Stability analysis of a fractional-order epidemics model with multiple equilibriums
- Davood Rostamy^{1} and
- Ehsan Mottaghi^{1}Email author
https://doi.org/10.1186/s13662-016-0905-4
© Rostamy and Mottaghi 2016
Received: 26 January 2016
Accepted: 20 June 2016
Published: 29 June 2016
Abstract
In this paper, we extend the SIR model with vaccination into a fractional-order model by using a system of fractional ordinary differential equations in the sense of the Caputo derivative of order \(\alpha\in(0,1]\). By applying fractional calculus, we give a detailed analysis of the equilibrium points of the model. In particular, we analytically obtain a certain threshold value of the basic reproduction number \(R_{0}\) and describe the existence conditions of multiple equilibrium points. Moreover, it is shown that the stability region of the equilibrium points increases by choosing an appropriate value of the fractional order α. Finally, the analytical results are confirmed by some numerical simulations for real data related to pertussis disease.
Keywords
1 Introduction
The study of epidemiology has attracted much interest during the recent years. In this direction, mathematical models have been developed to imply more realistic aspects of disease spreading [1, 2]. Most mathematical epidemic models descend from the classical SIR model of Kermack and McKendrick established in 1927 [3]. Recently, many researchers have discussed the SIR model allowing vaccination [4–6]. Epidemiologically, vaccines are extremely important and have been proved to be the most effective and cost-efficient method of preventing infectious diseases such as measles, polio, diphtheria, tetanus, pertussis, tuberculosis, etc. [7, 8]. Mathematically, the epidemic models containing vaccination lead to multiple equilibrium points and show bifurcation phenomena. For example, Kribs-Zaleta and Velasco-Hernández in [9] constructed a SIS model with vaccination and showed that it exhibits a backward bifurcation. Also, Allen et al. introduced a stochastic SIR model with vaccination and showed that their model may have multiple endemic equilibriums [10]. Most of the vaccination models have been established based on ordinary differential equations (ODEs) [11–14].
Recently, fractional calculus has been extensively applied in many fields [15–18]. Many mathematicians and applied researchers have tried to model real processes using the fractional calculus [19–22]. The fractional modeling is an advantageous approach which has been used to study the behavior of diseases because the fractional derivative is a generalization of the integer-order derivative. Also, the integer derivative is local in nature, while the fractional derivative is global. This behavior is very useful to model epidemics problems. In addition, the fractional derivative is used to increase the stability region of the system. Fractional calculus has previously been applied in epidemiological studies [23–26]. Previous works rarely discussed an epidemics model with vaccination strategies that lead to multiple equilibrium points. In this work, we extend the SIR model with vaccination to a fractional-order model and give a detailed analysis of the equilibrium points of the model by applying fractional calculus.
This paper is organized as follows. In the next section, we give the definition of the Caputo derivative and present the fractional-order SIR model with vaccination in the sense of the Caputo derivative of order \(\alpha\in(0,1]\). In Section 3, we analyze the existence and stability of disease-free and endemic equilibrium points. To verify our results, we provide some numerical simulations for real data related to pertussis disease in Section 4. Finally, conclusions are given in Section 5.
2 Fractional-order model
In this section, we introduce the definition of fractional-order integration and derivative. There are different definitions of the fractional derivative. Among them, Riemann-Liouville and Caputo’s fractional derivative have been used more than others [27]. Comparing these two fractional derivatives, one easily arrives at the fact that Caputo’s derivative of a constant is equal to zero, which is not the case for the Riemann-Liouville derivative. The main concern of the paper thus focuses on the Caputo derivative of order \(\alpha>0\), which is rather applicable in real application.
Definition 1
Definition 2
3 Existence and stability of equilibrium points
Theorem 1
3.1 Existence and stability of disease-free equilibrium point
Theorem 2
Proof
3.2 Existence conditions of endemic equilibrium points
Theorem 3
If \(\beta\le b + \mu\), then system (6) has no endemic equilibrium point.
Proof
Lemma 1
Proof
Since \(A<0\), the curve \(Q(I)\) has a minimum value. By direct calculation, we see that this minimum value occurs at the point \((I_{\min},R_{\min})\). □
Theorem 4
- (i)
If \(R_{0}>1\) or \(R_{0}=1\), \(B>0\) then system (6) has the unique endemic equilibrium point \(E_{u}^{*}\).
- (ii)
If \(R_{\min}=R_{0}<1\) and \(B>0\) then system (6) has the unique endemic equilibrium point \(E_{c}^{*}\).
- (iii)
If \(R_{\min}< R_{0}<1\) and \(B>0\) then system (6) has two endemic equilibrium points \(E_{1}^{*}\), \(E_{2}^{*}\).
- (iv)
If \(R_{0} < R_{\min}\) there is no endemic equilibrium point.
Proof
- (i)
If \(R_{0}>1\), the quadratic equation \(Q(I)=R_{0}\) has two real roots and one of them is non-negative and greater than \(I_{\min}\). If \(R_{0}=1\), the quadratic equation \(Q(I)=R_{0}\) has a non-zero real root such that it is non-negative and greater than \(I_{\min}\) when \(B>0\). So, system (6) has the unique endemic equilibrium point \(E_{u}^{*}\) such that \(I_{u}^{*}>I_{\min}\).
- (ii)
If \(R_{\min}=R_{0}<1\), the equation \(Q(I)=R_{0}\) has a repeated real root which is non-negative when \(B>0\). Thus, system (6) has the unique endemic equilibrium point \(E_{c}^{*}\) such that \(I_{c}^{*}=I_{\min}\).
- (iii)
If \(R_{\min}< R_{0}<1\), the equation \(Q(I)=R_{0}\) has two real roots \(I_{1}^{*}\) and \(I_{2}^{*}\). If \(B>0\), these roots are non-negative. Therefore, system (6) has two endemic equilibrium points \(E_{1}^{*}\) and \(E_{2}^{*}\) such that \(I_{1}^{*}< I_{\min}< I_{2}^{*}\).
3.3 Stability and α-stability of endemic equilibrium points
The following theorem shows that the stability region of endemic equilibrium points of system (6) can be increased by choosing an appropriate value of fractional order α.
Theorem 5
- (i)
The endemic equilibrium points \(E^{*}_{c}\) and \(E^{*}_{1}\) are unstable.
- (ii)
If \(\alpha\leq\frac{2}{3}\), the endemic equilibrium points \(E^{*}_{u}\) and \(E^{*}_{2}\) are locally asymptotically α-stable.
- (iii)
If \(\alpha> \frac{2}{3}\) and \(\vartheta\geq\theta\), \(E^{*}_{u}\) and \(E^{*}_{2}\) are locally asymptotically stable.
Proof
4 Numerical results
According to Theorem 5, we expect the equilibrium point \(E_{1}^{*}\) to be unstable for different values of α. Since \(\vartheta\geq\theta\), we envisage that the equilibrium point \(E_{2}^{*}\) is asymptotically stable for \(\alpha> 2/3\) and α-stable for \(\alpha\leq2/3\). Figures 3-5 show that the model presented here gradually approaches the steady state for different values of α but the dynamics of the model is governed by the distinct paths.
5 Conclusion
In this paper, we extended the classical SIR model with vaccination to a system of fractional ordinary differential equations (FODEs). For our fractional-order model, we determined the basic reproduction \(R_{0}\) and proved that if \(R_{0}<1\), the disease-free equilibrium is locally asymptotically stable. In the classical SIR model with vaccination, it is shown that \(R_{0}\) must be further reduced to less than a threshold value in order to ensure that the disease exterminates, but this value has not been obtained exactly [10]. In this work, we analytically obtained the threshold value of \(R_{0}\), denoted by \(R_{\min}\). Using the values of \(R_{0}\) and \(R_{\min}\), we established the existence conditions of endemic equilibrium points in Theorem 4. We proved Theorem 5 about the stability and α-stability of the endemic equilibrium points which are introduced in Theorem 4. Theorem 5 shows that the stability of endemic equilibrium points can be controlled by modifying the value of α. In fact, the fractional-order model can be achieved in the steady state by controlling the parameters which affect the value of α. Finally, the analytical results are confirmed by some numerical simulations for real data related to pertussis disease. In Figure 2, it is shown that pertussis disease can be contained by choosing appropriate values for σ and ϕ. The numerical simulations presented in Figures 3-5 are compatible by Theorems 4 and 5.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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