Dynamical Behavior of a New Epidemiological Model

R 0 is given. The model is studied in two cases: with and without time delay. For the model without time delay, the disease-free equilibrium is globally asymptotically stable provided that R 0 ≤ 1; if R 0 > 1, then there exists a unique endemic equilibrium, and it is globally asymptotically stable. For the model with time delay, a sufficient condition is given to ensure that the disease-free equilibrium is locally asymptotically stable. Hopf bifurcation in endemic equilibrium with respect to the time τ is also addressed.


Introduction
As we know, in the SIR model, the population is divided into three classes, , , and , where  denotes the number of susceptible individuals,  the number of infective individuals, and  the number of removed individuals, respectively.With respect to the SIR models, many studies have proposed several reasons for the nonlinearity of incidence rate at which susceptible individual becomes infective.In 1978, Capasso and Serio [1] found a saturated incidence rate ℎ() = /(1 + ), where ℎ() tends to a saturation level when  gets large.It is more reasonable than the bilinear incidence rate, because it reflects the behavioral change and crowding effect of the infective individuals and prevents the unboundedness of the contact rate by choosing suitable parameters.In 2007, Xiao and Ruan studied the global SIR model with nonmonotone incidence rate ℎ() = /(1 +  2 ) in [2].Recently, Yang and Xiao [3] extended this nonlinear incidence rate to ℎ() = /(1 +  ℎ ), ℎ ≥ 1, by using standard method.
However, the time period of immunity for many infectious diseases is short, or even they have no immunity.Furthermore, these diseases often lead to some other more dangerous diseases.For examples, Chagas disease, hepatitis C, gonorrhea, and other sexually transmitted diseases may advance through several infective stages and have different ability to transmit these infections in different stages of infection.Their infectivity usually depends on the parasite or viral loads in infected individuals or vectors [4].For instance, in the case of Chagas disease [5], the acute stage follows the invasion of the blood stream by the protozoan T. cruzi.This stage lasts from one to two months and infected individuals may or may not show symptoms of the disease.After the acute phase, the infected individuals enter the chronic stage and stay there for variable duration that lasts from 10 to 20 years.At its end, the disease may follow different paths: some individuals may develop mega syndromes and others may present myocarditis.Myocarditis is the terminal form which causes the highest mortality in the group of individuals between 20 and 50 years of age.In fact, there are some research achievements about this phenomenon [6][7][8][9][10].For example, Cai et al. [11] investigated a stage-structured epidemic model with a nonlinear incidence and a factor   .Motivated by the above discussions, we introduce a new epidemiological model in this paper.In this model, the population is divided into three classes, , , and , where () is the number of susceptible individuals, () the number of infective individuals, and () the number of individuals who are pathologically changed from the infective individuals not being cured for a certain time  at time , respectively.
Our model to be considered takes the following form: where  is the recruitment of the population,  is the natural death rate of the population,  is the proportional constant,  1 is the natural recovery rate of the infective individuals ,  2 is the natural recovery rate of the disease  (by , we also denote the disease induced by infective disease ), and  is the average time period from infectious disease  evolving to noninfectious disease . is the rate by which infective individuals  change into noninfectious individuals , and  is the parameter that measures the psychological or inhibitory effect. − is the probability that an individual during the incubation period time  survived to develop the disease  and did not emigrate [12].For simplicity, we set  =  − .We only consider the system in the first quadrant due to the epidemiological meaning.
The rest of this paper is organized as follows.In Section 2, we will study the stability of disease-free equilibrium and endemic equilibrium, respectively, of the system without time delay.In Section 3, the system with time delay is considered.The stability conditions for disease-free equilibrium and endemic equilibrium are investigated see Figures 1, 2, 3, and 4. Hopf bifurcation of the endemic equilibrium at a critical time  is studied.Finally, some conclusions and discussions will be given in the last section.

The System without Time Delay
If  = 0, system (1) becomes the following form: In this case, we neglect the time from infectious disease  evolving to noninfectious disease , and by  we denote the rate of disease  changed into disease .

Local Stability of Equilibria.
In order to examine local stability of an equilibrium, we should compute the eigenvalues of the linearized operator for system (6) at the equilibrium.Consider disease-free equilibrium  0 ; the characteristic equation is obtained by the standard method as follows: It is obvious that  1 = − < 0,  2 = − −  2 < 0, and  3 = ( 1 +  + )( 0 − 1) are the characteristic roots of (7).Thus, we have the following theorem.
Theorem 1.The local stability of disease-free equilibrium  0 has the following conclusions as  = 0.
Now, the local stability of the endemic equilibrium  * = ( * ,  * ,  * ) is considered.As we know, the endemic equilibrium  * exists if and only if  0 > 1.By computation, the characteristic equation of ( 6) at  * becomes where By calculation, we see that  > .Hence, all roots of (8) have negative real parts by the Routh-Hurwitz criterion.Therefore, we obtain the following result on the locally asymptotic stability of the endemic equilibrium.

Global Stability of Equilibria.
To study the global stability of an equilibrium, we first present two lemmas.
One has the following result regarding the nonexistence of periodic orbits in system (13), which implies the nonexistence of periodic orbits of system (6) by Lemmas 3 and 4.
We have The conclusion follows.
(i) If  0 ≤ 1, then system (1) has a unique equilibrium.The disease-free equilibrium  0 of system (1) is globally asymptotically stable in the interior of Λ.
(ii) If  0 > 1, then the system (1) has two equilibria,  0 and  * .Moreover, the endemic equilibrium  * is globally asymptotically stable in the interior of Λ.
(ii) By Theorem 1(i), Theorem 2, Lemma 3, and Theorem 5, we see that the endemic equilibrium  * is globally asymptotically stable in the interior of Λ.This completes the proof.

Local Stability of Equilibria.
In this section, we consider the system (1) with time delay  > 0. To derive the local stability of equilibrium, we should linearize the system (1) as the form where  and  are real matrixes.The characteristic equation [18] is By this method, we linearize the system (1) at the diseasefree equilibrium  0 and obtain the following characteristic equation: It is easy to see that  1 = − < 0 and  2 = − −  2 < 0 are two characteristic roots of ( 22).Hence, we only need to discuss the roots of the following equation: By discussing ( 22), we have the following result on the local asymptotic stability of the disease-free equilibrium.
Theorem 7. When  > 0, the disease-free equilibrium  0 of system (1) has the following conclusion.
Proof.(i) By implicit function theorem for complex variables, we know that the root of ( 23) is continuous on the parameter .
(iii) If  0 = 1, it is easy to know that  = 0 is a root of ( 23), for all  > 0, which leads to conclusion (iii).This completes the proof of the theorem.

Journal of Applied Mathematics
Now, the local stability of the endemic equilibrium  * = ( * ,  * ,  * ) is considered.As we know, the endemic equilibrium  * exists if and only if  0 > 1.By computation, the associated transcendental characteristic equation of (1) at  * becomes where When  = 0, (26) becomes where Thus, we obtain the following result on the local asymptotic stability of the endemic equilibrium.

Hopf Bifurcation.
In what follows, we consider the Hopf bifurcation of the disease-free equilibrium  0 .Theorem 9.If  > 0,  0 < 1, and  1 = ( + )/( 1  +  2 ) > 1, then there exists a pair of purely imaginary eigenvalues ± 0  as  =  0 , and the disease-free equilibrium of system (1) is locally asymptotically stable as 0 <  <  0 .On the other hand, system (1) can undergo a Hopf bifurcation if  >  0 , and a periodic orbit appears in the small neighborhood of the diseasefree equilibrium  0 .