A Design Method of PD Controller Based on the Spatiotemporal Propagation Model of Malware

Abstract

The present invention is a method for designing a PD controller based on a spatio-temporal propagation model of malware, including Step 1: introducing time delay and reaction diffusion terms on the basis of the malware propagation model to establish a partial differential spatio-temporal propagation model of malware with a non-linear saturated incidence rate; Step 2: applying a PD controller to the uncontrolled state of the malware spatio-temporal propagation model at the equilibrium point to obtain a controlled model of the malware spatio-temporal propagation; Step 3: linearizing the obtained controlled model at the equilibrium point to obtain a relevant characteristic equation; Step 4: selecting the time delay as the bifurcation parameter to conduct stability analysis and Hopf bifurcation analysis on the controlled model of the malware spatio-temporal propagation. Finally, appropriate controller parameters are selected, and through simulation, the influence of the controller on the system is verified. The present invention can not only effectively control the position of the bifurcation point, but also has an obvious influence on the control term, with a remarkable control effect, effectively reducing the harm of malware propagation.

Description

Technical Field

[0001] The present invention relates to the technical field of controllers, and specifically relates to a method for designing a PD controller based on a spatio-temporal propagation model of malware. Background Art

[0002] With the popularization of computer technology, malware is undoubtedly the most fatal unstable factor. The spread of malware among computer nodes has many similarities with the spread of infectious diseases among people. Therefore, many scholars use the same method to mathematically model both and conduct relevant dynamic analyses. Most of the existing malware spread models stay in the time domain. In fact, the spread of malware not only evolves over time but is also affected by spatial distribution. The introduction of the reaction-diffusion term can well depict the spread of malware in the spatial domain, making the established spatio-temporal propagation model more in line with reality.

[0003] Adding a control strategy based on the dynamic analysis of the spatio-temporal propagation model of malware can make the research more practically significant. However, the existing related research is only limited to time-delay feedback controllers, which are difficult to design and have a single adjustable control parameter. Summary of the Invention

[0004] To make up for the deficiencies of the prior art, the present invention provides a method for designing a PD controller based on a spatio-temporal propagation model of malware. This method greatly reduces the control difficulty of the spatio-temporal propagation model of malware and can effectively delay the large-scale outbreak of malware by adjusting the control parameters.

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

[0006] The present invention is a method for designing a PD controller based on a spatio-temporal propagation model of malware, and the method includes:

[0007] 1) Introduce time delay and reaction-diffusion term on the basis of the malware propagation model to establish a partial differential malware propagation model with a non-linear saturation incidence rate, and obtain information such as the basic reproduction number and equilibrium point;

[0008] 2) Apply a PD controller to the uncontrolled spatio-temporal propagation model of malware at the equilibrium point to obtain a controlled spatio-temporal propagation model of malware;

[0009] 3) Linearize the controlled model at the equilibrium point to obtain a relevant characteristic equation;

[0010] 4) Select the time delay as the bifurcation parameter to conduct a stability analysis and Hopf bifurcation analysis on the controlled spatio-temporal propagation model of malware;

[0011] Furthermore, it includes:

[0012] The described spatio - temporal propagation model of malware is expressed as:

[0013]

[0014] where \(t>0\), \(x\in\Omega\). The model satisfies the homogeneous Neumann boundary condition

[0015]

[0016] and the initial conditions

[0017]

[0018] In the model, \(S(t,x)\), \(I(t,x)\) and \(R(t,x)\) represent the distribution densities of susceptible nodes, infected nodes and recovered nodes at time \(t\) and spatial location \(x\) respectively. \(\tau_1\), \(\tau_2\) and \(\tau_3\) represent the latency delay, infectious period delay and immune period delay respectively. In the non - linear saturation incidence rate \(\beta SI / (1 + \alpha I)\), \(\alpha\) and \(\beta\) represent the prevention effect coefficient and the infection rate respectively. \(\varepsilon\) and \(\sigma\) represent the recovery rate and the regression rate respectively. \(d_1\), \(d_2\) and \(d_3\) represent the diffusion coefficients of susceptible nodes, infected nodes and recovered nodes respectively. represents the Laplace operator in one - dimensional space. Assume that \(\Omega=(0,\pi)\) is a bounded region with a smooth boundary and \(\eta\) is the outward unit normal vector on. The initial conditions \(S\) 0 (t,x), \(I\) 0 (t,x) and \(R\) 0 (t,x) are all non - negative continuous functions. The malware spatio - temporal propagation model considered in the present invention is one with a total number of nodes in dynamic equilibrium, so the node access rate and the exit rate are both \(\mu\). All the above - mentioned parameters are positive numbers.

[0019] By calculation, the basic reproduction number of the system is obtained

[0020]

[0021] and the extraordinary equilibrium point \(E\) * =(S * , \(I\) * , \(R\) * ), where,

[0022]

[0023]

[0024] Let \(\tau_1=\tau_2=\tau_3 = \tau\), the expression of adding a PD controller at the equilibrium point is as follows:

[0025]

[0026] Among them, the proportional controller parameter k p <[αI * (μ + σ)] / R0S * , and the derivative controller parameter k d <1.

[0027] Define -k 2 (k ∈ K0 = {0, 1, 2,...}) as the characteristic root of Δ. For convenient calculation, make the following substitution for the parameters:

[0028]

[0029]

[0030]

[0031] After linearizing the controlled malware spatio - temporal propagation model at the equilibrium point E * , the following characteristic equation is obtained:

[0032]

[0033] Among them,

[0034] a2(k 2 ) = m1(k 2 ) + m2(k 2 ) + m3(k 2 ),

[0035] a1(k 2 ) = m1(k 2 )m2(k 2 ) + m1(k 2 )m3(k 2 ) + m2(k 2 )m3(k 2 ),

[0036] a0(k 2 ) = m1(k 2 )m2(k 2 )m3(k 2 ),

[0037] b2(k 2 ) = ε + n2,

[0038] b1(k 2 ) = m1(k 2 )ε + m1(k 2 )n2 + m2(k 2 )ε + m3(k2 ) n2 + n1z2,

[0039] b0(k 2 ) = m1(k 2 ) m2(k 2 ) ε + m1(k 2 ) m3(k 2 ) n2 + n1z2m3(k 2 ),

[0040] c1(k 2 ) = n2ε,

[0041] c0(k 2 ) = m1(k 2 ) εn2 + n1z2ε - n1σε,

[0042] Here,

[0043]

[0044] Select the time delay τ as the bifurcation parameter. When the system is stable, the roots of its linearized characteristic equation all have negative real parts. Therefore, it is necessary to find the critical situation where the characteristic equation has pure imaginary roots.

[0045] 1) When the system has no time delay, that is, τ = 0, use the Routh - Hurwitz criterion to discuss whether the roots of the characteristic equation all have negative real parts;

[0046] 2) When the system has time delay, that is, τ > 0, use the Hopf bifurcation theory to discuss the existence of the bifurcation threshold of the controlled system.

[0047] The beneficial effects of the present invention are:

[0048] 1. The malicious software spatio - temporal propagation model proposed by the present invention fully considers the influence of the latency time delay, the infection time delay, the immunity time delay, and the spatial effect of reaction - diffusion on the model, and is more in line with the changes of computer nodes in the actual network environment;

[0049] 2. The PD controller designed in the present invention has excellent control effect and strong applicability, and is also applicable to other complex dynamic networks.

[0050] 3. The present invention can not only effectively control the position of the bifurcation point, but also has an obvious influence on the control term, with a significant control effect, effectively reducing the harm of malicious software propagation. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Figure 1 It is the flowchart of the method described in the present invention.

[0052] Figure 2The waveform diagram of the instability of the susceptible state node S(t,x) in the uncontrolled system (1) when τ = 9.3.

[0053] Figure 3 The waveform diagram of the instability of the infected state node I(t,x) in the uncontrolled system (1) when τ = 9.3.

[0054] Figure 4 The waveform diagram of the instability of the recovered state node R(t,x) in the uncontrolled system (1) when τ = 9.3.

[0055] Figure 5 For τ = 9.3, k p = -0.5, k d = 0.5, the waveform diagram of the susceptible state node S(t,x) in the controlled system (2) returning to stability.

[0056] Figure 6 For τ = 9.3, k p = -0.5, k d = 0.5, the waveform diagram of the infected state node I(t,x) in the controlled system (2) returning to stability.

[0057] Figure 7 For τ = 9.3, k p = -0.5, k d = 0.5, the waveform diagram of the recovered state node R(t,x) in the controlled system (2) returning to stability.

[0058] Figure 8 For τ = 13.5, k p = -0.5, k d = 0.5, the waveform diagram of the instability of the susceptible state node S(t,x) in the controlled system (2).

[0059] Figure 9 For τ = 13.5, k p = -0.5, k d = 0.5, the waveform diagram of the instability of the infected state node I(t,x) in the controlled system (2).

[0060] Figure 10 For τ = 13.5, k p = -0.5, k d = 0.5, the waveform diagram of the instability of the recovered state node R(t,x) in the controlled system (2).

[0061] Figure 11 For k d = 0, the curve of the variation of τ0 with k p .

[0062] Figure 12 For kd When it is 0.5, τ0 varies with k p The curve of the variation relationship.

[0063] Figure 13 For k p When it is 0, τ0 varies with k d The curve of the variation relationship.

[0064] Figure 14 For k p When it is -0.5, τ0 varies with k d The curve of the variation relationship. Specific implementation manner

[0065] The embodiments of the present invention will be disclosed below with reference to the drawings. For the sake of clarity, many practical details will be described together in the following description. 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.

[0066] The present invention is a PD controller design method based on a malware spatio-temporal propagation model. The method includes the following steps:

[0067] Step 1: Introduce time delay and reaction diffusion terms on the basis of the malware propagation model to establish a partial differential malware propagation model with a non-linear saturation incidence rate, and obtain information such as the basic reproduction number and the equilibrium point.

[0068] The malware spatio-temporal propagation model is expressed as:

[0069]

[0070] where t > 0, x ∈ Ω.

[0071] The model satisfies the homogeneous Neumann boundary condition

[0072]

[0073] and the initial condition

[0074]

[0075] In the model, S(t, x), I(t, x) and R(t, x) represent the distribution densities of susceptible nodes, infected nodes and recovered nodes at time t and spatial position x respectively. τ1, τ2 and τ3 represent the latency time delay, the infectious period time delay and the immunity period time delay respectively. In the non-linear saturation incidence rate βSI / (1 + αI), α and β represent the prevention effect coefficient and the infection rate respectively, ε and σ represent the recovery rate and the regression rate respectively, and d1, d2 and d3 represent the diffusion coefficients of susceptible nodes, infected nodes and recovered nodes respectively. The Laplace operator representing one-dimensional space. Assume that Ω = (0, π) is a bounded region with a smooth boundary , and η is the outward unit normal vector on, and the initial conditions S 0 (t, x), I 0 (t, x) and R 0 (t, x) are all non-negative continuous functions. The present invention considers a spatio-temporal propagation model of malware with the total number of nodes in dynamic equilibrium, so the node access rate and exit rate are both μ, and the above parameters are all positive numbers.

[0076] The basic reproduction number of the system is obtained by calculation

[0077]

[0078] and the extraordinary equilibrium point E of the system * = (S * , I * , R * ), where

[0079]

[0080]

[0081] Step 2: Apply a PD controller to the uncontrolled spatio-temporal propagation model of malware at the equilibrium point to obtain a controlled spatio-temporal propagation model of malware.

[0082] Let τ1 = τ2 = τ3 = τ, and the expression of adding a PD controller at the equilibrium point is as follows:

[0083]

[0084] Among them, the proportional controller parameter k p < [αI * (μ + σ)] / R0S * , and the derivative controller parameter k d < 1.

[0085] Step 3: Linearize the controlled model at the equilibrium point to obtain a relevant characteristic equation.

[0086] Define -k 2 (k ∈ K0 = {0, 1, 2,...}) as the characteristic root of Δ. For the convenience of calculation, make the following substitution for the parameters:

[0087]

[0088]

[0089]

[0090] After linearizing the controlled malware spatio-temporal propagation model at the equilibrium point \(E\) * , the following characteristic equation is obtained:

[0091]

[0092] where

[0093] \(a_2(k 2 ) = m_1(k 2 ) + m_2(k 2 ) + m_3(k 2 ),

[0094] \(a_1(k 2 ) = m_1(k 2 )m_2(k 2 ) + m_1(k 2 )m_3(k 2 ) + m_2(k 2 )m_3(k 2 ),

[0095] \(a_0(k 2 ) = m_1(k 2 )m_2(k 2 )m_3(k 2 ),

[0096] \(b_2(k 2 ) = \epsilon + n_2,

[0097] \(b_1(k 2 ) = m_1(k 2 )\epsilon + m_1(k 2 )n_2 + m_2(k 2 )\epsilon + m_3(k 2 )n_2 + n_1z_2,

[0098] \(b_0(k 2 ) = m_1(k 2 )m_2(k 2 )\epsilon + m_1(k 2 )m_3(k 2 )n_2 + n_1z_2m_3(k 2 ),

[0099] \(c_1(k 2 ) = n_2\epsilon,

[0100] \(c_0(k 2 ) = m_1(k 2 )\epsilon n_2 + n_1z_2\epsilon - n_1\sigma\epsilon,

[0101] Here

[0102]

[0103] Step 4: Select the time delay as the bifurcation parameter, and perform stability analysis and Hopf bifurcation analysis on the controlled model of the spatio-temporal propagation of malware.

[0104] Select the time delay τ as the bifurcation parameter and discuss as follows:

[0105] 1) When the system has no time delay, i.e., τ = 0

[0106] From (3), we get

[0107] λ 3 +V2(k 2 )λ 2 +V1(k 2 )λ+V0(k 2 )=0, (4)

[0108] where

[0109] V2(k 2 )=V 21 k 2 +V 20 ,

[0110] V1(k 2 )=V 12 k 4 +V 11 k 2 +V 10 ,

[0111] V0(k 2 )=V 03 k 6 +V 02 k 4 +V 01 k 2 +V 00 ,

[0112] Here,

[0113]

[0114]

[0115]

[0116]

[0117]

[0118]

[0119]

[0120]

[0121]

[0122] When \(k\in K_0\), obviously, \(V_2(k 2 )>0\), \(V_1(k 2 )>0\), if the following conditions hold:

[0123] (H1): \(V 00 >0\),

[0124] then \(V_0(k 2 )>0\), \(k\in K_0\).

[0125] Define \(y(k 2 ) = V_2(k 2 )V_1(k 2 ) - V_0(k 2 )\), we can get

[0126] \(y(k 2 ) = A_3k 6 +A_2k 4 +A_1k 2 +A_0\), \(k\in K_0\), (5)

[0127] where,

[0128] A_3 = V 21 V 12 -V 03 >0,

[0129] A_2 = V 21 V 11 +V 20 V 12 -V 02 >0,

[0130] A_1 = V 21 V 10 +V 20 V 11 -V 01 >0,

[0131] A_0 = V 20 V 10 -V 00 >0,

[0132] then \(y(k 2 )>0\), \(k\in K_0\).

[0133] Therefore, when (H1) is satisfied, \(V_2(k 2)V1(k 2 ) > V0(k 2 ) > 0, where k ∈ K0. According to the Routh-Hurwitz criterion, all the roots of equation (4) have negative real parts, and equation (2) is locally asymptotically stable at the equilibrium point E * .

[0134] 2) When the system has a time delay, i.e., τ > 0

[0135] Let λ = iω (ω > 0) be a solution of equation (3). Substituting it in and separating the real and imaginary parts, we get

[0136]

[0137]

[0138] where

[0139] p2(k 2 ) = b1(k 2 ) - a2(k 2 )b2(k 2 ),

[0140] p1(k 2 ) = a2(k 2 )b0(k 2 ) + a0(k 2 )b2(k 2 ) + b1(k 2 )c1(k 2 ) - a1(k 2 )b1(k 2 ) - b2(k 2 )c0(k 2 ),

[0141] p0(k 2 ) = b0(k 2 )c0(k 2 ) - a0(k 2 )b0(k 2 ),

[0142] q2(k 2 ) = b2(k 2 ),

[0143] q1(k 2 ) = a2(k 2 )b1(k 2 ) - b0(k 2 ) - a1(k 2 )b2(k 2 ) - b2(k 2 )c1(k 2 ),

[0144] q0(k 2 ) = a1(k 2 )b0(k 2 ) + b0(k 2 )c1(k 2 ) - a0(k 2 )b1(k 2 ) - b1(k 2 )c0(k 2 ),

[0145]

[0146]

[0147]

[0148] After squaring and adding both sides of Equation (6) and Equation (7), we get

[0149] ω 12 + l5(k 2 )ω 10 + l4(k 2 )ω 8 + l3(k 2 )ω 6 + l2(k 2 )ω 4 + l1(k 2 )ω 2 + l0(k 2 ) = 0, (8)

[0150] where

[0151]

[0152]

[0153]

[0154]

[0155]

[0156]

[0157] Define

[0158] g(ω) = ω 12 + l5(k 2 )ω 10 + l4(k 2 )ω 8 + l3(k2 )ω 6 +l2(k 2 )ω 4 +l1(k 2 )ω 2 +l0(k 2 ),

[0159] Obviously,

[0160] when k = 0, if the following conditions are satisfied:

[0161]

[0162] then g(0) < 0. Therefore, equation (8) has at least one positive root ω0, that is, equation (3) has a pair of pure imaginary roots ±iω0. At this time, equation (2) undergoes a Hopf bifurcation near the equilibrium point. Thus, from equation (6), we can obtain

[0163]

[0164] when k ≥ 1, if the following conditions are satisfied:

[0165] (H3): l κ > 0, κ ∈ {0, 1, 2, 3, 4, 5},

[0166] then when ω > 0, g(0) > 0 and g′(ω) > 0. Therefore, equation (8) has no positive roots, that is, equation (3) has no pure imaginary roots, and equation (2) is locally asymptotically stable at the equilibrium point.

[0167] Further verify the crossing condition for the bifurcation when k = 0. Assume that λ(τ) = υ(τ) + iω(τ) is a root of equation (6), satisfying υ(τ0) = 0 and ω(τ0) = ω0, where Thus, we have

[0168]

[0169] where,

[0170]

[0171]

[0172]

[0173]

[0174] Obviously,

[0175]

[0176] if the following conditions are satisfied:

[0177] (H4): M1N1 + M2N2 > 0,

[0178] Then the crossing condition is satisfied, that is, when ω = ω0, τ = τ0, Equation (2) undergoes a Hopf bifurcation at the equilibrium point E * in the vicinity.

[0179] In summary, if (H1)-(H4) are satisfied, the following conclusions can be obtained:

[0180] (i) When 0 ≤ τ < τ0, Equation (2) is locally asymptotically stable at the equilibrium point E * ;

[0181] (ii) When τ > τ0, Equation (2) is unstable at the equilibrium point E * ;

[0182] (iii) When τ = τ0, Equation (2) undergoes a Hopf bifurcation in the vicinity of the equilibrium point E * .

[0183] The following is a further illustration of the present invention by using examples.

[0184] The present invention is verified by using MATLAB simulation examples.

[0185] The first step: For the uncontrolled spatio-temporal propagation model of malicious software, the following parameters are selected:

[0186] μ = 0.05, β = 0.38, α = 0.1, ε = 0.15, σ = 0.1, d1 = 2, d2 = 1, d3 = 1,

[0187] The specific mathematical expression is as follows:

[0188]

[0189] Through program calculation, it can be obtained that the equilibrium point E * = (0.4103, 0.3932, 0.1966), and the bifurcation threshold τ1 of the uncontrolled system (9) is 9.2184.

[0190] As Figure 2-4 shown, when the time delay τ = 9.3 > τ1 is selected, the uncontrolled system (9) is unstable at the equilibrium point E * .

[0191] The second step: For the controlled spatio-temporal propagation model of malicious software, on the basis of the parameters selected in the first step, the controller parameters k p = -0.5, k d = 0.5 are selected. The specific mathematical expression is as follows:

[0192]

[0193] It can be obtained through program calculation that the bifurcation threshold τ0 of the controlled system (10) is 12.0923.

[0194] As Figure 5-7 shown, when the time delay τ = 9.3 < τ0 is selected, the controlled system (10) is stable at the equilibrium point E * there.

[0195] As Figure 8-10 shown, when the time delay τ = 13.5 > τ0 is selected, the controlled system (10) loses stability at the equilibrium point E * there.

[0196] The present invention can not only effectively control the position of the bifurcation point, but also has an obvious influence on the control term, with a remarkable control effect, effectively reducing the harm of malware propagation.

[0197] The above is only the implementation manner of the present invention and is 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 design method of a PD controller based on a spatio-temporal propagation model of malware, characterized in that: The PD controller design method includes the following steps: Step 1: Establish a partial differential malware spatio-temporal propagation model with a non-linear saturation incidence rate to obtain the basic reproduction number and equilibrium point information; Step 2: Apply a PD controller to the uncontrolled malware spatio-temporal propagation model at the equilibrium point to obtain a controlled malware spatio-temporal propagation model; Step 3: Linearize the controlled model obtained in Step 2 at the equilibrium point to obtain a relevant characteristic equation; Step 4: Select the time delay as the bifurcation parameter and perform stability analysis and Hopf bifurcation analysis on the controlled malware spatio-temporal propagation model; Wherein: The malware propagation model in Step 1 is expressed as: where \(t>0\), \(x\in\Omega\), in the model, \(S(t,x)\), \(I(t,x)\) and \(R(t,x)\) represent the distribution densities of susceptible nodes, infected nodes and recovered nodes at time \(t\) and spatial location \(x\) respectively, \(\tau_1\), \(\tau_2\) and \(\tau_3\) represent the latency delay, the infectious period delay and the immunity period delay respectively. In the nonlinear saturated incidence rate \(\beta SI / (1 + \alpha I)\), \(\alpha\) and \(\beta\) represent the prevention effect coefficient and the infection rate respectively, \(\varepsilon\) and \(\sigma\) represent the recovery rate and the regression rate respectively, and \(d_1\), \(d_2\) and \(d_3\) represent the diffusion coefficients of susceptible nodes, infected nodes and recovered nodes respectively. denotes the Laplace operator in one-dimensional space. Assume that \(\Omega=(0,\pi)\) is a bounded region with a smooth boundary . \(\eta\) is the outward unit normal vector on 0 . The initial conditions \(S\) 0 (t,x), \(I\) 0 (t,x) and \(R\) 0 (t,x) are all non-negative continuous functions, the node access rate and the exit rate are both \(\mu\), and all the above parameters are positive numbers. Let τ1 = τ2 = τ3 = τ, and the expression of adding a PD controller at the equilibrium point is as follows: Among them, the proportional controller parameter k p < [αI * (μ + σ)] / R0S * , and the derivative controller parameter k d < 1.

2. The PD controller design method based on the spatio-temporal propagation model of malware according to claim 1, wherein: The malware propagation model satisfies the homogeneous Neumann boundary condition and the initial condition 3. The design method of a PD controller based on a malware spatio-temporal propagation model according to claim 2, characterized in that: The basic reproduction number of the model in Step 1 is expressed as: Let β denote the infection rate, σ denote the fallback rate, and μ denote the node access rate and exit rate; The extraordinary equilibrium point E of the model in Step 1 * =(S * , I * , R * ), where Wherein: α represents the prevention effect coefficient, ε represents the recovery rate, σ represents the fallback rate, and μ is the node access rate and exit rate.

4. The PD controller design method based on the spatio-temporal propagation model of malware according to claim 1, characterized in that: The following characteristic equation is obtained after linearizing the controlled model in Step 3 at the equilibrium point: Wherein, a2(k 2 ) = m1(k 2 ) + m2(k 2 ) + m3(k 2 ), a1(k 2 ) = m1(k 2 )m2(k 2 ) + m1(k 2 )m3(k 2 ) + m2(k 2 )m3(k 2 ), a0(k 2 ) = m1(k 2 )m2(k 2 )m3(k 2 ), b2(k 2 ) = ε + n2, b1(k 2 ) = m1(k 2 )ε + m1(k 2 )n2 + m2(k 2 )ε + m3(k 2 )n2 + n1z2, b0(k 2 ) = m1(k 2 )m2(k 2 )ε + m1(k 2 )m3(k 2 )n2 + n1z2m3(k 2 ), c1(k 2 ) = n2ε, c0(k 2 ) = m1(k 2 )εn2 + n1z2ε - n1σε, Here, m1(k 2 ) = d1k 2 + z1 + μ > 0, m3(k 2 ) = d3k 2 + μ > 0, Let λ denote the root of the characteristic equation, and define -k 2 (k ∈ K0 = {0, 1, 2,...}) as the characteristic root of Δ, u1 = μ + z1, u3 = μ + ε。 5. The design method of a PD controller based on the spatio-temporal propagation model of malware according to claim 1, characterized in that: Select the time delay τ as the bifurcation parameter. When the system is stable, the roots of its linearized characteristic equation all have negative real parts. Therefore, it is necessary to find the critical situation where the characteristic equation has pure imaginary roots: 1) When the system has no time delay, i.e., τ = 0, use the Routh-Hurwitz criterion to discuss whether the roots of the characteristic equation all have negative real parts; 2) When the system has time delay, i.e., τ > 0, use the Hopf bifurcation theory to discuss the existence of the bifurcation threshold of the controlled system.

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