition for obtaining the hepatitis B virus propagation is the hepatitis B virus propagation threshold value
Figure FDA0003044046570000041
Greater than 1.
7. The method for distributing hepatitis B vaccine with priority based on human population heterogeneity according to claim 6, wherein said step five further comprises:
definition of
X＝{(S，I，V)：Si≥0，Ii≥0，Vi≥0，i∈N}，
Figure FDA0003044046570000042
Wherein the content of the first and second substances,
Figure FDA0003044046570000043
denotes a boundary, let Φ (t): x → X is the solution pattern of formula (2), i.e.,. phi. (t) (A)0) A (t), where Φ (t): x → X denotes a mapping of X to X, A0Representing the initial value of equation (2), A (t) represents the solution of equation (2), X is point dissipative with respect to Φ (t) since equation (2) is positive invariant and eventually bounded, as can be seen from the second equation of equation (2)
Figure FDA0003044046570000044
If Ii(0) > 0, then
Figure FDA0003044046570000045
For Φ (t), X0Is positive and constant, so
Figure FDA0003044046570000046
Is relatively closed in X, it is apparent that equation (2) has a globally asymptotically stable equilibrium point
Figure FDA0003044046570000047
Thus the disease-free balance point E0In that
Figure FDA0003044046570000048
Is the global attractor of phi (t),
order to
Figure FDA0003044046570000049
I (t) represents a vector (I)1(t)，I2(t)，…IN(t))T，
Then
Figure FDA00030440465700000410
Suppose passing A0∈X0Satisfies the solution A (t) for the normal number ε2Is provided with
Figure FDA00030440465700000411
Wherein lim inf represents the lower limit,
with the counter-syndrome method, assuming the conclusion is false, there is a certain time T1> 0 such that
Figure FDA0003044046570000051
For all T ≧ T1Is formed by
Figure FDA0003044046570000052
There is a sufficiently small constant epsilon3So that
Figure FDA0003044046570000053
Wherein
Figure FDA0003044046570000054
()N×NRepresenting a matrix of dimensions N x N,
if it is not
Figure FDA0003044046570000055
Wherein, WsIndicates the disease-free balance point E0The flow pattern of the water-soluble polymer (A),
Figure FDA0003044046570000056
representing an empty set, n represents an intersection symbol, then equation (2) has an initial value at X0Solution (S) of (1)i(t)，Ii(t)，Vi(t)), i ∈ N is such that when t → ∞ there are
Figure FDA0003044046570000057
If true, then there is a certain time T2> 0 such that T ≧ T2When there is
Figure FDA0003044046570000058
And Ii(t)＜ε2I ∈ N holds, as can be seen from the second equation of equation (2)
Figure FDA0003044046570000059
Because of (beta)ij)1≤i，j≤NIs irreducible, so
Figure FDA00030440465700000510
Is also irreducible, let (α)1，α2，…，αN) Is that
Figure FDA00030440465700000511
Corresponding to the radius of the spectrum
Figure FDA00030440465700000512
Positive left eigenvector of, satisfy
Figure FDA00030440465700000513
The Lyapunov function is defined as follows:
Figure FDA00030440465700000514
wherein alpha isiIs represented by (alpha)1，α2，…，αN) The i term in (1) calculates the derivative of L (T), obtained by equation (4), for all T ≧ T2Is provided with
Figure FDA0003044046570000061
It is true that T ≧ T is all2L (T) is not less than L (T)2) > 0, which is in accordance with
Figure FDA0003044046570000062
Conflict, therefore
Figure FDA0003044046570000063
From the above-mentioned demonstration, the disease-free balance point E0In X is an isolated invariant set, and
Figure FDA0003044046570000064
it is clear that,
Figure FDA0003044046570000065
the solution orbits of any of (1) converge to the disease-free equilibrium point E0And no disease balance point E0In that
Figure FDA0003044046570000066
Is acyclic, so equation (2) relates to
Figure FDA0003044046570000067
Is consistently sustained and has at least one point of endemic balance E*At this time I*＞＞0；
Therefore if hepatitis B virus transmission threshold
Figure FDA0003044046570000068
If the number of the hepatitis B virus particles is more than 1, the hepatitis B virus particles exist and spread in the group I all the time, and the infected patients are finally stabilized in a certain balance state, namely I*。
8. The population heterogeneity-based method for distributing hepatitis B vaccines with priority as claimed in claim 6, wherein the condition for hepatitis B virus extinction in step six is hepatitis B virus transmission threshold
Figure FDA0003044046570000069
Less than 1.
9. The method for distributing hepatitis B vaccine with priority based on human population heterogeneity according to claim 8, wherein said step six comprises:
let S (t) represent the vector (S)1(t)，S2(t)，…SN(t))TDue to (beta)ij)1≤i，j≤NIs irreducible, then M is also irreducible, so there is a constant ωi> 0, i ∈ N such that
(ω1，ω2，…，ωN)ρ(M)＝(ω1，ω2，…，ωN)M
Order to
Figure FDA0003044046570000071
Then there is
Figure FDA0003044046570000072
If ρ (M) < 1, then V'DFE0 if and only if i (t) is 0;
if ρ (M) ═ 1, then there is V'DFEIf 0 is true, it indicates that
Figure FDA0003044046570000073
If it is not
Figure FDA0003044046570000074
Then
Figure FDA0003044046570000075
Then equation (6) has only one trivial solution i (t) ═ 0, then V'DFE0 if and only if i (t) is 0 or ρ (M) ≦ 1,
Figure FDA0003044046570000076
of the set is the only invariant compact subset V'DFE0 is a single point set { E }0By the Lasell invariant principle, if rho (M) is less than or equal to 1, then the disease-free equilibrium point E0Is stable in the process of approaching to the target,
therefore if hepatitis B virus transmission threshold
Figure FDA0003044046570000077
Less than 1, the hepatitis B virus cannot be spread on a large scale, and the number of infected persons will decrease and eventually tend to 0, i.e., in this case, the hepatitis B virus will eventually disappear.
10. The method for distributing hepatitis B vaccine with priority based on human population heterogeneity according to claim 8, wherein said seventh step comprises:
by the formula
Figure FDA0003044046570000081
Obtaining N groups at tfThe total amount of immune control resources s required in a day,
by the formula
Figure FDA0003044046570000082
Obtaining daily allocatable immune control resources s*Wherein the number of susceptible persons of group 1S 1, the number of susceptible persons of group 2S 2, and the number of susceptible persons of group 3S 3 are known, then
Inoculum coverage η in cohort 33Is shown as
Figure FDA0003044046570000083
Substituting into equation (2) to solve the end time tfNumber of patients with lower infection I1(tf)、I2(tf)、I3(tf) I.e. the number of patients who finally get infected with the disease, and adjusting eta1，η2If the number of patients is the minimum, the regulation is finished and the final eta is obtained1，η2Calculating N groups at tfThe total amount of immune control resources required within a day and ultimately the allocable immune control resources for each day.
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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113409951A (en) * 2021-07-14 2021-09-17 医渡云（北京）技术有限公司 Method and device for calculating vaccination ratio, storage medium and electronic equipment
CN113539494A (en) * 2021-07-14 2021-10-22 医渡云（北京）技术有限公司 Method and device for calculating vaccination ratio, storage medium and electronic equipment
CN117253577A (en) * 2023-11-17 2023-12-19 四川省大数据技术服务中心 Multidimensional health data integrated processing system

Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111834015A (en) * 2020-07-16 2020-10-27 西北民族大学 Optimal vaccination strategy and screening strategy making method for HPV and cervical cancer caused by HPV
CN112382403A (en) * 2020-09-30 2021-02-19 哈尔滨工业大学 Method for constructing SEIQR infectious disease model based on personnel concentration

Patent Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111834015A (en) * 2020-07-16 2020-10-27 西北民族大学 Optimal vaccination strategy and screening strategy making method for HPV and cervical cancer caused by HPV
CN112382403A (en) * 2020-09-30 2021-02-19 哈尔滨工业大学 Method for constructing SEIQR infectious disease model based on personnel concentration

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
张贺婷: ""两组群传染病模型的动力学行为与最优控制"", 中国优秀硕士学位论文全文数据库基础科学辑》, pages 1 - 55 *

Cited By (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113409951A (en) * 2021-07-14 2021-09-17 医渡云（北京）技术有限公司 Method and device for calculating vaccination ratio, storage medium and electronic equipment
CN113539494A (en) * 2021-07-14 2021-10-22 医渡云（北京）技术有限公司 Method and device for calculating vaccination ratio, storage medium and electronic equipment
CN113409951B (en) * 2021-07-14 2023-08-18 医渡云（北京）技术有限公司 Vaccination proportion calculating method and device, storage medium and electronic equipment
CN113539494B (en) * 2021-07-14 2024-04-30 医渡云（北京）技术有限公司 Vaccination proportion calculating method and device, storage medium and electronic equipment
CN117253577A (en) * 2023-11-17 2023-12-19 四川省大数据技术服务中心 Multidimensional health data integrated processing system
CN117253577B (en) * 2023-11-17 2024-03-12 四川省大数据技术服务中心 Multidimensional health data integrated processing system

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