Dual-state feedback controller considering spatio-temporal diffusion effects and individual contact heterogeneity

Abstract

The present invention is a two-state feedback controller that takes into account the influence of spatio-temporal diffusion and individual contact heterogeneity. The controller design method includes: Step 1: Establish an SIR epidemic model; Step 2: Apply a two-state feedback controller to the SIR epidemic model; Step 3: Linearize the SIR epidemic model under the action of the two-state feedback controller at the equilibrium point to obtain the characteristic equation of the linearized controlled SIR epidemic model; Step 4: Select the time delay as the bifurcation parameter, and through stability analysis and bifurcation analysis of the characteristic equation of the linearized controlled SIR epidemic model, select appropriate state feedback parameters to make the model locally asymptotically stable near the equilibrium point. The two-state feedback controller of the present invention can perform feedback regulation control according to the state variables of susceptible individuals and infected individuals at the same time, and does not affect the characteristics of the original system, so as to efficiently and safely regulate the virus transmission process.

Description

Technical Field

[0001] The present invention belongs to the technical field of controllers, and specifically relates to a dual-state feedback controller for an SIR infectious disease model considering the influence of spatio-temporal diffusion and individual contact heterogeneity. Background Art

[0002] The spread of infectious diseases seriously threatens human life safety. At the same time, since large-scale experiments cannot be carried out on the spread of infectious diseases, establishing a mathematical model that accurately describes the transmission characteristics of the virus, conducting dynamic analysis on the mathematical model, exploring the virus transmission law, and predicting its transmission and development trend are one of the means to effectively control infectious diseases.

[0003] The Hopf bifurcation phenomenon is a typical dynamic bifurcation phenomenon, and its control is a research hotspot in the field of bifurcation control. As a control means, compared with previous control strategies, bifurcation control not only focuses on how to make the system with applied control better stabilize at the equilibrium state of the system, but also focuses on adjusting the controller gain to make the system exhibit bifurcation phenomena. Since the controller needs to solve more complex dynamic behaviors of the control system, the difficulty of the analysis process will also increase accordingly. So far, only some relatively simple control strategies are applicable to the bifurcation control of complex networks. Summary of the Invention

[0004] To solve the above problems, the present invention proposes a dual-state feedback controller for an SIR infectious disease model considering the influence of spatio-temporal diffusion and individual contact heterogeneity, which can perform feedback regulation control according to the state variables of susceptibles and infectives at the same time, and does not affect the characteristics of the original system, so as to efficiently and safely regulate the virus transmission process.

[0005] To achieve the above object, the present invention is realized by the following technical solutions:

[0006] The present invention is a dual-state feedback controller for an SIR infectious disease model considering the influence of spatio-temporal diffusion and individual contact heterogeneity, including the following steps:

[0007] Based on the traditional SIR infectious disease model, an SIR infectious disease model considering time delay and individual contact heterogeneity under partial differential equation description is established, the equilibrium point information is analyzed, and the basic reproduction number is calculated;

[0008] A dual-state feedback controller is applied to the uncontrolled SIR infectious disease model considering the influence of spatio-temporal diffusion and individual contact heterogeneity to obtain a controlled SIR infectious disease model;

[0009] The SIR infectious disease model under the action of the dual-state feedback controller is linearized at the equilibrium point to obtain the characteristic equation of the linearized controlled SIR infectious disease model;

[0010] Select the time delay as the bifurcation parameter. Through the stability analysis and bifurcation analysis of the characteristic equation of the controlled SIR epidemic model after linearization, select appropriate state feedback parameters to make the model locally asymptotically stable near the equilibrium point.

[0011] The established SIR epidemic model considering the influence of spatio-temporal diffusion and individual contact heterogeneity:

[0012]

[0013] Among them, S(x, t), I(x, t), and R(x, t) represent the numbers of susceptibles, infected individuals, and recovered individuals at position x and time t respectively. The non-negative constants d1, d2, and d3 represent the diffusion coefficients of susceptibles, infected individuals, and recovered individuals respectively. L is a positive bounded constant. is the Laplace operator in the space Ω, which is used to describe the random diffusion motion. The Neumann boundary condition indicates that this model is a closed space, and n is the outer unit normal vector with a smooth boundary Assume that the total population in this area is S(x, t)+I(x, t)+R(x, t) = 1, and the birth rate and natural death rate of the population in this area are both μ. β represents the virus infection rate, P0 represents the probability that an uninfected individual does not contact others, P1 represents the probability that an infected individual does not contact others, q0 = 1 - p0 represents the time proportion of an uninfected individual mixing with other individuals in the group, (q1 = 1 - p1) represents the time proportion of an infected individual mixing with other individuals in the group, and p r =(q0 - q1) / q0 represents the percentage reduction in the time of mixing with other individuals due to the infection of susceptibles. α is the recovery rate of infected individuals. According to the biological meaning of the system, these parameters are all non-negative constants and q1 ≤ q0.

[0014] It is calculated that the disease-free equilibrium point E0 = (1, 0, 0), and the endemic equilibrium point E1 = (S * , I * , R * ).

[0015]

[0016] The basic reproduction number is R0, where

[0017]

[0018] The expression of the dual-state feedback controller designed by the present invention is as follows:

[0019]

[0020] where m1, m2, m3, n1, n2, and n3 are the gain parameters of the state feedback controller, respectively. This controller can achieve bifurcation control without changing the original characteristics of the system.

[0021] The SIR epidemic model considering the influence of spatio-temporal diffusion and individual contact heterogeneity after adding the controller is:

[0022]

[0023] Linearizing the controlled model (5) at the endemic equilibrium point gives the characteristic equation:

[0024]

[0025] That is:

[0026]

[0027] where

[0028]

[0029] Select the time delay as the bifurcation parameter and conduct a stability analysis of the controlled system.

[0030] (1) When not considering the time delay τ = 0 of virus transmission, the characteristic equation of the controlled system (1) is:

[0031]

[0032] According to the Routh-Hurwitz criterion, the necessary and sufficient condition for the roots of the above equation to have negative real parts is:

[0033]

[0034] According to the characteristics of the system parameters, the parameter conditions that satisfy the inequality (9) are:

[0035]

[0036] Therefore, adjusting the controller parameters to satisfy the above two inequalities (H1), the endemic equilibrium point E1 of the controlled system (1) is locally asymptotically stable without time delay;

[0037] (2) When considering the time delay of virus transmission (τ > 0), substituting λ = iω into the characteristic equation (7) and separating the real and imaginary parts, we can obtain:

[0038]

[0039] where

[0040]

[0041] Squaring and adding the two equations of Equation (11) gives:

[0042]

[0043] When k = 0, the condition for the above Equation (13) to have at least one positive root ω0 is:

[0044] B 2 -C 2 <0 (14)

[0045] Differentiating the characteristic equation (7) with respect to τ and verifying the transversality condition gives (2Cω0 2 +Aa 23 ω0 2 )cosω0τ0+(ACω0 - 2a 23 ω0 3 )sinω0τ0+(Ca 23 -Ca 23 τ0 - a 23 2 )ω0 2 >0, a Hopf bifurcation will occur in the system, and the bifurcation point is

[0046]

[0047] When k ≥ 1, if the system parameters satisfy

[0048] A 2 a 23 2 +C 2 -2Ba 23 2 -a 23 4 >0

[0049] A 2 C 2 +a 23 2 B 2 -2BC 2 -2C 2 a 23 2 >0

[0050] B 2 C 2 -C 4 >0

[0051] At this time, the endemic equilibrium point of the system is locally asymptotically stable.

[0052] By observing the expression (15) of the bifurcation point, it can be seen that by adjusting the feedback parameters of the feedback state controller, the bifurcation point can be advanced or delayed, thereby achieving the control of the spread of infectious diseases.

[0053] The beneficial effects of the present invention are as follows:

[0054] 1. The present invention applies the controller to the SIR infectious disease model, improving the overall simulation accuracy. By setting the feedback parameters of two state nodes, the expansion of the model's stable region can be efficiently achieved, and the simulation effect is also better;

[0055] 2. The SIR infectious disease model proposed in the present invention, which considers the influence of spatio-temporal diffusion and individual contact heterogeneity, can better fit the actual virus transmission model. The introduced spatial diffusion term has important guiding significance for the study of infectious disease dynamics;

[0056] 3. The dual-state feedback controller designed in the present invention has strong applicability and is also applicable to other complex dynamic networks;

[0057] 4. Compared with other controllers, the controller of the present invention does not change the characteristics of the original system. By adjusting the corresponding controller parameters, the bifurcation time point can be effectively advanced or delayed, providing effective suggestions for stabilizing the spread of diseases. Description of the Drawings

[0058] Figure 1 It is the flowchart of the method described in the present invention.

[0059] Figure 2 It is the waveform diagram of the stability of susceptibles when τ = 5 for the uncontrolled model (20).

[0060] Figure 3 It is the waveform diagram of the stability of infecteds when τ = 5 for the uncontrolled model (20).

[0061] Figure 4 It is the waveform diagram of the stability of recovereds when τ = 5 for the uncontrolled model (20).

[0062] Figure 5 It is the waveform diagram of the instability of susceptibles when τ = 7.5 for the uncontrolled model (20).

[0063] Figure 6 It is the waveform diagram of the instability of infecteds when τ = 7.5 for the uncontrolled model (20).

[0064] Figure 7 It is the waveform diagram of the instability of recovereds when τ = 7.5 for the uncontrolled model (20).

[0065] Figure 8The waveform diagram of the susceptible population returning to stability when the controlled model (21) is under the controller parameters m1 = 0.5, m2 = 0.1, m3 = 0.1, n1 = -0.2, n2 = -0.2, n3 = -0.3 and τ = 7.5.

[0066] Figure 9 The waveform diagram of the infected population returning to stability when the controlled model (21) is under the controller parameters m1 = 0.5, m2 = 0.1, m3 = 0.1, n1 = -0.2, n2 = -0.2, n3 = -0.3 and τ = 7.5.

[0067] Figure 10 The waveform diagram of the recovered population returning to stability when the controlled model (21) is under the controller parameters m1 = 0.5, m2 = 0.1, m3 = 0.1, n1 = -0.2, n2 = -0.2, n3 = -0.3 and τ = 7.5. Detailed implementation manner

[0068] The embodiments of the present invention will be disclosed below with diagrams. For the sake of clarity, many practical details will be described together in the following narrative. However, it should be understood that these practical details are not used to limit the present invention. That is to say, in some embodiments of the present invention, these practical details are not necessary.

[0069] The present invention is a dual-state feedback controller for an SIR infectious disease model considering the influence of spatio-temporal diffusion and individual contact heterogeneity. The design method of the dual-state feedback controller includes the following steps:

[0070] Step 1: Based on the traditional SIR infectious disease model, establish an SIR infectious disease model considering time delay and individual contact heterogeneity under the description of partial differential equations, analyze the equilibrium point information, and calculate the basic reproduction number.

[0071] The SIR infectious disease model considering time delay and individual contact heterogeneity is expressed as:

[0072]

[0073] where: S(x, t), I(x, t) and R(x, t) respectively represent the numbers of susceptible, infected and recovered individuals at position x and time t. The non-negative constants d1, d2 and d3 respectively represent the diffusion coefficients of the susceptible, infected and recovered individuals. L is a positive bounded constant. is the Laplace operator in the space Ω, used to describe the random diffusion motion. The Neumann boundary condition indicates that this model is a closed space. n is a smooth boundary. The outward unit normal vector of, assuming that the total population in the area is S(x,t)+I(x,t)+R(x,t)=1 and the birth rate and natural death rate in the area are both μ, (1 - p r )q0βS(x,t)I(x,t - τ) / [1 - p r I(x,t - τ)] is the number of newly infected individuals at time t, β represents the virus transmission rate, P0 represents the probability that an uninfected individual does not come into contact with others, P1 represents the probability that an infected individual does not come into contact with others, q0 = 1 - p0 represents the time proportion of uninfected individuals mixing with other individuals in the group, (q1 = 1 - p1) represents the time proportion of infected individuals mixing with other individuals in the group, and p r =(q0 - q1) / q0 represents the percentage reduction in the time of mixing with other individuals due to the infection of susceptibles, α is the recovery rate of infected individuals. According to the meaning of systems biology, these parameters are all non - negative constants and q1 ≤ q0.

[0074] Calculations show that the disease - free equilibrium point E0=(1,0,0), and the endemic equilibrium point E1=(S * ,I * ,R * ), where

[0075]

[0076] The basic reproduction number is the threshold for determining whether an infectious disease is controllable. According to the infection compartment theory, the basic reproduction number is only related to the number of infected individuals carrying the virus in the body. Using the second - generation generation matrix method, the basic reproduction number at the disease - free equilibrium point E0 is obtained.

[0077]

[0078] Step 2: For the uncontrolled SIR epidemic model (1) considering the spatio - temporal diffusion effect and individual contact heterogeneity, add a two - state feedback controller (4) to the susceptibles and infecteds

[0079]

[0080] where m1, m2, m3, n1, n2 and n3 are the gain parameters of the state - feedback controller respectively. This kind of controller can achieve bifurcation control without changing the original characteristics of the system.

[0081] Step 3: For the controlled model, linearize it at the equilibrium point to obtain the characteristic equation of the controlled system:

[0082]

[0083] Let M(x,t)=S(x,t)-S *,N(x,t) = I(x,t) - I * . According to the theory of the infected compartment, it can be known that the dynamic properties of the virus transmission system are completely determined by the first two equations of model (5). Therefore, we have:

[0084]

[0085] Linearize the controlled system (6) at the endemic equilibrium point E1, and we get:

[0086]

[0087] where

[0088]

[0089] Substitute the specific values of S * , I * and R0 into (8), and we get

[0090]

[0091] The characteristic equation of the linearized controlled network is expressed as:

[0092]

[0093] That is:

[0094]

[0095] Step 4: Select the time delay as the bifurcation parameter. Through the stability analysis and bifurcation analysis of the characteristic equation of the controlled SIR epidemic model after linearization in Step 3, select appropriate state feedback parameters to make the model locally asymptotically stable near the equilibrium point.

[0096] (1) When the time delay τ = 0 of virus transmission is not considered, the characteristic equation of the controlled system (5) is:

[0097]

[0098] According to the Routh - Hurwitz criterion, the necessary and sufficient condition for the roots of the above equation to have negative real parts is:

[0099]

[0100] According to the characteristics of the system parameters, the parameter conditions satisfying the inequality (13) are:

[0101]

[0102] Therefore, we can get Conclusion 1:

[0103] A. Adjust the controller parameters to satisfy the above two inequalities (H1). When there is no time delay, the endemic equilibrium point E1 of the controlled system (5) is locally asymptotically stable.

[0104] (2) When considering the virus transmission time delay (τ > 0), substitute λ = iω into the characteristic equation (12). By separating the real and imaginary parts, we can obtain:

[0105]

[0106] where

[0107]

[0108] Square and add the two equations in (15) to get:

[0109]

[0110] Let

[0111] g(ω) = a 23 2 ω 6 +(A 2 a 23 2 +C 2 -2Ba 23 2 -a 23 4 )ω 4 +(A 2 C 2 +a 23 2 B 2 -2BC 2 -2C 2 a 23 2 )ω 2 +B 2 C 2 -C 4 Obviously

[0112] When k = 0, if the following conditions are satisfied:

[0113] (H2): (a 11 a 22 ) 2 -(a 11 a 23 +a 21 a 23 ) 2 <0 (18)

[0114] Then \(g(0)\lt0\), so there is at least one positive root \(\omega_0\) for (17), that is, the characteristic equation (11) has a pair of pure imaginary roots \(\pm i\omega_0\). At this time, the system (5) undergoes a Hopf bifurcation near the equilibrium point. The bifurcation point is

[0115]

[0116] Differentiate the characteristic equation (11) with respect to \(\tau\) and verify the transversality condition to obtain

[0117]

[0118] Obviously

[0119] If the system parameters satisfy the following:

[0120] (H3): \((2C\omega_0 2 +Aa 23 \omega_0 2 )\cos\omega_0\tau_0+(AC\omega_0 - 2a 23 \omega_0 3 )\sin\omega_0\tau_0+(-a 23 2 )\omega_0 2 >0 (20)

[0121] Then The crossing condition is satisfied, and the system will undergo a Hopf bifurcation.

[0122] When \(k\geq1\), if the following conditions are satisfied

[0123]

[0124] Obviously, when \(\omega\gt0\), \(g(0)\gt0\) and \(g'(\omega)\gt0\). At this time, the characteristic equation (11) has no positive roots, and the system (5) is locally asymptotically stable at the equilibrium point.

[0125] The bifurcation point is a critical value for the system to change from stable to unstable, and the roots of the corresponding characteristic equation cross from the left half-plane of the imaginary axis to the right half-plane. At the same time, the crossing condition (transversality condition) is satisfied: the system has a pair of conjugate complex roots \(\rho(\tau)\pm i\omega(\tau)\) for the characteristic equation at the equilibrium point, and at \(\tau = 0\), these conjugate complex roots satisfy \(\rho(0)=0,\omega(0)=\omega_0\gt0\), and the transversality condition \(\rho'\neq0\) is satisfied, that is, the trajectory of \(\rho(\tau)\pm i\omega(\tau)\) crosses the imaginary axis at \(\tau=\tau_0\).

[0126] By observing the expression (19) of the bifurcation point, it can be seen that by adjusting the feedback parameters of the feedback state controller, the bifurcation point can be advanced or postponed, so as to achieve the control of the spread of infectious diseases.

[0127] Therefore, if (H2)-(H4) are satisfied, the following conclusion 2 can be obtained:

[0128] B. When the time delay τ satisfies τ ∈ [0, τ0), the equilibrium point E1 = (S * , I * , R * ) of the controlled system (5) is locally asymptotically stable;

[0129] C. When the time delay satisfies τ = τ0, a Hopf bifurcation phenomenon occurs near the equilibrium point E1 = (S * , I * , R * ) of the system (5). When τ crosses τ0, the system (5) generates a set of periodic solutions.

[0130] The following uses Matlab simulation examples to verify.

[0131] The first step: Select the uncontrolled SIR epidemic model considering the influence of spatio-temporal diffusion and individual contact heterogeneity:

[0132]

[0133] Through Hopf bifurcation calculation, the bifurcation point of the controlled system (20) is τ0 1 = 6.42.

[0134] As Figure 2-4 shown, when the time delay τ = 5 < τ0 is selected, the endemic equilibrium point E * = (S * , I * , R * ) of the uncontrolled system (20) is locally asymptotically stable.

[0135] As Figure 5-7 shown, when the time delay τ = 7.5 > τ0 1 is selected, the endemic equilibrium point E * = (S * , I * , R * ) of the uncontrolled system (20) loses stability, meaning that the virus transmission in this area is in a large outbreak state.

[0136] The second step: Add a two-state feedback controller to the SIR epidemic model considering the influence of spatio-temporal diffusion and individual contact heterogeneity. The controller parameters are m1 = 0.5, m2 = 0.1, m3 = 0.1,

[0137] n1 = -0.2, n2 = -0.2, n3 = -0.3. The mathematical expression of the controlled system is as follows:

[0138]

[0139] It can be obtained through Hopf bifurcation calculation that the bifurcation point τ0 of the controlled system (21). 2 = 11.128.

[0140] As Figure 8-10 shown, when the time delay τ0 1 < τ = 7.5 < τ0 2 is selected, under the action of the two - state feedback controller, the endemic equilibrium point E * = (S * , I * , R * ) returns to stability.

[0141] The above examples show that the two - state feedback controller can effectively delay the time node of the epidemic outbreak by adjusting the controller coefficients m1 and n1.

[0142] Without changing the characteristics of the original system, the present invention can effectively advance or delay the time point of Hopf bifurcation of the SIR epidemic model by setting appropriate controller parameters.

[0143] The above are only the embodiments of the present invention and are not used to limit the present invention. For those skilled in the art, the present invention can have various changes and modifications. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the scope of the claims of the present invention.

Claims

1. A two-state feedback controller that takes into account the influence of spatio-temporal diffusion and individual contact heterogeneity, characterized in that: The design method of the dual-state feedback controller includes the following steps: Step 1: Based on the traditional SIR epidemic model, establish an SIR epidemic model considering time delay and individual contact heterogeneity under partial differential equation description, analyze the equilibrium point information, and calculate the basic reproduction number; Step 2: Apply a dual-state feedback controller to the uncontrolled SIR epidemic model considering spatio-temporal diffusion effects and individual contact heterogeneity to obtain a controlled SIR epidemic model; Step 3: Linearize the SIR epidemic model under the action of the dual-state feedback controller in Step 2 at the equilibrium point to obtain the characteristic equation of the linearized controlled SIR epidemic model; Step 4: Select the time delay as the bifurcation parameter, perform stability analysis and bifurcation analysis on the characteristic equation of the controlled SIR epidemic model after linearization in Step 3, and select appropriate state feedback parameters to make the model locally asymptotically stable near the equilibrium point; where: The SIR epidemic model considering time delay and individual contact heterogeneity in Step 1 is expressed as: where: S(x,t), I(x,t), and R(x,t) represent the numbers of susceptibles, infectives, and recovered individuals at location x and time t, respectively; the non-negative constants d1, d2, and d3 represent the diffusion coefficients of susceptibles, infectives, and recovered individuals, respectively; L is a positive bounded constant, is the Laplace operator in the space Ω, and the Neumann boundary condition indicates that this model is in a closed space. n is the outward unit normal vector with a smooth boundary . Assume that the total population in the area is S(x,t) + I(x,t) + R(x,t) = 1, and the birth rate and natural death rate of the population in the area are both μ. (1 - p r )q0βS(x,t)I(x,t - τ) / [1 - p r I(x,t - τ)] is the number of newly infected individuals at time t. β represents the virus infection rate, P0 represents the probability that an uninfected individual does not come into contact with others, P1 represents the probability that an infected individual does not come into contact with others, q0 = 1 - p0 represents the time proportion of uninfected individuals mixing with other individuals in the population, (q1 = 1 - p1) represents the time proportion of infected individuals mixing with other individuals in the population, and p r = (q0 - q1) / q0 represents the percentage reduction in the time of mixing with other individuals due to the infection of susceptibles. α is the recovery rate of infected individuals. According to the biological meaning of the system, these parameters are all non-negative constants and q1 ≤ q0; In step 1, it can be calculated from the SIR infectious disease model considering time delay and individual contact heterogeneity that the disease-free equilibrium point \(E_0=(1, 0, 0)\), and the endemic equilibrium point \(E_1=(S * ,I * ,R * ), where Using the second-generation generation matrix method, the basic reproduction number under the disease-free equilibrium point E0 is obtained. The dual-state feedback controller in Step 2 refers to applying state feedback controllers to both susceptibles and infected individuals in the SIR epidemic model: u1 = -m1(S(x,t) - S * ) - m2(S(x,t) - S * ) 2 - m3(S(x,t) - S * ) 3 , u2 = -n1(I(x,t) - I * ) - n2(I(x,t) - I * ) 2 - n3(I(x,t) - I * ) 3 , where m1, m2, m3, n1, n2, and n3 are respectively the gain parameters of the state feedback controller; The following controlled model is obtained after applying the dual-state feedback controller in Step 2: Let \(M(x,t)=S(x,t)-S\) * , \(N(x,t)=I(x,t)-I\) * , according to the theory of the infected compartment, the dynamic properties of the virus transmission system are determined by the first two equations of model (1), and the final controlled SIR epidemic model is expressed as: The calculation process of obtaining the characteristic equation of the linearized controlled SIR epidemic model in Step 3 is as follows: Linearize the controlled SIR epidemic model (2) at the equilibrium point to obtain: where Substitute S * , I * and the specific values of R0 into (4) to obtain The characteristic equation of the linearized controlled network is expressed as: That is:

2. The two-state feedback controller according to claim 1, which takes into account the influence of spatio-temporal diffusion and individual contact heterogeneity, is characterized in that: A sufficient condition for the linearized controlled SIR epidemic model after Step 3 to be locally asymptotically stable near the endemic equilibrium point is that the roots of the characteristic equation of the linearized controlled SIR epidemic model have negative real parts.

3. The two-state feedback controller according to claim 2, which considers the influence of spatio-temporal diffusion and individual contact heterogeneity, is characterized in that: The roots of the characteristic equation of the linearized controlled SIR epidemic model having negative real parts specifically include: (1) When the time delay τ of virus transmission is not considered, i.e., τ = 0, the characteristic equation of the controlled system (1) is: According to the Routh-Hurwitz criterion, the necessary and sufficient condition for the roots of the above equation to have negative real parts is: According to the characteristics of the system parameters, the parameter conditions satisfying the inequality (9) are: Therefore, by adjusting the controller parameters to satisfy the above two inequalities (H1), the endemic equilibrium point E1 of the controlled system (1) is locally asymptotically stable without time delay; (2) When considering the time delay of virus transmission (τ > 0), substitute λ = iω into the characteristic equation (7), and separate the real and imaginary parts to obtain: where Square and add the two equalities of equation (11) to get: When k = 0, the condition for the above equation (13) to have at least one positive root ω0 is: B 2 -C 2 <0 (14) Differentiate the characteristic equation (7) with respect to τ, and verify the transversality condition to obtain (2Cω0 2 +Aa 23 ω0 2 )cosω0τ0+(ACω0 - 2a 23 ω0 3 )sinω0τ0+(Ca 23 -Ca 23 τ0 - a 23 2 )ω0 2 > 0, the system will undergo a Hopf bifurcation, and the bifurcation point is When k ≥ 1, if the system parameters satisfy A 2 a 23 2 +C 2 -2Ba 23 2 -a 23 4 >0 A 2 C 2 +a 23 2 B 2 -2BC 2 -2C 2 a 23 2 >0 B 2 C 2 -C 4 > 0 At this time, the endemic equilibrium point of the system is locally asymptotically stable. By observing the expression (15) of the bifurcation point, it can be seen that by adjusting the feedback parameters of the feedback state controller, the bifurcation point can be advanced or delayed, so as to achieve the control of the spread of infectious diseases.

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