CN101777092A - Method for predicting epidemic situation by spatial heterogeneity-based infectious disease propagation model - Google Patents

Abstract

The invention provides a method for predicting epidemic situation by a spatial heterogeneity-based infectious disease propagation model. If subareas comprising an area 1 and an area 2 exist in a city, a common rule that crowd flow causes the propagation of infectious diseases and the statistical data (obtained in the form of questionnaire and the like) about the dynamic flow of the crowd in a day can be explained according to a formula, and a method with better operability is established on the basis. The invention aims at reasonably describing a fact, namely the flow that people go out early and come home late for work, shopping and the like does not cause the migration of population macroscopically, but can cause transregional diffusion of the infectious diseases. As the daily periodic flow of the population increases the number of infected persons in a net input subarea, diseases are diffused in space to change the development speed of the epidemic situation. Particularly, at the initial stage of the epidemic situation, the transregional diffusion has an extremely high speed. The thought has heuristic significance for searching regional economic problems such as transregional consumption, labor input and export, and the like.

Description

Based on considering spatial inhomogeneity distribution infectious disease propagation model epidemic prediction method

Technical field:

The present invention relates to belong to method control field based on considering spatial inhomogeneity distribution infectious disease propagation model epidemic prediction method.

Background technology:

Common infectious disease propagation model supposes that all kinds of crowds evenly distribute in the space, and therefore available ordinary differential equation group is described.But, then need to consider the heterogeneity in space for the bigger situation of propagation regions.

The propagation of infectious disease has two levels, and one is interior among a small circle spreading of exposed population group inside closely, and this can imagination be to be confined in the homogeneous sub-district, and velocity of propagation is proportional to the product of the crowd of catching an illness and Susceptible population.Another level is the propagation in space, is because the crowd carries due to pathogenic microorganisms moves in a big way.Practice shows, for the propagation of infectious disease in intensive crowd among a small circle, the ordinary differential equations model of using based on even supposition can obtain satisfied predicting the outcome.But, rationally describe infectious disease and then want the many of difficulty in the diffusion in space.Some researchers use the gas molecule diffusion notion in the physics, add the space diffusion term on ordinary differential model basis, set up Partial Differential Equation Model and describe.Space diffusion term supposition usually is the second derivative that is proportional to the space of attribute, and we think that this is worth having consultations with.So because the gas molecule diffusion only can be described with the second order space derivative, a basic premise is arranged, promptly the yardstick that is studied a question is much larger than the mean free path of molecular thermalmotion.We know that the crowd, does shopping etc. as on and off duty in the migration in space, and its typical Commuting Distance can reach several kilometers to tens kilometers, and this and our the big city space scale of research are same magnitudes.In addition, the intraday migration of big city crowd is not random motion, and it has than clear regularity, promptly compiles to the city from the suburb morning, and disperses to the suburb from the city at dusk.Therefore, it is not too rational using and describing the diffusion of infectious disease space based on the partial differential equation of second order of molecular thermalmotion analogy, and will to identify space coefficient of diffusion heterogeneous also be a difficult task.In fact, the thought of space diffusion model comes from the dynamic (dynamical) research of biotic population, supposes that biological random motion has caused the diffusion of population.But biological behavior may not resemble yet suppose in the model as the molecular diffusion at random, their migration has obvious seasonal and purpose, even foraging behavior also shows as near the to-and-fro movement nest in the daytime, with molecular diffusion and inequality.Diffusion model supposition in the past has isotropic coefficient of diffusion, and population spreads to low-density from high density, and in fact diffusion process is anisotropic, and may be spread to high density by low-density.In sum, traditional diffusion form of presentation has bigger defective, is necessary to develop the method that tallies with the actual situation more.

Below we will propose model more realistic and that be easy to use, it makes full use of crowd's household register statistics based on the big city administrative planning, and provides the practical parameter method of estimation.At present, yet there are no relevant report.

Summary of the invention:

The present invention proposes based on considering spatial inhomogeneity distribution infectious disease propagation model epidemic prediction method, improved the control of infectious disease or epidemic situation being propagated under the actual conditions, improved specific aim and efficient, can in time be reduced to infectious disease or epidemic situation propagation harm minimum.

At first, establish in the city and to have two subregions: 1,2 districts, i.e. i=1,2, N ₁, N ₂It is respectively the total number of persons in 1,2 districts; R ₁, R ₂It is respectively the number that shifts out in 1,2 districts; R is infectious rate (infectious rate of supposing 1,2 districts is identical); I ₁, I ₂It is the infected's number in 1,2 districts; S ₁, S ₂It is the susceptible number in 1,2 districts; λ ₁, λ ₂It is the rate of accepting for medical treatment (establishing the rate of accepting for medical treatment of only accepting one day 0 number of catching an illness constantly for medical treatment) in 1,2 districts; Promptly have: $\begin{array}{c}{N}_1=I_1+S_1+R_1\\ N_2=I_2+S_2+R_2\end{array};$

Therefore: the number of catching an illness in each subregion, susceptible number and the number that shifts out is intraday is changed to

$\frac{{\mathrm{dI}}_1}{\mathrm{dt}}={\mathrm{rI}}_1{S}_1/N_1-{\mathrm{\λ}}_1{I}_{10}+{\mathrm{FI}}_{2\→1}-{\mathrm{FI}}_{1\→2}$

$\frac{{\mathrm{dI}}_2}{\mathrm{dt}}={\mathrm{rI}}_2{S}_2/N_2-{\mathrm{\λ}}_2{I}_{20}-{\mathrm{FI}}_{2\→1}+{\mathrm{FI}}_{1\→2}$

$\frac{{\mathrm{dS}}_1}{\mathrm{dt}}=-r{I}_1{S}_1/N_1+{\mathrm{FS}}_{2\→1}-{\mathrm{FS}}_{1\→2}$

$\frac{{\mathrm{dS}}_2}{\mathrm{dt}}=-r{I}_2{S}_2/N_2-{\mathrm{FS}}_{2\→1}+{\mathrm{FS}}_{1\→2}$

$\frac{{\mathrm{dR}}_1}{\mathrm{dt}}={\mathrm{\λ}}_1{I}_{10}$

$\frac{{\mathrm{dR}}_2}{\mathrm{dt}}={\mathrm{\λ}}_2{I}_{20}$

I wherein ₁₀, I ₂₀Be respectively one day 0 number of catching an illness constantly, FI _{1 → 2}Be the number of catching an illness in inflow 2 districts in 1 district in the unit interval, FI _{2 → 1}Be the number of catching an illness in inflow 1 district in 2 districts in the unit interval, FS _{1 → 2}Be the susceptible number in inflow 2 districts in 1 district in the unit interval, FS _{2 → 1}Flow into the susceptible number in 1 district for 2 districts in the unit interval.T is one day from 0 o'clock to 24 o'clock any time;

And the formula in the system of equations (1) be in 1 district one day any time the catch an illness rate of change of number, be expressed as

Formula in the system of equations (2) be in 2 districts one day any time the catch an illness rate of change of number, be expressed as

Formula in the system of equations (3) be in 1 district one day any time the susceptible number rate of change, be expressed as

Formula in the system of equations (4) be in 2 districts one day any time the susceptible number rate of change, be expressed as

Formula in the system of equations (5) is the rate of change that any time is shifted out number in 1 district one day, is expressed as

Formula in the system of equations (6) is the rate of change that any time is shifted out number in 2 districts one day, is expressed as

Do not consider the crowd that the shifts out migration of (comprise and being in hospital, cure death etc.) herein, suppose that they are in isolation during epidemic situation.

Secondly: according to the mobile universal law that causes disease transmission of the soluble crowd of above formula, and the statistics of crowd's dynamic flow in one day scope (form such as questionnaire obtains by inquiry, belong to setting data), set up the method that has operability more on this basis, promptly differential equation group is carried out time integral and handles, be specially:

If be integrated to the T finish time from one day initial time 0, T is a constant, and general T is 24 hours;

Wherein the formula in the system of equations (1) integration obtains

$\underset{0}{\overset{T}{\∫}}\frac{{\mathrm{dI}}_1}{\mathrm{dt}}\mathrm{dt}=\underset{0}{\overset{T}{\∫}}{\mathrm{rI}}_1{S}_1/N_1\mathrm{dt}-\underset{0}{\overset{T}{\∫}}{\mathrm{\λI}}_{10}\mathrm{dt}+\underset{0}{\overset{T}{\∫}}({\mathrm{FI}}_{2\→1}-{\mathrm{FI}}_{1\→2})\mathrm{dt}$

The equation left end equals I _1T-I ₁₀

First of equation right-hand member can be written as by INTEGRAL THEOREM OF MEAN: rTI ₁(τ) S ₁(τ)/N ₁(τ), 0＜τ＜T wherein;

Second of equation right-hand member equals λ TI ₁₀,

The 3rd of equation right-hand member is net flux in the flow of personnel cause a day, and it can be considered equals zero, because the crowd in this model is clocklike among being flowing in one day, shopping waits the effect that causes because personnel go to work, and the sign in the morning and afternoon is opposite.Therefore average among one day, total effect is zero.

If but be chronomere with the day, then T=1 (day), and definition ${\tilde{I}}_1=I_1\left(\mathrm{\τ}\right),$ ${\tilde{S}}_1=S_1\left(\mathrm{\τ}\right),$ ${\tilde{N}}_1=N_1\left(\mathrm{\τ}\right),$ Wherein, can estimate by the sample survey result of flow of personnel

With

Can obtain in form simplifying with the system of equations before the upper integral; Promptly

$I_{1T}-I_{10}=r{\tilde{I}}_1{\tilde{S}}_1\left(\mathrm{\τ}\right)/{\tilde{N}}_1\left(\mathrm{\τ}\right)-{\mathrm{\λI}}_{10}$

Wherein: ${\tilde{I}}_1=I_{10}+{\mathrm{\ΔI}}_1,$ ${\tilde{S}}_1=S_{10}+{\mathrm{\ΔS}}_1,$ We are generalized to any one day initial time t ' to initial time 0, and change T into t '+1 constantly, then, ${\tilde{I}}_1=I_1\left(t^{\′}\right)+{\mathrm{\ΔI}}_1,$ ${\tilde{S}}_1=S_1\left(t^{\′}\right)+{\mathrm{\ΔS}}_1;$

And Δ I ₁With Δ S ₁Be that these numerical value can be estimated by following approach because flow of personnel causes the average increment of the number of catching an illness and susceptible number: sample survey obtains going to work, doing shopping from 2 districts to 1 district or be engaged in other movable population ratio α ₁₂, and they are on average in 1 district's residence time ratio beta ₁₂, definition transfer coefficient k ₁₂=α ₁₂β ₁₂, same definable 1 district is to the transfer coefficient k in 2 districts ₂₁=α ₂₁β ₂₁, then have

ΔI ₁＝-ΔI ₂＝k ₁₂I ₂-k ₂₁I ₁，ΔS ₁＝-ΔS ₂＝k ₁₂S ₂-k ₂₁S ₁；

Similarly, also can obtain each integrated form of other formula in the system of equations before the integration;

The formula in the system of equations (2) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{{\mathrm{dI}}_2}{\mathrm{dt}}\mathrm{dt}=\underset{0}{\overset{T}{\∫}}r{I}_2{S}_2/N_2\mathrm{dt}-\underset{0}{\overset{T}{\∫}}{\mathrm{\λI}}_{20}\mathrm{dt}+\underset{0}{\overset{T}{\∫}}({\mathrm{FI}}_{1\→2}-{\mathrm{FI}}_{2\→1})\mathrm{dt}$

Obtain: $I_2(t^{\′}+1)-I_2\left(t^{\′}\right)=r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2-{\mathrm{\λI}}_2\left(t^{\′}\right)$

The formula in the system of equations (3) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{{\mathrm{dS}}_1}{\mathrm{dt}}\mathrm{dt}=-\underset{0}{\overset{T}{\∫}}{\mathrm{rI}}_1{S}_1/N_1\mathrm{dt}+\underset{0}{\overset{T}{\∫}}({\mathrm{FS}}_{2\→1}-{\mathrm{FS}}_{1\→2})\mathrm{dt}$

Obtain: $S_1(t^{\′}+1)-S_1\left(t^{\′}\right)=-r{\tilde{I}}_1{\tilde{S}}_1/{\tilde{N}}_1$

The formula in the system of equations (4) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{{\mathrm{dS}}_2}{\mathrm{dt}}\mathrm{dt}=-\underset{0}{\overset{T}{\∫}}{\mathrm{rI}}_2{S}_2/N_2\mathrm{dt}+\underset{0}{\overset{T}{\∫}}({\mathrm{FS}}_{1\→2}-{\mathrm{FS}}_{2\→1})\mathrm{dt}$

Obtain: $S_2(t^{\′}+1)-S_2\left(t^{\′}\right)=-r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2$

The formula in the system of equations (5) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{{\mathrm{dR}}_1}{\mathrm{dt}}\mathrm{dt}=\underset{0}{\overset{T}{\∫}}{\mathrm{\λI}}_{10}\mathrm{dt}$

Obtain: R ₁(t '+1)-R ₁(t ')=λ I ₁(t ')

The formula in the system of equations (6) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{{\mathrm{dR}}_2}{\mathrm{dt}}\mathrm{dt}=\underset{0}{\overset{T}{\∫}}{\mathrm{\λI}}_{20}\mathrm{dt}$

Obtain: R ₂(t '+1)-R ₂(t ')=λ I ₂(t ')

Wherein,

${\tilde{I}}_1=I_1\left(t^{\′}\right)+k_{12}{I}_2-k_{21}{I}_1,$ ${\tilde{I}}_2=I_2\left(t^{\′}\right)-k_{12}{I}_2+k_{21}{I}_1$

${\tilde{S}}_1=S_1\left(t^{\′}\right)+k_{12}{S}_2-k_{21}{S}_1,$ ${\tilde{S}}_2=S_2\left(t^{\′}\right)-k_{12}{S}_2+k_{21}{S}_1$

Above derivation result is comprehensively as follows:

If any one day initial time is t ', then the situation of second day initial time t '+1 o'clock can be predicted by the DIFFERENCE EQUATIONS of following time stepping:

${\tilde{I}}_1=I_1\left(t^{\′}\right)+k_{12}{I}_2-k_{21}{I}_1$

${\tilde{I}}_2=I_2\left(t^{\′}\right)-k_{12}{I}_2+k_{21}{I}_1$

${\tilde{S}}_1=S_1\left(t^{\′}\right)+k_{12}{S}_2-k_{21}{S}_1$

${\tilde{S}}_2=S_2\left(t^{\′}\right)-k_{12}{S}_2+k_{21}{S}_1$

${\tilde{N}}_1={\tilde{I}}_1+{\tilde{S}}_1+R_1\left(t^{\′}\right)$

${\tilde{N}}_2={\tilde{I}}_2+{\tilde{S}}_2+R_2\left(t^{\′}\right)$

$I_1(t^{\′}+1)-I_1\left(t^{\′}\right)={r\tilde{I}}_1{\tilde{S}}_1/{\tilde{N}}_1-{\mathrm{\λI}}_1\left(t^{\′}\right)$

$I_2(t^{\′}+1)-I_2\left(t^{\′}\right)={r\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2-{\mathrm{\λI}}_2\left(t^{\′}\right)$

$S_1(t^{\′}+1)-S_1\left(t^{\′}\right)=-r{\tilde{I}}_1{\tilde{S}}_1/{\tilde{N}}_1,$

$S_2(t^{\′}+1)-S_2\left(t^{\′}\right)=-r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2,$

R ₁(t′+1)-R ₁(t′)＝λI ₁(t′)

R ₂(t′+1)-R ₂(t′)＝λI ₂(t′)

t′＝0，1，2，...

With the above system of equations of critical data substitution, we can obtain about infectious disease or epidemic situation propagation forecast numerical value.

Beneficial effect: the present invention is intended to reasonably describe this fact, i.e. the type of coming out early and return late such as personnel's working, shopping flows, and does not cause migrating of population on macroscopic view, but can cause the transregional diffusion of infectious disease.Why the diffusion in space takes place in disease, and key is to flow the diurnal periodicity of population to have increased personnel's the number of catching an illness in the clean input subregion, thereby has changed the epidemic situation speed of development.Especially epidemic situation initial stage, the effect as a single spark can start a prairie fire is played in this transregional diffusion.This thinking is for equal enlightening meanings of regional economics problem such as research as transregional consumption, labor service input and output.

Description of drawings:

Fig. 1 multi partition infects the time-evolution trend (transfer coefficient 0.015) of number

Fig. 2 transfer coefficient size infects the evolving trend influence (solid line: transfer coefficient 0.015 of number to multi partition; Round dot: transfer coefficient 0.0075)

Embodiment:

At first, establish in the city and to have two subregions: 1,2 districts, i.e. i=1,2, N ₁, N ₂It is respectively the total number of persons in 1,2 districts; R ₁, R ₂It is respectively the number that shifts out in 1,2 districts; R is infectious rate (infectious rate of supposing 1,2 districts is identical); I ₁, I ₂It is the infected's number in 1,2 districts; S ₁, S ₂It is the susceptible number in 1,2 districts; λ ₁, λ ₂It is the rate of accepting for medical treatment (establishing the rate of accepting for medical treatment of only accepting one day 0 number of catching an illness constantly for medical treatment) in 1,2 districts; Promptly have: $\begin{array}{c}{N}_1=I_1+S_1+R_1\\ N_2=I_2+S_2+R_2\end{array};$

Therefore: the number of catching an illness in each subregion, susceptible number and the number that shifts out is intraday is changed to

$\frac{{\mathrm{dI}}_1}{\mathrm{dt}}={\mathrm{rI}}_1{S}_1/N_1-{\mathrm{\λ}}_1{I}_{10}+{\mathrm{FI}}_{2\→1}-{\mathrm{FI}}_{1\→2}$

$\frac{{\mathrm{dI}}_2}{\mathrm{dt}}={\mathrm{rI}}_2{S}_2/N_2-{\mathrm{\λ}}_2{I}_{20}-{\mathrm{FI}}_{2\→1}+{\mathrm{FI}}_{1\→2}$

$\frac{{\mathrm{dS}}_1}{\mathrm{dt}}=-r{I}_1{S}_1/N_1+{\mathrm{FS}}_{2\→1}-{\mathrm{FS}}_{1\→2}$

$\frac{{\mathrm{dS}}_2}{\mathrm{dt}}=-r{I}_2{S}_2/N_2-{\mathrm{FS}}_{2\→1}+{\mathrm{FS}}_{1\→2}$

$\frac{{\mathrm{dR}}_1}{\mathrm{dt}}={\mathrm{\λ}}_1{I}_{10}$

$\frac{{\mathrm{dR}}_2}{\mathrm{dt}}={\mathrm{\λ}}_2{I}_{20}$

I wherein ₁₀, I ₂₀Be respectively one day 0 number of catching an illness constantly, FI _{1 → 2}Be the number of catching an illness in inflow 2 districts in 1 district in the unit interval, FI _{2 → 1}Be the number of catching an illness in inflow 1 district in 2 districts in the unit interval, FS _{1 → 2}Be the susceptible number in inflow 2 districts in 1 district in the unit interval, FS _{2 → 1}Flow into the susceptible number in 1 district for 2 districts in the unit interval.T is one day from 0 o'clock to 24 o'clock any time;

And the formula in the system of equations (1) be in 1 district one day any time the catch an illness rate of change of number, be expressed as

Formula in the system of equations (2) be in 2 districts one day any time the catch an illness rate of change of number, be expressed as

Formula in the system of equations (3) be in 1 district one day any time the susceptible number rate of change, be expressed as

Formula in the system of equations (4) be in 2 districts one day any time the susceptible number rate of change, be expressed as

Formula in the system of equations (5) is the rate of change that any time is shifted out number in 1 district one day, is expressed as

Formula in the system of equations (6) is the rate of change that any time is shifted out number in 2 districts one day, is expressed as

Do not consider the crowd that the shifts out migration of (comprise and being in hospital, cure death etc.) herein, suppose that they are in isolation during epidemic situation.

Secondly: flow according to the soluble crowd of above formula and to cause the universal law of disease transmission, and the statistics of crowd's dynamic flow in one day scope, set up the method that has operability more on this basis;

Statistics form such as questionnaire by inquiry obtains, because because crowd's flow direction is dynamic change, i.e. the input in the morning (output) changes the output (input) in afternoon into.These data need draw by sample survey result statistics.

In fact, the crowd in this model is clocklike among being flowing in one day, and shopping waits the effect that causes because personnel go to work, and the sign in the morning and afternoon is opposite.Therefore average among one day, total effect is zero.As shown in figure 11, within one day, because diffusion causes the number of catching an illness to increase existing the minimizing earlier, see on the macroscopic view, the fluctuation that with the day is the cycle does not so influence secular trend, but catches an illness crowd's mean value in having changed one day, thereby has accelerated the growth rate of infectious disease.Therefore, if set up difference equation model, only study the every day of constantly the variation of each crowd's number finally, then do not need to know detailed situation of change within a day, in expression formula the average effect of diffusion taken in when calculating the growth of crowd's number and get final product, the degree of dependence to observation data has just greatly alleviated like this.

Promptly differential equation group is carried out time integral and handles, be specially:

If be integrated to the T finish time from one day initial time 0, T is a constant, and general T is 24 hours;

Wherein the formula in the system of equations (1) integration obtains

$\underset{0}{\overset{T}{\∫}}\frac{{\mathrm{dI}}_1}{\mathrm{dt}}\mathrm{dt}=\underset{0}{\overset{T}{\∫}}{\mathrm{rI}}_1{S}_1/N_1\mathrm{dt}-\underset{0}{\overset{T}{\∫}}{\mathrm{\λI}}_{10}\mathrm{dt}+\underset{0}{\overset{T}{\∫}}({\mathrm{FI}}_{2\→1}-{\mathrm{FI}}_{1\→2})\mathrm{dt}$

The equation left end equals I _1T-I ₁₀

First of equation right-hand member can be written as by INTEGRAL THEOREM OF MEAN: rTI ₁(τ) S ₁(τ)/N ₁(τ), 0＜τ＜T wherein;

Second of equation right-hand member equals λ TI ₁₀,

The 3rd of equation right-hand member is net flux in the flow of personnel cause a day, and it can be considered equals zero, because the crowd in this model is clocklike among being flowing in one day, shopping waits the effect that causes because personnel go to work, and the sign in the morning and afternoon is opposite.Therefore average among one day, total effect is zero.

If but be chronomere with the day, then T=1 (day), and definition ${\tilde{I}}_1=I_1\left(\mathrm{\τ}\right),$ ${\tilde{S}}_1=S_1\left(\mathrm{\τ}\right),$ ${\tilde{N}}_1=N_1\left(\mathrm{\τ}\right),$ Wherein, can estimate by the sample survey result of flow of personnel

With Can obtain in form simplifying with the system of equations before the upper integral; Promptly

$I_{1T}-I_{10}=r{\tilde{I}}_1{\tilde{S}}_1\left(\mathrm{\τ}\right)/{\tilde{N}}_1\left(\mathrm{\τ}\right)-{\mathrm{\λI}}_{10}$

Wherein: ${\tilde{I}}_1=I_{10}+{\mathrm{\ΔI}}_1,$ ${\tilde{S}}_1=S_{10}+{\mathrm{\ΔS}}_1,$ We are generalized to any one day initial time t ' to initial time 0, and change T into t '+1 constantly, then, ${\tilde{I}}_1=I_1\left(t^{\′}\right)+{\mathrm{\ΔI}}_1,$ ${\tilde{S}}_1=S_1\left(t^{\′}\right)+{\mathrm{\ΔS}}_1;$

And Δ I ₁With Δ S ₁Be that these numerical value can be estimated by following approach because flow of personnel causes the average increment of the number of catching an illness and susceptible number: sample survey obtains going to work, doing shopping from 2 districts to 1 district or be engaged in other movable population ratio α ₁₂, and they are on average in 1 district's residence time ratio beta ₁₂, definition transfer coefficient k ₁₂=α ₁₂β ₁₂, same definable 1 district is to the transfer coefficient k in 2 districts ₂₁=α ₂₁β ₂₁, then have

ΔI ₁＝-ΔI ₂＝k ₁₂I ₂-k ₂₁I ₁，ΔS ₁＝-ΔS ₂＝k ₁₂S ₂-k ₂₁S ₁；

Similarly, also can obtain each integrated form of other formula in the system of equations before the integration;

The formula in the system of equations (2) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{{\mathrm{dI}}_2}{\mathrm{dt}}\mathrm{dt}=\underset{0}{\overset{T}{\∫}}r{I}_2{S}_2/N_2\mathrm{dt}-\underset{0}{\overset{T}{\∫}}{\mathrm{\λI}}_{20}\mathrm{dt}+\underset{0}{\overset{T}{\∫}}({\mathrm{FI}}_{1\→2}-{\mathrm{FI}}_{2\→1})\mathrm{dt}$

Obtain: $I_2(t^{\′}+1)-I_2\left(t^{\′}\right)=r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2-{\mathrm{\λI}}_2\left(t^{\′}\right)$

The formula in the system of equations (3) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{{\mathrm{dS}}_1}{\mathrm{dt}}\mathrm{dt}=-\underset{0}{\overset{T}{\∫}}{\mathrm{rI}}_1{S}_1/N_1\mathrm{dt}+\underset{0}{\overset{T}{\∫}}({\mathrm{FS}}_{2\→1}-{\mathrm{FS}}_{1\→2})\mathrm{dt}$

Obtain: $S_1(t^{\′}+1)-S_1\left(t^{\′}\right)=-r{\tilde{I}}_1{\tilde{S}}_1/{\tilde{N}}_1$

The formula in the system of equations (4) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{{\mathrm{dS}}_2}{\mathrm{dt}}\mathrm{dt}=-\underset{0}{\overset{T}{\∫}}{\mathrm{rI}}_2{S}_2/N_2\mathrm{dt}+\underset{0}{\overset{T}{\∫}}({\mathrm{FS}}_{1\→2}-{\mathrm{FS}}_{2\→1})\mathrm{dt}$

Obtain: $S_2(t^{\′}+1)-S_2\left(t^{\′}\right)=-r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2$

The formula in the system of equations (5) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{{\mathrm{dR}}_1}{\mathrm{dt}}\mathrm{dt}=\underset{0}{\overset{T}{\∫}}{\mathrm{\λI}}_{10}\mathrm{dt}$

Obtain: R ₁(t '+1)-R ₁(t ')=λ I ₁(t ')

The formula in the system of equations (6) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{{\mathrm{dR}}_2}{\mathrm{dt}}\mathrm{dt}=\underset{0}{\overset{T}{\∫}}{\mathrm{\λI}}_{20}\mathrm{dt}$

Obtain: R ₂(t '+1)-R ₂(t ')=λ I ₂(t ')

Wherein,

${\tilde{I}}_1=I_1\left(t^{\′}\right)+k_{12}{I}_2-k_{21}{I}_1,$ ${\tilde{I}}_2=I_2\left(t^{\′}\right)-k_{12}{I}_2+k_{21}{I}_1$

${\tilde{S}}_1=S_1\left(t^{\′}\right)+k_{12}{S}_2-k_{21}{S}_1,$ ${\tilde{S}}_2=S_2\left(t^{\′}\right)-k_{12}{S}_2+k_{21}{S}_1$

Above derivation result is comprehensively as follows:

If any one day initial time is t ', then the situation of second day initial time t '+1 o'clock can be predicted by the DIFFERENCE EQUATIONS of following time stepping:

${\tilde{I}}_1=I_1\left(t^{\′}\right)+k_{12}{I}_2\left(t^{\′}\right)-k_{21}{I}_1\left(t^{\′}\right)$

${\tilde{I}}_2=I_2\left(t^{\′}\right)-k_{12}{I}_2\left(t^{\′}\right)+k_{21}{I}_1\left(t^{\′}\right)$

${\tilde{S}}_1=S_1\left(t^{\′}\right)+k_{12}{S}_2\left(t^{\′}\right)-k_{21}{S}_1\left(t^{\′}\right)$

${\tilde{S}}_2=S_2\left(t^{\′}\right)-k_{12}{S}_2\left(t^{\′}\right)+k_{21}{S}_1\left(t^{\′}\right)$

${\tilde{N}}_1={\tilde{I}}_1+{\tilde{S}}_1+R_1\left(t^{\′}\right)$

${\tilde{N}}_2={\tilde{I}}_2+{\tilde{S}}_2+R_2\left(t^{\′}\right)$

$I_1(t^{\′}+1)-I_1\left(t^{\′}\right)=r{\tilde{I}}_1{\tilde{S}}_1/{\tilde{N}}_1-{\mathrm{\λI}}_1\left(t^{\′}\right)$

$I_2(t^{\′}+1)-I_2\left(t^{\′}\right)=r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2-{\mathrm{\λI}}_2\left(t^{\′}\right)$

$S_1(t^{\′}+1)-S_1\left(t^{\′}\right)=-r{\tilde{I}}_1{\tilde{S}}_1/{\tilde{N}}_1,$

$S_2(t^{\′}+1)-S_2\left(t^{\′}\right)=-r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2,$

R ₁(t′+1)-R ₁(t′)＝λI ₁(t′)

R ₂(t′+1)-R ₂(t′)＝λI ₂(t′)

t′＝0，1，2，...

With the above system of equations of critical data substitution, we can obtain about infectious disease or epidemic situation propagation forecast numerical value.

The solution procedure of following two subregion difference equation models that underdraw:

If a city is made up of two subregions, it is identical to establish two subregion movement of population parameters, gets the transregional proportion of flow α of population=0.1, mean residence time β=0.2 day, then k ₁₂=k ₂₁=α * β=0.02.Get infectious rate r=1, accept coefficient lambda=0.95 for medical treatment.

Initial t '=0 constantly, the initial Susceptible population that establishes each subregion is S ₁(0)=S ₂(0)=500,000 people, number of the infected I ₁(0)=50, I ₂(0)=0, shifts out number R ₁(0)=0, R ₂(0)=0.

Then, obtain by formula in the DIFFERENCE EQUATIONS (1)-(6)

${\tilde{I}}_1=I_1\left(0\right)+k_{12}{I}_2\left(0\right)-k_{21}{I}_1\left(0\right)=50+0.02\×0-0.02\×50=49$

${\tilde{I}}_2=I_2\left(0\right)-k_{12}{I}_2\left(0\right)+k_{21}{I}_1\left(0\right)=0-0.02\×0+0.02\×50=1$

${\tilde{S}}_1=S_1\left(0\right)+k_{12}{S}_2\left(0\right)-k_{21}{S}_1\left(0\right)=500000+0.02\×500000-0.02\×500000=500000$

${\tilde{S}}_2=S_2\left(0\right)-k_{12}{S}_2\left(0\right)+k_{21}{s}_1\left(0\right)=500000-0.02\×500000+0.02\×500000=500000$

${\tilde{N}}_1={\tilde{I}}_1+{\tilde{S}}_1+R_1\left(0\right)=49+500000+0=500049$

${\tilde{N}}_2={\tilde{I}}_2+{\tilde{S}}_2+R_2\left(0\right)=1+500000+0=500001$

By formula in the difference equation (7)-(12), obtain second day initial time, i.e. t '=1 situation constantly.

$I_1\left(1\right)=I_1\left(0\right)+r{\tilde{I}}_1{\tilde{S}}_1/{\tilde{N}}_1-{\mathrm{\λI}}_1\left(0\right)=50+1\×49\×500000/500049-0.9\×50=54$

$I_2\left(1\right)=I_2\left(0\right)+r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2-{\mathrm{\λI}}_2\left(0\right)=0+1\×1\×500000/500049-0.9\×0=1$

$S_1\left(1\right)=S_1\left(0\right)-r{\tilde{I}}_1{\tilde{S}}_1/{\tilde{N}}_1=500000-1\×49\×500000/500049=499951$

$S_2\left(1\right)=S_2\left(0\right)-r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2=500000-1\×1\×500000/500049=499999$

R ₁(1)＝R ₁(0)+λI ₁(0)＝0+0.9×50＝45

R ₂(1)＝R ₂(0)+λI ₂(0)＝0+0.9×0＝0

According to same step, can once calculate thereafter initial time every day (t '=2,3 ...) and the disease transmission situation.

According to two partition models, the author has proposed a kind of multi partition model again, is specially:

If the big city of studying or zone M administrative subregion arranged, the demographic data of each subregion is known, is N ₁, N ₂..., N _M

If certain day initial time t ', i subregion is S to the susceptible number of infectious disease _i, the number of catching an illness is I _i, the number of shifting out is R _i, i=1,2 .., M

Suppose that all kinds of crowds evenly distribute in each subregion.

In i the subregion, the number of catching an illness is owing to the increment that the mutual migration with other subregion causes is

Set up following difference equation model

$\left\{\begin{array}{c}{\tilde{I}}_i=I_i\left(t\right)+\underset{j=1}{\overset{M}{\mathrm{\Σ}}}[k_{\mathrm{ij}}{I}_j\left(t\right)-k_{\mathrm{ji}}{I}_i\left(t\right)],\\ {\tilde{S}}_i=S_i\left(t\right)+\underset{j=1}{\overset{M}{\mathrm{\Σ}}}[k_{\mathrm{ij}}{S}_j\left(t\right)-k_{\mathrm{ji}}{S}_i\left(t\right)],\\ N_i={\tilde{I}}_i\left(t\right)+{\tilde{S}}_i\left(t\right)+R_i\left(t\right)\\ I_i(t+1)-I_i\left(t\right)=r{\tilde{I}}_i\·{\tilde{S}}_i/{\tilde{N}}_i-\mathrm{\λ}{\tilde{I}}_i,\\ S_i(t+1)-S_i\left(t\right)=-r{\tilde{I}}_i\·{\tilde{S}}_i/{\tilde{N}}_i,\\ R_i(t+1)-R_i\left(t\right)=\mathrm{\λ}{\tilde{I}}_i,\\ i=\mathrm{1,2},...,M\end{array}\right.$

Wherein t and t+1 are respectively the adjacent two days finish times.

Be number of catching an illness and susceptible number at intraday mean value, it has reflected the statistics effect of behaviors such as crowd's working, shopping.

Coefficient k in the model _IjThe average effect that has reflected crowd's transregional mobile flux.If subregion j every day is proportional to be α _IjNumber move to subregion i, average waiting time is β _Ij(0＜β _Ij＜1), parameter alpha _Ij, β _IjAll can obtain by sample survey result statistics.Therefore can be by k _Ij=α _Ijβ _IjCalculate the by stages ac coefficient.

The numerical value example of multi partition:

If a sleeve configuration city is made up of four subregions, mutually transfer coefficient is identical between adjacent sectors, gets α=0.05, β=0.3, and k=α * β=0.015 then, non-conterminous by stages transfer coefficient is made as zero.The initial Susceptible population of supposing four subregions is 100,000 people.

Initial case appears at first subregion, gets to infect coefficient r=1, accepts coefficient lambda=0.95 for medical treatment

Analog result shows (as shown in Figure 1, 2), and four districts reach the infection peak value in succession, and finally are suppressed.The time lag of by stages and transfer coefficient size have direct relation.Transfer coefficient is more little, and the time lags behind more.

The number of initially catching an illness is many more, and then peak value more in advance.

It is big more to shift out rate, and peak value is low more, occurs also lagging behind more.When accepting coefficient greater than the infection coefficient for medical treatment, epidemic situation can not develop, and promptly directly be suppressed.

Consider that the crowd of catching an illness is relatively poor owing to energy, be in and have a rest or reasons such as isolation, its at the transfer coefficient of different by stages often less than the transfer coefficient of healthy population.Therefore the hysteresis of by stages infectious disease peak value will be more obvious.This also provides more approach and longer argin for the control of epidemic situation.These phenomenons, all be the homogeneous Epidemic Model can't reflect.

Claims (4)

1. based on considering spatial inhomogeneity distribution infectious disease propagation model epidemic prediction method, it is characterized in that:

At first, establish two subregions of existence in the city: 1,2 districts, N ₁, N ₂It is respectively the total number of persons in 1,2 districts; R ₁, R ₂It is respectively the number that shifts out in 1,2 districts; R is an infectious rate; I ₁, I ₂It is the infected's number in 1,2 districts; S ₁, S ₂It is the susceptible number in 1,2 districts; λ ₁, λ ₂It is the rate of accepting for medical treatment in 1,2 districts; Promptly have: $\begin{array}{c}{N}_1=I_1+S_1+R_1\\ N_2=I_2+S_2+R_2\end{array};$

Therefore: the number of catching an illness in each subregion, susceptible number and the intraday variation of the number that shifts out can utilize differential equation group to obtain; I wherein ₁₀, I ₂₀Be respectively one day 0 number of catching an illness constantly, FI _{1 → 2}Be the number of catching an illness in inflow 2 districts in 1 district in the unit interval, FI _{2 → 1}Be the number of catching an illness in inflow 1 district in 2 districts in the unit interval, FS _{1 → 2}Be the susceptible number in inflow 2 districts in 1 district in the unit interval, FS _{2 → 1}Flow into the susceptible number in 1 district for 2 districts in the unit interval; T is one day from 0 o'clock to 24 o'clock any time;

Secondly, flow according to the soluble crowd of above formula and to cause the universal law of disease transmission, and the statistics of crowd's dynamic flow in one day scope, on this basis differential equation group is carried out time integral and handles,

If any one day initial time is t ', then the situation of second day initial time t '+1 o'clock can be predicted by the DIFFERENCE EQUATIONS of following time stepping:

${\tilde{I}}_1=I_1\left(t^{\′}\right)+k_{12}{I}_2-k_{21}{I}_1$

${\tilde{I}}_2=I_2\left(t^{\′}\right)-k_{12}{I}_2+k_{21}{I}_1$

${\tilde{S}}_1=S_1\left(t^{\′}\right)+k_{12}{S}_2-k_{21}{S}_1$

${\tilde{S}}_2=S_2\left(t^{\′}\right)-k_{12}{S}_2+k_{21}{S}_1$

${\tilde{N}}_1={\tilde{I}}_1+{\tilde{S}}_1+R_1\left(t^{\′}\right)$

${\tilde{N}}_2={\tilde{I}}_2+{\tilde{S}}_2+R_2\left(t^{\′}\right)$

$I_1(t^{\′}+1)-I_1\left(t^{\′}\right)=r{\tilde{I}}_1{\tilde{S}}_1/{\tilde{N}}_1-\mathrm{\λ}{I}_1\left(t^{\′}\right)$

$I_2(t^{\′}+1)-I_2\left(t^{\′}\right)=r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2-\mathrm{\λ}{I}_2\left(t^{\′}\right)$

$S_1(t^{\′}+1)-S_1\left(t^{\′}\right)=-r{\tilde{I}}_1{\tilde{S}}_1/{\tilde{N}}_1,$

$S_2(t^{\′}+1)-S_2\left(t^{\′}\right)=-r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2,$

R ₁(t′+1)-R ₁(t′)＝λI ₁(t′)

R ₂(t′+1)-R ₂(t′)＝λI ₂(t′)

t′＝0，1，2，...

With the above system of equations of critical data substitution, obtain about infectious disease or epidemic situation propagation forecast numerical value.

2. according to claim 1 a kind of based on considering spatial inhomogeneity distribution infectious disease or epidemic situation propagation prediction method, it is characterized in that: differential equation group is carried out time integral handle, be specially:

If be integrated to the T finish time from one day initial time 0, T is a constant, and general T is 24 hours;

Wherein the formula in the system of equations (1) integration obtains

$\underset{0}{\overset{T}{\∫}}\frac{d{I}_1}{\mathrm{dt}}\mathrm{dt}=\underset{0}{\overset{T}{\∫}}r{I}_1{S}_1/N_1\mathrm{dt}-\underset{0}{\overset{T}{\∫}}\mathrm{\λ}{I}_{10}\mathrm{dt}+\underset{0}{\overset{T}{\∫}}(F{I}_{2\→1}-F{I}_{1\→2})\mathrm{dt}$

The equation left end equals I _1T-I ₁₀

First of equation right-hand member can be written as by INTEGRAL THEOREM OF MEAN: rTI ₁(τ) S ₁(τ)/N ₁(τ), 0＜τ＜T wherein;

Second of equation right-hand member equals λ TI ₁₀,

The 3rd of equation right-hand member is net flux in the flow of personnel cause a day, and it can be considered equals zero, because the crowd in this model is clocklike among being flowing in one day, shopping waits the effect that causes because personnel go to work, and the sign in the morning and afternoon is opposite; Therefore average among one day, total effect is zero;

If but be chronomere with the day, then T=1/ day, and definition ${\tilde{I}}_1=I_1\left(\mathrm{\τ}\right),$ ${\tilde{S}}_1=S_1\left(\mathrm{\τ}\right),$ ${\tilde{N}}_1=N_1\left(\mathrm{\τ}\right),$ Wherein, the sample survey result by flow of personnel estimates

With

Can obtain in form simplifying with the system of equations before the upper integral; Promptly $I_{1T}-I_{10}=r{\tilde{I}}_1{\tilde{S}}_1\left(\mathrm{\τ}\right)/{\tilde{N}}_1\left(\mathrm{\τ}\right)-\mathrm{\λ}{I}_{10}$

Wherein: ${\tilde{I}}_1=I_{10}+\mathrm{\Δ}{I}_1,$ ${\tilde{S}}_1=S_{10}+\mathrm{\Δ}{S}_1,$ We are generalized to any one day initial time t ' to initial time 0, and change T into t '+1 constantly, then, ${\tilde{I}}_1=I_1\left(t^{\′}\right)+\mathrm{\Δ}{I}_1,$ ${\tilde{S}}_1=S_1\left(t^{\′}\right)+\mathrm{\Δ}{S}_1;$

And Δ I ₁With Δ S ₁Be that these numerical value can be estimated by following approach because flow of personnel causes the average increment of the number of catching an illness and susceptible number: sample survey obtains going to work, doing shopping from 2 districts to 1 district or be engaged in other movable population ratio α ₁₂, and they are on average in 1 district's residence time ratio beta ₁₂, definition transfer coefficient k ₁₂=α ₁₂β ₁₂, same definable 1 district is to the transfer coefficient k in 2 districts ₂₁=α ₂₁β ₂₁, then have

ΔI ₁＝-ΔI ₂＝k ₁₂I ₂-k ₂₁I ₁，ΔS ₁＝-ΔS ₂＝k ₁₂S ₂-k ₂₁S ₁；

Similarly, also can obtain each integrated form of other formula in the system of equations before the integration;

The formula in the system of equations (2) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{d{I}_2}{\mathrm{dt}}\mathrm{dt}=\underset{0}{\overset{T}{\∫}}r{I}_2{S}_2/N_2\mathrm{dt}-\underset{0}{\overset{T}{\∫}}\mathrm{\λ}{I}_{20}\mathrm{dt}+\underset{0}{\overset{T}{\∫}}(F{I}_{1\→2}-F{I}_{2\→1})\mathrm{dt}$

Obtain: $I_2(t^{\′}+1)-I_2\left(t^{\′}\right)=r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2-\mathrm{\λ}{I}_2\left(t^{\′}\right)$

The formula in the system of equations (3) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{d{S}_1}{\mathrm{dt}}\mathrm{dt}=-\underset{0}{\overset{T}{\∫}}r{I}_1{S}_1/N_1\mathrm{dt}+\underset{0}{\overset{T}{\∫}}(F{S}_{2\→1}-F{S}_{1\→2})\mathrm{dt}$

Obtain: $S_1(t^{\′}+1)-S_1\left(t^{\′}\right)=-r{\tilde{I}}_1{\tilde{S}}_1/{\tilde{N}}_1$

The formula in the system of equations (4) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{d{S}_2}{\mathrm{dt}}\mathrm{dt}=-\underset{0}{\overset{T}{\∫}}r{I}_2{S}_2/N_2\mathrm{dt}+\underset{0}{\overset{T}{\∫}}(F{S}_{1\→2}-F{S}_{2\→1})\mathrm{dt}$

Obtain: $S_2(t^{\′}+1)-S_2\left(t^{\′}\right)=-r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2$

The formula in the system of equations (5) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{d{R}_1}{\mathrm{dt}}\mathrm{dt}=\underset{0}{\overset{T}{\∫}}\mathrm{\λ}{I}_{10}\mathrm{dt}$

Obtain: R ₁(t '+1)-R ₁(t ')=λ I ₁(t ')

The formula in the system of equations (6) integration wherein

$\underset{0}{\overset{T}{\∫}}\frac{d{R}_2}{\mathrm{dt}}\mathrm{dt}=\underset{0}{\overset{T}{\∫}}\mathrm{\λ}{I}_{20}\mathrm{dt}$

Obtain: R ₂(t '+1)-R ₂(t ')=λ I ₂(t ')

Wherein,

${\tilde{I}}_1=I_1\left(t^{\′}\right)+k_{12}{I}_2-k_{21}{I}_1,$ ${\tilde{I}}_2=I_2\left(t^{\′}\right)-k_{12}{I}_2+k_{21}{I}_1$

${\tilde{S}}_1=S_1\left(t^{\′}\right)+k_{12}{S}_2-k_{21}{S}_1,$ ${\tilde{S}}_2=S_2\left(t^{\′}\right)-k_{12}{S}_2+k_{21}{S}_1$

Above derivation result is comprehensively as follows:

If any one day initial time is t ', then the situation of second day initial time t '+1 o'clock can be predicted by the DIFFERENCE EQUATIONS of following time stepping:

${\tilde{I}}_1=I_1\left(t^{\′}\right)+k_{12}{I}_2-k_{21}{I}_1$

${\tilde{I}}_2=I_2\left(t^{\′}\right)-k_{12}{I}_2+k_{21}{I}_1$

${\tilde{S}}_1=S_1\left(t^{\′}\right)+k_{12}{S}_2-k_{21}{S}_1$

${\tilde{S}}_2=S_2\left(t^{\′}\right)-k_{12}{S}_2+k_{21}{S}_1$

${\tilde{N}}_1={\tilde{I}}_1+{\tilde{S}}_1+R_1\left(t^{\′}\right)$

${\tilde{N}}_2={\tilde{I}}_2+{\tilde{S}}_2+R_2\left(t^{\′}\right)$

$I_1(t^{\′}+1)-I_1\left(t^{\′}\right)=r{\tilde{I}}_1{\tilde{S}}_1/{\tilde{N}}_1-\mathrm{\λ}{I}_1\left(t^{\′}\right)$

$I_2(t^{\′}+1)-I_2\left(t^{\′}\right)=r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2-\mathrm{\λ}{I}_2\left(t^{\′}\right)$

$S_1(t^{\′}+1)-S_1\left(t^{\′}\right)=-r{\tilde{I}}_1{\tilde{S}}_1/{\tilde{N}}_1,$

$S_2(t^{\′}+1)-S_2\left(t^{\′}\right)=-r{\tilde{I}}_2{\tilde{S}}_2/{\tilde{N}}_2,$

R ₁(t′+1)-R ₁(t′)＝λI ₁(t′)

R ₂(t′+1)-R ₂(t′)＝λI ₂(t′)

t′＝0，1，2，...

With the above system of equations of critical data substitution, we can obtain about infectious disease or epidemic situation propagation forecast numerical value.

3. according to claim 1 and 2 a kind of based on considering spatial inhomogeneity distribution infectious disease or epidemic situation propagation prediction method, it is characterized in that: this method also is applicable to the prediction of multi partition.

4. according to claim 3 a kind of based on considering spatial inhomogeneity distribution infectious disease or epidemic situation propagation prediction method, it is characterized in that the prediction of multi partition is specially:

If the big city of studying or zone M administrative subregion arranged, the demographic data of each subregion is known, is N ₁, N ₂..., N _M

If certain day initial time t ', i subregion is S to the susceptible number of infectious disease _i, the number of catching an illness is I _i, the number of shifting out is R _i, i=1,2 ..., M

Suppose that all kinds of crowds evenly distribute in each subregion;

In i the subregion, the number of catching an illness is owing to the increment that the mutual migration with other subregion causes is

Set up following difference equation model

$\left\{\begin{array}{c}{\tilde{I}}_i=I_i\left(t\right)+\underset{j=1}{\overset{M}{\mathrm{\Σ}}}[k_{\mathrm{ij}}{I}_j\left(t\right)-k_{\mathrm{ji}}{I}_i\left(t\right)],\\ {\tilde{S}}_i=S_i\left(t\right)+\underset{j=1}{\overset{M}{\mathrm{\Σ}}}[k_{\mathrm{ij}}{S}_j\left(t\right)-k_{\mathrm{ji}}{S}_i\left(t\right)],\\ N_i={\tilde{I}}_i\left(t\right)+{\tilde{S}}_i\left(t\right)+R_i\left(t\right)\\ I_i(t+1)-I_i\left(t\right)=r{\tilde{I}}_i\·{\tilde{S}}_i/{\tilde{N}}_i-\mathrm{\λ}{\tilde{I}}_i,\\ S_i(t+1)-S_i\left(t\right)=-r{\tilde{I}}_i\·{\tilde{S}}_i/{\tilde{N}}_i,\\ R_i(t+1)-R_i\left(t\right)=\mathrm{\λ}{\tilde{I}}_i,\\ i=\mathrm{1,2},...,M\\ \\ \end{array}\right.$

Wherein t and t+1 are respectively the adjacent two days finish times;

Be that number of catching an illness and susceptible number are at intraday mean value;

Coefficient k in the model _IjThe average effect that has reflected crowd's transregional mobile flux; If subregion j every day is proportional to be α _IjNumber move to subregion i, average waiting time is β _Ij, wherein, 0＜β _Ij＜1, parameter alpha _Ij, β _IjAll can obtain by sample survey result statistics; Therefore can be by k _Ij=α _Ijβ _IjCalculate the by stages ac coefficient.