Stability and bifurcation analysis method and device for fractional-order discretized infectious disease model

Abstract

The present invention provides a stability and bifurcation analysis method and device for a fractional order discretized infectious disease model, the method comprising: establishing a fractional order infectious disease model; discretizing the fractional order infectious disease model to obtain a fractional order discretized infectious disease model, calculating the time step parameter of the fractional order discretized infectious disease model as h; calculating the equilibrium point of the fractional order discretized infectious disease model, including a disease-free equilibrium point and an endemic disease equilibrium point; using the time step parameter h and the fractional order as bifurcation parameters, analyzing the stability and bifurcation conditions of the disease-free equilibrium point and the stability and bifurcation conditions of the endemic disease equilibrium point, and obtaining the stability analysis result of the fractional order discretized infectious disease model. The present invention is conducive to revealing the epidemic law of infectious diseases by performing stability analysis on the fractional order discretized infectious disease model, and then formulating control strategies in advance, which is of great significance to the diagnosis and prevention of diseases.

Description

Technical Field

[0001] The present invention relates to the field of nonlinear dynamics of infectious disease models, and in particular to a stability and bifurcation analysis method and device for a fractional-order discretized infectious disease model. Background Art

[0002] After years of research and exploration, people have developed many methods to prevent and control infectious diseases. The research methods of infectious diseases can be roughly divided into four types: descriptive research, analytical research, experimental research and theoretical research. It is relatively unrealistic and difficult to conduct large-scale experiments on infectious diseases, so it is very necessary to conduct in-depth theoretical research. Infectious disease dynamics is an important method for theoretical quantitative research. Mathematical models play an extremely important role in the study of infectious disease dynamics. Using the method of dynamics to establish a mathematical model can be used to study whether a certain infectious disease will continue to spread in a certain area and become a "local disease" in the area or whether the infectious disease will eventually disappear. Using the knowledge of infectious disease dynamics to analyze the existence of system equilibrium points, local and global stability, whether bifurcation occurs, etc., the results of mathematical analysis can provide many powerful theoretical foundations and concepts. It has become a consensus to use mathematical models to discover the transmission mechanism of infectious diseases and predict the epidemic trend of infectious diseases. According to the above analysis, it is very necessary to establish appropriate mathematical models for theoretical analysis for different infectious diseases. Using mathematical models to study the spread dynamics of diseases in the population and formulate corresponding prevention and control strategies and analyze their effectiveness is playing an increasingly important role in the study of infectious diseases and is also a hot spot in the study of the laws of infectious disease transmission. With the further cross-integration of multiple disciplines, the research on infectious disease dynamics modeling has shown some new trends. These theoretical analyses can reveal the epidemic patterns of diseases and seek optimal strategies to control the spread of infectious diseases by analyzing the causes.

[0003] Therefore, it is necessary to establish a suitable mathematical model of infectious diseases and conduct theoretical analysis to study the epidemic patterns of infectious diseases and then adopt the optimal control strategy.

[0004] The above contents are only used to assist in understanding the technical solution of the present invention and do not constitute an admission that the above contents are prior art. Summary of the invention

[0005] The technical problem to be solved by the present invention is mainly to establish a suitable mathematical model of infectious diseases and analyze its dynamic behavior, so as to provide a theoretical basis for disease prevention.

[0006] In order to achieve the above objectives, the present invention provides a stability and bifurcation analysis method and device for a fractional-order discretized infectious disease model.

[0007] According to one aspect of the present invention, a stability and bifurcation analysis method for a fractional-order discretized infectious disease model is provided, comprising the following steps:

[0008] Establish a fractional-order infectious disease model;

[0009] Discretizing the fractional-order infectious disease model to obtain a fractional-order discretized infectious disease model, wherein the time step parameter of the fractional-order discretized infectious disease model is h;

[0010] Calculating and obtaining the equilibrium point of the fractional-order discretized infectious disease model, wherein the equilibrium point includes a disease-free equilibrium point and an endemic equilibrium point;

[0011] Taking the time step parameter h and the fractional order as bifurcation parameters, the stability and bifurcation conditions of the disease-free equilibrium point and the stability and bifurcation conditions of the endemic equilibrium point are analyzed, and the stability and bifurcation analysis results of the fractional-order discretized infectious disease model are obtained.

[0012] Furthermore, the mathematical expression of the fractional-order infectious disease model is:

[0013]

[0014] Among them, D α It represents the α-order differential, α represents the fractional order, 0<α<1, t>0 represents time, the number of susceptible persons is recorded as S(t), which indicates the number of individuals who are not infected with infectious diseases but are very susceptible to infection at time t due to low immunity, the number of infected persons is recorded as I(t), which indicates the number of individuals who are infected with infectious diseases at time t and are very susceptible to infecting other individuals, β represents the bilinear incidence rate, μ+γ represents the population mortality rate, μ represents the predicted mortality rate of infectious diseases, γ represents the additional mortality rate caused by external factors, and Λ represents the population birth rate.

[0015] Furthermore, the mathematical expression of the fractional-order discretized infectious disease model is:

[0016]

[0017] Where x represents dividing the continuous time t into x equal parts, the time step parameter h (h = t / x) refers to the difference between two adjacent time points, n represents the number of finite differences, S n+1 (x) represents the number of susceptible individuals obtained by performing n finite differences, I n+1 (x) represents the number of infected people obtained by performing n finite differences, S n (x) represents the number of susceptible individuals obtained by performing n-1 finite differences, I n (x) represents the number of infected people obtained by performing n-1 finite differences, Represents the Gamma function.

[0018] Furthermore, in the process of calculating the equilibrium point of the fractional-order discretized infectious disease model, it also includes:

[0019] Calculate the basic reproduction number of the fractional-order discretized epidemic model

[0020] According to the basic reproduction number The conditions for determining the existence of the disease-free equilibrium point and the endemic disease equilibrium point are specifically:

[0021] when When , the fractional-order discretized infectious disease model has only one disease-free equilibrium point;

[0022] when When , the fractional-order discretized infectious disease model has not only a disease-free equilibrium point but also an endemic equilibrium point.

[0023] Further, the disease-free equilibrium point is The endemic equilibrium point is

[0024] Furthermore, the step of analyzing the stability and bifurcation conditions of the disease-free equilibrium point specifically includes:

[0025] Calculate the disease-free equilibrium point The Jacob matrix of

[0026]

[0027] Then J(E 0 ) are

[0028]

[0029] And satisfy 0<α≤1;

[0030] Let |λ 1 |=1,|λ 2 |=1 to get the critical time step parameter value

[0031] When When the disease-free equilibrium point E 0 There are at least four different topology types for the parameter value:

[0032] (1) If h satisfies Then the disease-free equilibrium point is a node, which is locally asymptotically stable;

[0033] (2) If h satisfies Then the disease-free equilibrium point is the source point and is unstable;

[0034] (3) If h satisfies or Then the disease-free equilibrium point is a saddle point, which is unstable;

[0035] (4) If or Then the disease-free equilibrium point is a non-hyperbolic point.

[0036] Furthermore, the step of analyzing the stability and bifurcation conditions of the endemic equilibrium point specifically includes:

[0037] Calculate the endemic disease equilibrium point The Jacob matrix

[0038]

[0039] Let the characteristic equation be Let F(λ) = λ 2 -Trλ+Det, let λ 1 and λ 2 are the two roots of the characteristic equation, and we can calculate

[0040]

[0041]

[0042] The two roots of the characteristic equation are

[0043]

[0044] in

[0045]

[0046] Let Det = 1, and get the critical value h of the time step parameter 2 , and let |λ 1 |=1,|λ 2 |=1, calculate the critical value h of the time step parameter 1 and h 3 :

[0047]

[0048]

[0049]

[0050] Further

[0051]

[0052]

[0053]

[0054] When Sometimes, there is

[0055] (1) The endemic equilibrium is asymptotically stable when any of the following conditions are met:

[0056] Δ≥0 and h<h≤h 1 ; Δ<0 and 0<h<h 2 ;

[0057] (2) The endemic equilibrium is unstable when any of the following conditions are met;

[0058] Δ≥0 and h>h 3 ; Δ<0 and h>h 2 ;

[0059] (3) When Δ<0 and h=h 2 When , the endemic equilibrium point is unstable;

[0060] (4) The endemic equilibrium point is a non-hyperbolic point when any of the following conditions is met;

[0061] Δ≥0 and h=h 1 or 3 ; Δ<0 and h=h 2 ;

[0062] Definition of Ω 1 ,Ω 2 ,Ω 3 is the set of model parameters when the time step parameter h takes a certain value, then

[0063] (α,h,β,Λ,μ,γ)∈Ω 1 ∪Ω 2

[0064]

[0065]

[0066] When the time step parameter h is within h 1 The nearby neighborhood values ​​​​are taken, and the parameters satisfy (α,h,β,Λ,μ,γ)∈Ω 1 ; or the time step parameter h in h 3 The nearby neighborhood values ​​​​are taken, and the parameters satisfy (α,h,β,Λ,μ,γ)∈Ω 2 , the endemic equilibrium point flips and bifurcates;

[0067] (α,h,β,Λ,μ,γ)∈Ω 3

[0068]

[0069] When the time step parameter h is within h 2 When the nearby neighborhood takes values, and the parameters satisfy (α,h,β,Λ,μ,γ)∈Ω 3 , a secondary Hope bifurcation occurs at the endemic equilibrium point.

[0070] According to another aspect of the present invention, a stability and bifurcation analysis device for a fractional-order discretized infectious disease model is provided, comprising the following units:

[0071] A model building unit, used to build a fractional-order infectious disease model;

[0072] A discretization unit, used for discretizing the fractional-order infectious disease model to obtain a fractional-order discretized infectious disease model, wherein the time step parameter of the fractional-order discretized infectious disease model is h;

[0073] A calculation unit, used for calculating the equilibrium point of the fractional-order discretized infectious disease model, wherein the equilibrium point includes a disease-free equilibrium point and an endemic equilibrium point;

[0074] The stability analysis unit is used to use the step size parameter h and the fractional order as bifurcation parameters to analyze the stability and bifurcation conditions of the disease-free equilibrium point and the stability and bifurcation conditions of the endemic equilibrium point, and obtain the stability and bifurcation analysis results of the fractional-order discretized infectious disease model.

[0075] The technical solution provided by the present invention has the following beneficial effects:

[0076] A fractional-order discretized infectious disease model was established, and stability analysis and bifurcation analysis were performed on it. First, the fractional-order infectious disease model was established; the fractional-order infectious disease model was discretized to obtain the fractional-order discretized infectious disease model, and the equilibrium points of the fractional-order discretized infectious disease model were calculated, including the disease-free equilibrium point and the endemic equilibrium point; the influence of the critical time step parameter changing within a certain range on the stability was obtained by theorem calculation. When h is less than the critical value, the endemic equilibrium point tends to be stable; when h is greater than the critical value, a period doubling bifurcation (flip bifurcation) occurs at the endemic equilibrium point, and as h increases within the range, its periodic orbit gradually appears; when h continues to increase, the system tends to be chaotic. When the time step parameter is kept fixed, the effect of the fractional order on the number of susceptible and infected persons is studied, and the fractional order is increased within a certain range: when the fractional order is less than the critical value, the endemic disease equilibrium point is stable; when the fractional order increases to the critical value, the endemic disease equilibrium point loses its stability, and a limit cycle is generated at the endemic disease equilibrium point; when it continues to increase, the system tends to chaos, and finally, numerical simulation is used to verify the correctness of the inference, and the epidemic law of the disease is revealed through mathematical models, and the optimal strategy is sought to control the epidemic of infectious diseases by analyzing the causes. BRIEF DESCRIPTION OF THE DRAWINGS

[0077] The present invention will be further described below with reference to the accompanying drawings and embodiments, in which:

[0078] Figure 1 It is a flow chart of a stability and bifurcation analysis method of a fractional-order discretized infectious disease model of the present invention;

[0079] Figure 2 This is the bifurcation diagram of the periodic doubling of the number of susceptible people when α=0.99 of the present invention;

[0080] Figure 3 This is the bifurcation diagram of the periodic doubling of the number of infected persons when α=0.99 of the present invention;

[0081] Figure 4 The 2-period orbit phase diagram of the present invention when h=2.5;

[0082] Figure 5 This is the 8-period orbit phase diagram of the present invention when h=2.8;

[0083] Figure 6 The 16-period orbit phase diagram of the present invention when h=2.82;

[0084] Figure 7 This is the chaotic phase diagram of the present invention when h=2.95;

[0085] Figure 8 This is the bifurcation diagram of the present invention when α=0.82;

[0086] Fig. 9 This is the bifurcation diagram of the present invention when α=0.8266;

[0087] Fig.10 is the limit cycle generated when α=0.8266 in the present invention;

[0088] Fig.11 This is the chaotic phase diagram of the instability of the equilibrium point when α=0.913;

[0089] Fig.12 This is a structural diagram of a stability and bifurcation analysis device for a fractional-order discretized infectious disease model of the present invention. DETAILED DESCRIPTION

[0090] To make the objectives, technical solutions and advantages of the present invention more clear, the embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0091] It should be understood that the specific embodiments described herein are only used to explain the present invention, and are not used to limit the present invention.

[0092] The fractional order model is a generalization of the classic integer order model. It is beneficial and worthwhile to introduce it into the infectious disease model. The knowledge of infectious disease dynamics is used to analyze the existence of system equilibrium points, local and global stability, and whether bifurcation occurs. The results of mathematical analysis can provide many powerful theoretical foundations and concepts. It has become a consensus to use mathematical models to discover the transmission mechanism of infectious diseases and predict the epidemic trend of infectious diseases.

[0093] Firstly, a continuous-time infectious disease model is established and discretized by backward difference method. Then the disease-free equilibrium point and endemic equilibrium point of the model as well as the basic reproduction number are calculated.

[0094] Determine the stability and bifurcation conditions of the endemic disease equilibrium point with the change of the time step parameter when the fractional order remains unchanged, and the stability and bifurcation conditions of the endemic disease equilibrium point with the change of the fractional order when the time step parameter is fixed.

[0095] When studying the stability of the system, we first proved that any solution of the model under given initial conditions is positive and ultimately bounded. If the disease-free equilibrium point is locally asymptotically stable, the disease will not break out. For the endemic equilibrium point, we established conditions to ensure local stability and gave the conditions for the existence of bifurcation. Finally, we used numerical simulation to verify the conjecture.

[0096] Reference Figure 1 , Figure 1 The flowchart of the stability and bifurcation analysis method of a fractional order discretized infectious disease model of the present invention is as follows. In this embodiment, a stability and bifurcation analysis method of a fractional order discretized infectious disease model is provided, comprising the following steps:

[0097] Step S1: Establish a fractional-order infectious disease model:

[0098] Given an integer-order epidemic model:

[0099]

[0100] The number of susceptible people is recorded as S(t), which indicates the number of individuals who are not infected by infectious diseases but are very susceptible to infection at time t due to low immunity. The number of infected people is recorded as I(t), which indicates the number of individuals who are infected by infectious diseases at time t and are very susceptible to infecting other individuals. β is the bilinear incidence rate, μ+γ refers to the population mortality rate, μ is the predicted mortality rate of infectious diseases, γ is the additional mortality rate caused by external factors, and the population birth rate is Λ.

[0101] With the rapid development of fractional-order differential equations, fractional-order differential models can well describe the memory and genetic material of matter, which exist in most biological systems. They can also overcome the defects of integer-order differential models, such as the poor agreement between theory and experimental results.

[0102] Therefore, in this embodiment, the integer-order infectious disease model is converted into a fractional-order integer-order infectious disease model.

[0103]

[0104] Where α represents the fractional order, 0<α<1, t>0.

[0105] Step S2: discretize the fractional-order infectious disease model to obtain a fractional-order discretized infectious disease model with a time step parameter of h:

[0106] The fractional-order epidemic model is discretized using the finite difference equation, and the right side of equation (2) is approximated by D a X(t)=f(x(t)),t>0,α∈(0,1),assuming S(0)=S 0 ,I(0)=I 0 is the initial condition of the system. Discretize the entire system process into x small processes, that is, divide the continuous time into x equal parts. After discretization, the time step parameter h (h = t / x) refers to the difference between the two previous and next time points, that is, the length of time for each small equal part. The discretization of the equation is given by the following formula.

[0107]

[0108] When t∈[0,h),t / h∈[0,1), we have

[0109]

[0110]

[0111] When t∈[h,2h),t / h∈[1,2], we have

[0112]

[0113]

[0114] When t∈[nh,(n+1)h), the above process is iterated n times to obtain the fractional-order discretized epidemic model

[0115]

[0116] Step S3, calculate the equilibrium points of the fractional-order discretized infectious disease model, including the disease-free equilibrium point and the endemic equilibrium point:

[0117] Substituting I(t) = 0 into equation (2), we get the disease-free equilibrium point Let the left side of equation (2) be 0, and the endemic disease equilibrium point is calculated as

[0118] In epidemiology, there is an important concept called the basic reproduction number. It indicates the number of new patients infected by a patient during the average illness period when he is placed in a susceptible population at the early stage of the disease. Obviously, as a threshold for distinguishing whether a disease is prevalent or not, its practical meaning is very obvious. When , that is, the number of new patients infected by a patient during the average illness period is less than 1, the model has only one globally asymptotically stable disease-free equilibrium point, and the disease will eventually die out; when When βS > 1, the number of new patients infected by a patient during the average illness period is greater than 1. In this case, the model has an endemic equilibrium point in addition to a disease-free equilibrium point. The disease will always exist, thus forming an endemic disease. The infection matrix of the system is F = βS 0 , the regeneration matrix of the system is V = μ + γ.

[0119] The basic reproduction number is equal to the reproduction matrix (FV -1 ) spectral radius, the basic reproduction number of this fractional-order discretized epidemic model is

[0120]

[0121] At this time, the endemic equilibrium point of the fractional-order discretized infectious disease model can be expressed as

[0122] Take a fixed point E in the fractional-order discretized epidemic model * (S * , I *), the Jacob matrix of the fractional-order discretized epidemic model is

[0123]

[0124] Step S4: Using the step size parameter h and the fractional order as bifurcation parameters, analyze the stability and bifurcation conditions of the disease-free equilibrium point and the stability and bifurcation conditions of the endemic equilibrium point, and obtain the stability and bifurcation analysis results of the fractional-order discretized infectious disease model.

[0125] Step S4.1: Stability of disease-free equilibrium

[0126] Conclusion 1 If Then the disease-free equilibrium point E 0 There are at least four different topological types for the parameter values

[0127] (1) If the time step parameter h satisfies Then the disease-free equilibrium point is the node.

[0128] (2) If h satisfies Then the disease-free equilibrium point is the source point.

[0129] (3) If h satisfies or Then the disease-free equilibrium point is a saddle point.

[0130] (4) If or Then the disease-free equilibrium point is a non-hyperbolic point.

[0131] Proof will Substituting into formula (9), the Jacob matrix of the disease-free equilibrium point is:

[0132]

[0133] Then J(E 0 ) are

[0134]

[0135] And satisfy 0<α≤1;

[0136] In order to study the stability of the fixed point of the system, the following lemma is used

[0137] Lemma 1 Let λ 1 and λ 2 are the two roots of the matrix J(M) and are defined as follows:

[0138] (1) If |λ 1 |<1 and |λ 2 |<1, then M(x * ,y* ) is a locally asymptotically stable equilibrium point (node)

[0139] (2) If |λ 1 |>1 and |λ 2 |>1, then M(x * ,y * )’s equilibrium point is unstable (source point).

[0140] (3) If |λ 1 |<1 and |λ 2 |>1(or |λ 1 |>1 and |λ 2 |<1), then M(x * ,y * ) is an unstable equilibrium point (saddle point).

[0141] (4) If |λ 1 |=1or|λ 2 |=1, then M(x * ,y * ) is called a non-hyperbolic equilibrium point.

[0142] Substituting the values ​​of the two characteristic roots into |λ 1 |=1,|λ 2 |=1 to get the critical time step parameter value

[0143] That’s proof When the disease-free equilibrium point E 0 The parameter values ​​satisfy the following conditions

[0144] (1) If the time step parameter h satisfies Then the disease-free equilibrium point is the node, which is locally asymptotically stable.

[0145] (2) If h satisfies Then the disease-free equilibrium point is the source point and is unstable.

[0146] (3) If h satisfies or Then the disease-free equilibrium point is a saddle point, which is unstable.

[0147] (4) If or Then the disease-free equilibrium point is a non-hyperbolic point.

[0148] Step S4.1: Stability of endemic equilibrium

[0149] Conclusion 2 Sometimes, there are

[0150] 1. When any of the following conditions is met, the endemic equilibrium point E 1 is asymptotically stable (node)

[0151] Δ≥0 and h<h≤h 1 ; Δ<0 and 0<h<h 2 .

[0152] 2. When any of the following conditions is met, the endemic equilibrium point E 1 is unstable (source)

[0153] Δ≥0 and h>h 3 ; Δ<0 and h>h 2 .

[0154] 3. When Δ<0 and h=h 2 When the endemic equilibrium point E 1 is unstable (saddle point).

[0155] 4. When any of the following conditions is met, the endemic equilibrium point E 1 is a non-hyperbolic point

[0156] Δ≥0 and h=h 1 or 3 ; Δ<0 and h=h 2 .

[0157] Prove that E 1 Substituting into formula (9) we can get the Jacob matrix of endemic equilibrium point:

[0158]

[0159] Let the characteristic equation be Let F(λ) = λ 2 -Trλ+Det, let λ 1 and λ 2 are the two roots of the characteristic equation, and we can calculate

[0160]

[0161] The two eigenvalues ​​of the equation are

[0162]

[0163] in

[0164]

[0165] Lemma 2 Let the characteristic equation be Let F(λ) = λ 2 -Trλ+Det, let λ 1 and λ 2 are the two roots of the characteristic equation. Let F(1)>0, then we have

[0166] 1. |λ 1 |<1 and |λ 2 |<1 if and only if F(–1)>0 and Det<1;

[0167] 2. |λ 1 |<1 and |λ 2 |>1 or (|λ 1 |>1 and |λ 2 |<1) if and only if F(–1)<0;

[0168] 3. |λ 1 |>1 and |λ 2 |>1 if and only if F(–1)>0 and Det>1;

[0169] 4. Lambda 1 =–1 and λ 2 ≠1 if and only if F(–1)=0 and Tr≠0,2;

[0170] 5. Lambda 1 and λ 2 is a complex number, and |λ 1 |=|λ 2 |If and only if Tr 2 -4Det<0 and Det=1.

[0171] In Lemma 2, Det=1 is calculated by equation (12), and the critical value h is obtained 2 . And substituting the two characteristic roots of the characteristic equation (13) into Lemma 2 |λ 1 |=1,|λ 2 |=1, calculate the critical value h of the time step parameter 1 and h 3 .

[0172]

[0173] Substituting equation (14) into the denominator of equation (15), we obtain

[0174]

[0175] That’s proof Sometimes

[0176] 1. E1 is asymptotically stable when any of the following conditions is met (node)

[0177] Δ≥0 and h<h≤h 1 ; Δ<0 and 0<h<h 2 .

[0178] 2. When h satisfies one of the conditions E 1 is unstable (source)

[0179] Δ≥0 and h>h 3 ; Δ<0 and h>h 2 .

[0180] 3. When Δ<0 and h=h 2 , E1 is unstable (saddle point).

[0181] 4. When any of the following conditions is met, E1 is a non-hyperbolic point.

[0182] Δ≥0 and h=h 1 or 3 ; Δ<0 and h=h 2 .

[0183] Definition of Ω 1 ,Ω 2 ,Ω 3 is the set of model parameters when the variable h takes certain values, the above conclusion can be expressed as

[0184] (α,h,β,Λ,μ,γ)∈Ω 1 ∪Ω 2

[0185]

[0186]

[0187] When the variable h is in h 1 The nearby neighborhood values, and the system parameters satisfy (α,h,β,Λ,μ,r)∈Ω 1 ; or variable h in h 3 The nearby neighborhood values, and the system parameters satisfy (α,h,β,Λ,μ,γ)∈Ω 2 , E 1 The equilibrium point flips and bifurcates.

[0188] (α,h,β,Λ,μ,γ)∈Ω 3

[0189]

[0190] When the variable h is in h 2 The nearby neighborhood values, and the system parameters satisfy (α,h,β,Λ,μ,γ)∈Ω 3 , E 1 A secondary Hop bifurcation occurs at the equilibrium point.

[0191] Step S5: Numerical simulation

[0192] In this embodiment, simulation is performed by using MATLAB software, and a simulation diagram is drawn.

[0193] Let the system parameters be (α,h,β,Λ,μ,γ)=(0.99,2.305,0.09,3.5,0.145,0.12)∈Ω 1 First, we select h as the variable and the initial condition (S 0 ,I 0 )=(3.7,7.8), take α=0.99, when h<2.305, E 1 is gradually stable.

[0194] refer to Figure 2 and Figure 3 , when h = 2.305, a flip bifurcation occurs and the stability is lost. When h∈(2.305,2.83), periodic orbits appear in the range as h increases; refer to Figure 4-Figure 7 When h=2.5, a 2-period orbit appears; when h=2.75, a 4-period orbit appears; when h=2.8, an 8-period orbit appears; when h=2.82, a 16-period orbit appears; when h=2.9, a 12-period orbit appears; when h=2.95, the system tends to chaos.

[0195] Let the system parameters be (α,h,β,Λ,μ,γ)=(0.826,8.1,0.1,1.5,0.2,0.3)∈Ω 3 , the time step parameter h takes a fixed value of h = 8.1, and the initial condition (S 0 ,I 0 )=(5.1,0.9), select variable as α, reference Figure 8-Figure 11 , it can be observed from the simulation graph that its value range is 0.8≤α≤0.99, at which time a secondary Hopp bifurcation occurs. When α<0.826, E 1 is stable, and loses stability when α = 0.826. When the fractional order increases within the range of (0.826, 0.87), a limit cycle appears, and chaos occurs when α increases further within the range of values.

[0196] The present invention mainly establishes a discretized fractional-order infectious disease model based on nonlinear dynamics knowledge, and further performs stability and bifurcation analysis. First, the integer-order infectious disease model is converted into a fractional-order infectious disease model through Caputo's theorem, and then it is discretized by Euler's backward difference method to obtain the final model expression. Then, the disease-free equilibrium point, endemic disease equilibrium point, basic reproduction number and other epidemiological professional concepts are calculated to judge its local and global stability. The influence of the critical time step parameter changing within a certain range on stability is obtained by theorem calculation. When h is less than the critical value, the endemic disease equilibrium point tends to be stable; when h is greater than the critical value, a periodic doubling bifurcation (flip bifurcation) occurs at the endemic disease equilibrium point, and as h increases within the range, its periodic orbit gradually appears; when h continues to increase, the system tends to be chaotic. When the time step parameter is kept fixed, the effect of the fractional order on the number of susceptible and infected persons is studied, and the fractional order is increased within a certain range: when the fractional order is less than the critical value, the endemic equilibrium point is stable; when the fractional order increases to the critical value, the endemic equilibrium point loses its stability, and a limit cycle is generated at the endemic equilibrium point; when it continues to increase, the system tends to chaos.

[0197] It can be concluded that the discrete-time model is obtained from the discretization of the fractional-time system integration model. The discretization process provides key terms such as h (time step parameter) and α (fractional order), and then these variables are changed to describe the dynamic behavior of the model. When h and α are changed, the model exhibits several complex dynamic behaviors, including secondary Hopkins bifurcation and flip bifurcation. The study shows that the fractional-order discrete epidemic model exhibits rich dynamic characteristics.

[0198] As an optional implementation, the present invention also provides a stability and bifurcation analysis device for a fractional-order discretized infectious disease model, referring to Fig.12 , the device comprises:

[0199] Model building unit 1, used to build a fractional-order infectious disease model;

[0200] A discretization unit 2 is used to discretize the fractional-order infectious disease model to obtain a fractional-order discretized infectious disease model, wherein the time step parameter of the fractional-order discretized infectious disease model is h;

[0201] A calculation unit 3 is used to calculate the equilibrium point of the fractional-order discretized infectious disease model, wherein the equilibrium point includes a disease-free equilibrium point and an endemic equilibrium point;

[0202] The stability analysis unit 4 is used to use the step size parameter h and the fractional order as bifurcation parameters to analyze the stability and bifurcation conditions of the disease-free equilibrium point and the stability and bifurcation conditions of the endemic equilibrium point, and obtain the stability analysis result of the fractional-order discretized infectious disease model.

[0203] It should be noted that, in this article, the terms "include", "comprises" or any other variations thereof are intended to cover non-exclusive inclusion, so that a process, method, article or system including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or system. In the absence of further restrictions, an element defined by the sentence "comprises a ..." does not exclude the existence of other identical elements in the process, method, article or system including the element.

[0204] The serial numbers of the embodiments of the present invention are for descriptive purposes only and do not represent the superiority or inferiority of the embodiments. In a unit claim that lists several means, several of these means may be embodied by the same hardware item. The use of the words first, second, and third, etc. does not indicate any order and these words may be interpreted as identifiers.

[0205] The above are only preferred embodiments of the present invention, and are not intended to limit the patent scope of the present invention. Any equivalent structure or equivalent process transformation made using the contents of the present invention specification and drawings, or directly or indirectly applied in other related technical fields, are also included in the patent protection scope of the present invention.

Claims

1. A stability and bifurcation analysis method for a fractional-order discretized infectious disease model, characterized in that: It includes the following steps: Establish a fractional-order infectious disease model; Discretize the fractional-order infectious disease model to obtain a fractional-order discretized infectious disease model, where the time step parameter of the fractional-order discretized infectious disease model is h; Calculate the equilibrium points of the fractional-order discretized infectious disease model, where the equilibrium points include a disease-free equilibrium point and an endemic equilibrium point; Take the time step parameter h and the fractional-order as bifurcation parameters, analyze the stability and bifurcation conditions of the disease-free equilibrium point, and the stability and bifurcation conditions of the endemic equilibrium point, to obtain the stability and bifurcation analysis results of the fractional-order discretized infectious disease model; The mathematical expression of the fractional-order infectious disease model is: Among them, D α represents the α-order differential, α represents the fractional order, 0<α<1, t>0 represents time, the number of susceptible persons is recorded as S(t), which indicates the number of individuals who are not infected by infectious diseases but are very susceptible to infection at time t due to low immunity, the number of infected persons is recorded as I(t), which indicates the number of individuals who are infected by infectious diseases at time t and are very susceptible to infecting other individuals, β represents the bilinear incidence rate, μ+γ represents the population mortality rate, μ represents the predicted mortality rate of infectious diseases, γ represents the additional mortality rate caused by external factors, and Λ represents the population birth rate; The mathematical expression of the fractional-order discretized infectious disease model is: Where x represents dividing the continuous time t into x equal parts, the time step parameter h (h = t / x) refers to the difference between two adjacent time points, n represents the number of finite differences, S n+1 (x) represents the number of susceptible individuals obtained by performing n finite differences, I n+1 (x) represents the number of infected people obtained by performing n finite differences, S n (x) represents the number of susceptible individuals obtained by performing n-1 finite differences, I n (x) represents the number of infected people obtained by performing n-1 finite differences, represents the Gamma function; In the process of calculating the equilibrium points of the fractional-order discretized infectious disease model, it also includes: Calculate the basic reproduction number of the fractional-order discretized epidemic model According to the basic reproduction number The conditions for determining the existence of the disease-free equilibrium point and the endemic disease equilibrium point are specifically: when When , the fractional-order discretized infectious disease model has only one disease-free equilibrium point; when When , the fractional-order discretized infectious disease model has not only a disease-free equilibrium point, but also an endemic equilibrium point; The disease-free equilibrium point is The endemic equilibrium point is The steps of analyzing the stability and bifurcation conditions of the disease-free equilibrium point specifically include: Calculate the disease-free equilibrium point The Jacob matrix of Then the two roots of J(E0) are And satisfy 0<α≤1; Let |λ1|=1, |λ2|=1, and get the critical time step parameter value When When , the parameter value of the disease-free equilibrium point E0 has at least four different topological types: (1) If h satisfies Then the disease-free equilibrium point is a node, which is locally asymptotically stable; (2) If h satisfies Then the disease-free equilibrium point is the source point and is unstable; (3) If h satisfies or Then the disease-free equilibrium point is a saddle point, which is unstable; (4) If or Then the disease-free equilibrium point is a non-hyperbolic point; The steps of analyzing the stability and bifurcation conditions of the endemic equilibrium point specifically include: Calculate the endemic disease equilibrium point The Jacob matrix of Let the characteristic equation be Let F(λ) = λ 2 -Trλ+Det, let λ1 and λ2 be the two roots of the characteristic equation, and we can get The two roots of the characteristic equation are respectively Where Let Det = 1 to obtain the critical value h2 of the time step parameter, and let |λ1| = 1, |λ2| = 1, and calculate the critical values h1 and h3 of the time step parameter: Further obtain When Sometimes, there is (1) When any of the following conditions is satisfied, the endemic equilibrium point is asymptotically stable; Δ≥0 and h < h ≤ h1; Δ < 0 and 0 < h < h2; (2) When any of the following conditions is satisfied, the endemic equilibrium point is unstable; Δ≥0 and h > h3; Δ < 0 and h > h2; (3) When Δ < 0 and h = h2, the endemic equilibrium point is unstable; (4) When any of the following conditions is satisfied, the endemic equilibrium point is a non-hyperbolic point; Δ≥0 and h = h1 or h3; Δ < 0 and h = h2; Define Ω1, Ω2, Ω3 as the model parameter sets when the time step parameter h takes certain values respectively, then there is (α, h, β, Λ, μ, γ) ∈ Ω1UΩ2 When the time step parameter h takes values in the neighborhood of h1 and the parameters satisfy (α, h, β, Λ, μ, γ) ∈ Ω1; or when the time step parameter h takes values in the neighborhood of h3 and the parameters satisfy (α, h, β, Λ, μ, γ) ∈ Ω2, the endemic equilibrium point undergoes a flip bifurcation; (α, h, β, Λ, μ, γ) ∈ Ω3 When the time step parameter h takes values in the neighborhood of h2 and the parameters satisfy (α, h, β, Λ, μ, γ) ∈ Ω3, the endemic equilibrium point undergoes a secondary Hopf bifurcation.

2. A stability and bifurcation analysis control device for a fractional-order discretized infectious disease model, used to implement a stability and bifurcation analysis method for a fractional-order discretized infectious disease model as claimed in claim 1, characterized in that: It includes the following units: A model establishment unit for establishing a fractional-order infectious disease model; A discretization unit for discretizing the fractional-order infectious disease model to obtain a fractional-order discretized infectious disease model, where the time step parameter of the fractional-order discretized infectious disease model is h; A calculation unit for calculating the equilibrium points of the fractional-order discretized infectious disease model, where the equilibrium points include a disease-free equilibrium point and an endemic equilibrium point; The stability analysis unit is used to use the step size parameter h and the fractional order as bifurcation parameters to analyze the stability and bifurcation conditions of the disease-free equilibrium point and the stability and bifurcation conditions of the endemic equilibrium point, and obtain the stability and bifurcation analysis results of the fractional-order discretized infectious disease model.

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