Research Article Dynamics of SCIR Modeling for COVID-19 with Immigration

In this study, for COVID-19, we divide people into four categories: susceptible S ( t ) , closely contacted C ( t ) , infective I ( t ) , and removed R ( t ) according to the current epidemic situation and then investigate two models: the SCIR models with immigration (Model (2) and without immigration (Model 1). For the former, Model 1, we obtain the condition for global stability of its disease-free equilibrium. For the latter, Model 2, we establish the local asymptotic stability of its endemic equilibrium by constructing Lyapunov function. Afterwards, by the bifurcation theory, we qualitatively analyze the properties of its Hopf bifurcations of the latter. Finally, numerical simulations are given to illustrate the obtained results of two models. The results imply the importance of ﬁnding closely contacted and overseas imports on epidemic control. It indicates that not only the incubation delay τ is crucial for the containment of the COVID-19 but also the scientiﬁc and rigorous containment measures are the key factors of the success of the containment.

In order to prevent the spread of the epidemic, the suspected cases and occasional local cases should be checked, including nasopharyngeal and oropharyngeal swabs, CT scans, serum testing, bronchoalveolar lavage, and other different methods of inspection [27]. In [28], their model is based on the usual SIR model, dividing the total proportion of infected individuals into two parts: patients who have not yet been detected and patients who have been detected by tests. In addition, because a large number of people travel from one country to another, the disease is mainly spread between different countries through air travel. For countries with better epidemic control, imported cases have become the main source of cases. erefore, it is important to consider foreign inputs.
Based on the global pandemic COVID-19, the epidemic lasts for such a long time; personnel exchanges between countries are inevitable. In this study, we discuss the impact of immigration between countries on the prevention and control of epidemic. e total population of each group is divided into four compartments, that is, susceptible, closely contacted, infective, and removed compartments. We will investigate two models: the SCIR models with immigration (Model (2) and without immigration (Model 1), respectively. e arrangement of this study is as follows. In Section 2, we establish the global stability of disease-free equilibrium for Model 1. In Section 3, we obtain the local asymptotic stability of endemic equilibrium and analyze the Hopf bifurcations for Model 2. In Section 4, this study is concluded.

Model 1
2.1. Construction of Model 1. Take one country as a whole, not taking into account the number of imported people, the birth rate, or the normal death rate in the countries. e model (Model 1) reads where S(t) means the number of susceptible individuals, C(t) means the closely contacted individuals, that is the exposed, I(t) means the infected at time t, R(t) is removed from the infected system including recovered and dead, and τ indicates the incubation delay, i.e., the duration between the time when the people who is exposed to the infected cases, but not yet infectious, and later the time when they become infectious. And a means transition rate of closely contacted individuals to susceptible by testing and quarantining, b means contact rate of transmission per contact from infected class, β means transition rate of closely contacted individuals to the infected case, and r means removal rate including mortality of infectious disease and recovery rate of infected individuals. Furthermore, N is nearly a constant, the total population of a country or region.

Global Asymptotical Stability for Model 1. System
erefore, when b < r/2, then the unique equilibrium, i.e., the disease-free equilibrium.
E 0 has global asymptotical stability.

Construction of Model 2.
With the joint efforts of the whole country, the epidemic in some countries has been brought under control. However, due to the immigration of imported cases and the virus on imported cold-chain food, the epidemic has occurred many times. For the convenience of discussion, we will collectively refer to imported 2 Complexity personnel and imported cold-chain food as imported from abroad. e corresponding model (Model 2) is where Λ 1 , Λ 2 , and Λ 3 present the immigration rates of susceptible class, closely contacted class, and infected class from the foreign countries, respectively. d S , d C , d I , and d R present the death rates of susceptible individuals, closely contacted individuals, infected, and removed, respectively. Moreover, N � S + C + I + R is a variable, which keeps increasing with the input of Λ 1 , Λ 2 , and Λ 3 and holds decreasing with output of the population which we do not consider. erefore, combining the two cases into consideration, we can assume that the population is nearly a constant, the total population of a country or region. System (8) has no disease-free equilibrium (0, 0, 0, 0) or other boundary equilibrium; there exists only the endemic equilibrium E * (S * , C * , I * , R * ). From (8), it follows that where S * satisfies We analyze the root of (10), by calculation: When Δ > 0, equation (10) has two positive roots S 1 and . When Δ > 0, equation (10) has two positive roots S 1 and S 2 . However, if Λ 1 + Λ 2 − d S S 1 < 0, or Λ 1 + Λ 2 − d S S 2 < 0, then system (8) has the unique equilibrium, i.e., the positive . When Δ � 0, equation (10) has one positive root S * . Furthermore, if Λ 1 + Λ 2 − d S S * > 0, then system (8) has the unique equilibrium, i.e., the positive equilibrium E * (S * , C * , I * , R * ).

Locally Asymptotically Stability for Model 2.
e Jacobian matrix [29] on E * of sysem (8) is Hence, we can get the characteristic equation as So, we have the characteristic root λ � − d R , and the other characteristic roots satisfying with 4 Complexity By (14), when τ � 0, we have with Assume Using Routh-Hurwotz criterion, when en, all roots of (16) have negative real parts. So, we can obtain the following result. (19) holds, all roots of (16) have negative real parts. e endemic equilibrium E * of system (8) is asymptotically stable for τ � 0 if condition (19) holds.

Theorem 2. If condition
Next, we suppose condition (19) is established. In order to determine if the real parts of some roots of (9) reaches to zero and ultimately become positive as τ varies continuously, let λ � iω(ω ≠ 0) be the root; we have Assume z � ω 2 , equation (21) can be converted to We have the following result.
is contradicts with eorem 4, so we have the following conclusion.

Local Asymptotic Stability of Model 2. Assume
For the simplicity of the symbol, we omit the tildes; system (31) can be rewritten as We rewrite the second equation of system (32) as By the same method, from the third equation of the linearized system (32), we have

Complexity
In order to prove the local asymptotic stability of the equilibrium (0, 0, 0, 0) of system (32), it is sufficient to consider the existence of a strict Lyapunov functional. Let Considering V 21 (t) � C 2 (t), from (33) and the inequality ab ≤ a 2 + b 2 /2, we obtain (36) In addition, considering Similarly, from (34) and ab ≤ a 2 + b 2 /2, let Complexity erefore, when then V ′ is negative definite. Hence, we have the following theorem.
Let z(t) � 〈q * , u t 〉 and On the center manifold C 0 , we obtain With and for center manifold C 0 , the local coordinates in the direction of q * and q * are z and z, respectively. We know if u t is real, then W is real. Here, we only consider real solutions.

Complexity 13
With From (52), (68), and (64), we can obtain with If we expand the above series and compare the corresponding coefficients, then we can obtain By (67), assume F 0 (z, z) � F z 2 z 2 /2 + F zz zz + F z 2 z 2 /2 + F z 2 z z 2 z/2 + · · · and (68); we obtain In addition, we have From and (76), we compare the coefficients with (69) and have g 20 , g 11 , and g 02 can be directly computed, but we need to compute W 20 (θ) and W 11 (θ) for θ ∈ [− 1, 0) in the following in order to compute g 21 .
Firstly, we study the situation of θ ∈ [− 1, 0) of (70). By (70), we obtain Complexity If we put (76) into the above equation and compare the coefficient of z 2 /2 and zz, then we obtain By AW 20 (0) � 2iω 0 τ 0 W 20 (0) − H 20 (0), we obtain If (86) and (92) are put into (93), then we can obtain By We obtain So, By AW 11 (0) � − H 11 (0), we obtain If (89) and (92) are put into (98), then we can obtain Complexity By (95), we obtain So, From K 1 and K 2 , we have W 11 (θ) and W 20 (θ). Hence, we can determine g 21 . On the basis of the above analysis, we know each g ij in (78) is decided by the parameters and delays in system (8). erefore, we can have the following quantities: It determines the properties of bifurcating periodic solutions near the critical value τ 0 , and we get the following theorem.

Theorem 7.
e following results hold for the expression of (102).

Conclusions
In this study, we divided people into four categories: S(t) susceptible, C(t) closely contacted, I(t) infective, and R(t) removed according to the current situation in the middle and late stages of COVID-19, investigating the SCIR models with immigration and without immigration, and obtained the following main results: (1) If b < r/2, then the disease-free equilibrium E 0 (S 0 , 0, 0) of Model 1 is globally asymptotically stable (2) If Δ 1 > 0, Δ 2 > 0, Δ 3 > 0, the endemic equilibrium E * of Model 2 is asymptotically stable for τ ∈ [0, τ 0 ) (3) If condition (42) holds, then the positive equilibrium E * (S * , C * , I * , R * ) of Model 2 is locally asymptotically stable by constructing Lyapunov function (4) By the bifurcation formulae, in [31], we qualitatively analyze the properties for the occurring Hopf bifurcations of Model 2 Moreover, considering neither the immigration nor the death rate during the outbreak, Model 1 has only the diseasefree equilibrium. However, considering both the immigration and the death rate during late stage, Model 2 has only one or two endemic equilibria. Both indicate the importance of finding closely contacted and overseas imports on epidemic control, respectively.
Data Availability e numerical simulation data used to support the findings of the study are included within the article.