Bifurcation and Chaotic Behavior of a Discrete-Time SIS Model

The discrete-time epidemic model is investigated, which is obtained using the Euler method. It is verified that there exist some dynamical behaviors in this model, such as transcritical bifurcation, flip bifurcation, Hopf bifurcation, and chaos. The numerical simulations, including bifurcation diagrams and computation of Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behaviors.


Introduction
Epidemic models have been widely used in different forms for studying epidemiological processes such as the spread of HIV [1], SARS [2], and influenza [3].It is well known that dynamical systems with simple dynamical behavior in the constant parameter case display very complex behaviors including chaos when they are periodically perturbed [4,5].The continuous-time epidemic models have been widely investigated in many articles (e.g., [6][7][8][9][10] and the references cited therein).In recent years, we have found that more attention is paid to the discrete-time epidemic models (see [11][12][13][14][15] and the references cited therein).The reasons are as follows: first, difference models are more realistic than continuous differential ones because the epidemic statistics are compiled from given time intervals and are discontinuous.Second, the discrete-time models can provide natural simulators for the continuous cases.One can thus not only study the behaviors of the continuous-time model with good accuracy, but also assess the effect of lager time steps.At last, the use of discrete -time models makes it possible to use the entire arsenal of methods recently developed for the study of mappings and lattice equations, either from the integrability and/or chaos points of view.
On the other hand, daily treatments are frequently done for some infections, such as the group of those being responsible for the common cold, which do not confer any long lasting immunity.Such infections do not have a recovered state and individuals become susceptible again after infection.
For such reasons, according to [16], we firstly consider the SIS epidemic model with nonlinear incidence rate: where  denotes the susceptible population,  is the infected population, and  is the intrinsic birth rate constant.,  are the carrying capacity and the infection rate, respectively. ≥ 0 is the recovery rate constant (1/ is the average infective time).Let We obtain the following system analogous to (1): where Applying Euler scheme to (3), we obtain the following equation: This paper is organized as follows.In Section 2, we give sufficient conditions of existence for transcritical bifurcation, flip bifurcation, and Hopf bifurcation.In Section 3, a series of numerical simulations show that there are bifurcation and chaos in the discrete-time epidemic model.Finally, we give remarks to conclude this paper in Section 4.
Proof.The Jacobian matrix of (5) at  1 takes the form ) .
( 1 ) has eigenvalues equation (5) becomes where By the following transformation: equation ( 8) becomes where Then, we can consider which must satisfy Thus, we have And ( 8) is restricted to the center manifold, which is given by Since system (5) undergoes a transcritical bifurcation at  1 .This proves the theorem.Theorem 2. Equation (5) undergoes a flip bifurcation at  1 when  0 = 2,  1 ̸ =  0 .Furthermore, the stable periodic-2 point bifurcates from this fixed point.
system (5) becomes )   +  1 (  ,   ,   ) , where By the following transformation: equation ( 19) becomes where Then, we can consider Thus, we have   = 0,  = 1,2, 3. We obtain the center manifold as follows: And ( 19) is restricted to the center manifold, which is given by Direct calculations show that (27) Hence, system (5) undergoes a flip bifurcation at  1 .This completes the proof.
The positive fixed point is so important to the biological system that people usually are very interested in it.We will next pay attention to the only positive fixed point  2 of (5).Theorem 3. Equation (5) undergoes a transcritical bifurcation at  2 when  1 =  0 ,  1 ̸ = 2, and Δ > 0, where Since the analysis is similar to the case at  1 , the previously mentioned proofs are omitted.
We next give the condition of existence of Hopf bifurcation by using the hopf bifurcation theorem in [17].

Numerical Simulations
With development of scientific computation, computer becomes a powerful tool to study nonlinear systems, especially for a system without explicit solution.It not only is able to explore new complex dynamical behaviors, for example, periodic orbits and chaos in different regions [16,19], but also can do rigorous analysis by combining modern dynamical systems theory and reliable computation, for example, topological horseshoes [20][21][22][23].In this section, we use the bifurcation diagrams, Lyapunov exponents, and phase portraits to illustrate the previous analytic results and find new dynamics of the model (5) as the parameters vary.The attractors of ( 5) are also given by using the method in [20].
For case (I).The bifurcation diagram of ( 5) in  0 -  and  0 -  space for  1 = 0.9,  3 = 0.01 is given in Figures 1(a) and 1(b) to show the dynamical changes of susceptible and infective, respectively, as  0 varies.There is obvious phenomenon of bifurcation when we select the stepsize to be 10 −4 .The spectrum of Lyapunov exponents of the system (5) with respect to parameter  1 is given in Figure 1(c).Moreover, we can see that the orbit with initial values (1, 1) approaches to the stable fixed point  2 for  0 < 1.8 approximately, and Hopf bifurcation occurs at  0 ≈ 1.8.When increased to  0 ≈ 2.05, (5) becomes stable.In Figures 1(b) and 1(c), we observe the period -4,8 windows within the chaotic regions and boundary crisis at  0 ≈ 2.05.For  0 ∈ (1.95, 2.05) the maximum Lyapunov exponents are positive which correspond to chaotic region.To well see the dynamics, the attractor in the system (5) and time series of   and   are given in Figures 1(d) and 1(e), respectively.
For case (II).Figure 2(a) is the bifurcation diagram of (5) for   and   and depicts that there are period -4,5 windows within the chaotic regions and boundary crisis at  1 ≈ 1. Figure 2(b) shows the spectrum of Lyapunov exponents of the system (5) with respect to parameter  1 .For  1 ∈ (0.85, 0.95) the maximum Lyapunov exponents are positive which correspond to chaotic region.
For case (III).Figures 3(a) and 3(b) are the bifurcation diagrams of (5) for   and   , respectively, and depict that there are period -2,4 windows within the chaotic regions and boundary crisis at  2 = 0.5.Figure 3(c) shows the spectrum of Lyapunov exponents of (5) with respect to parameter  2 .Figure 3(d) is the attractor of (5) with  2 = 0.46.

Conclusion
In this paper, we investigate the behaviors of a discrete-time SIS epidemic model with nonlinear incidence rate, and find many complex and new interesting dynamical phenomena.Without the recovery rate of infectious hosts, (1) becomes the SI model (see [16]).Our theoretical analysis and numerical simulations have demonstrated that the model exhibits the variety of dynamical behaviors, which includes that the discrete epidemic model undergoes transcritical bifurcation,     flip bifurcation, Hopf bifurcation, and chaos.Furthermore, chaos can cause the population to run a higher risk of extinction due to the unpredictability [24,25].Thus, how to control chaos in the epidemic model is very important, which needs further consideration.