Global dynamics for a class of discrete SEIRS epidemic models with general nonlinear incidence.

In this paper, a class of discrete SEIRS epidemic models with general nonlinear incidence is investigated. Particularly, a discrete SEIRS epidemic model with standard incidence is also considered. The positivity and boundedness of solutions with positive initial conditions are obtained. It is shown that if the basic reproduction number R 0 ≤ 1 , then disease-free equilibrium is globally attractive, and if R 0 > 1 , then the disease is permanent. When the model degenerates into SEIR model, it is proved that if R 0 > 1 , then the model has a unique endemic equilibrium, which is globally attractive. Furthermore, the numerical examples verify an important open problem that when R 0 > 1 , the endemic equilibrium of general SEIRS models is also globally attractive. © Fan et al. 2016.

Introduction

As is well known, many infectious diseases possess a latent period, such as Hepatitis, HIV, SARS, Ebola, MERS, etc. When a susceptible individual is infected at the beginning, the disease incubates inside the susceptible for a period of time, then the susceptible becomes an exposed individual before becoming infectious. For such infectious diseases, the resulting model is SEIR (susceptible S, exposed E, infectious I, removed R) epidemic type. The study on SEIR-type epidemic dynamical models is a very important subject in the mathematical theory of epidemiology, and in the last two decades there have been a number of researches on modeling, theoretical analysis, and applications. Continuous SEIR-type epidemic models described by the differential equations have been widely studied. Many important and interesting results can be found in [1-9] and the references therein.

As we all know, it is very difficult to accurately solve a nonlinear differential equation with a given initial condition. Therefore, for many practical requirements, such as numerical calculation, it is often necessary to discretize a continuous model to obtain the corresponding discrete model. At the present time, there are various discretization methods to discretize a continuous model, including the standard methods, such as Euler method, Runge-Kutta method, and some other standard finite difference schemes, and the nonstandard finite difference (NSFD) scheme, which is originally developed by Mickens [10-12].

In recent years, discrete epidemic models have been widely studied. The basic and important research subjects for these models are the computing of the thresholds values and basic reproduction numbers, the local and global stability of disease-free equilibrium and the endemic equilibrium, the persistence, permanence, and extinction of the disease, and bifurcations and chaos phenomena of the models when some parameters of the models vary, and so on. Many important and interesting results can be found in [13-36] and the references therein. Particularly, we see that in [14, 17, 18, 20–22, 27, 36] discrete SI-type epidemic models are investigated, and in [19, 24, 25, 31, 32, 35] discrete SIR-type epidemic models are discussed.

However, we see that up to now there have been fewer research works on discrete SEI- and SEIR-type epidemic models, where the disease has a latent period. Cao and Zhou [16] formulated and studied a discrete age-structured SEIT epidemic model, and as an application, discussed the tuberculosis transmission in China. In [26], the authors applied Micken’s discretization method to obtain a discrete SEIR epidemic model. The positivity of solutions and the existence and stability of equilibrium are discussed. The design of a state observer for the model is tackled. Some sufficient conditions to ensure the asymptotic stability of the observer are provided in terms of a matrix inequality. In [28], the authors studied a discrete plant virus disease model with roguing and replanting, which is derived from the continuous case by using the backward Euler method. The basic reproduction number is obtained. It is showed that the disease-free equilibrium is globally attractive if , and otherwise, the disease is permanent if . In [29, 30], the authors proposed a class of discrete SEIS epidemic models with bilinear incidence, which is established from the corresponding continuous SEIS epidemic model by applying the well-known backward difference scheme. The positivity of solutions and the permanence of the model are established. Furthermore, using the Lyapunov function method, the authors proved that if the basic reproduction number , then the disease-free equilibrium is globally asymptotically stable, and if , then the endemic equilibrium exists and is globally asymptotically stable.

Consider the following continuous SEIRS epidemic model with general nonlinear incidence: Some particular cases for this model have been investigated in [3, 4, 26], where the basic reproduction number is calculated, and the dynamical properties, such as the local and global stability of disease-free equilibrium and endemic equilibrium and the extinction and persistence of the disease are established. Motivated by this work, in this paper, we propose the following discrete SEIRS epidemic model with general nonlinear incidence established by using the backward difference scheme to discretize model (1): where .

Our purpose in this paper is to investigate the dynamical behaviors of model (2). The basic reproduction number is defined. We will prove by using the linearization method and Lyapunov function that if , then disease-free equilibrium is globally attractive, and as a result, the disease is also extinct, and by using the theory of persistence for dynamical systems that if , then the disease is permanent. Furthermore, when model (2) degenerates into the particular case and , by constructing the suitable discrete type Lyapunov function we also will prove that if , then model (2) has a unique endemic equilibrium, which is globally attractive.

The organization of this paper is as follows. In Section 2, the model description and some basic properties are given. Section 3 deals with the global attractivity of disease-free equilibrium of model (2). In Section 4, the criterion on the permanence of the disease for model (2) is stated and proved. In Section 5, the criterion on the global attractivity of the endemic equilibrium for model (2) in the particular case and is stated and proved. Furthermore, in Section 6, some numerical examples are provided to illustrate the validity of main results obtained in this paper and verify the interesting open problem given in Remark 5.1. Lastly, a discussion is given in Section 7.

Basic properties

In model (2), , , , and denote the numbers of susceptible, exposed, infectious, and recovered classes at nth generation, respectively, Λ is the recruitment rate of the susceptible, () are the death rates of susceptible, exposed, infectious, and recovered individuals, respectively. Particularly, includes the natural death rate and the disease-related death rate of the infectious class. δ is the translation rate from exposed to infectious, γ is the recovery rate of the infectious individuals, and σ is the rate of losing immunity of the recovered; indicates that the recovered individuals possess the provisional immunity, and predicates that the recovered individuals acquire permanent immunity. The incidence rate of the infectious is described by a nonlinear function .

In this paper, we always assume that the parameters Λ, (), δ, and γ are positive constants, σ is a nonnegative constant, and . We set For a nonlinear incidence , we introduce the following assumption.

is continuously differentiable with respect to , is increasing with respect to and decreasing with respect to and , and is nonincreasing with respect to . Furthermore, and , where .

Remark 2.1

When or , where , and , , , , and are constants, (H) naturally holds. Furthermore, when , (H) degenerates into the following form:

and are continuously differentiable with respect to and , respectively, is increasing for , and is nonincreasing for . Furthermore, and .

The initial condition for model (2) is given by We have the following result on the positivity and ultimate boundedness of solutions.

Theorem 2.1

Model (2) has a unique positive solution for all with initial condition (3), and this solution is ultimately bounded.

Proof

We can prove this theorem by using an argument similar to that introduced in [33], Theorem 2.2. In fact, we only need to prove by induction that, for any integer , if exists and , , , and , then also exists, and , , , and .

From model (2) by calculating we can obtain and where and . Since , we obtain and .

Let . By the second equation of model (2) and by (4) and (5), y satisfies the equation where and Let . Since from (H) we obtain that is increasing with respect to . Then, we obtain and Therefore, has a unique positive solution . This shows that exists and .

By (4), when exists, then also exists, and . By the fourth equation of model (2) we further have that exists and .

Let . By the first equation of model (2) it follows that By (H), when exists, then is increasing for . Since and , we obtain that has a unique positive solution x̄. Therefore, exists, and .

From the previous discussions we finally obtain that exists and is positive. Therefore, solution uniquely exists and is positive for all .

From model (2) we have When , where , we have for all . In a general way, for any we can obtain . Therefore, is ultimately bounded. This completes the proof. □

Remark 2.2

From the previous discussion we see that the region is a positive invariable set for model (2) and absorbs all nonnegative solutions of model (2). Therefore, we can assume in the rest of this paper that , , , and for all .

The basic reproduction number for model (2) is given by Particularly, when and , becomes of the following forms, respectively, On the existence of equilibria of model (2), we have the following result.

Theorem 2.2

If , then model (2) has only a disease-free equilibrium , where .

If , then model (2) has a unique endemic equilibrium , except for .

Any equilibrium of model (2) satisfies the equations Hence, we have and Thus, Let . Then , , and is decreasing for . From the second equation of (6) we have Define By (H), is decreasing for , , and

If , then . Hence, has no positive roots. This shows that model (2) has only a disease-free equilibrium .

If , then . Hence, has a unique positive root . This shows that model (2) has a unique endemic equilibrium , where and This completes the proof. □

We have the following result on the local stability of the disease-free equilibrium and endemic equilibrium.

Theorem 2.3

When , the disease-free equilibrium of model (2) is locally asymptotically stable, and when , is unstable.

The linearization system of model (2) at equilibrium is From the second and third equations of system (7) we have where Since , we easily prove that two eigenvalues () of the matrix A satisfy . Therefore, two eigenvalues () of the matrix satisfy .

From the first and fourth equations of system (7) we have where Obviously, the matrix has eigenvalues () satisfying . Therefore, equilibrium of system (7) is asymptotically stable. Consequently, when , the equilibrium of model (2) is locally asymptotically stable.

When , we easily prove that two eigenvalues () of the matrix are real numbers and and . Hence, the equilibrium of system (8) is unstable. This shows that the equilibrium is unstable when . □

Remark 2.3

It is unfortunate that we do not establish the local asymptotic stability of endemic equilibrium of model (2). In fact, the linearization system of model (2) at endemic equilibrium is where In order to obtain the local asymptotic stability of endemic equilibrium , we only need to prove that all eigenvalues λ of matrix satisfy . However, it is a pity that here we do not obtain this.

Therefore, when , whether the endemic equilibrium of model (2) also is locally asymptotically stable still is an interesting open problem.

Global attractivity of disease-free equilibrium

In this section, we discuss the global attractivity of disease-free equilibrium of model (2). We have the following result.

Theorem 3.1

The disease-free equilibrium of model (2) is globally attractive if and only if .

The necessity is obvious because when , model (2) has an endemic equilibrium . Now, we prove the sufficiency. When , we can choose a constant such that

Let be any positive solution of model (2). Choosing the Lyapunov function we have From (9) we have . It is clear that . When , from the third equation of model (2) we have . From the fourth equation of model (2) we further have . From the first equation of model (2) we also have . This shows that the maximal invariable set in is a disease-free equilibrium .

Therefore, using the theorems of stability of difference equations (see Theorem 6.3 in [37]), we finally obtain that the disease-free equilibrium of model (2) is globally attractive. This completes the proof. □

Remark 3.1

When (standard incidence), by Theorem 3.1, if , then the disease-free equilibrium in model (2) is globally attractive.

Permanence of disease

For model (2), disease is said to be permanent if there exists constants such that for any solution of model (2) with initial condition (3), . We have the following result.

Theorem 4.1

Disease in model (2) is permanent if and only if .

The necessity is obvious. In fact, if , then by Theorem 3.1 the disease-free equilibrium is globally attractive.

Now, we prove the sufficiency. When , we can choose constants and such that

We will use the persistence theory of dynamical systems (see [35], Section 1.3 in Chapter 1) to prove the theorem. Define the sets and Let be the solution of model (2) with initial condition . Define the set It is clear that the solution of model (2) with initial condition has the form . Hence, we have Suppose that there is such that . Then, we have or . Let be the solution of model (2) with initial condition . If , then from the second equation of model (2) we have Hence, for all . From the third equation of model (2) we further have Hence, for all . This shows that a solution for all . If , then from third equation of model (2) we have for all . Since and for all , from the second equation of model (2) we further have for all . This also shows that the solution for all . Hence, , which leads to a contradiction. Thus, we also have Therefore, .

It is clear that model (2) restricted to has a globally attractive equilibrium . This shows that in is isolated invariable and acyclic. Now, we prove that where which is said to be a stable set of . Suppose that there is a point such that . Since for the above , there is such that when , , , and , we have We can choose an integer such that , , , and for all .

Consider the Lyapunov function We have that, for , where Hence, we finally have , which leads to a contradiction with . It follows that . Thus, by the theorems of uniform persistence for dynamical systems given in [38], we obtain that model (2) is permanent. This completes the proof. □

Remark 4.1

When , by Theorem 4.1, if , then the disease in model (2) is permanent.

Remark 4.2

Theorem 4.1 only obtains the permanence of the disease for model (2). However, whether we can also prove that an endemic equilibrium is globally attractive for model (2) when ? In the following section, we will give a partial positive answer. We will prove that, for special case of model (2), an endemic equilibrium is globally attractive only when .

Global attractivity of endemic equilibrium in a particular case

In this section, we consider a particular case of model (2), that is, and in model (2). Model (2) becomes of the form Because does not appear in the first three equations of model (2), we only need to consider the equivalent system We have the following result on the global attractivity of the endemic equilibrium for model (12).

Theorem 5.1

If , then the endemic equilibrium of model (12) is globally attractive.

Let be an endemic equilibrium of model (12). Then Let be any positive solution of system (12). Define the functions and From (H) we easily obtain that when , Since for and , we obtain that when and , and . Computing , we have From (H) it follows that, for any η between and , We have Therefore, from (13) we obtain

Calculating , we obtain Using the inequality for , we have Therefore,

Similarly, calculating , we obtain

Choose the Lyapunov function For convenience of calculations, we denote , , and . Computing , from (14)-(16) we obtain From (H) we obtain for any , and implies for all . From and the third equation of model (12) it follows that for all . Furthermore, from the second equation of model (12) we obtain that for all .

Therefore, using the theorems of stability of difference equations, we finally obtain that the endemic equilibrium of model (12) is globally attractive. This completes the proof. □

Remark 5.1

In Remark 4.2, we indicated that for SEIRS-type model (2), an important problem is to prove that the endemic equilibrium is globally attractive only when . From Theorem 5.1 we see that only for the particular case of model (2), that is, SEIR-type model, we get a positive answer. Therefore, an interesting open problem for general SEIRS model (2) is whether the endemic equilibrium is also globally attractive only when .

Remark 5.2

From the proofs of Theorem 3.1, Theorem 4.1, and Theorem 5.1 we easily see that the condition is not used. In fact, this condition is only used in Theorem 2.1 to obtain the positivity of solutions of model (2). Therefore, an interesting question is whether the condition can be taken out in the proof of the positivity of solutions of model (2).

Numerical examples

Now, we give numerical examples to show that for SEIRS-type model (2), the endemic equilibrium may be globally attractive for different incidence function , which satisfies (H) only when the basic reproduction number .

Example 6.1

In model (2), we take , , , , , , , , and . The parameters , α, and σ will be chosen later.

By calculating we have the basic reproduction number . We further take , , and . Then the endemic equilibrium . From the numerical simulations (see Figure 1) we obtain that may be globally attractive.

Figure 1

Time series of , , , and in Example

Example 6.2

In model (2), we take , , , , , , , and . The parameters , α, and σ will be chosen later.

By calculating we have the basic reproduction number . We further take , , and . Then the endemic equilibrium . By numerical simulations (see Figure 2) we obtain that may be globally attractive.

Figure 2

Time series of , , , and in Example

Example 6.3

In model (2), we take , , , , , , , , and . The parameters , α, and σ will be chosen later.

By calculating we have the basic reproduction number . We further take , , and . Then the endemic equilibrium . By numerical simulations (see Figure 3) we obtain that may be globally attractive.

Figure 3

Time series of , , , and in Example

Example 6.4

In model (2), we take , , , , , , , , and . The parameters , α, and σ will be chosen later.

By calculating we have the basic reproduction number . We further take , , and . Then the endemic equilibrium . By numerical simulations (see Figure 4) we obtain that may be globally attractive.

Figure 4

Time series of , , , and in Example

All these examples of numerical simulations show that when , no matter sufficiently greater than one or closer to one but still greater than one, we always obtain that the endemic equilibrium is globally attractive, which may offer an affirmative conjecture to the open problem given in Remark 5.1, that is, for the general SEIRS model (2) the endemic equilibrium is globally attractive only when . Therefore, in our future work, we expect to obtain the corresponding theoretical results for this open problem.

Discussion

In this paper, we proposed a discrete SEIRS epidemic model (2) with general nonlinear incidence, which is described by the backward difference scheme. By our discussions presented in this paper, necessary and sufficient conditions for the global attractivity of the disease-free equilibrium and the permanence of the disease are established, that is, if the basic reproduction number , then the disease-free equilibrium is globally attractive, and if , then the disease is permanent. Furthermore, when the model degenerates into SEIR model, it is proved that when , the model has a unique globally attractive endemic equilibrium.

Unfortunately, for SEIRS model (2), when the basic reproduction number is greater than one, we do not obtain the local asymptotic stability and global attractivity of the endemic equilibrium. But the numerical examples given in Section 5 show that the endemic equilibrium for general SEIRS model (2) may be globally attractive. Therefore, it is still an important and interesting open problem how to apply the linearization method to establish the local asymptotic stability of the endemic equilibrium and how to construct the discrete analogue Lyapunov functions to study the global attractivity of the endemic equilibrium for general SEIRS model (2).

In addition, the dynamical behaviors for the nonautonomous discrete SEIRS epidemic models, discrete SEIRS epidemic models with vaccination, stage-structured discrete SEIRS epidemic models, and delayed discrete SEIRS epidemic models with nonlinear incidence described by the backward difference scheme are rarely considered. Whether similar results on the permanence and extinction of the disease and the global attractivity of the disease-free equilibrium for these models can be obtained is also an interesting open question.

On the other hand, corresponding to continuous model (1), we also have the following discrete SEIRS epidemic models with general nonlinear incidence described by the forward difference scheme or Micken’s nonstandard finite difference scheme: where , and with the denominator function , and is the time-step size. An important open problem is whether the results obtained in this paper for model (2) can also be extended to these models.