Dynamics analysis and optimal control of delayed SEIR model in COVID-19 epidemic

The coronavirus disease 2019 (COVID-19) remains serious around the world and causes huge deaths and economic losses. Understanding the transmission dynamics of diseases and providing eﬀective control strategies play important roles in the prevention of epidemic diseases. In this paper, to investigate the eﬀect of delays on the transmission of COVID-19, we propose a delayed SEIR model to describe COVID-19 virus transmission, where two delays indicating the incubation and recovery periods are introduced. For this system, we prove its solutions are nonnegative and ultimately bounded with the nonnegative initial conditions. Furthermore, we calculate the disease-free and endemic equilibrium points and analyze the asymptotical stability and the existence of Hopf bifurcations at these equilibrium points. Then, by taking the weighted sum of the opposite number of recovered individuals at the terminal time, the number of exposed and infected individuals during the time horizon, and the system cost of control measures as the cost function, we present a delay optimal control problem, where two controls represent the social contact and the pharmaceutical intervention. Necessary optimality conditions of this optimal control problem are exploited to characterize the optimal control strategies. Finally, numerical simulations are performed to verify the theoretical analysis of the stability and Hopf bifurcations at the equilibrium points and to illustrate the eﬀectiveness of the obtained optimal strategies in controlling the COVID-19 epidemic.


Introduction
The coronavirus disease 2019 (COVID- 19) is an epidemic caused by the SARS-CoV-2 virus, which affects mostly the human respiratory system.Due to the strong infectivity, fast transmission speed, and long incubation period, it has brought significant losses to human life and the global economy [1,2].Although many prevention and control measures have been taken to mitigate the disease spread, the world is still suffering from the infection and death cases of COVID-19.According to the World Health Organization report, as of 2 November 2023, the total number of infections caused by the COVID-19 epidemic has reached 697,318,367, with more than 6,934,066 deaths [3].Thus, there is an urgent need to study the COVID-19 virus transmission behavior and provide effective control measures to prevent the spread of the disease.
Mathematical models have been proven effective means in describing and understanding the complex transmission dynamics of epidemics [4][5][6].In this regard, several mathematical models have been investigated to formulate and control COVID-19 spread [7][8][9].SIR (Susceptible, Infectious, Removed) models have been proposed to simulate and predict the COVID-19 pandemic in [10,11].The stability of the SEIR (Susceptible, Exposed, Infectious, Removed) model is analyzed in [12].Stability analysis of a spatial extension of the SEIR model is carried out in [13].Optimal control of both non-pharmacological and pharmacological interventions for a modified SEIR epidemiological model is studied in [14].Optimal control of COVID-19 spread considering quarantine effect on people with diabetes is investigated in [15].Stability analysis and optimal control of a COVID-19 system, including susceptible, exposed, symptomatically infected, asymptomatically infected, hospitalized, and recovered individuals, are discussed in [16].Bifurcation analysis and optimal control of a discrete SIR system are studied in [17].Optimal control of the Omicron and Delta strains in COVID-19 is investigated in [18].Although the aforementioned results are certainly valid and interesting, time delays are ignored in the dynamics analysis and optimal control problems.In fact, time delays exist in COVID-19 virus transmission since there is an incubation period for the exposed individual to manifest COVID-19 symptoms and signs, and a recovery period is required for the infected individual to become a recovered individual [19,20].
Motivated by the above issue, in the current paper, we propose a delayed SEIR model to describe COVID-19 virus transmission in which two delays representing the incubation period and the recovery period are introduced.For this delayed SEIR system, we prove that its solutions with the nonnegative initial conditions are nonnegative and ultimately bounded.Furthermore, we calculate the disease-free and endemic equilibrium points and analyze the locally asymptotical stability and Hopf bifurcations of these two equilibrium points.Then, by taking the weighted sum of the opposite number of recovered individuals at the terminal time, the number of exposed and infected individuals during the time horizon, and the system cost of control measures as the cost function, we present a delay optimal control problem, where two controls denote the social contact and the pharmaceutical intervention.We also derive the necessary optimality conditions to characterize the optimal controls.Based on the necessary optimality conditions, we develop a numerical approach for solving the delay optimal control problem, which is different from the reported methods in [21][22][23][24][25][26].Finally, numerical simulations are performed to verify the theoretical analysis of the stability and the existence of Hopf bifurcations at equilibrium points and demonstrate the effectiveness of computed optimal strategies in controlling the COVID-19 epidemic.Compared with the existing literature, the main contributions and innovations of this paper include: (i) a novel SEIR system with incubation and recovery delays is proposed for COVID-19 virus transmission; (ii) the stability and Hopf bifurcation of the equilibria in the delayed SEIR model are thoroughly analyzed; and (iii) an effective solution approach is developed for solving the delayed optimal control problem with both the terminal and integral costs.
The paper is organized as follows: In Sect.2, the delayed SEIR system is proposed, and some important properties are proved.In Sect.3, the stability and existence of Hopf bifurcations at the disease-free and endemic equilibrium points are analyzed.In Sect.4, we present the delay optimal control problem and derive the corresponding necessary optimality conditions.In Sect.5, we illustrate the numerical simulation results.In Sect.6, we provide conclusions.

Delayed SEIR model
In the COVID-19 virus transmission process, the primary source of transmission is social contact between susceptible and exposed individuals who are usually asymptomatic.Another transmission source is direct contact between frontline workers and infected individuals.Some exposed individuals usually recover due to their strong natural immune systems or use over-the-counter medications, while others require hospitalization after a certain incubation period.Many infected individuals recover from the disease after extensive treatment, but unfortunately, for some, the infection costs their lives.As is wellknown, a susceptible individual does not show the corresponding symptoms quickly after infection with the virus; that is, there is an incubation period for the exposed individual to be an infective individual.Besides, it is required for the infected individual to spend some time to be a recovered individual.Thus, time delays must be incorporated into the COVID-19 virus transmission process.Based on SEIR system [14], we propose the following delayed SEIR system to describe the COVID-19 virus transmission: where t is the time of process; S(t), E(t), I(t), and R(t) are, respectively, the susceptible, exposed, infected, and recovered individuals; N(t) = S(t) + E(t) + I(t) + R(t) is the total number of individuals affected by the outbreak; is the recruitment in the human population; d is the natural mortality rate; β 1 is the transmission rate due to social contact; β 2 is the transmission rate due to frontline contact; ε is the infection rate; γ 1 is the recovery rate of infectious individuals; γ 2 is the immune recovery rate; μ is the probability of death due to COVID-19; τ 1 and τ 2 are two delay arguments that indicate the incubation and recovery periods, respectively.A flow chart of the delayed SEIR system (1) is shown in Fig. 1.
Proof It is easy to show by Theorem 5.2.1 [27] that S(t) ≥ 0, E(t) ≥ 0, I(t) ≥ 0 and R(t) ≥ 0 for t ≥ 0. Hence, the nonnegative cone R 4 + is invariant for system (1) with nonnegative which gives Then, the delayed SEIR system (1) has a biologically feasible range as indicated below: which implies the ultimate boundedness of the solutions.

Equilibrium points, stability and Hopf bifurcation
In this section, we will analyze the local stability of system (1) at the equilibrium points and establish the existence of Hopf bifurcations at the equilibrium points.

Stability and Hopf bifurcation of endemic equilibrium point
The Jacobian matrix of the system (1) at E 1 is given by where ).For (27), one negative eigenvalue is given by λ 1 = -d.The other eigenvalues are determined by the following characteristic equation: where εγ 1 + (Y + d)εγ 1 .Now, we investigate the following cases for τ 1 and τ 2 at the endemic equilibrium point E 1 .
Recall that the positive endemic equilibrium point E 1 exists if and only if R 0 > 1.Thus, by the Routh-Hurwitz theorem [29], we have the following theorem: Theorem 6 Suppose that τ 1 = τ 2 = 0.If R 0 > 1, then E 1 is locally asymptotically stable.
During the COVID-19 spread, social contacts can be mitigated and suppressed by nonpharmacological interventions, such as wearing masks, maintaining social distancing, testing and isolation, and closing businesses, while the length of hospital stay can be minimized by pharmacological interventions and using new treatment modalities.Thus, using both non-pharmacological and pharmacological interventions as the control strategies, our goal is to maximize the number of recovered individuals at the terminal time, as well as to minimize the number of exposed and infected individuals during the time horizon, and the system cost of control measures.Therefore, the cost function in controlling the COVID-19 epidemic can be stated as where q i > 0, i = 1, 2, 3, r 1 > 0 and r 2 > 0 are weighting coefficients.As a result, we propose the following delay optimal control problem: (DOCP) min J(u)
To explore the effects of each control means, we set up the following control scheme.For the first set of parameters (i.e., the second row in Table 1), we solve Problem (DOCP) and obtain the optimal control strategies u * 1 and u * 2 shown in Fig. 16.The state trajectories corresponding to different control strategies, i.e., u 1 = u 2 = 0, u 1 = u * 1 and u 2 = 0, u 1 = 0 Figure 5 When τ 2 = 0, E 0 is asymptotically stable for τ 1 = 0.9 < τ 0 1 = 1.4683 and u 2 = u * 2 , and u 1 = u * 1 and u 2 = u * 2 , are plotted in Fig. 17.Similarly, for the second set of parameters (i.e., the third row in Table 1), we also solve Problem (DOCP) and obtain the optimal control strategies ū * 1 and ū * 2 depicted in Fig. 18.The state trajectories corresponding to different control strategies, i.e., u 1 = u 2 = 0, u 1 = ū * 1 and u 2 = 0, u 1 = 0 and u 2 = ū * 2 , and u 1 = ū * 1 and u 2 = ū * 2 , are plotted in Fig. 19.From Figs. 16 and 18, we can see that, for the social contact u 1 , it begins with the maximal value of 0.07, keeps the maximal value for a period of time, and then decreases to zero.This is mainly because, in the case of a sudden outbreak of COVID-19, quarantine measures are effective in stopping the spread of the disease by cutting off the route of transmission in the real world.As for the pharmacological intervention u 2 , since the pharmacological intervention for the infected individuals is not immediately administered, it starts from the minimal value of zero, rapidly increases to the maximal value of 0.1, maintains at this maximal value for a period of time, and ultimately reduces to zero.
From Figs. 17(b), 17(c), 19(b), and 19(c), it can be seen that the number of exposed and infected individuals under no control is the highest, while the lowest number of exposed and infected individuals is under strategy C.Moreover, the implementation of u 1 can decrease the number of both exposed and infectious individuals, while the implementation of u 2 is particularly effective in reducing the number of infected individuals.Nevertheless, relying solely on one control measure (strategies A and B) or not implementing any control measures at all is less effective than the optimal control strategy C.
For strategy C, we also solved the optimal control problem without delay.The computed optimal control strategies under two sets of parameter values are also illustrated in Figs.16 and 18, respectively.Under the optimal control strategies in Figs.16 and 18,  and 21, we observe that the peaks of infected individuals for the cases with τ 1 = 0.5 and τ 2 = 0.7 are higher than those without delay.This implies that time delays could aggravate the transmission of COVID-19.As a result, to minimize the number of infections, effective control strategies should be implemented as soon as possible.

Conclusion
In this paper, we have studied the dynamics analysis and optimal control of the delayed SEIR model in the COVID-19 epidemic.Two delays representing the incubation and recovery periods in COVID-19 virus transmission have been introduced.A key issue with delays describing the incubation and recovery periods is whether they cause sustained oscillations.This was investigated through Hopf bifurcation analysis.By using the charac- teristic equations of delay differential equations, the existence of Hopf bifurcations at the disease-free and endemic equilibrium points was established.It has been shown that under some conditions, delays representing the incubation and recovery periods may destabilize the disease-free and endemic equilibrium points and cause the population to fluctuate.From Theorems 3, 4, 5, 7, 8, and 9, we see that thresholds for two delays were identified to characterize the existence of Hopf bifurcations at the disease-free and endemic equilibrium points.In addition, two controls representing the social contact and pharmaceutical intervention have also been introduced.The delay optimal control problem was formulated, and its necessary optimality conditions were exploited to solve the optimal controls.Numerical simulation results indicate that the pharmacological intervention was more effective for hospitalized patients, whereas suppression of social contact reduced the num- We note that the effect of vaccination is ignored in systems ( 1) and (47).Vaccination has played a major role in preventing the spread of COVID-19 [36][37][38].Moreover, the fractional derivative can be regarded as the generalization of the traditional integer derivative, which shows many important properties that the integer derivative does not have [39][40][41][42][43][44].Many scholars have applied fractional derivative differential equations to study the spread of COVID-19 [45][46][47].As a result, it is worthwhile to investigate the fractional optimal control of the COVID-19 epidemic by incorporating vaccination.This will be left for future research work.

Figure 1
Figure 1 Flow chart of delayed SEIR model

Figure 17 Figure 18
Figure 17 State trajectories with different control strategies under the first set of parameter values

Figure 19 Figure 20
Figure 19 State trajectories with different control strategies under the second set of parameter values

Figure 21
Figure 21 Optimal state trajectories under the second set of parameter values

Table 1
Two sets parameter values in the simulations