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The dynamics of SEIR epidemic model with saturated incidence rate and saturated treatment function are explored in this paper. The basic reproduction number that determines disease extinction and disease survival is given. The existing threshold conditions of all kinds of the equilibrium points are obtained. Sufficient conditions are established for the existence of backward bifurcation. The local asymptotical stability of equilibrium is verified by analyzing the eigenvalues and using the Routh-Hurwitz criterion. We also discuss the global asymptotical stability of the endemic equilibrium by autonomous convergence theorem. The study indicates that we should improve the efficiency and enlarge the capacity of the treatment to control the spread of disease. Numerical simulations are presented to support and complement the theoretical findings.

In recent years, various epidemic models have been proposed and explored to prevent and control the spread of the infectious diseases, such as measles, tuberculosis, and flu (see e.g., [

It is well known that treatment is an important and effective method to prevent and control the spread of various infectious diseases. In classical epidemic models, the treatment rate of the infection is assumed to be proportional to the number of the infective individuals, but in general, the recovery rate depends on the medical resources, such as drugs, vaccines, hospital beds, isolation places, and efficiency of the treatment. Noting that every community or country has limited capacity for the treatment of a disease, therefore, it is very important to adopt a suitable treatment function. Wang and Ruan [

Besides this, we know that the efficiency of treatment will be seriously affected if the infective individuals are delayed for treatment. In [

Although the dynamics of SIR or SIS epidemic models with the saturated incidence rate have been frequently used in many literatures [

Motivated by these points, to better understand their effects on the spreading of infectious diseases, in this paper, we will discuss the SEIR model with the saturated incidence rate and the saturated treatment function. We suppose that, in incubation period, the hosts who have been infected by viruses do not have the ability to infect other hosts and the recovered individuals and vaccinated-treated individuals have gained permanent immunity and can no longer be infected.

The paper is organized as follows. In Section

In [

In [

Based on the above motivations, in this paper, we further explore the SEIR epidemic model with saturated incidence rate

Since the first three equations in (

Denote

The system (

An endemic equilibrium always satisfies

By some simple calculation, we have

For the endemic equilibrium to exist, the solutions of (

We note

The following results hold.

Let

Let

system (

system (

system (

system (

system (

From Theorem

System (

For sufficiency, let us consider the graph of

The necessary is obvious, since, if

Under the condition of Theorem

When

When

So a backward bifurcation occurs at

In order to verify the bifurcation curve (the graph of

From (

The figure of infective sizes at equilibria versus

In this section, we will examine the local stability of the equilibria by analyzing the eigenvalues of the Jacobian matrices of (

The disease-free equilibrium

The Jacobian matrix of (

Clearly,

When

The Jacobian matrix of (

The characteristic equation is

From the second and third equations of (

Let

In fact, we have

By a direct calculation, we have that

In this section, we analyze the global stability of the disease-free and endemic steady states. Firstly, we consider the global stability of the disease-free equilibrium.

Define

If

If

Consider the following Lyapunov function:

In the following, we will discuss the global stability of the endemic equilibrium when

Let

There exists a compact absorbing set

Equation (

The basic idea of this method is that if the equilibrium

Li and Muldowney showed that if

The analysis of the global stability of the endemic equilibrium may be usefully approached by means of the Poincare-Bendixson trichotomy. If the endemic equilibrium is globally asymptotically stable, then the disease will permanently be present in the population in case of infinitesimal initial prevalence. Here we will provide an analytical proof of global stability of

Assume that conditions

Under the condition

The Jacobian matrix of system (

Let

Let

Next calculating

Take a maximum of two diagonal elements of

To demonstrate the theoretical results obtained in this paper, we will give some numerical simulations. We consider the hypothetical set of parameter values as the following.

(a)–(d) show that system (

(a)–(d) show that system (

(a)–(d) show that system (

In this paper, we consider the SEIR epidemic model with saturated incidence and saturated treatment function to understand the effect of delayed treatment on the disease transmission. Generally speaking, in many epidemic models, the basic reproduction number, which is the key concept in epidemiology, can be decreased below unity to eradicate the disease. However, in our model, the basic reproduction number below unity is not enough to eradicate the disease. According to our analysis in this paper, we find that a backward bifurcation occurs when the capacity of the treatment is low (i.e.,

Lastly, a numerical simulation provided that when

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the anonymous referees for their careful reading of the original paper and their many valuable comments and suggestions that greatly improve the presentation of this work. This work is supported by the Natural Science Foundation of Shanxi Province (2013011002-2).