Abstract
A new epidemiological model is introduced with nonlinear incidence, in which the infected disease may lose infectiousness and then evolves to a chronic noninfectious disease when the infected disease has not been cured for a certain time . The existence, uniqueness, and stability of the disease-free equilibrium and endemic equilibrium are discussed. The basic reproductive number 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 ; if , 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 is also addressed.
1. 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 , where 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 in [2]. Recently, Yang and Xiao [3] extended this nonlinear incidence rate to , , 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 [6–10]. 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, is the natural recovery rate of the infective individuals , 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.
It is easy to see that system (1) always has a disease-free equilibrium . To find the other positive equilibria, set By direct computation, we have Then, system (1) has exactly a positive equilibrium if and only if with
For system (1) with , define the basic reproduction number [13] as follows: By (2) and (3), it is easy to see that(i)if , then system (1) has no positive equilibrium;(ii)if , then system (1) has a unique positive equilibrium , which is called the endemic equilibrium;(iii)the equilibrium and the equillibrium are on the hyperplane .
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.
| (a) When , then the solution converges to the endemic equilibrium as |
| (b) When , then a periodic orbit appears in the small neighborhood of the endemic equilibrium |
2. The System without Time Delay
If , 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 .
2.1. 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 ; the characteristic equation is obtained by the standard method as follows: It is obvious that , , and are the characteristic roots of (7). Thus, we have the following theorem.
Theorem 1. The local stability of disease-free equilibrium has the following conclusions as .(i)If , then the disease-free equilibrium is unstable.(ii)If , then the disease-free equilibrium is locally asymptotically stable.(iii)If , then the disease-free equilibrium is degenerated.
Now, the local stability of the endemic equilibrium is considered. As we know, the endemic equilibrium exists if and only if . 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.
Theorem 2. If , then the endemic equilibrium is locally asymptotically stable.
2.2. Global Stability of Equilibria
To study the global stability of an equilibrium, we first present two lemmas.
Lemma 3. There exists a positively invariant set for system (6), where That is, if the initial value , then the solutions of (6) are in the set , for all .
Proof. If the initial value , then the solution of (6) locally exists and is unique by standard theory on ordinary differential equations [14].
By (6), it is easy to see that on the surface , , if , on the surface , , if , and on the surface , , if . Hence, no solution of system (6) can exit from the boundary , , , if [15].
By adding the first three equations of system (6), we obtain
where . Thus,
If the initial value and , then . Hence, the solution of system (6) cannot blow up to infinite for all .
In summary, if the initial value , then the solution of (6) is in , for all .
It is clear that the limit set of system (6) is on the plane . Thus, we have the following lemma.
Lemma 4. The limit set of system (6) in is equivalent to that of the following reduced system: in the triangle region .
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.
Theorem 5. System (13) does not have nontrivial periodic orbits.
Proof. Consider system (13) for and . Take a Dulac function We have The conclusion follows.
Theorem 6. When , system (1) has the following global dynamics.(i)If , then system (1) has a unique equilibrium. The disease-free equilibrium of system (1) is globally asymptotically stable in the interior of .(ii)If , then the system (1) has two equilibria, and . Moreover, the endemic equilibrium is globally asymptotically stable in the interior of .
Proof. (i) It is easy to check the conclusion by Theorem 1(ii), Lemma 3, and Theorem 5 as . Therefore, we only consider the case . For simplicity, we rescale (13) by
Then, we obtain
where
Clearly, if and only if , and system (17) has a unique equilibrium in the region . Hence, there exists a small neighborhood of such that the dynamics of system (17) is equivalent to that of
where . By Theorem 7.1 in [16] (pp. 114) or Theorem in [17] (pp. 1500), we know that is a saddle-node.
(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 . This completes the proof.
3. System with Time Delay and Hopf Bifurcation
3.1. Local Stability of Equilibria
In this section, we consider the system (1) with time delay . 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 disease-free equilibrium and obtain the following characteristic equation:
It is easy to see that and 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 , the disease-free equilibrium of system (1) has the following conclusion.(i)If , then the disease-free equilibrium of system (1) is locally asymptotically stable.(ii)If , then the disease-free equilibrium of system (1) is unstable.(iii)If , then the disease-free equilibrium of system (1) is degenerated.
Proof. (i) By implicit function theorem for complex variables, we know that the root of (23) is continuous on the parameter .
If , then 0 is not a root of (23), for all . Note that all complex roots of (23) must come in conjugate pairs and all roots of (23) are negative for . Thus, all roots of (23) have negative real parts for small ; that is, . Assume that there exists a positive number such that (23) has a pair of purely imaginary roots , where is a positive number. Then,
Adding up the square of both equations in (24), we obtain
When , then . It is a contradiction with which leads to the nonexistence of .
(ii) We set . When and , then and . Therefore, (23) must have a positive real root for all .
(iii) If , it is easy to know that is a root of (23), for all , which leads to conclusion (iii). This completes the proof of the theorem.
Now, the local stability of the endemic equilibrium is considered. As we know, the endemic equilibrium exists if and only if . By computation, the associated transcendental characteristic equation of (1) at becomes where When , (26) becomes where Thus, we obtain the following result on the local asymptotic stability of the endemic equilibrium.
Theorem 8. If , , , and , then the endemic equilibrium of system (1) is locally asymptotically stable for .
Proof. By implicit function theorem for complex variables, we know that the root of (23) is continuous on the parameter . If , then is not a root of (26) for all . Note that all complex roots of (26) must come in conjugate pairs and all roots of (26) have negative real parts as by Theorem 2. Thus, all roots of (26) have negative real parts for very small , . Suppose that there exists a positive such that (26) has a pair of purely imaginary roots , . Then, satisfies
Separating the real and imaginary parts, we have
which implies that
Let . Then, (33) becomes
where , , and . By computation, we have
where , , and
It is easy to check that if and , then , , and .
Now, we examine the sign of as follows:
By (2), we know that . Now, we set ; then it follows that
Now that , , and under the conditions of this theorem, (34) has no positive roots, which implies the nonexistence of . Thus, all roots of (26) have negative real parts for . The proof is finished.
3.2. Hopf Bifurcation
In what follows, we consider the Hopf bifurcation of the disease-free equilibrium .
Theorem 9. If , , and , then there exists a pair of purely imaginary eigenvalues as , and the disease-free equilibrium of system (1) is locally asymptotically stable as . On the other hand, system (1) can undergo a Hopf bifurcation if , and a periodic orbit appears in the small neighborhood of the disease-free equilibrium .
Proof. When , we obtain from (25). Let
By the implicit function theorem for complex variables, the first conclusion is completely proved.
Suppose that is a root of (23) as . Differentiating (23) with respect to , we obtain
Note that . We consider
By (24), we have that . Thus,
which leads to as . The proof is finished.
Now, we consider the Hopf bifurcation of the endemic equilibrium .
By Theorem 9, we know that the endemic equilibrium is locally asymptotically stable for all if , , and . Now, we consider the case that , , if , .
Lemma 10. If , , , and then there exists a pair of purely imaginary eigenvalues as , and the endemic equilibrium of system (6) is locally asymptotically stable as .
Proof. By Theorem 9, we know that characteristic equation (26) may have a pair of purely imaginary roots for some when and (34) has positive real roots.
We know that (34) has at least a positive root if , where
It is easy to check that if , . From Theorem 9, if
then . Hence, (34) has at least a positive root and at most three different positive real roots under the conditions of this lemma.
Assume that (34) has positive real roots, denoted by , . Then, (33) has positive roots . Let
We set
Then, the characteristic equation (26) of equilibrium has a pair of purely imaginary roots as . Note that all roots of (26) have negative real parts as . Therefore, all roots of (26) have negative real parts as by continuity of roots on parameter . This completes the proof of Lemma 10.
Theorem 11. Assume that the condition of Lemma 10 holds. Then, system (1) can undergo a Hopf bifurcation if , and a periodic orbit appears in the small neighborhood of the endemic equilibrium if either or .
Proof. We consider the transversal conditions [19] for the complex eigenvalues of the endemic equilibrium at .
Suppose that is a root of (26) as . Differentiating (26) with respect to , we obtain [3]
where .
Since and , if either or . Therefore we have
if either or .
By Lemma 10 and Hopf bifurcation theorem in [18], system (6) can undergo a Hopf bifurcation as , and a periodic orbit appears in the small neighborhood of the endemic . This completes the proof.
4. Conclusion
In this paper, we propose the SID model with time delays. Our purpose is to comprehend the number of disease which is caused by the infectious disease in a certain area. For example, hepatitis B causes liver cancer and we want to know the number of liver cancer caused by hepatitis B as . Our analytical results and numerical simulations with different parameter values show that this SID model has some interesting dynamics as follows. When , the basic reproductive number plays a role in the behaviors of this system. If , then the disease will become extinct as ; when , the endemic equilibrium is globally asymptotically stable. Namely, the number of disease will tend to a certain number and will not die out as . But, for the stability of the time delayed model, we cannot give complete analysis because of the complexity of the system and it is the work in the future. Here, we give some stability analysis in some special case. When , , and , the Hopf bifurcation appeared which implies that the disease will appear again even it dies out at this time. We have shown that if , , , and then there exists a positive time period such that the model has a periodic solution as which does not appear by bilinear incidence rate [20].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are very grateful to the anonymous referees for careful reading and helpful suggestions which led to the improvement of their original paper. This work was partially supported by the National Natural Science Foundation of China, Program for Changjiang Scholars and Innovative Research Team in University (IRT1226), and Research Fund for the Doctoral Program of Higher Education of China (no. 20124410110001).