CN113450924B - Novel coronavirus propagation model establishing method and system - Google Patents
Novel coronavirus propagation model establishing method and system Download PDFInfo
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- CN113450924B CN113450924B CN202110562893.9A CN202110562893A CN113450924B CN 113450924 B CN113450924 B CN 113450924B CN 202110562893 A CN202110562893 A CN 202110562893A CN 113450924 B CN113450924 B CN 113450924B
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Abstract
The invention relates to a method and a system for establishing a novel coronavirus propagation model, wherein the method comprises the following steps: step S1: establishing a differential kinetic equation set according to the inoculation rate, the isolation period and the recovery rate to construct a new coronavirus transmission model; step S2: according to a differential kinetic equation set, acquiring a disease-free balance point and a disease balance point of a new coronavirus transmission model by utilizing kinetic modeling; and step S3: and (3) carrying out stability analysis on the new coronavirus transmission model, and determining conditions required for enabling the disease-free balance point and the disease balance point to achieve local stability by using the basic regeneration number. The invention provides a novel coronavirus propagation model, which simulates and analyzes the influence of an isolation period and a vaccination rate on a virus propagation process, proves that strict control measures have a better control effect on the propagation of new coronavirus, and verifies that the model has better fitting goodness by acquiring actual case data and fitting the model and the actual data.
Description
Technical Field
The invention relates to the field of virus propagation, in particular to a method and a system for establishing a novel coronavirus propagation model.
Background
Mathematical models are important tools for testing the spread and control of different infectious diseases. Recently, researchers have proposed many models to study the dynamics of new coronaviruses, but the epidemic situation remains constantly changing, and there is still a need for more comprehensive studies on the transmission of new coronaviruses.
Disclosure of Invention
In order to solve the above technical problems, the present invention provides a method and a system for establishing a novel coronavirus propagation model.
The technical solution of the invention is as follows: a novel coronavirus transmission model building method comprises the following steps:
step S1: establishing a differential kinetic equation set according to the inoculation rate, the isolation period and the recovery rate so as to construct a new coronavirus propagation model;
step S2: according to the differential kinetic equation set, utilizing kinetic modeling to obtain a disease-free balance point and a disease balance point of the new coronavirus transmission model;
and step S3: and performing stability analysis on the new coronavirus transmission model, and determining conditions required for locally stabilizing the disease-free balance point and the disease balance point by using a basic regeneration number.
Compared with the prior art, the invention has the following advantages:
based on the epidemic propagation theory, the invention considers that the isolation period and the vaccination rate of susceptible people can influence the virus propagation process, provides a novel coronavirus propagation model, simulates and analyzes the influence of the isolation period and the vaccination rate on the virus propagation process, proves that strict control measures have better control effect on the propagation of the new coronavirus, and verifies that the model has better fitting goodness by acquiring actual case data and fitting the model and the actual data.
Drawings
FIG. 1 is a flow chart of a method for establishing a novel coronavirus propagation model according to an embodiment of the present invention;
FIG. 2 is a diagram illustrating the state transition of various types of people in a model according to an embodiment of the present invention;
fig. 3 shows a step S2 in the method for establishing a novel coronavirus propagation model according to the embodiment of the present invention: according to a differential kinetic equation set, utilizing kinetic modeling to obtain a flow chart of disease-free balance points and disease balance points of a new coronavirus transmission model;
fig. 4 is a flowchart of a step S3 in the method for establishing a novel coronavirus propagation model according to the embodiment of the present invention: performing stability analysis on the new coronavirus propagation model, and determining a flow chart of conditions required for enabling a disease-free balance point and a disease balance point to achieve local stability by using a basic regeneration number;
FIG. 5 is a graph of density of various populations over time at a disease free balance point for the novel coronavirus transmission model in an embodiment of the invention;
FIG. 6 is a graph of the density of various populations of the novel coronavirus transmission model at the diseased balance point over time in an embodiment of the invention;
FIG. 7 is a graph of infected population density over time at different quarantine periods in an example of the invention;
FIG. 8 is a graph of the time course of the immunized population at different vaccination rates in an example of the invention;
fig. 9 is a block diagram of a system for establishing a novel coronavirus propagation model according to an embodiment of the present invention.
Detailed Description
The invention provides a novel coronavirus propagation model establishing method, which utilizes the novel coronavirus propagation model to simulate and analyze the influence of the isolation period and the vaccination rate on the virus propagation process, and proves that strict control measures have better control effect on the propagation of new coronavirus.
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings.
Example one
As shown in fig. 1, a method for establishing a novel coronavirus propagation model provided in an embodiment of the present invention includes the following steps:
step S1: establishing a differential kinetic equation set according to the inoculation rate, the isolation period and the recovery rate to construct a new coronavirus transmission model;
step S2: according to a differential kinetic equation set, acquiring a disease-free balance point and a disease balance point of a new coronavirus transmission model by utilizing kinetic modeling;
and step S3: and (3) carrying out stability analysis on the new coronavirus transmission model, and determining conditions required for enabling the disease-free balance point and the disease balance point to achieve local stability by using the basic regeneration number.
In one embodiment, the step S1: establishing a differential kinetic equation set according to the inoculation rate, the isolation period and the recovery rate, wherein the differential kinetic equation set specifically comprises the following steps:
wherein S (t), I (t) and R (t) respectively represent the densities of a susceptible person, an infected person and an immunized person at the time t, mu represents the migration rate and the migration rate of a user,
indicates the period of isolation, beta indicates the inoculation rate, and gamma indicates the recovery rate.
In the embodiment of the invention, the transmission rule of the new coronavirus is defined as follows:
(1) Dynamic changes of people are considered, and the dynamic changes are not propagated in a closed system;
(2) The population is divided into three categories, namely susceptible persons, infected persons and immunized persons;
(3) Considering the influence of the isolation time of susceptible people on the spread of new coronavirus, defining that in the process of converting susceptible people into infected people, the infection rate is reduced due to the increase of the isolation time;
(4) Considering the influence of the vaccination rate on the spread of new coronavirus, it is defined that the number of immunized people will increase due to the increase of vaccination rate in the process of transforming susceptible into immunized people.
Fig. 2 is a diagram illustrating state transition of various types of people in a model according to an embodiment of the present invention.
As shown in fig. 3, in one embodiment, the step S2: according to a differential kinetic equation set, by utilizing kinetic modeling, acquiring a disease-free balance point and a disease balance point of a new coronavirus transmission model, which specifically comprises the following steps:
step S21: according to the formula (1), calculating to obtain a disease-free balance point
Is denoted by P 1 ＝(S 1 ，I 1 ，R 1 )；
Step S22: according to the formula (1), calculating to obtain the balance point with diseases
Is denoted by P 2 ＝(S 2 ，I 2 ，R 2 )。
As shown in fig. 4, in one embodiment, the step S3: carrying out stability analysis on the new coronavirus transmission model, and determining conditions required for enabling a disease-free balance point and a disease balance point to achieve local stability by using a basic regeneration number, wherein the conditions specifically comprise the following steps:
step S31: calculating the basic regeneration number R by adopting a regeneration matrix spectrum radius method 0 ；
First, the novel coronavirus transmission model is represented by the following formula:
wherein,
secondly, the matrix
At P from the matrix psi 1 The jacobian matrices are:
wherein,
it is possible to obtain,
obtaining FV -1 Spectral radius of (2) is the basic regeneration number R 0 ：
Step S32: judging conditions through Route-Hurwitz, and judging according to R 0 And point of disease-free equilibrium P 1 ＝(S 1 ，I 1 ，R 1 ) It can be seen that the condition R is satisfied for the disease-free equilibrium point to be locally stable 0 ≤1；
First, according to the formula (1) and P 1 The jacobian matrix can be obtained as:
order to
The eigenvalues were found to be as follows:
λ 1 ＝-μ-γ
λ 2 ＝-μ
obtained according to the discrimination condition of Router-Hurwitz, when R is 0 When the number of the characteristic values is less than or equal to 1, the 3 characteristic values are all less than 0, namely all the characteristic values have negative real parts, and the disease-free balance point P is at the moment 1 Is locally stable;
step S33: judging conditions through Route-Hurwitz, and judging according to R 0 And a diseased equilibrium point P 2 ＝(S 2 ，I 2 ，R 2 ) It is known that the condition R is satisfied when the disease equilibrium point reaches local stability 0 >1；
According to the formula (1) and P 2 The jacobian matrix can be obtained as:
order to
The eigenvalues were found to be as follows:
λ 1 ＝-μ-γ
obtained according to the discrimination condition of Route-Hurwitz when R 0 >1, all the 3 eigenvalues are less than 0, namely all the eigenvalues have negative real parts, and the sick equilibrium point P is at the moment 2 Is locally stable.
By using the new coronavirus propagation model, the change of the density of various types of crowds at a disease-free balance point along with time is simulated and analyzed, and the result is shown in fig. 5.
By using the new coronavirus transmission model, the change of various populations at the diseased balance point along with time is simulated and analyzed, and the result is shown in fig. 6.
By utilizing the new coronavirus propagation model, the situation that the density of infected people changes along with time in different isolation periods is simulated and analyzed, and the result is shown in fig. 7, wherein fig. 7 shows the influence of the change of population isolation time on the number of infected people with the new coronavirus, and the number of infected people decreases along with the increase of the isolation time, which shows that the isolation has great influence on the change of the number of infected people with the new coronavirus.
By using the new coronavirus propagation model, the time-dependent change of the density of infected people at different vaccination rates was analyzed in a simulation manner, and the results are shown in fig. 8, wherein fig. 8 shows the influence of the change of the vaccination rate on the number of people infected with new coronavirus. It can be seen that the higher the vaccination rate, the more the immunized population, and the less the infected population. The method shows that the variation of the number of new coronavirus immune crowds can be influenced by the difference of the vaccination rate, but the new coronavirus becomes popular only in 1 month of 2021, so that the collected data cannot reflect the influence of the vaccination rate on the spread of the new coronavirus, but the numerical simulation result can show that the number of immune crowds can be effectively increased along with the increase of the vaccination rate.
According to the actual data of the new coronavirus epidemic situation in Beijing City released by the health Committee of Beijing City, the new coronavirus propagation model disclosed by the embodiment of the invention is fitted with the actual data, and relevant parameters in the model are estimated according to the acquired actual data. And the least square method is used for data fitting, so that the goodness of fit of the novel coronavirus propagation model in the embodiment of the invention is verified.
Based on the epidemic propagation theory, the invention considers that the isolation period and the vaccination rate of susceptible people can influence the virus propagation process, provides a novel coronavirus propagation model, simulates and analyzes the influence of the isolation period and the vaccination rate on the virus propagation process, proves that strict control measures have better control effect on the propagation of the new coronavirus, and verifies that the model has better fitting goodness by acquiring actual case data and fitting the model and the actual data.
Example two
As shown in fig. 9, an embodiment of the present invention provides a novel coronavirus propagation model establishing system, which includes the following modules:
a new coronavirus propagation model building module 41 for building a differential kinetic equation set according to the inoculation rate, the isolation period and the recovery rate so as to build a new coronavirus propagation model;
an acquire healthy equilibrium point and healthy equilibrium point module 42 for acquiring a healthy equilibrium point and a healthy equilibrium point of the new coronavirus transmission model by using kinetic modeling according to a differential kinetic equation set;
and a local balance point determination for disease-free balance point and disease balance point module 43, configured to perform stability analysis on the new coronavirus propagation model, and determine conditions required for local stabilization of the disease-free balance point and disease balance point by using the basic regeneration number.
The above examples are provided only for the purpose of describing the present invention, and are not intended to limit the scope of the present invention. The scope of the invention is defined by the appended claims. Various equivalent substitutions and modifications can be made without departing from the spirit and principles of the invention, and are intended to be within the scope of the invention.
Claims (3)
1. A method for establishing a novel coronavirus propagation model is characterized by comprising the following steps:
step S1: establishing a differential kinetic equation set according to the inoculation rate, the isolation period and the recovery rate to construct a new coronavirus transmission model: establishing a differential kinetic equation set according to the inoculation rate, the isolation period and the recovery rate, wherein the differential kinetic equation set specifically comprises the following steps:
wherein S (t), I (t) and R (t) respectively represent the densities of a susceptible person, an infected person and an immunized person at the time t, mu represents the migration rate and the migration rate of a user,
represents the quarantine period, beta represents the inoculation rate, and gamma represents the recovery rate;
step S2: according to the differential kinetic equation set, utilizing kinetic modeling to obtain the disease-free balance point and the disease balance point of the new coronavirus transmission model, the method specifically comprises the following steps:
step S21: calculating the disease-free balance point P according to the formula (1) 1 ＝(S 1 ，I 1 ，R 1 ) Wherein
step S22: calculating to obtain the ill balance point P according to the formula (1) 2 ＝(S 2 ，I 2 ，R 2 ) Wherein, in the process,
and step S3: and performing stability analysis on the new coronavirus transmission model, and determining conditions required for locally stabilizing the disease-free balance point and the disease balance point by using a basic regeneration number.
2. The method for establishing a novel coronavirus propagation model according to claim 1, wherein the step S3: performing stability analysis on the new coronavirus propagation model, and determining conditions required for locally stabilizing the disease-free balance point and the disease balance point by using a basic regeneration number, wherein the conditions comprise the following specific steps:
step S31: by regenerationCalculating basic regeneration number R by matrix spectrum radius method 0 ；
Step S32: judging conditions through Route-Hurwitz, and judging according to R 0 And the disease-free balance point P 1 ＝(S 1 ，I 1 ，R 1 ) It is found that the condition R is satisfied when the disease-free equilibrium point is locally stabilized 0 ≤1；
Step S33: judging conditions through Route-Hurwitz, and judging according to R 0 And the diseased equilibrium point P 2 ＝(S 2 ，I 2 ，R 2 ) It is found that the local stabilization of the diseased equilibrium point is required to satisfy the condition R 0 >1。
3. A novel coronavirus propagation model building system is characterized by comprising the following modules:
and constructing a new coronavirus propagation model module for establishing a differential kinetic equation set according to the inoculation rate, the isolation period and the recovery rate so as to construct a new coronavirus propagation model: establishing a differential kinetic equation set according to the inoculation rate, the isolation period and the recovery rate, wherein the differential kinetic equation set specifically comprises the following steps:
wherein S (t), I (t) and R (t) respectively represent the densities of a susceptible person, an infected person and an immunized person at the time t, mu represents the migration rate and the migration rate of a user,
represents the quarantine period, beta represents the inoculation rate, and gamma represents the recovery rate;
the module for obtaining the disease-free balance point and the disease balance point is used for obtaining the disease-free balance point and the disease balance point of the new coronavirus propagation model by dynamic modeling according to the differential kinetic equation set, and specifically comprises the following steps:
step S21: calculating the disease-free balance point P according to the formula (1) 1 ＝(S 1 ，I 1 ，R 1 ) Which isIn (1),
step S22: calculating to obtain the ill balance point P according to the formula (1) 2 ＝(S 2 ，I 2 ，R 2 ) Wherein
and the module for determining the local balance condition of the disease-free balance point and the disease balance point is used for carrying out stability analysis on the new coronavirus transmission model and determining the condition required for enabling the local stability of the disease-free balance point and the disease balance point to be achieved by utilizing the basic regeneration number.
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