^{1}

The global stability of a multigroup SEIR epidemic model with general latency distribution and general incidence rate is investigated. Under the given assumptions, the basic reproduction number

Mathematical models have become important tools in analyzing the spread and control of infectious diseases. The SIR model is one of the most popular ones in this field, for which the total population is subdivided into three compartments: susceptible, infectious, and removed. For some diseases, it is reasonable to include a latent (or exposed) class for those susceptible individuals who are infected with the disease but are not yet infectious, which leads to SEIR model [

(

In fact, the integral term in model (

Heterogeneity in the host population can result from different contact modes such as those among children and adults for childhood diseases (e.g., measles and mumps) or different behaviors such as the numbers of sexual partners for some sexually transmitted infections (e.g., herpes and condyloma acuminatum). Taking into consideration different contact patterns, distinct number of sexual partners, or different geography and so forth, it is more proper to divide individual hosts into groups. Therefore, lots of multigroup models have been proposed in the literature to describe the transmission of infectious disease in heterogeneity environment (see [

In multigroup epidemic models, a heterogeneous host population is divided into several homogeneous groups according to modes of transmission, contact patterns, or geographic distributions, so that within-group and intergroup interactions can be modeled separately. In this paper, we formulate a multigroup SEIR epidemic model with general exposed distribution and general incidence rates. The population is divided into

In [

Motivated by these facts, in this paper, we incorporate the general incidence rate presented in [

The organization of this paper is as follows; in the next section, we give some preliminaries of our main model. In Section

Since the variables

The incidence function

assume that the functions

For model (

It is clear that system (

Let

For finite time

Since the limiting system (

Let

An equilibrium

For system (

Assume that the functions

If

If

It follows from the Perron-Frobenius theorem (see Theorem

If

Denote

To get the global stability of

Assume that the functions

Define a Lyapunov functional as

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 and the editor for very helpful suggestions and comments which led to improvements of our original paper. J. Wang is supported by National Natural Science Foundation of China (no. 11201128), the Science and Technology Research Project of the Department of Education of Heilongjiang Province (no. 12531495), the Natural Science Foundation of Heilongjiang Province (no. A201211), and the Science and Technology Innovation Team in Higher Education Institutions of Heilongjiang Province.