Complexity

Complexity / 2022 / Article

Research Article | Open Access

Volume 2022 |Article ID 9182830 | https://doi.org/10.1155/2022/9182830

Haiyin Li, Yan Wu, "Dynamics of SCIR Modeling for COVID-19 with Immigration", Complexity, vol. 2022, Article ID 9182830, 22 pages, 2022. https://doi.org/10.1155/2022/9182830

Dynamics of SCIR Modeling for COVID-19 with Immigration

Academic Editor: Toshikazu Kuniya
Received10 Sep 2021
Revised30 Dec 2021
Accepted14 Mar 2022
Published11 May 2022

Abstract

In this study, for COVID-19, we divide people into four categories: susceptible , closely contacted , infective , and removed 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 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.

1. Introduction

The 2019 novel coronavirus disease (COVID-19) is running rampantly in the world, which has caused a great concern on the global public health. Research on infectious diseases has never stopped [110]. In [11], the authors investigated global stability for the SEIR model in epidemiology. The dynamical behavior of an epidemic model with nonlinear incidence was studied in Ruan and Wang [12]. Peng [13] considered asymptotic curve of the positive steady state of the SIS epidemic reaction diffusion system. From 2020 to 2021, to defeat the epidemic, scientists in different fields cooperated to investigate COVID-19 from different points of view. Sex ratio, media reports, super-spreaders, cytokine storm, mask wearing, healthcare, vaccination, age structure, impact on energy demand and consumption, etc., are covered [1426].

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. Therefore, 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. The 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.

The 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.

2. 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.

The model (Model 1) readswhere means the number of susceptible individuals, means the closely contacted individuals, that is the exposed, means the infected at time , is removed from the infected system including recovered and dead, and 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 means transition rate of closely contacted individuals to susceptible by testing and quarantining, means contact rate of transmission per contact from infected class, means transition rate of closely contacted individuals to the infected case, and means removal rate including mortality of infectious disease and recovery rate of infected individuals. Furthermore, is nearly a constant, the total population of a country or region.

Since does not intervene in the dynamics of , and , the following system can be separated from system (1):

The initial conditions of the delayed system (2) arewhere is the initial function, is the Banach space of continuous functions mapping the interval into , and , and is any norm in . Usually, we use regular symbols for .

2.2. Global Asymptotical Stability for Model 1

System (2) does not have the endemic equilibrium and has only one equilibrium, i.e., the disease-free equilibrium , where is the number of the initial susceptible individuals.

Theorem 1. If , then of system (2) has global asymptotical stability.

Proof . Assume . If , , we haveIf , we haveBecause of if , we can get .
If , we haveBecause of if , we can get .
Therefore, when , then the unique equilibrium, i.e., the disease-free equilibrium.
has global asymptotical stability.

Example 1. Let then, system (2) reads

For system (7), apparently , by Theorem 1, the equilibrium has global asymptotical stability. We have Figure 1 by simulations. Note that, in Figure 1, the time histories of in system (7), for , with initial functions and and for and are presented, respectively, and all converge to as tends to .

3. Model 2

3.1. 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 personnel and imported cold-chain food as imported from abroad.

The corresponding model (Model 2) iswhere present the immigration rates of susceptible class, closely contacted class, and infected class from the foreign countries, respectively. present the death rates of susceptible individuals, closely contacted individuals, infected, and removed, respectively. Moreover, is a variable, which keeps increasing with the input of , and and holds decreasing with output of the population which we do not consider. Therefore, 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 or other boundary equilibrium; there exists only the endemic equilibrium . From (8), it follows thatwhere satisfies We analyze the root of (10), by calculation:

When , equation (10) has two positive roots and . Furthermore, if then system (8) has two positive equilibria and .

When , equation (10) has two positive roots and . However, if or then system (8) has the unique equilibrium, i.e., the positive equilibrium or .

When , equation (10) has one positive root . Furthermore, if then system (8) has the unique equilibrium, i.e., the positive equilibrium .

3.2. Locally Asymptotically Stability for Model 2

The Jacobian matrix [29] on of sysem (8) is

Hence, we can get the characteristic equation as

So, we have the characteristic root , and the other characteristic roots satisfyingwith

By (14), when , we havewith

Assume

Using Routh–Hurwotz criterion, when

Then, all roots of (16) have negative real parts. So, we can obtain the following result.

Theorem 2. If condition (19) holds, all roots of (16) have negative real parts. The endemic equilibrium of system (8) is asymptotically stable for if condition (19) holds.

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 varies continuously, let be the root; we have

Thus, , in other words,

Assume , equation (21) can be converted towith .

Assume . We have the following result.

Lemma 1 (see [30]). (i) When , equation (22) has at least one positive root.(i)When and , equation (22) has no positive root.\(ii)When and , equation (22) has positive root if and only if and

Therefore, the following theorem is obtained by Lemma 1.

Theorem 3. When

Equation (22) has at least one positive root.

Moreover, we suppose condition (23) is established. Equation (14) has at least one positive root, i.e., equation (22) has at least one positive root. And equation (22) is a cubic, so it may have one, two, or three positive roots. Without loss of generality, we suppose that it has three positive roots, expressed as z1, z2, and z3 Then, equation (21) has three positive roots, that is, , and .

Assume

With , meetingso are a pair of purely imaginary roots of equation (14) with obviously,

So, we can assume

When condition (23) is established, owing to Theorem 2, equation (14) always has positive root and has roots having zero real part for , and is the minimum of . Therefore, for all , the roots of equation (14) have negative real parts. We can have the following theorem.

Theorem 4. When , all roots of equation (14) have negative real parts. That is, when , the endemic equilibrium of system (8) is locally asymptotically stable.

Assume is the root of equation (14) satisfying . We suppose in order to ensure that are simple pure imaginary roots of equation (14) for and satisfies the transversality condition.

The two sides of Equation (14) are differentiated with respect to ; we can obtainwith

Putting into the above equation, separating the real and imaginary parts, and by Equation (20) and the definition of , we obtain

When for and close to , we obtain equation (14) has a root with . This contradicts with Theorem 4, so we have the following conclusion.

Theorem 5. When , a pair of simple purely imaginary roots of equation (14) are and all other roots have negative real parts. In addition, .

Hence, of system (8) is asymptotically stable (unstable) when , respectively, and satisfying Equation (27).

3.3. Local Asymptotic Stability of Model 2

Assume ; then, system (8) is linearized as

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

In order to prove the local asymptotic stability of the equilibrium of system (32), it is sufficient to consider the existence of a strict Lyapunov functional.

Let ; the full-time derivative of is

Considering , from (33) and the inequality , we obtain

Considering , we have

In addition, considering and , we obtain

Similarly, from (34) and , let , , , and ; we obtain

Assume and . Obviously, and if and only if . Additionally, if , then . Furthermore, using , we have

Therefore, whenthen is negative definite. Hence, we have the following theorem.

Theorem 6. When the following conditions,hold, then equilibrium is locally asymptotically stable for system (32), that is, the positive equilibrium is also locally asymptotically stable for system (8).

Example 2. Assume ; then, system (8) readsThrough calculation, , and . Because condition (42) for Theorem 6 is satisfied, is locally asymptotically stable, which is also obtained from Figure 2. Note that, in Figure 2, the time histories of in system (43), for , with initial functions and , and , respectively, all converge to as tends to . The time histories are presented as follows.

3.4. Hopf Bifurcation for Model 2

Simply, system (8) was nondimensionalized using the scaling , and ; then, we have