Study on SEAI Model of COVID-19 Based on Asymptomatic Infection

Abstract

In this paper, an SEAI epidemic model with asymptomatic infection is studied under the background of mass transmission of COVID-19. First, we use the next-generation matrix method to obtain the basic reproductive number $R_0$ and calculate the equilibrium point. Secondly, when $R_0<1$, the local asymptotic stability of the disease-free equilibrium is proved by Hurwitz criterion, and the global asymptotic stability of the disease-free equilibrium is proved by constructing the Lyapunov function. When $R_0>1$, the system has a unique endemic equilibrium point and is locally asymptotically stable, and it is also proved that the system is uniformly persistent. Then, the application of optimal control theory is carried out, and the expression of the optimal control solution is obtained. Finally, in order to verify the correctness of the theory, the stability of the equilibrium point is numerically simulated and the sensitivity of the parameters of $R_0$ is analyzed. We also simulated the comparison of the number of asymptomatic infected people and symptomatic infected people before and after adopting the optimal control strategy. This shows that the infection of asymptomatic people cannot be underestimated in the spread of COVID-19 virus, and an isolation strategy should be adopted to control the spread speed of the disease.

1. Introduction

Infectious diseases [1,2] are one of the biggest threats to human survival. The frequent outbreaks and epidemics of infectious diseases not only affect people’s health and hinder economic development, but also threaten social stability and damage social interests to a greater extent. Therefore, experts in the field of epidemiology and related biology are deeply studying infectious diseases, and their goal is to expose the spread dynamics and patterns of diseases and predict the future development trend by analyzing and understanding various infectious disease models [3,4,5].

Since the end of 2019, COVID-19 [6], caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), has spread all over China and the world. This virus shows symptoms such as dry cough, fever and fatigue, and may further develop into pneumonia and renal failure, which may lead to death in severe cases [7,8,9,10]. The disease is mainly spread by bodily contact and sneezing and coughing droplets [11], and more and more evidence shows that asymptomatic infected people play an important driving role in its rapid spread [12]. Therefore, in order to prevent the rapid spread of COVID-19, many researchers in the field of biology aimed to understand it, and wanted to expose the epidemic law of diseases through the study of models, so as to provide theoretical basis and strategies for the detection, prevention and management of disease outbreaks. For example, Zhang et al. [13] combined the transmission mechanism of COVID-19 with preventive measures, put forward a new stochastic dynamic model, and estimated and predicted the epidemic trend and control opportunity abroad. In addition, Tang et al. [14] considered medical follow-up quarantine, hospital isolation, treatment and other control and management measures, and evaluated the impact of public health interventions on the spread of COVID-19.

Asymptomatic infected people are contagious, but have no clinical symptoms. They are hidden in the crowd and are not easy to find found. Therefore, this group plays a great role in the spread of COVID-19. This characteristic of asymptomatic infected people is the main reason why COVID-19 has become a pandemic. Several studies have found that asymptomatic infection accounts for about $40\%$ of patients in COVID-19, and individuals with asymptomatic infection are more likely to cause a larger epidemic than imported cases [15,16,17]. In order to understand more deeply how asymptomatic infected people promote the rapid spread of new pneumonia, many experts and scholars have established relevant infectious disease models. For example, Khan et al. [18] studied the dynamics of a stochastic SAIR mathematical model. Tan et al. [19] estimated the spread of symptomatic and asymptomatic COVID-19 with contact information. Dobrovolny [20] researched the role of asymptomatic individuals in the spread of infection. This study shows that even if asymptomatic infection does not necessarily account for a large proportion of infections, it can still change the scale and lethality of epidemics. Sun et al. [21] established an SCIRA model to estimate the impact of asymptomatic infected people. Their research shows that the potential impact of these hidden cases greatly promoted the outbreak of COVID-19 because asymptomatic infected people are contagious.

As an important method to study the transmission mechanism of infectious diseases, dynamic modeling has always been one of the hot issues in the field of infectious diseases. At present, the spread theory and modeling of infectious diseases have been extensively studied [22,23]. Scholars have established an infectious disease model based on the characteristics of COVID-19 to predict the epidemic spread trend, but there are still some limitations. In order to make up for the deficiency of the existing research, this paper used the actual transmission characteristics of COVID-19 to improve the classic SEAI infectious disease model, so as to accurately reveal the transmission mechanism of asymptomatic infected people in COVID-19, explore how asymptomatic infected people affect the outbreak of COVID-19 epidemic, and put forward some strategies to control the spread of COVID-19.

In this paper, we establish an SEAI model of COVID-19 with asymptomatic infection, and make a dynamic analysis to determine the influence of asymptomatic infection in the spread of COVID-19. The rest is organized as follows: Section 1 introduces the research background of COVID-19. Section 2 establishes the model. Section 3 and Section 4 calculate the basic reproduction number and the equilibrium point of the model and prove the stability of the equilibrium point. In Section 5, the optimal control theory is put forward and the optimal control solution is obtained. In Section 6, the stability of the equilibrium point is numerically simulated, the sensitivity of the parameters affecting $R_0$ is analyzed, and the number of asymptomatic infected people and symptomatic infected people before and after adopting the optimal control strategy is further simulated. Section 7 gives the conclusion of this paper.

2. Model Formulation

At present, scholars at home and abroad have begun to study the influence of limited resources on the spread of diseases. For example, Zhou et al. [24] proposed a continuously differentiable treatment function

$$h(I)=\frac{rI}{1+\alpha I},$$

to describe the “saturation” phenomenon of limited treatment. r represents the maximum cure rate, $\alpha$ describes the effect of the delayed treatment of the infected person. Obviously, $r/\alpha$ represents the maximum supply of therapeutic resources, while $1/(1+\alpha I)$ describes the supply efficiency of medical resources, which has an important impact on the spread and control of diseases.

In this paper, the total population N is divided into four different parts: S stands for susceptible person, E stands for the infiltrator, A stands for asymptomatic infected person, I stands for symptomatic infected person. Considering that the treatment of COVID-19 is saturated, the saturated treatment function $h(I)$ is introduced into the classical infectious disease model, and the SEAI model is established. A flow chart of the model is shown in Figure 1.

Figure 1. Flow chart of SEAI transmission.

The propagation dynamics model corresponding to the flow chart is as follows:

$$\left\{\begin{array}{c}\frac{dS}{dt}=\mathsf{\Lambda }-{\beta }_{1}SA-{\beta }_{2}SI-dS+\frac{a_{1}A}{1+b_{1}A}+\frac{a_{2}I}{1+b_{2}I},\hfill \\ \frac{dE}{dt}={\beta }_{1}SA+{\beta }_{2}SI-(\delta +d)E,\hfill \\ \frac{dA}{dt}=k\delta E-(p+d)A-\frac{a_{1}A}{1+b_{1}A},\hfill \\ \frac{dI}{dt}=(1-k)\delta E+pA-(d+\mu )I-\frac{a_{2}I}{1+b_{2}I}.\hfill \end{array}\right.$$

(1)

Assuming that all the parameters and variables involved in the above model are non-negative, please refer to Table 1 for the dynamic significance of infectious diseases of the corresponding parameters. Note that the expressions $\frac{a_{1}A}{1+b_{1}A}$ and $\frac{a_{2}I}{1+b_{2}I}$ represent limited medical resources.

Table 1. Parameter definitions for model (1).

Parameter	Definition	Unit
$\mathsf{\Lambda }$	the constant input of population	${\mathrm{people}}^{-1}{\mathrm{day}}^{-1}$
d	the natural death rate of population	${\mathrm{day}}^{-1}$
$\mu$	the morbidity and mortality of symptomatic infected persons	${\mathrm{day}}^{-1}$
p	the rate of transformation from asymptomatic infection to symptomatic infection	${\mathrm{day}}^{-1}$
$a_1$	the maximum cure rate for asymptomatic patients	${\mathrm{day}}^{-1}$
$a_2$	the maximum cure rate for an infected person	${\mathrm{day}}^{-1}$
$b_1$	resource constraints for treating asymptomatic patients	${\mathrm{day}}^{-1}$
$b_2$	resource constraints for treating infected people	${\mathrm{day}}^{-1}$
${\beta }_1$	the infection rate of asymptomatic infected people to susceptible people	${\mathrm{people}}^{-1}{\mathrm{day}}^{-1}$
${\beta }_2$	the infection rate of infected people with symptoms to susceptible people	${\mathrm{people}}^{-1}{\mathrm{day}}^{-1}$
$\delta$	the transfer rates of latent to infected persons	${\mathrm{day}}^{-1}$
$k\delta$	the rate at which latent persons develop asymptomatic infections	${\mathrm{day}}^{-1}$
$(1-k)\delta$	the rate at which latent persons become infected with symptoms	${\mathrm{day}}^{-1}$

It follows from model (1) that $N=S+E+A+I$ and $\frac{dN}{dt}=\mathsf{\Lambda }-dN-\mu I\le \mathsf{\Lambda }-dN$. When $N>\frac{\mathsf{\Lambda }}{d}$ approaches infinity, such that $\frac{dN}{dt}<0$, for the sake of generality, the feasible domain is

$$X=\left\{(S,E,A,I)\in R_{+}^4:S+E+A+I\le \frac{\mathsf{\Lambda }}{d}\right\}.$$

(2)

3. Basic Regeneration Number and Equilibrium

By simple calculation, the only disease-free equilibrium of system (1) is $E_0=(\frac{\mathsf{\Lambda }}{d},0,0,0)$, where $S_0=\frac{\mathsf{\Lambda }}{d}, E_0=A_0=I_0=0$.

By the next generation matrix method [25,26], we will derive the basic regeneration number of model (1). Let

$$\mathcal{F}=\left[\begin{array}{c}{\beta }_{1}SA+{\beta }_{2}SI\\ 0\\ 0\end{array}\right], \mathcal{V}=\left[\begin{array}{c}{Q}_{1}E\\ -k\delta E+Q_{2}A+\frac{a_{1}A}{1+b_{1}A}\\ -(1-k)\delta E-pA+Q_{3}I+\frac{a_{2}I}{1+b_{2}I}\end{array}\right]$$

be the input rate of newly infected individuals and the rate of transfer of individuals, where

$$Q_1=\delta +d, Q_2=p+d, Q_3=d+\mu .$$

Respectively, then we obtain

$$F=\left[\begin{array}{ccc}0& {\beta }_1\frac{\mathsf{\Lambda }}{d}& {\beta }_2\frac{\mathsf{\Lambda }}{d}\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],V=\left[\begin{array}{ccc}{Q}_1& 0& 0\\ -k\delta & Q_2+a_1& 0\\ -(1-k)\delta & -p& Q_3+a_1\end{array}\right].$$

Thus, we have

$$F{V}^{-1}=\left[\begin{array}{ccc}\frac{{\beta }_1\mathsf{\Lambda }k\delta }{d{Q}_1(Q_2+a_1)}+\frac{{\beta }_2\mathsf{\Lambda }k\delta p}{d{Q}_1(Q_2+a_1)(Q_3+a_2)}+\frac{{\beta }_2\mathsf{\Lambda }(1-k)\delta }{d{Q}_1(Q_3+a_2)}& \frac{{\beta }_1\mathsf{\Lambda }(Q_3+a_2)+{\beta }_2\mathsf{\Lambda }p}{d(Q_2+a_1)(Q_3+a_2)}& \frac{{\beta }_2\mathsf{\Lambda }}{d(Q_3+a_2)}\\ 0& 0& 0\\ 0& 0& 0\end{array}\right].$$

Therefore, the basic regeneration number is

$$R_0=\rho (F{V}^{-1})=r_1+r_2+r_3,$$

(3)

where

$$r_1=\frac{k\delta {\beta }_1\mathsf{\Lambda }}{d{Q}_1(Q_2+a_1)}, r_2=\frac{(1-k)\delta {\beta }_2\mathsf{\Lambda }}{d{Q}_1(Q_3+a_2)}, r_3=\frac{k\delta p{\beta }_2\mathsf{\Lambda }}{d{Q}_1(Q_2+a_1)(Q_3+a_2)}.$$

The endemic equilibrium of system (1) is denoted as $E^{*}=(S_1,E_1,A_1,I_1)$. To solve the endemic equilibrium, we set the right side of model (1) to 0, that is

$$\left\{\begin{array}{c}\mathsf{\Lambda }-{\beta }_1{S}_1{A}_1-{\beta }_2{S}_1{I}_1-d{S}_1+\frac{a_1{A}_1}{1+b_1{A}_1}+\frac{a_2{I}_1}{1+b_2{I}_1}=0,\hfill \\ {\beta }_1{S}_1{A}_1+{\beta }_2{S}_1{I}_1-(\delta +d)E_1=0,\hfill \\ k\delta E_1-(p+d)A_1-\frac{a_1{A}_1}{1+b_1{A}_1}=0,\hfill \\ (1-k)\delta E_1+p{A}_1-(d+\mu )I_1-\frac{a_2{I}_1}{1+b_2{I}_1}=0,\hfill \end{array}\right.$$

(4)

where

$$\left\{\begin{array}{ccc}{A}_1\hfill & =\hfill & A_1,\hfill \\ E_1\hfill & =\hfill & \frac{Q_2(1+b_1{A}_1)+a_1}{(1+b_1{A}_1)k\delta }{A}_1,\hfill \\ I_1\hfill & =\hfill & \frac{(R_0-1)(\mathsf{\Lambda }-dN)W}{R_0\mu [W-d{Q}_1(Q_2+a_1)(Q_3+a_2)]},\hfill \\ S_1\hfill & =\hfill & \frac{\mu Q_1[Q_2(1+b_1{A}_1)+a_1]A_1}{(1+b_1{A}_1)k\delta [\mu {\beta }_1{A}_1+{\beta }_2(\mathsf{\Lambda }-dN)]},\hfill \end{array}\right.$$

and $W=k\delta {\beta }_1\mathsf{\Lambda }(Q_3+a_2)+(1-k)\delta {\beta }_2\mathsf{\Lambda }(Q_2+a_1+p)$.

If $R_0>1$, then $I_1$ is positive, and $A_1$ satisfies the following equation

$$w_1{A}_1^2+w_2{A}_1+w_3=0,$$

where

$$\begin{array}{c}{w}_1=b_{1}kp+(1-k)Q_2{b}_1>0,w_2=kp+(1-k)Q_2+(1-k)a_1>0,\hfill \\ w_3=-H<0,H=\frac{a_2(\mathsf{\Lambda }-dN)}{\mu +b_2(\mathsf{\Lambda }-dN)}>0,A_1=-w_2/(2w_1),\hfill \\ \Delta =w_2^2-4w_1{w}_3={[kp+(1-k)Q_2+(1-k)a_1]}^2+4(b_{1}kp+(1-k)Q_2{b}_1)H>0.\hfill \end{array}$$

So there is only root $A_1>0$. When $R_0>1$, $S_1,E_1,A_1,I_1$ are positive and system (1) has a unique endemic equilibrium $E^{*}=(S_1,E_1,A_1,I_1)$.

3.1. Stability of Disease-Free Equilibrium

Theorem 1. 

When $R_0<1$, the disease-free equilibrium $E_0$ of system (1) is locally asymptotically stable in X; When $R_0>1$, the disease-free equilibrium is unstable.

Proof of Theorem 1. 

The Jacobian matrix of system (1) at $E_0$ is

$$J_{*}=\left[\begin{array}{cccc}-d& 0& -{\beta }_1\frac{\mathsf{\Lambda }}{d}+a_1& -{\beta }_2\frac{\mathsf{\Lambda }}{d}+a_2\\ 0& -Q_1& {\beta }_1\frac{\mathsf{\Lambda }}{d}& {\beta }_2\frac{\mathsf{\Lambda }}{d}\\ 0& k\delta & -Q_2-a_1& 0\\ 0& (1-k)\delta & p& -Q_3-a_2\end{array}\right].$$

The characteristic equation of matrix $J_{*}$ is

$$(\lambda +d)({\lambda }^3+{\mathcal{A}}_1{\lambda }^2+{\mathcal{A}}_2\lambda +{\mathcal{A}}_3)=0.$$

(5)

Obviously, ${\lambda }_1=-d$ is a negative real root of equation, and

$$\begin{array}{cc}{\mathcal{A}}_1\hfill & =Q_1+Q_2+Q_3+a_1+a_2>0,\hfill \\ {\mathcal{A}}_2\hfill & =(Q_3+a_2)(Q_1+Q_2+a_1)+Q_1(Q_2+a_1)-{\beta }_2(1-k)\delta \frac{\mathsf{\Lambda }}{d}-{\beta }_{1}k\delta \frac{\mathsf{\Lambda }}{d}\hfill \\ \hfill & =Q_1(Q_3+a_2)(1-r_2)+Q_1(Q_2+a_1)(1-r_1)+(Q_3+a_2)(Q_2+a_1)>0,\hfill \\ {\mathcal{A}}_3\hfill & =Q_1(Q_2+a_1)(Q_3+a_2)-(Q_2+a_1){\beta }_2\frac{\mathsf{\Lambda }}{d}(1-k)\delta -(Q_3+a_2){\beta }_1\frac{\mathsf{\Lambda }}{d}k\delta -k\delta p{\beta }_2\frac{\mathsf{\Lambda }}{d}\hfill \\ \hfill & =Q_1(Q_2+a_1)(Q_3+a_2)(1-R_0)>0.\hfill \end{array}$$

Then, we obtain

$$\begin{array}{ccc}{\mathcal{A}}_1{\mathcal{A}}_2-{\mathcal{A}}_3\hfill & =\hfill & Q_1^2[(Q_3+a_2)(1-r_2)+(Q_2+a_1)(1-r_1)]\hfill \\ \hfill & \hfill & +(Q_2+Q_3+a_1+a_2)[Q_1(Q_3+a_2)(1-r_2)+Q_1(Q_2+a_1)(1-r_1)\hfill \\ \hfill & \hfill & +(Q_3+a_2)(Q_2+a_1)]+Q_1(Q_2+a_1)(Q_3+a_2)R_0>0.\hfill \end{array}$$

Therefore, according to the Hurwitz criterion [27,28], when $R_0<1,{\mathcal{A}}_3>0$, system (1) is locally asymptotically stable at the disease-free equilibrium point $E_0$. When $R_0>1$ ${\mathcal{A}}_3<0$, the disease-free equilibrium point is unstable. The proof is complete. □

Theorem 2. 

When $R_0\le 1-\frac{W}{{\beta }_{1}A+{\beta }_{2}I}<1$, the disease-free equilibrium point $E_0$ of system (1) is globally asymptotically stable in X, where

$$W=\left[\frac{{\beta }_1(Q_3+a_2)+p{\beta }_2}{(Q_2+a_1)(Q_3+a_2)}\frac{a_1{b}_1{A}^2}{1+b_{1}A}+\frac{{\beta }_2}{Q_3+a_2}\frac{a_2{b}_2{I}^2}{1+b_{2}I}\right].$$

Proof of Theorem 2. 

Construction of Lyapunov function

$$\begin{array}{ccc}V\hfill & =\hfill & \frac{\mathsf{\Lambda }}{d}\left[\frac{k\delta }{Q_1(Q_2+a_1)}{\beta }_1+\frac{k\delta p+(1-k)\delta (Q_2+a_1)}{Q_1(Q_2+a_1)(Q_3+a_2)}{\beta }_2\right]E\hfill \\ \hfill & \hfill & +\frac{\mathsf{\Lambda }}{d}\left[\frac{{\beta }_1}{Q_2+a_1}+\frac{p{\beta }_2}{(Q_2+a_1)(Q_3+a_2)}\right]A+\frac{\mathsf{\Lambda }}{d}\frac{{\beta }_2}{Q_3+a_2}I>0.\hfill \end{array}$$

When $R_0\le 1-\frac{W}{{\beta }_{1}A+{\beta }_{2}I}<1$,

$$\begin{array}{ccc}\frac{dV}{dt}\hfill & =\hfill & \frac{\mathsf{\Lambda }}{d}\left[\frac{k\delta }{Q_1(Q_2+a_1)}{\beta }_1+\frac{k\delta p+(1-k)\delta (Q_2+a_1)}{Q_1(Q_2+a_1)(Q_3+a_2)}{\beta }_2\right]E^{\prime }\hfill \\ \hfill & \hfill & +\frac{\mathsf{\Lambda }}{d}\left[\frac{{\beta }_1}{Q_2+a_1}+\frac{p{\beta }_2}{(Q_2+a_1)(Q_3+a_2)}\right]A^{\prime }+\frac{\mathsf{\Lambda }}{d}\frac{{\beta }_2}{Q_3+a_2}{I}^{\prime }\hfill \\ \hfill & =\hfill & \frac{\mathsf{\Lambda }}{d}\left[\frac{k\delta }{Q_1(Q_2+a_1)}{\beta }_1+\frac{k\delta p+(1-k)\delta (Q_2+a_1)}{Q_1(Q_2+a_1)(Q_3+a_2)}{\beta }_2\right]\left[{\beta }_{1}SA+{\beta }_{2}SI-(\delta +d)E\right]\hfill \\ & \hfill & +\frac{\mathsf{\Lambda }}{d}\left[\frac{{\beta }_1}{Q_2+a_1}+\frac{p{\beta }_2}{(Q_2+a_1)(Q_3+a_2)}\right][k\delta E-(p+d)A-\frac{a_{1}A}{1+b_{1}A}]\hfill \\ \hfill & \hfill & +\frac{{\beta }_2}{Q_3+a_2}[(1-k)\delta E+pA-(d+\mu )I-\frac{a_{2}I}{1+b_{2}I}]\frac{\mathsf{\Lambda }}{d}\hfill \\ \hfill & =\hfill & (R_0-1)({\beta }_{1}A+{\beta }_{2}I)+\left[\frac{{\beta }_1(Q_3+a_2)+p{\beta }_2}{(Q_2+a_1)(Q_3+a_2)}\frac{a_1{b}_1{A}^2}{1+b_{1}A}+\frac{{\beta }_2}{Q_3+a_2}\frac{a_2{b}_2{I}^2}{1+b_{2}I}\right]\hfill \\ \hfill & =\hfill & (R_0-1)({\beta }_{1}A+{\beta }_{2}I)+W\le 0.\hfill \end{array}$$

Let $\mathsf{\Omega }=\left\{(S,E,A,I)|\frac{dV}{dt}=0\right\}=\left\{(S,E,A,I)\right|A=I=0\}$, within $\mathsf{\Omega }$, when $t\to \infty$, there is $S\to \frac{\mathsf{\Lambda }}{d}$. Therefore, $E_0$ is the maximal w-invariant set of $\mathsf{\Omega }$. According to LaSalle’s invariant set principle [29], any trajectory within X converges to $E_0$, where $E_0$ is the disease-free equilibrium and is globally asymptotically stable within X. The proof is complete. □

3.2. Stability of the Endemic Equilibrium

Theorem 3. 

When $R_0>1$, if ${\mathcal{B}}_1{\mathcal{B}}_2-{\mathcal{B}}_3>0$ and ${\mathcal{B}}_3({\mathcal{B}}_1{\mathcal{B}}_2-{\mathcal{B}}_3)>{\mathcal{B}}_1^2{\mathcal{B}}_4$, then the endemic equilibrium $E^{*}$ of system (1) is locally asymptotically stable in X. Among them,

$$\begin{array}{ccc}{\mathcal{B}}_1\hfill & =\hfill & a_{11}+a_{12}+d,\hfill \\ {\mathcal{B}}_2\hfill & =\hfill & a_{11}(a_{12}+d)+a_{13}+a_{14}-a_{15},\hfill \\ {\mathcal{B}}_3\hfill & =\hfill & (a_{12}+d)(a_{13}+a_{14}-a_{15})+a_{12}{a}_{17}+a_{16}+a_{18}-k\delta p{\beta }_2{S}_1,\hfill \\ {\mathcal{B}}_4\hfill & =\hfill & (a_{12}+d)(a_{16}-{\beta }_2{S}_{1}k\delta p-a_{18})\hfill \\ \hfill & \hfill & +a_{12}\{\delta ({\beta }_2{S}_1+N)[kp+(1-k)(Q_2+M)]+k\delta (Q_3+N)({\beta }_1{S}_1+M)\}\hfill \end{array}$$

and have

$$\begin{array}{c}{a}_{11}=Q_1+Q_2+Q_3+M+N,a_{12}={\beta }_1{A}_1+{\beta }_2{I}_1,\hfill \\ a_{13}=(Q_2+M)(Q_3+N),a_{14}=Q_1(Q_2+Q_3+M+N),\hfill \\ a_{15}=[{\beta }_2(1-k)-k{\beta }_1]S_1\delta ,a_{16}=Q_1(Q_2+M)(Q_3+N),\hfill \\ a_{17}=(1-k)\delta ({\beta }_2{S}_1+N)+k\delta ({\beta }_1{S}_1+M),\hfill \\ a_{18}=(1-k)\delta {\beta }_2{S}_1(Q_2+M)+k\delta {\beta }_1{S}_1(Q_3+N),\hfill \end{array}$$

where

$$M=\frac{a_1}{{(1+b_1{A}_1)}^2},N=\frac{a_2}{{(1+b_2{I}_1)}^2}.$$

Proof of Theorem 3. 

The Jacobian matrix of system (1) at $E^{*}$ is

$$J_{|E_{*}}=\left[\begin{array}{cccc}-{\beta }_1{A}_1-{\beta }_2{I}_1-d& 0& -{\beta }_1{S}_1-\frac{a_1}{{(1+b_1{A}_1)}^2}& -{\beta }_2{S}_1-\frac{a_2}{{(1+b_2{I}_1)}^2}\\ \beta A_1+{\beta }_2{I}_1& -Q_1& {\beta }_1{S}_1& {\beta }_2{S}_1\\ 0& k\delta & -Q_2-\frac{a_1}{{(1+b_1{A}_1)}^2}& 0\\ 0& (1-k)\delta & p& -Q_3-\frac{a_2}{{(1+b_2{I}_1)}^2}\end{array}\right],$$

then

$$\lambda Identity-J_{|E_{*}}=\left[\begin{array}{cccc}\lambda +{\beta }_1{A}_1+{\beta }_2{I}_1+d& 0& {\beta }_1{S}_1+\frac{a_1}{{(1+b_1{A}_1)}^2}& {\beta }_2{S}_1+\frac{a_2}{{(1+b_2{I}_1)}^2}\\ -\beta A_1-{\beta }_2{I}_1& \lambda +Q_1& -{\beta }_1{S}_1& -{\beta }_2{S}_1\\ 0& -k\delta & \lambda +Q_2+\frac{a_1}{{(1+b_1{A}_1)}^2}& 0\\ 0& -(1-k)\delta & -p& \lambda +Q_3+\frac{a_2}{{(1+b_2{I}_1)}^2}\end{array}\right],$$

then

$$det(\lambda Identity-J_{|E_{*}})={\lambda }^4+{\mathcal{B}}_1{\lambda }^3+{\mathcal{B}}_2{\lambda }^2+{\mathcal{B}}_3\lambda +{\mathcal{B}}_4=0.$$

If $R_0>1$, it can be determined that

$${\mathcal{B}}_1,{\mathcal{B}}_2,{\mathcal{B}}_3,{\mathcal{B}}_4>0,$$

(6)

then we have

$${\mathcal{B}}_1{\mathcal{B}}_2-{\mathcal{B}}_3>0,{\mathcal{B}}_3({\mathcal{B}}_1{\mathcal{B}}_2-{\mathcal{B}}_3)>{\mathcal{B}}_1^2{\mathcal{B}}_4.$$

(7)

The proofs of (6) and (7) are given in Appendix A and Appendix B, respectively. Therefore, according to Hurwitz criterion [30], the equilibrium $E^{*}$ of system (1) is locally asymptotically stable. The proof is complete. □

4. Persistence

Theorem 4. 

When $R_0>1$, the system (1) is uniformly persistent.

Proof of Theorem 4. 

To prove persistence, we first give the following notation and definition

$$X_0=\left\{(S,E,A,I)\in X|E,A,I>0\right\},\partial X_0=X\setminus X_0.$$

We obtain from system (1)

$$E(t)\ge E(t_0)e^{-(\sigma +d)(t-t_0)},A(t)\ge A(t_0)e^{-(p+d)(t-t_0)},I(t)\ge I(t_0)e^{-(\mu +d)(t-t_0)}.$$

(8)

Therefore, X and $X_0$ are positively invariant sets, where $\partial X_0$ is a relatively closed set of X. Next, we will prove that system (1) is uniformly persistent. Let

$$M_{\partial }=\left\{(S(0),E(0),A(0),I(0))|(S(t),E(t),A(t),I(t))\in \partial X_0,\forall t\ge 0\right\}.$$

Now we prove

$$M_{\partial }=(S(0),0,0,0)|S(t)\ge 0,$$

we clearly have

$$\left\{(S(0),0,0,0)|S(t)\ge 0\right\}\subseteq M_{\partial },$$

and now we just need proof

$$M_{\partial }\subseteq \left\{(S(0),0,0,0)|S(t)\ge 0\right\}.$$

Let $(S(0),E(0),A(0),I(0))\in M_{\partial }$ be a statement. We need to prove that for statement $\forall t\ge 0$, have $E(t)=0,A(t)=0,I(t)=0$, there exists the statement $E(t)=0,A(t)=0,$ $I(t)=0$. By using proof by contradiction, let us assume that the conclusion is not true. In that case, there exists the statement $t_0\ge 0$ such that one of the following equations holds:

$$(i)E(t_0)>0;(ii)A(t_0)>0;(iii)I(t_0)>0.$$

For case $(i)$, solving Equation (8) for all $t>t_0$ yields statement $A(t)>0$. Furthermore, from system (1), we have $E(t)>0,I(t)>0$. Hence, we have statement $(S(t),E(t),A(t),I(t))\notin \partial X_0$, which leads to a contradiction. For case $(ii)$, a similar approach leads to a contradiction with statement $(S(0),E(0),A(0),I(0))\in M_{\partial }$. In the case of $(iii)$, that is $E(t_0)>0$, when $t>t_0$ holds, we can obtain

$$A(t)=A(t_0)e^{-(p+d)(t-t_0)}+{\int }_{t_0}^t(k\delta E-\frac{a_{1}A}{1+b_{1}A})e^{p+d}dt.$$

Clearly, when $E(t_0)>0$, for $\forall t>t_0$ have $A(t)>0$ holds. Similarly, by applying the same method, the formal solutions for $I(t)$ can be obtained. When $t>t_0$, we have $I(t)>0$. Therefore, statement $(S(t),E(t),A(t),I(t))\notin \partial X_0$ leads to a contradiction. Thus, it is concluded that $M_{\partial }=\left\{(S(0),0,0,0)\right|S(0)\ge 0\}$ holds. The system (1) has a globally asymptotically stable disease-free equilibrium $E(0)$, and there is only one equilibrium $E(0)$ in $M_{\partial }$.

The following will prove that $E_0$ is weakly repulsive with respect to the set $X_0$, that is, proving $\underset{t\to \infty }{lim}supd(\mathsf{\Phi }(t),E_0)>0$ is sufficient by demonstrating $W^s(E_0)\cap X_0=⌀$. Using proof by contradiction, let us assume this conclusion is not true. Then there exists a positive solution $(S(t),E(t),A(t),I(t))$ for system (1) such that

$$\underset{t\to \infty }{lim}(S(t),E(t),A(t),I(t))=(S^0,0,0,0).$$

Define $M=F-V$. Due to $R_0>1$, therefore $s(M)>0$. For a small enough value of $\epsilon >0$, there exists $s(M-M_{\epsilon })>0$, where

$$M_{\epsilon }=\left[\begin{array}{ccc}0& {\beta }_1\epsilon & {\beta }_2\epsilon \\ 0& 0& 0\\ 0& 0& 0\end{array}\right].$$

There exists $T>0$ such that for any $t>T$, have $S^0-\epsilon <S(t)<S^0+\epsilon$. Thus, the following differential equation inequality can be obtained:

$$\left\{\begin{array}{c}\frac{dE}{dt}\ge {\beta }_1(S^0-\epsilon )A+{\beta }_2(S^0-\epsilon )I-(\delta +d)E,\hfill \\ \frac{dA}{dt}=k\delta E-(p+d)A-\frac{a_{1}A}{1+b_{1}A},\hfill \\ \frac{dI}{dt}=(1-k)\delta E+pA-(d+\mu )I-\frac{a_{2}I}{1+b_{2}I}.\hfill \end{array}\right.$$

Auxiliary system

$$\left\{\begin{array}{c}\frac{dE}{dt}={\beta }_1(S^0-\epsilon )A+{\beta }_2(S^0-\epsilon )I-(\delta +d)E,\hfill \\ \frac{dA}{dt}=k\delta E-(p+d)A-\frac{a_{1}A}{1+b_{1}A},\hfill \\ \frac{dI}{dt}=(1-k)\delta E+pA-(d+\mu )I-\frac{a_{2}I}{1+b_{2}I}.\hfill \end{array}\right.$$

(9)

Because of $s(M-M_{\epsilon })>0$, when $t\to \infty$, $E(t)\to \infty ,A(t)\to \infty ,I(t)\to \infty$. This contradicts the assumption that when $t\to \infty$, $E(t)\to 0,A(t)\to 0,I(t)\to 0$. Thus, it is proved that $W^s(E_0)\cap X_0=⌀$ holds. In summary, it can be concluded that the system (1) with respect to $(X_0,\partial X_0)$ is uniformly persistent. The proof is complete. □

5. Optimal Control Strategy

The maximum principle proposed by Pontryagin is one of the three cornerstones of optimal control theory. It can be applied to solve interdisciplinary problems, formulate rational and effective control strategies in mathematical models, and effectively prevent the spread of COVID-19. Given the complexity of infectious disease modeling, which typically involves numerous parameters, the use of optimal control measures to analyze the dynamics of diseases is crucial. The application of optimal control theory contributes to improving the predictive accuracy of models and designing the most effective strategies to reduce the impact of diseases on populations [31].

In this section, the Pontryagin maximum principle is employed to identify optimal control strategies for addressing COVID-19, with the aim of minimizing the number of infected individuals and minimizing the associated control costs. Applying the Pontryagin maximum principle from optimal control theory allows for the identification of rational and effective strategies for controlling infectious diseases, necessitating the exploration of optimal control conditions within the model.

Rewrite system (1) as the following set of nonlinear differential equations:

$$\left\{\begin{array}{c}\frac{dS}{dt}=\mathsf{\Lambda }-{\beta }_1(1-u_1)SA-{\beta }_2(1-u_2)SI-dS+\frac{a_{1}A}{1+b_{1}A}+\frac{a_{2}I}{1+b_{2}I},\hfill \\ \frac{dE}{dt}={\beta }_1(1-u_1)SA+{\beta }_2(1-u_2)SI-(\delta +d)E,\hfill \\ \frac{dA}{dt}=k\delta E-(p+d+u_3)A-\frac{a_{1}A}{1+b_{1}A},\hfill \\ \frac{dI}{dt}=(1-k)\delta E+pA-(d+\mu +u_4)I-\frac{a_{2}I}{1+b_{2}I},\hfill \end{array}\right.$$

(10)

where $u_1$ signifies the strategy of isolating to reduce interaction between susceptible and asymptomatic infected individuals, $u_2$ indicates the strategy of isolating to lessen contact between susceptible and symptomatic infected individuals, $u_3$ denotes the approach to enhance treatment and recovery of asymptomatic infected individuals, and $u_4$ corresponds to the strategy of improving treatment and recovery of symptomatic infected individuals.

Define the control set

$$U=\left\{(u_1(t),u_2(t),u_3(t),u_4(t))\right|0\le u_i(t)\le 1,i=1,2,3,4\},$$

and $u_i(t)$ is Lebesgue measurable at $[0,1]$.

By applying the method of Ahmad et al. [32] to construct the objective function, we have completed additional research on the basis of their work and constructed the following objective function

$$J(u_1,u_2,u_3,u_4)={\int }_0^T(P_{1}A+P_{2}I+\frac{w_1}{2}{u}_1^2+\frac{w_2}{2}{u}_2^2+\frac{w_3}{2}{u}_3^2+\frac{w_4}{2}{u}_4^2)dt,$$

(11)

where: $P_1$ and $P_2$, respectively, represent the weight coefficients of asymptomatic infected persons and symptomatic infected persons; $w_1,w_2,w_3,w_4$ respectively represent the weight coefficients corresponding to each control strategy; $\frac{w_1}{2}{u}_1^2,\frac{w_2}{2}{u}_2^2,\frac{w_3}{2}{u}_3^2,\frac{w_4}{2}{u}_4^2$ respectively indicate the cost required for the corresponding control policy.

The optimal control strategy problem is now described as

$$OCP:\left\{\begin{array}{c}minJ(u),\hfill \\ s.t.u=(u_1,u_2,u_3,u_4)\in U.\hfill \end{array}\right.$$

(12)

5.1. Existence of Optimal Control Solutions

Theorem 5. 

System (10) has optimal control $\overrightarrow{u^{*}}=(u_1^{*},u_2^{*},u_3^{*},u_4^{*})\in U$, such that

$$J({u_1}^{*},{u_2}^{*},{u_3}^{*},{u_4}^{*})=\underset{u_1,u_2,u_3,u_4\in U}{min}J(u_1,u_2,u_3,u_4).$$

(13)

Proof of Theorem 5. 

According to the theory of optimal existence [33], it is established that

(1)

For any control variable $u_i\in U$, the initial values of system (10) are all negative;

(2)

The control set is a closed and convex set;

(3)

The right-hand linear function of system (10) satisfies the initial conditions, ensuring boundedness on the control set U;

(4)

The integrand of the objective function (11) is convex on the control set U, and there exist constants $c_1$ and $c_2$, such that

$$P_{1}A+P_{2}I+\frac{w_1}{2}{u}_1^2+\frac{w_2}{2}{u}_2^2+\frac{w_3}{2}{u}_3^2+\frac{w_4}{2}{u}_4^2\ge c_1{\Vert u\Vert }^{c_2},$$

where $P_{1}A+P_{2}I\ge 0$, $c_1=\frac{1}{2}min(B_i)(i=1,2,3,4)$, $c_2=2$.

Therefore, the optimal control solution for system (10) exists. The proof is complete. □

5.2. Optimal Control Solution

Next, the Pontryagin’s Maximum Principle is employed to determine the solution for optimal control [34]. For $\forall t\in [0,T]$, the Hamiltonian function H is defined as

$$\begin{array}{ccc}H(Y,U,\lambda )\hfill & =\hfill & P_{1}A+P_{2}I+\frac{w_1}{2}{u}_1^2+\frac{w_2}{2}{u}_2^2+\frac{w_3}{2}{u}_3^2+\frac{w_4}{2}{u}_4^2+{\lambda }_1\frac{dS}{dt}+{\lambda }_2\frac{dE}{dt}+{\lambda }_3\frac{dA}{dt}+{\lambda }_4\frac{dI}{dt}\hfill \\ \hfill & =\hfill & P_{1}A+P_{2}I+\frac{w_1}{2}{u}_1^2+\frac{w_2}{2}{u}_2^2+\frac{w_3}{2}{u}_3^2+\frac{w_4}{2}{u}_4^2\hfill \\ \hfill & \hfill & +{\lambda }_1[\mathsf{\Lambda }-{\beta }_1(1-u_1)SA-{\beta }_2(1-u_2)SI-dS+\frac{a_{1}A}{1+b_{1}A}+\frac{a_{2}I}{1+b_{2}I}]\hfill \\ \hfill & \hfill & +{\lambda }_2[{\beta }_1(1-u_1)SA+{\beta }_2(1-u_2)SI-(\delta +d)E]\hfill \\ \hfill & \hfill & +{\lambda }_3[k\delta E-(p+d+u_3)A-\frac{a_{1}A}{1+b_{1}A}]\hfill \\ \hfill & \hfill & +{\lambda }_4[(1-k)\delta E+pA-(d+\mu +u_4)I-\frac{a_{2}I}{1+b_{2}I}],\hfill \end{array}$$

where ${\lambda }_1(t), {\lambda }_2(t), {\lambda }_3(t), {\lambda }_4(t)$ represent the adjoint variables corresponding to each state. And there is the following:

• ${\mathcal{H}}_1$: control system satisfaction

  $$\frac{dS}{dt}=\frac{\partial H}{\partial {\lambda }_1},\frac{dE}{dt}=\frac{\partial H}{\partial {\lambda }_2},\frac{dA}{dt}=\frac{\partial H}{\partial {\lambda }_3},\frac{dI}{dt}=\frac{\partial H}{\partial {\lambda }_4};$$
• ${\mathcal{H}}_2$: adjoint system satisfaction

  $$\frac{d{\lambda }_1}{dt}=-\frac{\partial H}{\partial S},\frac{d{\lambda }_2}{dt}=-\frac{\partial H}{\partial E},\frac{d{\lambda }_3}{dt}=-\frac{\partial H}{\partial A},\frac{d{\lambda }_4}{dt}=-\frac{\partial H}{\partial I};$$
• ${\mathcal{H}}_3$: minimum condition

  $$H(Y^{*}(t),U^{*}(t),{\mathsf{\Lambda }}^{*}(t))=\underset{0\le u_i\le u_{imax}}{min}H(Y^{*}(t),U^{*}(t),{\mathsf{\Lambda }}^{*}(t)).$$

In addition, the following cross-sectional conditions also hold true.

$${\lambda }_i(T)=0,i=1,2,3,4.$$

Theorem 6. 

There exists an optimal control $U(t)$, suppose $(S^{*}(t),E^{*}(t),A^{*}(t),I^{*}(t))$ is the optimal control solution for the system (11), where ${\lambda }_1^{*}(t),{\lambda }_2^{*}(t),{\lambda }_3^{*}(t),{\lambda }_4^{*}(t)$ is the adjoint function. According to Pontryagin’s Maximum Principle, the derivative of the adjoint variables is obtained as

$$\left\{\begin{array}{c}\frac{d{\lambda }_1^{*}}{dt}=[{\beta }_1(1-u_1)A+{\beta }_2(1-u_2)I]({\lambda }_1^{*}-{\lambda }_2^{*})+d{\lambda }_1^{*},\hfill \\ \frac{d{\lambda }_2^{*}}{dt}=(\delta +d){\lambda }_1^{*}-k\delta {\lambda }_3^{*}-(1-k)\delta {\lambda }_4^{*},\hfill \\ \frac{d{\lambda }_3^{*}}{dt}=-P_1+{\beta }_1(1-u_1)S({\lambda }_1^{*}-{\lambda }_2^{*})+\frac{a_1}{{(1+b_{1}A)}^2}({\lambda }_3^{*}-{\lambda }_1^{*})+(p+d+u_3){\lambda }_3^{*}-p{\lambda }_4^{*},\hfill \\ \frac{d{\lambda }_4^{*}}{dt}=-P_2+{\beta }_2(1-u_2)S({\lambda }_1^{*}-{\lambda }_2^{*})+\frac{a_2}{{(1+b_{2}I)}^2}({\lambda }_4^{*}-{\lambda }_1^{*})+(d+\mu +u_4){\lambda }_4^{*},\hfill \end{array}\right.$$

(14)

and there is a cross-sectional condition ${\lambda }_i^{*}(T)=0,i=1,\dots ,4$. Additionally, the optimal control must satisfy

$$\left.\frac{\partial H}{\partial u_1}\right|u_1^{*}=0,\left.\frac{\partial H}{\partial u_2}\right|u_2^{*}=0,\left.\frac{\partial H}{\partial u_3}\right|u_3^{*}=0,\left.\frac{\partial H}{\partial u_4}\right|u_4^{*}=0.$$

(15)

Furthermore, the optimal control solution is expressed in the following form:

$$u_1^{*}(t)=min\left\{max\left\{0,\frac{({\lambda }_1^{*}-{\lambda }_1^{*}){\beta }_1{S}^{*}{A}^{*}}{w_1}\right\},1\right\},$$

$$u_2^{*}(t)=min\left\{max\left\{0,\frac{({\lambda }_2^{*}-{\lambda }_1^{*}){\beta }_2{S}^{*}{I}^{*}}{w_1}\right\},1\right\},$$

$$u_3^{*}(t)=min\left\{max\left\{0,\frac{{\lambda }_3^{*}{A}^{*}}{w_1}\right\},1\right\},$$

$$u_4^{*}(t)=min\left\{max\left\{0,\frac{{\lambda }_4^{*}{I}^{*}}{w_1}\right\},1\right\}.$$

6. Numerical Simulation

6.1. Stability of Balance Point

Assuming $\mathsf{\Lambda }=100\mathrm{people}/\mathrm{day}$, ${\beta }_1=0.01/\mathrm{people}/\mathrm{day}$, ${\beta }_2=0.014/\mathrm{people}/\mathrm{day}$, $d=0.05/\mathrm{day}$, $a_1=0.15/\mathrm{day}$, $b_1=0.25/\mathrm{day}$, $a_2=0.2/\mathrm{day}$, $b_2=0.3/\mathrm{day}$, $\delta =0.4/\mathrm{day}$, $p=0.3/\mathrm{day}$, $k=0.6$, $\mu =0.07/\mathrm{day}$, we obtain $R_0\approx 0.588\mathrm{people}<1$, as shown in Figure 2, which verifies that the disease-free equilibrium point is globally asymptotically stable.

Figure 2. Time series of $S(t),E(t),I(t),A(t)$.

Assuming $\mathsf{\Lambda }=100\mathrm{people}/\mathrm{day}$, ${\beta }_1=0.015/\mathrm{people}/\mathrm{day}$, ${\beta }_2=0.0145/\mathrm{people}/\mathrm{day}$, $d=0.05/\mathrm{day}$, $a_1=0.15/\mathrm{day}$, $b_1=0.25/\mathrm{day}$, $a_2=0.2/\mathrm{day}$, $b_2=0.3/\mathrm{day}$, $\delta =0.4/\mathrm{day}$, $p=0.3/\mathrm{day}$, $k=0.6$, $\mu =0.066/\mathrm{day}$, we obtain $R_0\approx 1.527\mathrm{people}>1$, as shown in Figure 3, the endemic equilibrium point is globally asymptotically stable.

Figure 3. Time series of $S(t),E(t),I(t),A(t)$.

6.2. Sensitivity Analysis

Sensitivity analysis [35] is a method to evaluate the influence of various input parameters on the output. Simply put, it is a technique for determining which input parameters have a significant impact on the results of the model. Through sensitivity analysis, we can find out the most important factors, which can help to optimize model design, improve model precision and provide a more reliable basis for decision making.

Specific methods and applications of sensitivity analysis may vary in different fields and applications, but the core objective is to better understand the behavior and predictive results of models. Sensitivity analysis is a common tool for model verification, risk assessment and decision support in the fields of statistics, engineering, economics, and environmental science. One of the most important methods is the Partial Rank Correlation Coefficient method, which is a very useful statistical sampling technique and has been widely used in the analysis of infectious disease models [36,37].

This method first generates a large number of parameter combinations through Latin Hypercube Sample (LHS), and then calculates the model output corresponding to these parameter combinations (e.g., base generation $R_0$). Finally, the influence of each parameter on the model output is evaluated by calculating the partial correlation coefficient between each parameter and the model output. This method can reveal the key parameters that affect the output of the model and provide important information for the further study and application of the model.

This article employs Latin hypercube sampling to conduct sensitivity analysis on various parameters affecting the basic reproduction number $R_0$ of the system (1), thereby determining the degree of influence of each parameter on $R_0$. Through the analysis, it is found that among the selected parameters, $d,a_2$ and $\mu$ are significantly negatively correlated with $R_0$, while $\mathsf{\Lambda }$, ${\beta }_1$ and ${\beta }_2$ are significantly positively correlated with $R_0$.

As can be seen from Figure 4, among these parameters that have different degrees of influence on the outbreak of COVID-19, the infection rate of asymptomatic infected people to susceptible people is ${\beta }_1$. It has a significant effect on $R_0$. Therefore, it can be reduced by reducing ${\beta }_1$ to reduce and control the spread of COVID-19; one can also increase the maximum cure coefficients $a_1$ and $a_2$ of asymptomatic infected persons and symptomatic infected persons can shorten the cure cycle of COVID-19; or, the constant input $\mathsf{\Lambda }$ of the reduced population can reduce the population mobility, thus reducing the spread speed of COVID-19.

Figure 4. Correlation between $R_0$ and parameters. The units of each parameter are shown in Table 1.

6.3. Optimal Control

Suppose $\mathsf{\Lambda }=100\mathrm{people}/\mathrm{day}$, ${\beta }_1=0.0015/\mathrm{people}/\mathrm{day}$, ${\beta }_2=0.002/\mathrm{people}/\mathrm{day}$, $d=0.06/\mathrm{day}$, $a_1=0.15/\mathrm{day}$, $b_1=0.25/\mathrm{day}$, $a_2=0.02/\mathrm{day}$, $b_2=0.03/\mathrm{day}$, $\delta =0.0068/\mathrm{day}$, $p=0.94/\mathrm{day}$, $k=0.86,\mu =0.07/\mathrm{day}$. Each control strategy has its limitations, so the maximum values of $u_1$, $u_2$, $u_3$ and $u_4$ are $0.8,0.9,0.6$ and $0.7$ respectively. As shown in Figure 5, and all control strategies will gradually decrease with the change of time. In Figure 6 and Figure 7, we obtain the comparison of the number of asymptomatic infected people and symptomatic infected people before and after adopting the optimal control strategy. Obviously, due to adopting the control strategy, the number of infected people quickly approaches zero. This shows that the control strategy studied can play a very good role in controlling the spread of COVID-19, which proves the effectiveness of the control strategy. This practice keeps both asymptomatic and symptomatic infected people in COVID-19 at a relatively low level.

Figure 5. The functional relationship between the control variable $u(t)$ and time t.

Figure 6. Comparing the number of asymptomatic infectives before and after implementing the control measures.

Figure 7. Comparing the number of symptomatic infectives before and after implementing the control measures.

7. Conclusions

This paper proposes an SEAI model of COVID-19 with asymptomatic infection. This paper mainly studies the influence of asymptomatic infected people on the rapid spread of novel coronavirus. Firstly, the basic regeneration number $R_0$ and equilibrium point of the model are calculated. Then, the local asymptotic stability of disease-free equilibrium and endemic equilibrium is proved by Hurwitz criterion, and the stability of disease-free equilibrium is proved by constructing Laplace function, and the uniform persistence of the system is proved. Then, the Pontryagin maximum principle is applied to solve the optimal control problem. Finally, the theoretical results are verified by numerical simulation, and the sensitivity of parameters is analyzed by PRCC technology, which shows that the influence of asymptomatic infected people on the spread of COVID-19 should not be underestimated.

In the current COVID-19 epidemic, isolation and keeping social distance are the main measures to control the spread of the virus. The state can start with this aim and implement a perfect isolation policy by taking appropriate measures. For asymptomatic infected people, the state can use advanced technology to identify them and then isolate them to prevent the outbreak of diseases. In addition, the state can also reduce the mobility of the population by raising people’s awareness of COVID-19, including prevention measures such as washing hands frequently, wearing masks when going out, avoiding gatherings of crowds, reducing the contact between susceptible people and infected people, and reducing the constant input of population, so as to slow down and eventually control the spread of this disease.