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 [1–10]. 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 [14–26].
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 .
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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.
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3.4. Hopf Bifurcation for Model 2
Simply, system (8) was nondimensionalized using the scaling , and ; then, we havewith
Assume , then the Hopf bifurcation value of (44) is . When , let
Using the Riesz representation theorem, there is a matrix , whose elements are functions of bounded variation such that
Actually, we choosewhich satisfies (47).
When , we let the operator beandwhere
Therefore, system (8) is equal to the following operator equation:with and for .
When , let
And a bilinear form iswith . Hence, and are adjoint operators.
We know the eigenvalues of are and has the same eigenvalues. Furthermore, we can verify that the eigenvectors of and corresponding to the eigenvalues and are vectors and , respectively, and
Namely,
It means that , and follow as
IfEquation (57) has not trivial solution. Assume , and we obtain
By the same method, we have
, and have no trivial solution. Assume ; we obtain
In order to satisfy the normalized condition and the orthogonal condition , we need to determine the value . From (54), we know
So, we choose satisfying
With the same notations in [31], we first calculate the coordinates describing the center manifold at . Assume is the solution of system (8) for .
Let and
On the center manifold , we obtain
Withand for center manifold , the local coordinates in the direction of and are and , respectively. We know if is real, then is real. Here, we only consider real solutions.
To the solution of system (8), owing to ,which can be rewritten as
With
From (52), (68), and (64), we can obtainwith
If we expand the above series and compare the corresponding coefficients, then we can obtain
By (67), assume and (68); we obtain
By (64), we can get , sowith
In addition, we have
Fromand (76), we compare the coefficients with (69) and have
, and can be directly computed, but we need to compute and for in the following in order to compute .
Firstly, we study the situation of of (70). By (70), we obtain
and
Hence,
Therefore,and
By (82), we obtain
From (76), hence,
If we solve the above equation for , then we have
By the same method, from (83), we can obtain
From (76), hence,
If we solve the above equation for , then we have
Secondly, to obtain and , we study the situation of of (70). By (70), we obtain
So,
If we put (76) into the above equation and compare the coefficient of and , then we obtain
By , we obtain
If (86) and (92) are put into (93), then we can obtain
By
We obtain
So,
By , we obtain
If (89) and (92) are put into (98), then we can obtain
By (95), we obtain
So,
From and , we have and . Hence, we can determine . On the basis of the above analysis, we know each in (78) is decided by the parameters and delays in system (8). Therefore, we can have the following quantities:
It determines the properties of bifurcating periodic solutions near the critical value , and we get the following theorem.
Theorem 7. The following results hold for the expression of (102).(i)The direction of the Hopf bifurcation is determined by the sign of , that is, Hopf bifurcations are forward (backward) for , respectively(ii)The stability of bifurcating periodic solutions is determined by the sign of , that is, the bifurcation periodic solutions on the center manifold are stable (unstable) for , respectively(iii)The periods of the bifurcating periodic solutions is determined by the sign of , that is, the periods increase (decrease) for , respectively
Example 3. Assume , then system (8) readsThrough calculation, system (103) has a positive equilibrium . , satisfying condition (19). Furthermore, , so the Hopf critical value by the calculation process in Section 3.2.
From Theorems 3 and 4, this equilibrium of system (103) is asymptotically stable for and unstable for , which can also be found from Figures 3–6 for the values of interest, , respectively, since to the best knowledge of the authors, the most incubation delay is 24 days. Note that, in Figures 3–6, four orbits start from initial point for , respectively, and all converge to .
As tends to , for the greater values , we do not simulate it any more.
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4. Conclusions
In this study, we divided people into four categories: susceptible, closely contacted, infective, and 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 , then the disease-free equilibrium of Model 1 is globally asymptotically stable(2)If , the endemic equilibrium of Model 2 is asymptotically stable for (3)If condition (42) holds, then the positive equilibrium 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 disease-free 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
The numerical simulation data used to support the findings of the study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work was partially supported by “the National Natural Science Foundation of China (Grant nos. 11701250 and 12171221)” and by “the Natural Science Foundation of Shangdong Province (Grant no. ZR2019YQ04).