A fractional order SIR epidemic model with nonlinear incidence rate
 Abderrahim Mouaouine^{1}Email authorView ORCID ID profile,
 Adnane Boukhouima^{1},
 Khalid Hattaf^{1, 2} and
 Noura Yousfi^{1}
https://doi.org/10.1186/s136620181613z
© The Author(s) 2018
Received: 12 January 2018
Accepted: 25 April 2018
Published: 3 May 2018
Abstract
In this paper, a fractional order SIR epidemic model with nonlinear incidence rate is presented and analyzed. First, we prove the global existence, positivity, and boundedness of solutions. The equilibria are calculated and their stability is investigated. Finally, numerical simulations are presented to illustrate our theoretical results.
Keywords
1 Introduction
Fractional calculus is a generalization of integral and derivative to noninteger order that was first applied by Abel in his study of the tautocrone problem [1]. Therefore, it has been largely applied in many fields such as mechanics, viscoelasticity, bioengineering, finance, and control theory [2–6].
As opposed to the ordinary derivative, which is a local operator, the fractional order derivative has the main property called memory effect. More precisely, the next state of fractional derivative for any given function f depends not only on their current state, but also upon all of their historical states. Due to this property, the fractional order derivative is more suited for modeling problems involving memory, which is the case in most biological systems [7, 8]. Also, another advantage for using fractional order derivative is enlarging the stability region of the dynamical systems.
2 Properties of solutions
Lemma 2.1
 (1)
\(F(X)\) and \(\frac{\partial F}{\partial X}\) are continuous.
 (2)
\(\Vert F(X)\Vert \leq \omega + \lambda \Vert X\Vert \ \forall X\in \mathbb{R}^{2}\), where ω and λ are two positive constants.
Theorem 2.2
Proof
 Case 1::

If \(\alpha_{1}\neq 0\), we haveThen$$\begin{aligned} F(X) &=\varepsilon +A_{1} X+\frac{\alpha_{1}S}{1+\alpha_{1}S+\alpha _{2}I+\alpha_{3}SI}A_{2}X. \end{aligned}$$$$\begin{aligned} \bigl\Vert F(X)\bigr\Vert & \leq \Vert \varepsilon \Vert +\Vert A_{1}X\Vert + \Vert A_{2}X\Vert \\ &=\Vert \varepsilon \Vert + \bigl(\Vert A_{1}\Vert +\Vert A_{2}\Vert \bigr) \Vert X\Vert . \end{aligned}$$
 Case 2::

If \(\alpha_{2}\neq 0\), we haveThen$$\begin{aligned} F(X) &=\varepsilon +A_{1} X+\frac{\alpha_{2}I}{1+\alpha_{1}S+\alpha _{2}I+\alpha_{3}SI}A_{3}X. \end{aligned}$$$$\begin{aligned} \bigl\Vert F(X))\bigr\Vert & \leq \Vert \varepsilon \Vert + \bigl( \Vert A_{1}\Vert + \Vert A_{3}\Vert \bigr) \Vert X \Vert . \end{aligned}$$
 Case 3::

If \(\alpha_{3}\neq 0\), we getThen$$\begin{aligned} F(X) &=\varepsilon +A_{1}X+\frac{\alpha_{3}SI}{1+\alpha_{1}S+\alpha _{2}I+\alpha_{3}SI}A_{4}. \end{aligned}$$$$\begin{aligned} \bigl\Vert F(X)\bigr\Vert & \leq \Vert \varepsilon \Vert +\Vert A_{4} \Vert + \Vert A_{1} \Vert \Vert X\Vert . \end{aligned}$$
 Case 4::

If \(\alpha_{1}=\alpha_{2}=\alpha_{3}= 0\), we obtainThen$$\begin{aligned} F(X) &=\varepsilon +A_{1}X+IA_{5}X. \end{aligned}$$$$\begin{aligned} \bigl\Vert F(X)\bigr\Vert & \leq \Vert \varepsilon \Vert +\bigl(\Vert A_{1}\Vert + \Vert I \Vert \Vert A_{5}\Vert \bigr) \Vert X \Vert . \end{aligned}$$
3 Equilibria and their local stability
Theorem 3.1
 (i)
If \(R_{0}\leq 1 \), then system (2) has a unique diseasefree equilibrium of the form \(E_{0} ( S_{0},0 ) \), where \(S_{0}=\frac{\Lambda }{\mu }\).
 (ii)If \(R_{0}>1 \), the diseasefree equilibrium is still present and system (2) has a unique endemic equilibrium of the form \(E^{*}(S^{*},\frac{\Lambda \mu S^{*}}{a}) \), wherewith \(a=\mu +d+r \) and \(\Delta = (\beta \alpha_{1}a+\alpha_{2} \mu \alpha_{3}\Lambda)^{2}+4\alpha_{3}\mu (a+\alpha_{2}\Lambda)\).$$S^{*}=\frac{2(a+\alpha_{2}\Lambda)}{\beta \alpha_{1}a+\alpha_{2} \mu \alpha_{3}\Lambda +\sqrt{\Delta }}, $$
Theorem 3.2
The diseasefree equilibrium \(E_{0} \) is locally asymptotically stable if \(R_{0}<1 \) and unstable whenever \(R_{0}>1 \).
Proof
Theorem 3.3
If \(R_{0}>1 \), then the endemic equilibrium \(E^{*} \) is locally asymptotically stable.
4 Global stability
In this section, we investigate the global stability of both equilibria.
Theorem 4.1
The diseasefree equilibrium \(E_{0} \) is globally asymptotically stable whenever \(R_{0}\leq 1 \).
Proof

If \(R_{0}< 1 \), then \(I=0 \).

If \(R_{0}= 1 \), from the first equation in (2) and \(S=S_{0} \), we havewhich implies that \(\frac{\beta S_{0}I}{1+\alpha_{1}S_{0}+\alpha_{2}I+ \alpha_{3}S_{0}I}=0\). Consequently, we get \(I=0\). From the above discussions, we conclude that the largest invariant set of \(\lbrace (S,I)\in \mathbb{R}^{2}_{+}: D^{\alpha }L_{0}(t)=0\rbrace \) is the singleton \(\lbrace E_{0}\rbrace \). Consequently, from [24, Lemma 4.6], \(E_{0} \) is globally asymptotically stable.$$0=\Lambda \mu S_{0}\frac{\beta S_{0}I}{1+\alpha_{1}S_{0}+\alpha_{2}I+ \alpha_{3}S_{0}I}, $$
Theorem 4.2
Assume that \(R_{0}>1 \). Then the endemic equilibrium \(E^{*} \) is globally asymptotically stable.
Proof
5 Numerical simulations
In this section, we give some numerical simulations to illustrate our theoretical results. Here, we solve the nonlinear fractional system (2) by applying the numerical method presented in [25]. System (2) can be solved by other numerical methods for fractional differential equations [26–29].
In the above Figs. 1 and 2, we show that the solutions of (2) converge to the equilibrium points for different values of α, which confirms the theoretical results. In addition, the model converges rapidly to its steady state when the value of α is very small. This result was also observed in [20, 29].
6 Conclusion
In this paper, we have presented and studied a new fractional order SIR epidemic model with the Caputo fractional derivative and the specific functional response which covers various types of incidence rate existing in the literature. We have established the existence and the boundedness of nonnegative solutions. After calculating the equilibria of our model, we have proved the local and the global stability of the diseasefree equilibrium when \(R_{0}\leq 1\), which means the extinction of the disease. However, when \(R_{0}>1\), the diseasefree equilibrium becomes unstable and system (2) has an endemic equilibrium which is globally asymptotically stable. In this case, the disease persists in the population.
From our numerical results, we can observe that the different values of α have no effect on the stability of both equilibria but affect the time to reach the steady states.
Declarations
Acknowledgements
The authors would like to express their gratitude to the editor and the anonymous referees for their constructive comments and suggestions which have improved the quality of the manuscript.
Authors’ contributions
All authors contributed equally to the writing of this paper. They read and approved the final version of the manuscript.
Competing interests
The authors declare that they have no competing interests.
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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