Since the outbreak of coronavirus disease 2019 (COVID-19), the global public health system has faced very serious challenges for a long time. Moreover, the widespread spread of the virus will not only cause serious problems of human health and life safety, but also have an irreparable impact on global economic development and social stability. According to the mathematical modeling method, it is the core direction with important practical value in the study of infectious disease dynamics to deeply explore the dynamic law of the spread of novel coronavirus, accurately predict the evolution trend of the epidemic situation, and design scientific and effective prevention and control intervention programs accordingly. This paper systematically sorts out and summarizes the research results in related fields at home and abroad. The existing academic research provides a solid theoretical basis and research reference for the study of Hopf bifurcation of fractional-order multi-delay COVID-19 propagation model from two aspects: model system construction and dynamic analysis method. In the related research of biodynamic models, especially infectious disease transmission models, many scholars have achieved rich results. Among them, Reference [1] constructed an integer-order SEIR dynamic model with dual delay parameters of incubation period and recovery period based on the real transmission characteristics of the novel coronavirus epidemic, and focused on the mechanism of time delay on the stability of the equilibrium point and the Hopf bifurcation dynamic behavior of the system. This study further integrates control strategies and drug intervention-related variables, constructs an optimal control analysis framework with time-delay characteristics, and establishes the time-delay disturbance problem as the core research context, which provides an important reference for the research ideas of this paper. Based on the traditional SEIR model, Reference [2] extended the model into a spatial reaction-diffusion system. At the same time, it also introduced many realistic factors such as the randomness of population spatial migration and nonlinear infection rate, which enriched the research ideas of infectious disease modeling coupled with multi-dimensional realistic variables, and provided a new reference direction for the optimization and improvement of the model in this paper.

Fractional differential equations are widely used in epidemic modeling because they can describe the memory effect and historical cumulative dependence of the system. In the literature [3]-[9], etc., combined with the actual epidemic characteristics of asymptomatic infection, multiple physiological delays, saturated incidence, multi-group warehouse division, etc., different forms of fractional-order delay SEIR-like dynamic systems are constructed. It is fully confirmed that fractional-order operators combined with discrete delays can be more suitable for the complex evolution of infectious diseases. In addition, Reference [10] explored the bifurcation characteristics of the double-delay SEIR model under the dual propagation path. Reference [11] carried out the quantitative calculation of Hopf bifurcation based on the combination of saturation incidence and time delay. Reference [12] completed the threshold determination and global stability proof of the classical SEIR model. Reference [13] introduced the complex network topology to describe the non-uniform contact mode of the crowd, and improved the extended form of the traditional SEIR model from multiple angles. In [14]-[16], the optimal control problem of vaccination with state constraints is added to the SEIR model, and the optimal intervention scheme is solved by distinguishing different constraints, which provides an important reference for this paper to build a time-delay optimal control model with dual control variables of social contact control and drug treatment.

In terms of theoretical analysis and numerical simulation methodology, the existing research has formed a complete and mature technical system. In many literatures, the fixed point theory is widely used to prove the existence and uniqueness of the solution of the fractional order system. The next generation matrix method is used to solve the basic reproduction number, and the Lyapunov direct method is used to complete the local and global stability of the equilibrium point. The stability switching condition is determined by the root distribution of the characteristic equation. Based on the center manifold theorem and the normal form method, the direction, period and stability of Hopf bifurcation are derived. In the numerical simulation stage, Adams-Bashforth-Moulton predictor-corrector algorithm, finite difference method, network topology simulation and other tools are widely used to verify the theoretical derivation results. The above complete qualitative analysis theory and numerical calculation scheme lay a methodological foundation for the equilibrium point analysis, Hopf bifurcation condition derivation and numerical verification of the optimal control strategy of the fractional-order multi-delay new crown propagation model.

We propose the following delayed SEIR model to describe the spread of the COVID-19 virus.

Here, the fractional order parameter q ∈ ( 0 , 1 ] , S ( t ) , E ( t ) , I ( t ) , and R ( t ) denote the susceptible, exposed, infected, and recovered at time t , respectively, and N ( t ) = S ( t ) + E ( t ) + I ( t ) R ( t ) denotes the total population tree. τ 1 , τ 2 represents the time delay parameters of two different physiological processes, where τ 1 represents the latent period delay after infection, and τ 2 represents the recovery period delay of infected individuals. The biological interpretation of the remaining parameters in the system Equation (1) is summarized in Table 1.

Table 1. Parameter descriptions of the fractional-order delayed SEIR epidemic model.

In order to facilitate the subsequent stability analysis and Hopf bifurcation study of system (1), the definition of Caputo fractional derivative and several basic lemmas are given.

Definition 2.1 (Caputo Fractional derivative [17]-[19]).The Caputo definition of fractional derivatives can be written as:

where f ( t ) ∈ C m ( [ a , ∞ ) , ℝ ) . In particular, if 0 < α ≤ 1 , a = 0 , Equation (2) can be written as:

Definition 2.2 (Laplace Transform [20][21] of Fractional Derivative). The Laplace transform of Caputo fractional derivative of order α ( n − 1 < α ≤ n ) for the function f ( t ) ∈ C n ( [ a , ∞ ) , ℝ ) is

where F ( s ) is the Laplace transform of f ( t ) , and f ( k ) ( a ) ( k = 0 , 1 , ⋯ , n − 1 ) are the initial conditions. Obviously, if f ( k ) ( a ) = 0 for k = 0 , 1 , ⋯ , n − 1 , Equation (4) can be written as

Definition 2.3.[22] Consider the following n -dimensional fractional-order system with time delay:

where 0 < α ≤ 1 and the time delay τ ≥ 0 . System (5) undergoes Hopf bifurcation at the equilibrium u * = ( u 1 * , u 2 * , ⋯ , u n * ) when τ = τ 0 if the following three conditions are satisfied:

i) All the eigenvalues λ j ( j = 1 , 2 , ⋯ , n ) of the coefficient matrix J of the linearized system of (5) with τ = 0 satisfy

ii) The characteristic equation of the linearized system of (5) has a pair of purely imaginary roots ± i ω 0 when τ = τ 0 .

iii)

where Re [ ⋅ ] denotes the real part of the complex number.

Using the next-generation matrix method proposed by van den Driessche and Watmough [23], the basic reproduction number of the system (1) can be expressed as:

For system (1), the disease-free and the endemic equilibrium points are defined as

where

We know that when the basic reproduction number R 0 < 1 , only the disease-free equilibrium exists and the disease dies out; R 0 = 1 corresponds to the critical bifurcation case; when R 0 > 1 , an endemic equilibrium exists and the disease persists.

For Caputo fractional-order systems with 0 < q < 1 , the equilibria and the Jacobian matrix evaluated at each equilibrium are completely determined by the right-hand-side vector field of the system and are independent of the fractional order q . Accordingly, the eigenvalues of the Jacobian matrix can be solved by exactly the same algebraic procedures as those used for integer-order systems. Nevertheless, local asymptotic stability of an equilibrium for fractional-order systems is no longer equivalent to the condition that all eigenvalues have negative real parts. Instead, the fractional-order eigenvalue criterion should be adopted: an equilibrium is locally asymptotically stable if and only if every eigenvalue λ of the Jacobian matrix satisfies

For the delay-free counterpart of our system, all eigenvalues of the Jacobian at the disease-free equilibrium satisfy | arg ( λ ) | > π 2 whenever R 0 < 1 . Combined with 0 < q < 1 , we have q π 2 < π 2 , so the fractional-order stability condition

| arg ( λ ) | > q π 2 is automatically fulfilled. Hence, R 0 < 1 guarantees the local

asymptotic stability of the disease-free equilibrium in the delay-free case. It is worth noting that this property holds only in the absence of time delays. When time delays τ 1 , τ 2 are involved, the characteristic equation becomes transcendental, and the eigenvalue analysis based on the constant Jacobian matrix is invalid. We have to investigate the stability switches and Hopf bifurcation induced by time delays.

At the equilibrium point E 1 = ( S * , E * , I * , R * ) , by applying the transformations

system (1) can be converted into

where

In this section, we thoroughly investigate the local stability of the system (1) at its equilibrium points. Taking the time delay as the bifurcation parameter, we further demonstrate the existence of Hopf bifurcation at the equilibrium points of the system through case-by-case analysis.

Firstly, we need to discuss the stability of system (1) at τ 1 = τ 2 = 0 .

Case 3.1.1 If τ 1 = τ 2 = 0 , We can get the characteristic equation of system (1) is

where

According to Definition condition i), to guarantee the local asymptotic stability of the equilibrium point E 1 = ( S * , E * , I * , R * ) of system (1) at τ 1 = τ 2 = 0 , the following Assumption (A1) needs to be satisfied.

(A1) The coefficients a i of Equation (8) satisfy the following conditions

i) a i > 0 , i = 0 , 1 , 2 , 3 ;

ii) a 3 a 2 > a 1 ;

iii) a 3 a 2 a 1 > a 1 2 + a 3 a 0 .

Claim 1.According to the Routh-Hurwitz criterion, if Assumption (A1) holds, all roots of Equation (8) have negative real parts, which implies that | arg ( λ ) | > π 2

is satisfied for all characteristic roots. Therefore, the equilibrium E 1 ( S * , E * , I * , R * ) is locally asymptotically stable under the delay-free condition.

To analyze the stability and Hopf bifurcation behavior of system (1), this paper performs the Laplace transform on both sides of the linearized equation (7), and further derives the corresponding characteristic matrix of the system

It follows from det ( Δ ( s ) ) = 0 that the characteristic equation of the fractional-order system with two delays is given by

where the polynomial functions P 0 ( s , q ) , P 1 ( s , q ) , P 2 ( s , q ) , P 3 ( s , q ) are defined as

where

Next, we investigate the following cases for the delay parameters τ 1 and τ 2 at the equilibrium point E 1 .

Case 3.1.2 If τ 1 = 0 , τ 2 > 0 , then Equation (9) becomes

Assume that s = i ω ( ω > 0 ) is a purely imaginary root of Equation (6). Substitute s = ( cos π 2 + i sin π 2 ) ω into P k ( s , q ) for k = 0 , 1 , 2 , 3 . Let

P k ( i ω , q ) = A k + i B k . By separating the real and imaginary parts of the characteristic Equation (10), we derive the following system:

where

For the convenience of subsequent calculations, we perform equivalent transformation on Equation (10) and obtain the following system of equations

where

We can readily derive

By the identity cos 2 ω τ 2 + sin 2 ω τ 2 = 1 , we have

Assume that Equation (11) has at least one positive real root ω , which can be computed by the symbolic computation software Maple. Accordingly, the critical delay τ 20 corresponding to the bifurcation point is defined as

where ω is the root of Equation (11).

Finally, we consider the transversality condition, namely Condition iii in Definition 2.3. Let s ( τ ) = γ ( τ ) + i ω ( τ ) denote a root of Equation (6) in the neighborhood of τ = τ 20 , which satisfies γ ( τ 20 ) = 0 and ω ( τ 20 ) = ω 20 > 0 . Differentiating both sides of Equation (10) with respect to τ 2 , we obtain

where P ′ i ( s , q ) stands for the derivative of the function P i ( s , q ) ( i = 0 , 1 , 2 , 3 ). Hence,

Let P k ( i ω 0 , q ) = μ k + i ε k for k = 1 , 2 , 3 , and P ′ l ( i ω 0 , q ) = ℏ l + i ℓ l for l = 0 , 1 , 2 , 3 . Substituting these expressions into Equation (8), we obtain

where

In summary, we propose the second assumption and obtain the following theorem1.

(A2):

Theorem 1.Assume that conditions (A1) and (A2) hold, then the equilibrium point ( S * , E * , I * , R * ) of the system (1) is asymptotically stable when τ 2 ∈ [ 0 , τ 20 ) ; when τ 2 = τ 20 , Hopf bifurcation occurs near the equilibrium point.

Case 3.1.3 If τ 1 > 0 , τ 2 = 0 , then Equation (9) becomes

Separating the real and imaginary parts, we obtain the following system of trigonometric equations:

For simplicity, we denote

Then the equation is rewritten as

By Cramer’s rule, we solve for cos ( ω τ 1 ) and sin ( ω τ 1 ) :

Using the fundamental trigonometric identity

we arrive at the constraint

Equation (14) guarantees the existence of a positive real root ω . The critical delay τ 10 can be computed numerically with Maple 13 via the formula

The parameters are defined the same as those in Subsection case 3.1.2. Taking the partial derivative on both sides of Equation (13), we finally obtain:

where