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Bifurcations and chaos in a discrete SI epidemic model with fractional order
Advances in Difference Equations volume 2018, Article number: 44 (2018)
Abstract
In this study, a new discrete SI epidemic model is proposed and established from SI fractionalorder epidemic model. The existence conditions, the stability of the equilibrium points and the occurrence of bifurcation are analyzed. By using the center manifold theorem and bifurcation theory, it is shown that the model undergoes flip and Neimark–Sacker bifurcation. The effects of step size and fractionalorder parameters on the dynamics of the model are studied. The bifurcation analysis is also conducted and our numerical results are in agreement with theoretical results.
1 Introduction
Mathematical modeling plays an important role in understanding the dynamics of many infectious diseases. Thus, the use of modeling is crucial in order to analyze the spread, control strategies and the mechanisms of transmission of diseases. Over the years, numerous epidemiological models have been formulated mathematically (see, e.g., [1–6]). Although most of these models have been restricted to integerorder differential equations (IDEs), in the last three decades, it has turned out that many problems in different fields such as sciences, engineering, finance, economics and in particular epidemiology can be described successfully by the fractionalorder differential equations (FDEs) (see, e.g., [7–12]). A property of these fractionalorder models is their nonlocal property which does not exist in IDEs. Nonlocal property means that the next state of a model depends not only upon its current state but also upon all of its historical states [13]. The transformation of an integerorder model into a fractionalorder model needs to be precise as to the order of differentiation α: a small change in α may cause a big change in the final results [14]. FDEs can be used to model certain phenomena which cannot be adequately modeled by IDEs [15]. FDEs are commonly used on biological systems since they are related in a natural way to systems with memory. Similar to a nonlinear differential system, a nonlinear fractional differential system may also have complex dynamics, such as chaos and bifurcation. Studying chaos in fractionalorder dynamical systems is an attractive and interesting topic [16].
Discrete models constructed by the discretization of continuous models have been used to describe some epidemic models. Many significant and meaningful types of such research can be found in [17–27] and the references therein. The reason for the use of discrete models is that statistical data on epidemics are collected in discrete times and hence comparing data with output of a discrete model may be easier [28]. In general, the dynamical behavior of discrete fractionalorder epidemic models may exhibit phenomena such as the perioddoubling and chaotic behavior. Iwami et al. [29] have proposed a mathematical model in a continuoustime version to interpret a model of the spread of avian–human influenza epidemic. In [29], the dynamics of the spread into bird population and between bird and human populations have been thoroughly studied. The main objective of the current study is to extend the research done in [29] by employing fractional models with discretetime systems. In this paper, a discretetime SI is established by SI fractionalorder epidemic model. This model contains two parameters in addition to those already existing in the original SI model proposed in [29]; time step parameter and fractionalorder parameter. The occurrence of bifurcations as these parameters are varied is shown in Section 7. The stability of fixed points, the emergence of flip bifurcation and Neimark–Sacker bifurcation are also studied. Using the discretizedtime SI model with fractional order is a new topic, thus this paper provides a new contribution to the literature.
This paper is organized as follows. In Section 2, the fractionaltime SI epidemic model from its continuoustime counterpart is constructed. In Section 3, the discretizedtime SI model with fractionalorder parameter is established. Section 4 discusses the existence and stability of fixed points of the discretizedtime SI model with fractional order. In Sections 5 and 6, flip and Neimark–Sacker bifurcations are demonstrated, respectively. In Section 7, some numerical examples are shown. A brief discussion of our results is given in Section 8.
2 The FDE epidemic model
Let us consider the following IDE epidemic model:
This model is constructed by Iwami et al. [29] to explain the spreads of avian influenza through the bird world and describes the interactions between them. The population in this model is divided into susceptible birds with size S and infected birds with size I. The new birds birth rate is expressed by the parameter Λ. Susceptible birds die at the rate μ and infected birds die at the rate \(\mu+r\), where r is the additional death rate mediated by avian influenza. The parameter β is the bilinear incidence rate.
There are several definitions of fractional derivatives [30, 31]. One of the most common definitions is the Caputo definition [32]. This definition is often used in real applications and shown in Definition 1.
Definition 1
The fractional integral of order \(\beta\in \mathbb{R} ^{+}\) of the function \(f(t)\), \(t>0\), is defined by
and the fractional derivative of order \(\alpha\in ( n1,n ) \) of \(f ( t ) \), \(t>0\), is defined by
where \(f^{(n)}\) represents the nth order derivative of \(f(t)\), \(n=[\alpha]\) is the value of α rounded up to the nearest integer, \(I^{\beta}\)is the βth order RiemannLiouville integer operator and \(\Gamma(\cdot)\) is Euler’s Gamma function. The operator \(D^{\alpha}\) is called the ‘αth order Caputo differential operator’.
Now, the fractionalorder form of the SI epidemic model (2.1) can be formulated as follows:
where \(D_{t}^{\alpha}\) represents the Caputo fractional derivative, \(t>0\), and α is the fractional order satisfying \(\alpha\in ( 0,1 ] \).
3 Discretization process
There are many discretization methods that have been used to construct the discretetime model using continuoustime methods such as explicit and implicit Euler’s method, Runge–Kutta method, predictorcorrector method and nonstandard finite difference methods [33–37]. Some of them are approximation for the derivative and some for the integral. In [38, 39] a discretization process was introduced to discretize FDEs. This discretization method is an approximation for the right hand side of the differential equation has the formula \(D^{\alpha}x ( t ) =f ( x ( t ) ) \), \(t>0\), \(\alpha\in ( 0,1 ) \). This method is now applied to the fractionalorder SI model (2.2). Assume that \(S ( 0 ) =S_{0}\) and \(I ( 0 ) =I_{0}\) are the initial conditions of system (2.2). So, the discretization of the system (2.2) is given by the following formulas:
First, let \(t\in[ 0,h ) \), \(t/h\in [ 0,1 ) \). Then
and the solution of (3.2) reduces to
Second, let \(t\in [ h,2h ) \), which makes \(1\leqslant t/h<2\). Thus, we obtain
which have the following solutions:
Repeating the discretization process n times yields
where \(t\in [ nh, ( n+1 ) h ) \). For \(t\rightarrow ( n+1 ) h\), system (3.6) is reduced to
Remark 3.1
It should be noticed that if \(\alpha\rightarrow1\) in (3.7), the Euler discretization of SI model is obtained.
4 Stability of fixed points
In this section, an approach as in [40] is employed. The stability of the system (3.7) is studied around its fixed points. It is clear that the model (3.7) has always a disease free equilibrium point \(E_{0}=(\frac{\Lambda}{\mu},0)\) and an endemic equilibrium point \(E_{1}= ( \frac{\mu+r}{\beta},\frac{\Lambda}{\mu+r}\frac{\mu}{\beta} ) \). Furthermore, the system (3.7) has basic reproduction number \(\Re_{0}=\frac{\beta\Lambda}{\mu ( \mu+r ) }>1\), then \(E_{1}\) can be reformulated as \(E_{1}= ( \frac{\mu+r}{\beta},\frac{\mu ( \Re _{0}1 ) }{\beta} ) \). It can be seen that the free equilibrium point \(E_{0}\) always exists while the positive endemic \(E_{1}\) exists only when \(\Re_{0}>1\). So as to analyze the dynamical properties of (3.7), we compute the Jacobian matrix J of (3.7), and we evaluate at the fixed point \(E= ( S^{\ast},I^{\ast} ) \)
In order to study the stability of the fixed points of the system (3.7), the two following lemmas are employed.
Lemma 4.1
Let \(\lambda_{1}\) and \(\lambda_{2}\) be the two roots of matrix \(J ( M ) \), we have the following definitions:

(i)
If \(\vert \lambda_{1} \vert <1\) and \(\vert \lambda _{2} \vert <1\), then the equilibrium point of \(M ( x^{\ast },y^{\ast } ) \) is locally asymptotically stable (sink).

(ii)
If \(\vert \lambda_{1} \vert >1\) and \(\vert \lambda _{2} \vert >1\), then the equilibrium point of \(M ( x^{\ast },y^{\ast } ) \) is unstable (source).

(iii)
If \(\vert \lambda_{1} \vert <1\) and \(\vert \lambda _{2} \vert >1\) (or \(\vert \lambda_{1} \vert >1\) and \(\vert \lambda_{2} \vert <1\)), then the equilibrium point of \(M ( x^{\ast},y^{\ast} ) \) is unstable (saddle).

(iv)
If \(\vert \lambda_{1} \vert =1\) or \(\vert \lambda _{2} \vert =1\), then the equilibrium point of \(M ( x^{\ast },y^{\ast } ) \) is called nonhyperbolic.
Lemma 4.2
Let \(F ( \lambda ) =\lambda^{2}\operatorname{Tr}\lambda+\mathrm{Det}\). Suppose that \(F ( 1 ) >0\), \(\lambda_{1}\), \(\lambda_{2}\) are the two roots of \(F ( \lambda ) =0\). Then

(i)
\(\vert \lambda_{1} \vert <1\) and \(\vert \lambda _{2} \vert <1\) if and only if \(F ( 1 ) >0\) and \(\mathrm{Det}<1\);

(ii)
\(\vert \lambda_{1} \vert <1\) and \(\vert \lambda _{2} \vert >1\) or (\(\vert \lambda_{1} \vert >1\) and \(\vert \lambda_{2} \vert <1 \)) if and only if \(F ( 1 ) <0\);

(iii)
\(\vert \lambda_{1} \vert >1\) and \(\vert \lambda _{2} \vert >1\) if and only if \(F ( 1 ) >0\) and \(\mathrm{Det}>1\);

(iv)
\(\lambda_{1}=1\) and \(\lambda_{2}\neq1\) if and only if \(F ( 1 ) =0\) and \(\mathrm{Tr}\neq0,2\);

(v)
\(\lambda_{1}\) and \(\lambda_{2}\) are complex and \(\vert \lambda _{1} \vert = \vert \lambda_{2} \vert \) if and only if \(\mathrm{Tr}^{2}4\mathrm{Det}<0\) and \(\mathrm{Det}=1\).
Based on Lemmas 4.1 and 4.2, the following results can be achieved.
Theorem 4.3
If \(\Re_{0}<1\), then the freeequilibrium point \(E_{0}\) has at least four different topological types for all its values of parameters

(i)
\(E_{0}\) is a sink if \(0< h<\min \{ \sqrt[\alpha]{\frac{2\Gamma (1+\alpha)}{\mu}},\sqrt[\alpha]{\frac{2\Gamma(1+\alpha)}{ ( \mu +r ) ( 1\Re_{0} ) }} \} \).

(ii)
\(E_{0}\) is a source if \(h>\max \{ \sqrt[\alpha]{\frac{2\Gamma (1+\alpha)}{\mu}},\sqrt[\alpha]{\frac{2\Gamma(1+\alpha)}{ ( \mu +r ) ( 1\Re_{0} ) }} \} \).

(iii)
\(E_{0}\) is a saddle if \(\sqrt[\alpha]{\frac{2\Gamma(1+\alpha )}{ ( \mu+r ) ( 1\Re_{0} ) }}< h<\sqrt[\alpha]{\frac {2\Gamma(1+\alpha)}{\mu}}\) or \(\sqrt[\alpha]{\frac{2\Gamma(1+\alpha )}{\mu}}< h<\sqrt[\alpha]{\frac{2\Gamma(1+\alpha)}{ ( \mu+r ) ( 1\Re_{0} ) }}\).

(iv)
\(E_{0}\) is nonhyperbolic if \(h=\sqrt[\alpha]{\frac{2\Gamma (1+\alpha )}{\mu}}\) or \(h=\sqrt[\alpha]{\frac{2\Gamma(1+\alpha)}{ ( \mu +r ) ( 1\Re_{0} ) }}\).
Proof
The Jacobian matrix of \(E_{0}\) is
The eigenvalues of \(J ( E_{0} ) \) are \(\lambda _{1}=1\frac{h^{\alpha}\mu}{\Gamma ( 1+\alpha ) }\) and \(\lambda _{2}=1\frac{h^{\alpha}(\mu+r) ( 1\Re_{0} ) }{\Gamma ( 1+\alpha ) }\) where \(0<\alpha\leqslant1\) and h, \(\frac{h^{\alpha}}{\Gamma ( 1+\alpha ) }>0\). Hence applying the stability conditions using Lemma 4.1 the results (i)–(iv) can be achieved. □
Theorem 4.4
If \(\Re_{0}>1\), we have

(i)
\(E_{1}\) is asymptotically stable (sink) if one of the following conditions holds:

(i.1)
\(\Delta\geqslant0\) and \(0< h< h_{1}\).

(i.2)
\(\Delta<0\) and \(0< h< h_{2}\).

(i.1)

(ii)
\(E_{1}\) is unstable (source) if one of the following conditions holds:

(ii.1)
\(\Delta\geqslant0\) and \(h>h_{3}\).

(ii.2)
\(\Delta<0\) and \(h>h_{2}\).

(ii.1)

(iii)
\(E_{1}\) is unstable (saddle) if \(\Delta\geqslant0\) and \(h_{1}< h< h_{3}\).

(iv)
\(E_{1}\) is nonhyperbolic if one of the following conditions holds:

(iv.1)
\(\Delta\geqslant0\) and \(h=h_{1}\) or \(h_{3}\),

(iv.2)
\(\Delta<0\) and \(h=h_{2}\),
where
$$\begin{gathered} h_{1}=\sqrt[\alpha]{\frac{ ( \mu\Re_{0}\sqrt{\Delta} ) \Gamma(1+\alpha)}{\mu ( \mu+r ) ( \Re_{0}1 ) }}, \\ h_{2}=\sqrt[\alpha]{\frac{\Re_{0}\Gamma(1+\alpha)}{ ( \mu+r ) ( \Re_{0}1 ) }}, \\ h_{3}=\sqrt[\alpha]{\frac{ ( \mu\Re_{0}+\sqrt{\Delta} ) \Gamma(1+\alpha)}{\mu ( \mu+r ) ( \Re_{0}1 ) }},\end{gathered} $$and
$$\Delta= \bigl[ \mu ( \Re_{0}2 ) \bigr] ^{2}4\mu r ( \Re_{0}1 ) .$$ 
(iv.1)
Proof
The Jacobian matrix of \(E_{1}\) can be written as
The characteristic equation of \(J ( E_{1} ) \) has the form
where
and
Then the characteristic equation \(J ( E_{1} ) \) has two eigenvalues, which are
By applying Lemmas 4.1, 4.2 and the Jury conditions [45], the stability conditions (i)–(iv) can be achieved. □
From the above analysis if the statement (iv.1) of Theorem 4.4 holds, then one of the eigenvalues of \(J ( E_{1} ) \) is −1 and the other is neither 1 nor −1. The statement (iv.1) can be reformulated as follows:
where
and
If the parameter h varies in the neighborhood of \(h_{1}\) and \(( \alpha,h,\beta,\Lambda,\mu,r ) \in\Omega_{1}\) or \(h_{3}\) and \(( \alpha,h,\beta,\Lambda,\mu,r ) \in\Omega_{2}\), the system (3.7) may undergo a flip bifurcation of equilibrium \(E_{1}\).
When the statement (iv.2) of Theorem 4.4 holds, then the two eigenvalues of \(J ( E_{1} ) \) are a pair of conjugate complex numbers and the modules of each of them equals 1. The statement (iv.2) can be reformulated as follows:
where
If the parameter h varies in the neighborhood of \(h_{2}\) and \(( \alpha,h,\beta,\Lambda,\mu,r ) \in\Omega_{3}\), the system (3.7) may undergo a Neimark–Sacker bifurcation of equilibrium \(E_{1}\).
5 Flip bifurcation analysis
Flip bifurcation of the equilibrium point \(E_{1}\) when parameters \(( \alpha,h,\beta,\Lambda,\mu,\gamma ) \) vary in the small neighborhood of \(\Omega_{1}\) or \(\Omega_{2}\) is discussed in this section. Let \(A_{1}=\frac{h^{\alpha}}{\Gamma ( 1+\alpha ) }\) and \(A_{\ast}\) be a perturbation of bifurcation parameter, then a perturbed form of model (3.7) can be formulated as follows:
We translate \(E_{1} ( S^{\ast},I^{\ast} ) \) to the origin by using transformations \(X_{n}=S_{n}S^{\ast}\) and \(Y_{n}=I_{n}I^{\ast}\). Then (5.1) can reformulated as follows:
where
and \(A=A_{1}\).
Let \(T_{1}=\bigl( {\scriptsize\begin{matrix}{}a_{11} & a_{12}\cr a_{21} & a_{22}\end{matrix}} \bigr) \), then the generalized eigenvectors of \(T_{1}\) corresponding to the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) where \(\lambda_{1}=1\) and \(\vert \lambda_{2} \vert \neq1\) are \(\bigl( {\scriptsize\begin{matrix}{} K_{1}\cr K_{2}\end{matrix}} \bigr) =K_{2}\bigl( {\scriptsize\begin{matrix}{} \frac{a_{12}}{1+a_{11}}\cr 1 \end{matrix}} \bigr) \) and \(\bigl( {\scriptsize\begin{matrix}{} K_{3}\cr K_{4}\end{matrix}} \bigr) =K_{4}\bigl( {\scriptsize\begin{matrix}{} \frac{a_{12}}{a_{11}\lambda_{2}}\cr 1 \end{matrix}} \bigr) \), respectively. Here, we choose \(K_{2}= ( 1+a_{11} ) \) and \(K_{4}= ( a_{11}\lambda_{2} ) \). Then we have an invertible matrix \(T_{2}=\bigl( {\scriptsize\begin{matrix}{} K_{1} & K_{3}\cr K_{2} & K_{4}\end{matrix}} \bigr) =\bigl( {\scriptsize\begin{matrix}{} a_{12} & a_{12}\cr 1a_{11} & \lambda_{2}a_{11}\end{matrix}} \bigr) \). Consider the following transformation:
Taking \(T_{2}^{1}\) on both sides of (5.2), we obtain
where
and
Now, the center manifold \(W^{c} ( 0,0,0 ) \) of (5.3) at the fixed point \(( 0,0 ) \) in a small neighborhood of \(A_{\ast}=0\) can be formulated. Hence, based on the center manifold theorem, we know there exists a center manifold
for \(u_{n}\), \(A_{\ast}\) sufficiently small. We assume a center manifold of the form
which must satisfy
By equating coefficients of like powers in (5.4) to zero, we obtain
Therefore, the map f, which is model (3.7) restricted to the center manifold \(W^{c} ( 0,0,0 ) \) takes the form
where
According to the flip bifurcation, the discriminatory quantities \(\chi _{1}\) and \(\chi_{1}\) are given by
Thus, \(\chi_{1}=\varphi_{1}\) and \(\chi_{2}=\varphi_{5}+\varphi_{3}^{2}\). Therefore according to flip bifurcation conditions in [46], the following theorem can be stated.
Theorem 5.1
If \(\chi_{2}\neq0\), and the parameter \(A_{\ast}\) alters in the limiting region of the point \(( 0,0 ) \), then the system (5.1) passes through a flip bifurcation at the point \(E_{1} ( S^{\ast},I^{\ast} ) \). Further, the period2 points that bifurcate from the fixed point \(E_{1} ( S^{\ast},I^{\ast} ) \) are stable if \(\chi_{2}>0\) and unstable if \(\chi_{2}<0\).
6 Neimark–Sacker bifurcation
A Neimark–Sacker bifurcation of the equilibrium point \(E_{1}\) occurs when parameters \(( \alpha,h,\beta,\Lambda,\mu,\gamma ) \) vary in the small neighborhood of \(\Omega_{3}\). A perturbation form of model (3.7) can be written as follows:
where \(\vert A^{\ast} \vert \ll 1\) is a limited perturbation parameter. Let \(X_{n}=S_{n}S^{\ast}\), \(Y_{n}=I_{n}I^{\ast}\), then the fixed point \(E_{1} ( S^{\ast},I^{\ast} ) \) to \(( 0,0 ) \) can be retranslated and (6.1) can be reformulated as follows:
where
and \(A=A_{2}\).
The characteristic equation associated with the linearization system of model (6.2) at \(( 0,0 ) \) is
where
Since the parameters \(( \alpha,h,\beta,\Lambda,\mu,r ) \in \Omega_{3}\) and \(A^{\ast}\) varies in a small neighborhood of \(A^{\ast}=0\), and the roots of (6.3) are pair of complex conjugate numbers \(\omega_{1}\) and \(\omega _{2}\) denoted by
we have
In addition, it is required that, when \(A^{\ast }=0\), \(\overline{\omega}^{n},\omega^{n}\neq1\) (\(n=1,2,3,4 \)), which is equivalent to \(p ( 0 ) \neq2,0,1,2\). Since \(p^{2}(0)4q(0)<0\) and \(q(0)=1\), we have \(p^{2}(0)<4\); then \(p(0)\neq\pm2\). It is only required that \(p(0)\neq0,1\), which leads to
Therefore, the eigenvalues \(\omega_{1,2}\) of fixed point \(( 0,0 ) \) of (6.2) does not lie on the intersection of the unit circle with the coordinate axes when \(A^{\ast}=0\).
Next, the normal form of model (6.2) when \(A^{\ast}=0\) is studied. Let \(\theta=\operatorname{Re} ( \omega_{1,2} ) \), \(\eta=\operatorname {Im} ( \omega_{1,2} ) \) and
Consider the translation below
Taking \(T^{1}\) on both sides of (6.2), we obtain
where
Now, we obtain
According to the Neimark–Sacker bifurcation, the discriminatory quantity ℏ is given by
where
From the above analysis and the theorem in [47], Theorem 6.1 can be stated.
Theorem 6.1
If conditions (6.4) and (6.10) hold, then the system (3.7) undergoes Neimark–Sacker bifurcation at the positive fixed point \(E_{1} ( S^{\ast },I^{\ast} ) \) when the parameter \(A^{\ast}\) varies in the small neighborhood of \(A_{2}\). Furthermore if (\(\hslash<0\), \(\hslash >0 \)) then (an attracting, a repelling) invariant closed curve bifurcates from the fixed point \(E_{1} ( S^{\ast},I^{\ast} ) \) for (\(A_{2}>A^{\ast}\), \(A_{2}< A^{\ast}\)), respectively.
7 Numerical examples
This section shows the bifurcation diagrams, phase portraits and maximum Lyapunov exponents for the model (3.7) to confirm the above theoretical analysis and to illustrate the complex dynamics of our model using numerical continuation. Bifurcation occurs when the stability of an equilibrium point changes [48]. In general, the dynamics of a discrete SI model with integerorder has been examined by Hu et al. [42]. As discussed earlier in Section 1, this paper focuses on varying the time step size parameter h and the fractionalorder parameter α in the model (3.7), which can be seen as extension of the corresponding results given in [29, 42]. Based on the previous analysis, the parameters of the model (3.7) can be examined in the following two cases.
Case 1. Varying h in range \(2\leqslant h\leqslant2.85\) and fixing \(\Lambda=3.5\), \(\mu=0.145\), \(r=0.12\), \(\beta=0.09\), \(\alpha=0.99\) with initial conditions \(( S_{0},I_{0} ) = ( 3.1,7.8 ) \).
Case 2. Varying α in range \(0.8\leqslant\alpha\leqslant0.99\) and fixing \(\Lambda=1.5\), \(\mu=0.2\), \(r=0.3\), \(\beta=0.1\), \(h=8.1\), \(( S_{0},I_{0} ) = ( 5.1,0.9 ) \).
In Case 1, the basic reproduction number \(\Re_{0}=8.198>1\), thus the model (3.7) has one positive endemic equilibrium point \(E_{1} ( 2.944,11.596 ) \). Since \(\Delta=0.307>0\) and \(h_{1}=2.305\), according to Theorem 4.4, \(E_{1}\) is asymptotically stable (sink) when \(h<2.305\). When \(h=h_{1}\), flip bifurcation emerges from the equilibrium point \(E_{1}\) with \(\chi_{1}=0.871\), \(\chi_{2}=0.015>0\) and \(( \alpha,h,\beta,\Lambda,\mu,r ) = ( 0.99,2.305,0.09,3.5,0.145,0.12 ) \in\Omega_{1}\). The occurrence of these bifurcations is illustrated in Figure 1(a)–(c). These figures show that \(E_{1}\) is stable when \(h<2.305\) and loses its stability through the flip bifurcation when \(h=2.305\). Period2, 4, 8, 16 orbits appear as h increases in the range when \(h\in ( 2.305,2.83 ) \). The phase portraits for \(h\in ( 2.305,2.83 ) \) are plotted in Figure 2(a)–(d) to illustrate these observations further. The emergence of period2, 4, 8 and 16 orbits are observed when \(h=2.5,2.75,2.8\) and 2.82 in Figure 2(a)–(d). Some interesting phenomena are also seen when h increases further: for instance, when \(h=2.9\) (Figure 2(e)), period12 orbit appears. The occurrence of chaotic regions is also observed in Figure 1(a)–(c): these phenomena can be illustrated by the phase portrait in Figure 2(f) (e.g. when \(h=2.95\)). Maximal Lyapunov exponents (LEs) are computed in Figure 3 corresponding to observations in Figure 1(a)–(c). It is observed that some LE values are positive and some are negative, so there exist stable fixed points or stable period windows in the chaotic regions. Generally, a positive LE is considered to be one of the characteristics which imply the existence of chaos (e.g. when \(h>2.83\)).
In Case 2, the basic reproduction number \(\Re_{0}=1.5>1\), thus, model (3.7) has one positive endemic equilibrium point \(E_{1} ( 5,1 ) \). Since \(\Delta=0.11<0\), according to Theorem 4.4 \(E_{1}\) is asymptotically stable (sink) when \(\alpha<0.826\) and unstable when \(\alpha>0.826\). The Neimark–Sacker (NS) bifurcation emerges from \(E_{1}\) at \(\alpha =0.826\) with \(\hslash=0.778<0\) and \(( \alpha,h,\beta,\Lambda,\mu,r ) = ( 0.826,8.1,0.1,1.5,0.2,0.3 ) \in\Omega_{3}\). The occurrence of an NS bifurcation is illustrated in Figure 4(a)–(c). Figure 4 shows that \(E_{1}\) is stable for \(\alpha<0.826\) and loses its stability through an NS bifurcation at \(\alpha=0.826\). Attracting invariant circle appears as fractionalorder parameter increases in the range of \(\alpha\in ( 0.826,0.87 ) \). The phase portraits for various αvalues corresponding to Figure 4 are plotted in Figure 5(a)–(i) to illustrate these observations. Furthermore, the quasiperiodic orbits (\(\alpha=0.913\)) and periodic13 orbits (\(\alpha=0.921\)) are observed within the chaotic regions in Figure 5(f) and (g), respectively. Attracting chaotic sets are also seen when α increases further and these observations are plotted in Figure 5(h)–(i). Maximal Lyapunov exponents corresponding to observations in Figure 4(a)–(c) are computed in Figure 6. It is observed that some LE values are positive and some are negative. So there exist stable fixed points or stable period windows in the chaotic regions (e.g. when \(\alpha>0.921\)).
8 Discussion and conclusion
A new discretetime SI epidemic model has been discussed in this paper. Such a discretetime model is obtained from the discretization of the fractionaltime SI model. The discretization process provides crucial terms such as h (time step parameter) and α (fractionalorder parameter), which are then varied in order to describe the dynamical behaviors of the model. As h and α are varied, the model exhibits several complicated dynamical behaviors including the emergence of flip and NS bifurcations, period2, 4, 8, 12, 13, 16 orbits, quasiperiodic orbits, attracting invariant circle and chaotic sets. Analytically, necessary and sufficient conditions on the parameters for the occurrence of flip and NS bifurcations are derived. Moreover, numerical continuation is carried out to illustrate the validity of the analytical results and it is observed that both numerical and analytical findings are in good agreement. In conclusion, the proposed fractionaltime SI model can engender more complex dynamical behaviors than its continuous model counterpart. Studying the dynamical behaviors of the full avian–human influenza SIR epidemic model described by a discretetime with fractionalorder version will be our future work.
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The authors would like to thank the editor and the referees for their helpful comments and suggestions. The authors acknowledge financial support from FRGS grant 203/PMATHS/6711570.
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The main idea of this paper was proposed by MAMA. The manuscript prepared initially and all steps of the proof performed by MAMA, AII, FAA, and MHM. All authors read and approved the final manuscript.
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Abdelaziz, M.A.M., Ismail, A.I., Abdullah, F.A. et al. Bifurcations and chaos in a discrete SI epidemic model with fractional order. Adv Differ Equ 2018, 44 (2018). https://doi.org/10.1186/s1366201814816
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DOI: https://doi.org/10.1186/s1366201814816
Keywords
 Discrete SI epidemic model with fractional order
 Center manifold theorem
 Flip bifurcation
 Neimark–Sacker bifurcation
 Chaos