Dynamic Analysis of an SEIR Model with Distinct Incidence for Exposed and Infectives

An SEIR model with vaccination strategy that incorporates distinct incidence rates for the exposed and the infected populations is studied. By means of Lyapunov function and LaSalle's invariant set theorem, we proved the global asymptotical stable results of the disease-free equilibrium. The sufficient conditions for the global stability of the endemic equilibrium are obtained using the compound matrix theory. Furthermore, the method of direct numerical simulation of the system shows that there is a periodic solution, when the system has three equilibrium points.


Introduction
Mathematical models have become important tools in analyzing the spread and the control of infectious diseases. Many infectious diseases in nature, such as measles, HIV/AIDS, SARS, and tuberculosis (see [1][2][3][4][5][6]), incubate inside the hosts for a period of time before the hosts become infectious. Li and Fang (see [7]) studied the global stability of an age-structured SEIR model with infectivity in latent period. Yi et al. (see [8]) discussed the dynamical behaviors of an SEIR epidemic system with nonlinear transmission rate. Li and Zhou (see [9]) considered the global stability of an SEIR model with vertical transmission and saturating contact rate.
In this paper, we will consider an SEIR model that the diseases can be infected in the latent period and the infected period. The population size is divided into four homogeneous classes: the susceptible ( ), the exposed (in the latent period) ( ), the infective ( ), and the recovered ( ). It is assumed that all the offsprings at birth are susceptible to the disease. The inflow rate (including birth and immigration) and outflow rate (including natural death and emigration) are denoted by and , respectively. The rate of disease-caused death is taken as . We assume that susceptible individuals are vaccinated at a constant per capita rate . Due to the partial efficiency of the vaccine, only fraction of the vaccinated susceptibles goes to the recovered class. The remained 1 − fraction of the vaccinated susceptibles has no immunity at all and goes to the exposed class after infected by contact with the infectives. If = 0, it means that the vaccine has no effect at all, and if = 1, the vaccine is perfectly effective. The positive parameter is the rate at which the exposed individuals become infectious. is the constant rate, at which the infectious individuals recover with acquiring permanent immunity. The transfer mechanism from the class ( ) to the class ( ) is guided by the function ( + )/ , where is the force of infection. denotes the relative measure of infectiousness for the asymptomatic class ( ).
Based on these considerations, and with reference to [10][11][12], the SEIR model is given by the following system of differential equations:  (2) The system (2) is equivalent to (1). From biological considerations, we study (2) in the following closed set: where 3 + denotes the nonnegative cone of 3 including its lower dimensional faces.

Equilibria and Global Stability
It is easy to visualize that (2) always has a disease-free equilibrium 0 ( /( + ), 0, 0). The Jacobian matrix of (2) at an arbitrary point ( , , ) takes the following form: Proof. Let We calculate the characteristic equation of ( 0 ) as follows: The stability of 0 is equivalent to all eigenvalues of (8) being with negative real parts, which can be guaranteed by 0 < 1. Consequently, the disease-free equilibrium is local asymptotical stability. This proves the theorem.

(12)
Proof. Let the right side of each of the first three differential equations equal to zero in (2); we obtain the following: * (1 − ) ( * + * ) − ( + ) * + * * = 0, * − ( + + ) * + * 2 = 0, The When the three equations of (14) are multiplied together, we obtain the following: Define the following: where Since Δ > 1, the linear function ( ) has exactly one intersection with the function ( ) where lies in the interval (0, 1). Furthermore, * and * can be uniquely determined from * by the following: From this, we can easily see that (2) has a unique endemic equilibrium. This completes the proof.
Denote the interior of by ∘ . In this paper, we obtain sufficient conditions that the equilibrium is globally asymptotically stable using the geometrical approach of Li and Muldowney in [14].