Abstract
A fractional-order epidemic model with time delay is considered. Firstly, stability of the disease-free equilibrium point and endemic equilibrium point is studied. Then, by choosing the time delay as a bifurcation parameter, the existence of Hopf bifurcation is studied. Finally, numerical simulations are given to illustrate the effectiveness and feasibility of theoretical results.
1. Introduction
Mathematical model plays an important role in describing the dynamics of biological system [1–3]. The dynamics of the epidemic models have received much attention during the recent years, and to explain the disease spreading and control strategies a series of epidemic models [4–7] was proposed. A stochastic SIRS epidemic model was formulated in [8]; it investigated the effect of stochastic environmental variability on interpandemic transmission dynamics of influenza A. In [9], an age-structured SEIR epidemic model was considered. The authors investigated an SEIR model with varying population size and vaccination strategy in [10], and different threshold parameters were obtained to govern the disease eradication. Many models in biological mathematics involve some time delays. In biological dynamics, time delay was widely applied to reflect some biological facts, such as immunity period [11] and latent period of the disease [12]. An epidemic model with time delay was proposed in [13], and the model is shown as follows:where , represent the number of susceptible and infected population. represents the intrinsic birth rate constant, represents carrying capacity of susceptible population, represents the force of infection or the rate of transmission, represents immigration coefficient, represents death coefficient of , and is the latent period of the disease.
Fractional calculus is a generalization of classical differentiation and integration to arbitrary (noninteger) order [14]. In the past decades, fractional-order calculus garnered considerable attention and it was applied to various fields [15–21]. Recently, many investigators started to study the fractional-order biological models [22–24]. The main reason is that fractional-order models are naturally related to systems with memory which exists in most biological systems [25, 26]. In [27], the authors introduced a fractional-order prey-predator model and deal with the biological behaviors of the model. A fractional-order SIS model with variable population size is considered in [28], and the stability of equilibrium points is studied. A fractional-order model of two-species facultative mutualism with harvesting was presented in [29], and stability of the model was analyzed. In [30], the authors introduced a fractional-order epidemic model with vaccination; it shows that the stability region of the model is related to threshold-value and value of the fractional-order . A delayed fractional-order differential model of HIV infection of CD4+ was investigated in [31]. In [32], a fractional-order prey-predator model with time delay and Monod-Haldane function was studied.
In this paper, a fractional-order epidemic model with time delay is studied. We investigate stability and bifurcation of the model with respect to basic reproduction number , fractional-order and time delay . We provide theoretical analysis, using the eigenvalues method and linearization techniques and bifurcation method. The model is depicted as follows:where , , and is in the sense of Caputo fractional derivatives. , , and .
The corresponding linearized system of (2) at any equilibrium point is defined asTaking Laplace transform [33] on both sides of (3), one obtains the characteristic matrix as follows:The properties of eigenvalues of characteristic equation indicate the stability of system (2).
The rest of the paper is organized as follows. In Section 2, some necessary definitions and notions are presented. In Section 3, stability and Hopf bifurcation of the equilibrium point are analyzed. Numerical simulations are given in Section 4 and some conclusions are given in Section 5.
2. Preliminary
There are three main definitions of fractional-order differential, that is, Riemann-Liouville, Grünwald-Letnikov, and Caputo’s definitions. This paper is based on Caputo’s definition.
Definition 1 (see [34]). The Caputo fractional derivative with order of a continuous function is defined as follows:where .
Lemma 2 (see [35]). Considering the fractional differential system with the Caputo derivative,where , , and . The characteristic equation of system (6) is . If the real parts of all the eigenvalues of are negative, then the zero solution to system (6) is locally asymptotically stable.
Lemma 3 (see [33]). Considering the fractional delayed differential system with the Caputo derivative,where , , , and . The characteristic equation of the system (7) is . If all the roots of the characteristic equation have negative real parts, then the zero solution of system (7) is locally asymptotically stable.
3. Main Results
3.1. Basic Production Number and the Existence of the Equilibrium Point
Following from [16], system (2) has a disease-free equilibrium point and the basic reproduction number for the model is . Endemic equilibrium point is . Obviously, . Then we know that model (2) has an endemic equilibrium point when .
3.2. Stability of the Disease-Free Equilibrium Point
Theorem 4. The disease-free equilibrium point of system (2) is locally asymptotically stable if .
Proof. The characteristic matrix of system (3) evaluated at the equilibrium point isand the characteristic equation isLet ; we can rewrite (9) asClearly, , . When , we get . According to Lemma 2, the disease-free equilibrium point is locally asymptotically stable. This completes the proof.
3.3. Stability of the Endemic Equilibrium Point
The characteristic matrix of system (3) evaluated at the equilibrium point isfrom which we have the characteristic equationwhere
Theorem 5. When , the endemic equilibrium point of system (2) is locally asymptotically stable if .
Proof. Let ; we can rewrite (12) asIf , one obtains . Obviously, the two roots of (14) are negative. According to Lemma 2, the endemic equilibrium point is locally asymptotically stable. This completes the proof.
When , (12) can be rewritten as whereAssume that (15) has a pair of pure imaginary roots , and then substitute into equation (15); one obtainsand then separating the real and imaginary parts of (17) one hasSquaring and adding the two equations in (18), we obtainDenote , where ; therefore (19) has one positive root at least. If , , , are the roots of , we assume is positive. Substituting into (18), one obtainsSquaring and adding the two equations in (20), one obtainsFrom (21), can be obtained
Theorem 6. When and , the endemic equilibrium point of system (2) is locally asymptotically stable if and unstable if , where .
Proof. Let and ; (12) reduces toDenote ; one hasDifferentiating both sides of (24) with respect to , we obtainDifferentiating both sides of (24) with respect to yieldsFrom (26), one obtainsWith , the above equality becomesWhen , one gets . Hence, the endemic equilibrium point of system (2) is locally asymptotically stable if and unstable if . This completes the proof.
Remark 7. It is worth noting that there will be some future directions to apply our main results to more complex ones like models with time varying delay [36] and models with perturbed parameters [37] or to study the Hopf bifurcation of models with discrete and distributed delays [38].
4. Numerical Simulations
In this section, we give some numerical simulations for system (2) by using the method mentioned in [39, 40].
In Figure 1, we select parameters as , , , , , and , with initial conditions , . After calculation, one obtains disease-free equilibrium point and . In (a), we take , and in (b) we take . According to Theorem 4, the disease-free equilibrium point of system (2) is locally asymptotically stable when .
(a)
(b)
In Figure 2, the selected parameters are , , , , , , and , with initial conditions , After calculation, one obtains endemic equilibrium and . According to Theorem 5, if and , the endemic equilibrium point of system (2) is locally asymptotically stable. The numerical simulation results are shown in Figure 2.
In Figures 3(a) and 3(b), we plotted the effect measure of immigration coefficient on susceptible and infected populations. The selected parameters are same as Figure 2 with initial conditions , . Values of are shown in the legend. From Figures 3(a) and 3(b), we observe that the number of susceptible individuals increases as increases at the beginning but is finally stable at the same fixed value. The number of infected individuals increases as increases. It shows that after the endemic formation, the number of the susceptible individuals increases as the number of floating population increases in the short term, but in the long run the number of susceptible individuals is the same, and only the number of infected individuals increases.
(a)
(b)
Figure 4 depicts the Hopf bifurcation of the endemic equilibrium. The parameters are taken as , , , , and , with initial conditions , . After calculation, one obtains , , and . When , is calculated. In (a), we let , and in (b) . (a) and (b) show Hopf bifurcation occurs at . Then one selects different order , we get different time delay, and the results are shown in (c). Figure 4(c) shows that as the value of becomes smaller, the stability domain becomes larger. When , , and , Figure 4(d) shows that the endemic equilibrium point becomes stable.
(a)
(b)
(c)
(d)
5. Conclusion
In this paper, a fractional-order epidemic model with time delay is studied and stability and bifurcation of the model are analyzed. The results show that when , the disease-free equilibrium point is locally asymptotically stable for . And we get that when and , the endemic equilibrium point is locally asymptotically stable. According to Theorem 6, when and , the stability of the endemic equilibrium point changes at bifurcation point . Some numerical simulations are given to verify the correctness of the theory, and stability region of model is related to the value of , , and fractional-order .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (nos. 61573008, 61473178), the Natural Science Foundation of Shandong Province (no. ZR201709230160), Post-Doctoral Applied Research Projects of Qingdao (no. 2016115), and SDUST Research Fund (no. 2014TDJH102).