Bifurcation and Chaotic Behavior of a Discrete-Time SIS Model

Abstract

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.

1. 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., [6–10] and the references cited therein). In recent years, we have found that more attention is paid to the discrete-time epidemic models (see [11–15] 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: 𝑑𝑆𝑆𝑑𝑡=𝑏𝑆1−𝐾−𝛽𝑆²𝐼+𝛾𝐼,𝑑𝐼𝑑𝑡=𝛽𝑆²𝐼−𝑑𝐼−𝛾𝐼, (1) where 𝑆 denotes the susceptible population, 𝐼 is the infected population, and 𝑏 is the intrinsic birth rate constant. 𝐾, 𝛽 are the carrying capacity and the infection rate, respectively. 𝛾≥0 is the recovery rate constant (1/𝛾 is the average infective time).

Let 𝑆=𝑑+𝛾𝛽𝑥,𝐼=𝑑+𝛾𝛽𝑦,𝑑𝜏=𝑑+𝛾𝑑𝑡. (2) We obtain the following system analogous to (1): 𝑑𝑥𝑑𝜏=𝑎₀𝑥−𝑎₁𝑥²−𝑥²𝑦+𝑎₂𝑦,𝑑𝑦𝑑𝜏=𝑥²𝑦−𝑦, (3) where 𝑎₀=𝑏𝑑+𝛾,𝑎₁=𝑏𝐾𝛽𝑑+𝛾,𝑎₂=𝛾.𝑑+𝛾 (4) Applying Euler scheme to (3), we obtain the following equation: 𝑥_𝑛+1=𝑎₀+1𝑥_𝑛−𝑎₁𝑥²_𝑛−𝑥²_𝑛𝑦_𝑛+𝑎₂𝑦_𝑛,𝑦_𝑛+1=𝑥²_𝑛𝑦_𝑛. (5)

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.

2. Bifurcations

It is easy to visualize that system (5) has three fixed points 𝑃₀(0,0), 𝑃₁(𝑎₀/𝑎₁,0), and 𝑃₂(1,(𝑎₀−𝑎₁)/(1−𝑎₂)) when 𝑎₁, 𝑎₀ are fixed. We can see that the fixed point 𝑃₀(0,0) is a saddle. In the following paper, we focus on investigating the bifurcations of 𝑃₁, 𝑃₂.

Theorem 1. If 𝑎₀=𝑎₁ and 𝑎₀≠2, (5) undergo a transcritical bifurcation at 𝑃₁. Furthermore, the system has three fixed points when 𝑎₀>𝑎₁ and has two fixed points when 𝑎₀≤𝑎₁.

Proof. The Jacobian matrix of (5) at 𝑃₁ takes the form 𝐽𝑃₁=1−𝑎₀𝑎₂−𝑎²₀𝑎²₁0𝑎²₀𝑎²₁. (6)𝐽(𝑃₁) has eigenvalues 𝜆₁=1−𝑎₀, 𝜆₂=𝑎²₀/𝑎²₁. And 𝑎₀≠2 implies that |𝜆₁|≠1.
Let 𝜉_𝑛=𝑥_𝑛−𝑎₀𝑎₁,𝜂_𝑛=𝑦_𝑛,𝑐_𝑛=𝑎₀−𝑎₁, (7) equation (5) becomes 𝜉_𝑛+1=1−𝑎₁𝜉_𝑛+𝑎₂−1𝜂_𝑛+𝐹₁𝜉_𝑛,𝜂_𝑛,𝑐_𝑛,𝜂_𝑛+1=𝜂_𝑛+𝐹₂𝜉_𝑛,𝜂_𝑛,𝑐_𝑛,𝑐_𝑛+1=𝑐_𝑛, (8) where 𝐹₁𝜉_𝑛,𝜂_𝑛,𝑐_𝑛=−𝑎₁+𝜂_𝑛𝜉²_𝑛−2𝜉_𝑛𝜂_𝑛𝑐_𝑛𝑎₁−2𝜂_𝑛+𝑐_𝑛𝜉_𝑛−𝜂_𝑛𝑐²_𝑛𝑎²₁−2𝑐_𝑛𝜂_𝑛𝑎₁,𝐹₂𝜉_𝑛,𝜂_𝑛,𝑐_𝑛=2𝜉_𝑛𝜂_𝑛+2𝑐_𝑛𝜂_𝑛𝑎₁+𝜉²_𝑛𝜂_𝑛+𝜂_𝑛𝑐²_𝑛𝑎²₁+2𝜉_𝑛𝜂_𝑛𝑐_𝑛𝑎₁. (9) By the following transformation: 𝜉_𝑛𝜂_𝑛𝑐_𝑛=1𝑎110₁𝑎₂0𝑝−1001^𝑛𝑞_𝑛𝜎_𝑛, (10) equation (8) becomes 𝑝_𝑛+1𝑞_𝑛+1𝜎_𝑛+1=1−𝑎₁00𝑝010001^𝑛𝑞_𝑛𝜎_𝑛+𝑓₁𝑝_𝑛,𝑞_𝑛,𝜎_𝑛𝑓₂𝑝_𝑛,𝑞_𝑛,𝜎_𝑛0, (11) where 𝑓₁𝑝_𝑛,𝑞_𝑛,𝜎_𝑛=𝑎₁+𝑎₂−11−𝑎₂𝑞_𝑛−𝑎₁𝑝²_𝑛+2𝑎₁+𝑎₂−1𝜎_𝑛+1𝑞_𝑛+𝑞²_𝑛𝑎₁1−𝑎₂−𝜎_𝑛−2𝑎₁𝑞_𝑛𝑝_𝑛+2𝑎₁+𝑎₂−1𝑞³_𝑛𝑎₁1−𝑎₂+2𝑎₁+𝑎₂−1𝜎_𝑛+𝑎₁𝑎₁1−𝑎₂−𝑎₁𝑞²_𝑛+𝑎₁𝑎₂+2𝑎₂−2𝜎_𝑛𝑎₁1−𝑎₂+2𝑎₁+𝑎₂−1𝜎²_𝑛𝑎³₁1−𝑎₂𝑞_𝑛,𝑓₂𝑝_𝑛,𝑞_𝑛,𝜎_𝑛=𝑞³_𝑛+21+𝑝_𝑛+𝜎_𝑛𝑎₁𝑞²_𝑛+2𝑝_𝑛𝑞_𝑛+2𝑞_𝑛𝜎_𝑛𝑎₁+𝑎₁𝑝_𝑛+𝜎_𝑛²𝑞_𝑛𝑎²₁. (12) Then, we can consider 𝑝_𝑛=𝑔𝑞_𝑛,𝜎_𝑛=𝜀₁𝑞²_𝑛+𝜀₂𝑞_𝑛𝜎_𝑛+𝜀₃𝜎²_𝑛+𝑜𝑝_𝑛+𝑞_𝑛³, (13) which must satisfy 𝑔𝑞_𝑛+1,𝜎_𝑛+1=𝑔𝑞_𝑛+𝑓₂𝑔𝑞_𝑛,𝜎_𝑛,𝑞_𝑛,𝜎_𝑛,𝜎_𝑛+1=1−𝑎₁𝑔𝑞_𝑛,𝜎_𝑛+𝑓₁𝑔𝑞_𝑛,𝜎_𝑛,𝑞_𝑛,𝜎_𝑛. (14) Thus, we have 𝜀₁=𝑎₁𝑎₂+2𝑎₂−2𝑎₁1−𝑎₂,𝜀₂=𝑎₁𝑎₂+2𝑎₂−2𝑎²₁1−𝑎₂,𝜀₃=0. (15) And (8) is restricted to the center manifold, which is given by 𝑓∶𝑞_𝑛+1=2𝑎₁𝑎₂+2𝑎₂−2𝑎₁1−𝑎₂𝑞³_𝑛+𝑞²_𝑛𝜎_𝑛𝑎₁+𝑞³_𝑛+2𝑞²_𝑛+𝑞_𝑛+2𝑞²_𝑛𝜎_𝑛𝑎₁+2𝑞_𝑛𝜎_𝑛𝑎₁+𝑜𝑞_𝑛+𝜎_𝑛⁴. (16) Since 𝑓0,𝜎_𝑛=0,𝜕𝑓𝜕𝑞_(0,0)𝜕=1,²𝑓𝜕𝑞²_(0,0)𝜕=4,₂𝑓𝜕𝑞𝜕𝜎_(0,0)2=−𝑎₁≠0, (17) system (5) undergoes a transcritical bifurcation at 𝑃₁. This proves the theorem.

Theorem 2. Equation (5) undergoes a flip bifurcation at 𝑃₁ when 𝑎₀=2, 𝑎₁≠𝑎₀. Furthermore, the stable periodic-2 point bifurcates from this fixed point.

Proof. Let 𝜉_𝑛=𝑥_𝑛−𝑎₀𝑎₁,𝜌_𝑛=𝑦_𝑛,𝑐_𝑛=𝑎₀−2, (18) system (5) becomes 𝜉_𝑛+1=−𝜉_𝑛+𝑎₂−4𝑎²₁𝜌_𝑛+𝐹₁𝜉_𝑛,𝑐_𝑛,𝜌_𝑛,𝑐_𝑛+1=−𝑐_𝑛,𝜌_𝑛+1=4𝑎²₁𝜌_𝑛+𝐹₂𝜉_𝑛,𝑐_𝑛,𝜌_𝑛, (19) where 𝐹₁𝜉_𝑛,𝑐_𝑛,𝜌_𝑛=−𝑎₁+𝜌_𝑛𝜉²_𝑛−𝑐_𝑛+2𝑐_𝑛+2𝜌_𝑛𝑎₁𝜉_𝑛−4𝑐_𝑛+𝑐²_𝑛𝑎²₁𝜌_𝑛,𝐹₂𝜉_𝑛,𝑐_𝑛,𝜌_𝑛=𝜌_𝑛𝜉²_𝑛+4𝑐_𝑛+𝑐²_𝑛𝑎²₁𝜌_𝑛+2𝑐_𝑛+2𝜌_𝑛𝑎₁𝜉_𝑛. (20) By the following transformation: 𝜉_𝑛𝑐_𝑛𝜌_𝑛=𝑎10101000²¹+4𝑎₂𝑎²₁𝜙−4^𝑛𝑢_𝑛𝑣_𝑛, (21) equation (19) becomes 𝜙_𝑛+1𝑢_𝑛+1𝑣_𝑛+1=4−1000−1000𝑎²₁𝜙_𝑛𝑢_𝑛𝑣_𝑛+𝑓₁𝜙_𝑛,𝑢_𝑛,𝑣_𝑛0𝑓₂𝜙_𝑛,𝑢_𝑛,𝑣_𝑛, (22) where 𝑓₁𝜙_𝑛,𝑢_𝑛,𝑣_𝑛=𝑎²₁+𝑎₂𝑎²₁4−𝑎₂𝑎²₁𝑣_𝑛−𝑎₁𝜙²_𝑛+21+𝑎₂𝑎²₁𝑣²_𝑛4−𝑎₂𝑎²₁+2𝑎₁𝑢_𝑛−2+𝑎₂𝑎²₁+𝑎₂𝑢_𝑛+2𝑎₂𝑣_𝑛4−𝑎₂𝑎²₁−𝑢_𝑛𝜙_𝑛+𝑎²₁+𝑎₂𝑎²₁4−𝑎₂𝑎²₁𝑣³_𝑛+𝑎₁𝑎₂𝑎²₁+2𝑎₂𝑢_𝑛+4𝑎₂+2𝑢_𝑛4−𝑎₂𝑎²₁𝑣²_𝑛+1+𝑎₂𝑢²_𝑛𝑣_𝑛4−𝑎₂𝑎²₁+𝑎₂4+𝑎²₁𝑢_𝑛𝑣_𝑛4−𝑎₂𝑎²₁,𝑓₂𝜙_𝑛,𝑢_𝑛,𝑣_𝑛=𝑣³_𝑛+2𝜙_𝑛+𝑢_𝑛+4𝑎₁𝑣²_𝑛+𝑎₁𝜙_𝑛+𝑢_𝑛𝑎₁𝜙_𝑛+𝑢_𝑛+4𝑣_𝑛𝑎²₁. (23) Then, we can consider 𝑣_𝑛=𝑔(𝜙_𝑛,𝑢_𝑛)=𝜀₁𝜙²_𝑛+𝜀₂𝜙_𝑛𝑢_𝑛+𝜀₃𝑢²_𝑛+𝑜((|𝜙_𝑛|+|𝑢_𝑛|)³), which must satisfy 𝑔−𝜙_𝑛+𝑓₁𝜙_𝑛,𝑔𝜙_𝑛,𝑢_𝑛,𝑣_𝑛,𝑢_𝑛+1=4𝑎²₁𝑔𝜙_𝑛,𝑢_𝑛+𝑔³𝜙_𝑛,𝑢_𝑛+2𝜙_𝑛+𝑢_𝑛+4𝑎₁𝑔²𝜙_𝑛,𝑢_𝑛+𝑎₁𝜙_𝑛+𝑢_𝑛𝑎₁𝜙_𝑛+𝑢_𝑛+4𝑔𝜙_𝑛,𝑢_𝑛𝑎²₁. (24) Thus, we have 𝜀_𝑘=0, 𝑘=1,2,3. We obtain the center manifold as follows: 𝑣_𝑛=𝑔𝜙_𝑛,𝑢_𝑛=𝜀₄𝜙²_𝑛𝑢_𝑛+𝜀₅𝜙_𝑛𝑢²_𝑛+𝜀₆𝑢³_𝑛+𝑜𝜙_𝑛+𝑢_𝑛⁴. (25) And (19) is restricted to the center manifold, which is given by 𝑓∶𝜙_𝑛+1=−𝜙_𝑛+𝑓₁𝜙_𝑛,𝑔𝜙_𝑛,𝑢_𝑛,𝑣_𝑛. (26) Direct calculations show that 𝜕𝑓𝜕𝜕𝜎²𝑓𝜕𝑝²𝜕+2²𝑓𝜕𝑝𝜕𝜎_(0,0)1=−2,2𝜕²𝑓𝜕𝑝²²+13𝜕³𝑓𝜕𝑝³_(0,0)=𝑎²₁2. (27) Hence, system (5) undergoes a flip bifurcation at 𝑃₁. 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 𝑃₂ of (5).

Theorem 3. Equation (5) undergoes a transcritical bifurcation at 𝑃₂ when 𝑎₁=𝑎₀, 𝑎₁≠2, and Δ>0, where Δ=1+𝑎₂²𝑎²₀1−𝑎₂²+8𝑎₁−𝑎₀+4𝑎²₁𝑎²₂−4𝑎₂𝑎₂+11−𝑎₂². (28)

Theorem 4. If Δ>0, 𝑎₁≠𝑎₀, and 𝑎₀=(2−2𝑎₂−𝑎₁+3𝑎₁𝑎₂)/2𝑎₂, (5) undergoes a flip bifurcation at 𝑃₂.

Since the analysis is similar to the case at 𝑃₁, the previously mentioned proofs are omitted.

We next give the condition of existence of Hopf bifurcation by using the hopf bifurcation theorem in [17].

The characteristic equation of the Jacobian matrix 𝐽(𝑃₂) can be written as 𝜆²−𝑡₂𝜆+𝑑₂=0, where 𝑡₂=2+𝑎₀−2𝑎₁−2𝑎₀−𝑎₁1−𝑎₂,𝑑₂=𝑡₂−1+2𝑎₀−2𝑎₁. (29)

The eigenvalues of the Jacobian matrix of (5) at 𝑃₂ are 𝜆_1,2=(𝑡₂±𝑡²₂−4𝑑₂)/2. The eigenvalues 𝜆_1,2 are complex conjugates for Δ<0. We translate the fixed point 𝑃₂(1,(𝑎₀−𝑎₁)/(1−𝑎₂)) to the origin by 𝑥_𝑛=𝜑_𝑛−1, 𝑦_𝑛=𝛿_𝑛−((𝑎₀−𝑎₁)/(1−𝑎₂)), and the system (5) becomes 𝜑_𝑛+1=𝑡₂−1𝜑_𝑛+𝑎₂−1𝛿_𝑛−𝛿_𝑛+𝑎₀−𝑎₁𝑎₂1−𝑎₂𝜑²_𝑛−2𝜑_𝑛𝛿_𝑛,𝛿_𝑛+1=2𝑎₀−𝑎₁1−𝑎₂𝜑_𝑛+𝛿_𝑛+2𝜑_𝑛𝛿_𝑛+𝛿_𝑛+𝑎₀−𝑎₁1−𝑎₂𝜑²_𝑛. (30) The eigenvalues of the matrix associated with the linearized map (30) at fixed point (0,0) are complex conjugates which are written as 𝜆,𝜆=(𝑡₂±𝑖4𝑑₂−𝑡²₂)/2, where |𝜆|=1+3𝑎₀−4𝑎₁−2𝑎₀−𝑎₁1−𝑎₂. (31) Now assume that 3𝑎₂<1, and let 𝑎₀₀=(2𝑎₁(1−2𝑎₂))/(1−3𝑎₂). Then we have 𝑑(|𝜆|)𝑑𝑎_{0𝑎0=𝑎00}=1−3𝑎₂21−𝑎₂>0,𝜆𝑎₁₀𝜆𝑎=1,₀₀𝑎=1+¹𝑎₂−𝑎₁1−3𝑎₂±𝑖𝑎₁1−𝑎₂2−𝑎₁+𝑎₁𝑎₂−6𝑎₂1−3𝑎₂,𝜆^𝑗≠1,𝑗=1,2,3,4. (32) Let 𝑎T=²−102−𝑡₂2−4𝑑₂−𝑡²₂2,𝜑_𝑛𝛿_𝑛𝜙=T^𝑛𝜎_𝑛. (33) The system (30) becomes 𝜙_𝑛+1𝜎_𝑛+1=𝑡₂2−4𝑑₂−𝑡²₂24𝑑₂−𝑡²₂2𝑡₂2𝜙_𝑛𝜎_𝑛+𝑓₁𝜙_𝑛,𝜎_𝑛𝑓₂𝜙_𝑛,𝜎_𝑛, (34) where 𝑓₁𝜙_𝑛,𝜎_𝑛=𝑎₀+𝑎₀𝑎₂2−𝑎₁𝑎₂𝜙³_𝑛+𝑎₂−14𝑑₂−𝑡²₂2𝜎_𝑛+𝑎₂2𝑎₀−𝑎₁−𝑎₁𝑎₂𝑎₂𝜙−1^2𝑛+4𝑑₂−𝑡²₂𝜎_𝑛𝜙_𝑛,𝑓₂𝜙_𝑛,𝜎_𝑛=2−𝑡₂2𝑎₂−𝑡₂𝑎₂−124𝑑₂−𝑡²₂𝜙³_n+𝑎₂−1𝑎₂−𝑡₂2𝜙²_𝑛𝜎_𝑛−2−𝑡₂2−𝑡₂+𝑎₁𝑎₂−𝑎₀4𝑑₂−𝑡²₂𝜙²_𝑛−2𝑎₂−12−𝑡₂−𝑎₀+𝑎₁4𝑑₂−𝑡²₂𝜙²_𝑛+2𝑎₂−𝑡₂𝜎_𝑛𝜙_𝑛. (35) Notice that (34) is exactly on the center manifold in the form, in which the coefficient 𝑙 [18] is given by 𝑙=−Re[((1−2𝜆)𝜆²/(1−𝜆))𝑙₁₁𝑙₂₀]−(1/2)|𝑙₁₁|²−|𝑙₀₂|²+Re(𝜆𝑙₂₁), where 𝑙₁₁=14𝑓_1𝜙𝜙+𝑓_1𝜎𝜎+𝑓_2𝜙𝜙+𝑓_2𝜎𝜎𝑖=3𝑎₀𝑎₂−2𝑎₁𝑎²₂−𝑎₀4𝑎₂−1−444𝑑₂−𝑡²₂×4−2𝑡₂2−𝑡₂+𝑎₁𝑎₂−𝑎₀+𝑎₂−14−2𝑡₂−2𝑎₀+2𝑎₁𝑙𝑖,₂₀=18𝑓_1𝜙𝜙−𝑓_1𝜎𝜎+2𝑓_2𝜙𝜎+𝑓_2𝜙𝜙−𝑓_2𝜎𝜎−2𝑓_1𝜙𝜎𝑖=14𝑎₂−𝑎₁𝑎₂−2+2𝑎₀−184𝑑₂−𝑡²₂×4−2𝑡₂2−𝑡₂+𝑎₁𝑎₂−𝑎₀+𝑎₂×−14−2𝑡₂−2𝑎₀+2𝑎₁−8𝑑₂+2𝑡²₂𝑙𝑖,₀₂=18𝑓_1𝜙𝜙−𝑓_1𝜎𝜎−2𝑓_2𝜙𝜎+𝑓_2𝜙𝜙−𝑓_2𝜎𝜎+2𝑓_1𝜙𝜎𝑖=182𝑎₀−2𝑎₀𝑎₂−2−4𝑎₂+3𝑡₂+182𝑎₀−𝑎₁𝑎₂−2+𝑡₂+24𝑑₂−𝑡²₂𝑙𝑖,₂₁=1𝑓16^{1𝜙𝜙𝜙}+𝑓_{1𝜙𝜎𝜎}+𝑓_{2𝜙𝜙𝜎}+𝑓_{2𝜎𝜎𝜎}+𝑓_{2𝜙𝜙𝜙}+𝑓_{2𝜙𝜎𝜎}−𝑓_{1𝜙𝜙𝜎}−𝑓_{1𝜎𝜎𝜎}𝑖=181−𝑎₂3+𝑎₂−2𝑡₂+184𝑑₂−𝑡²₂6𝑎₂−3𝑡₂+3𝑎₂𝑡₂+8𝑑₂+𝑡²₂𝑖. (36) From the previous analysis, we have Theorem 5.

Theorem 5. System (5) undergoes a Hopf bifurcation at fixed point 𝑃₂, if 𝑙≠0, Δ<0 and 𝑎₁≠21−𝑎₂+2𝑎₂𝑎₁1+𝑎₃. (37)

3. 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 [20–23]. 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]. The bifurcation parameters are considered in the following three cases:(I)varying 𝑎₀ in range 1.6≤𝑎₀≤2.2 and fixing 𝑎₁=0.9, 𝑎₃=0.01;(II)varying 𝑎₁ in range 0.8≤𝑎₁≤1.2 and fixing 𝑎₀=2, 𝑎₃=0.01;(III)varying 𝑎₂ in range 0.3≤𝑎₂≤0.6 and fixing 𝑎₀=2, 𝑎₁=0.9.

For case (I). The bifurcation diagram of (5) in  𝑎₀-𝑥_𝑛 and 𝑎₀-𝑦_𝑛 space for 𝑎₁=0.9, 𝑎₃=0.01 is given in Figures 1(a) and 1(b) to show the dynamical changes of susceptible and infective, respectively, as  𝑎₀ 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 𝑎₁ is given in Figure 1(c).

(a)

(b)

(c)

(d)

(e)

Figure 1 

(a) Bifurcation diagram in (𝑎0,𝑥𝑛,𝑦𝑛) space. (b) The bifurcation diagram in 𝑎0-𝑥𝑛 plane and in 𝑎0-𝑦𝑛 plane. (c) Spectrum of Lyapunov exponents corresponding to (b). (d) The attractor of (5) with 𝑎0=2 (e) Time series of 𝑥𝑛 and 𝑦𝑛 of the same parameters.

Moreover, we can see that the orbit with initial values (1,1) approaches to the stable fixed point 𝑃₂ for 𝑎₀<1.8 approximately, and Hopf bifurcation occurs at 𝑎₀≈1.8. When increased to 𝑎₀≈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 𝑎₀≈2.05. For 𝑎₀∈(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 𝑥_𝑛 and 𝑦_𝑛 are given in Figures 1(d) and 1(e), respectively.

For case (II). Figure 2(a) is the bifurcation diagram of (5) for 𝑥_𝑛 and 𝑦_𝑛 and depicts that there are period -4,5 windows within the chaotic regions and boundary crisis at 𝑎₁≈1. Figure 2(b) shows the spectrum of Lyapunov exponents of the system (5) with respect to parameter 𝑎₁. For 𝑎₁∈(0.85,0.95) the maximum Lyapunov exponents are positive which correspond to chaotic region.

(a)

(b)

Figure 2 

For case (III). Figures 3(a) and 3(b) are the bifurcation diagrams of (5) for 𝑥_𝑛 and 𝑦_𝑛, respectively, and depict that there are period -2,4 windows within the chaotic regions and boundary crisis at 𝑎₂=0.5. Figure 3(c) shows the spectrum of Lyapunov exponents of (5) with respect to parameter 𝑎₂. Figure 3(d) is the attractor of (5) with 𝑎₂=0.46.

(a)

(b)

(c)

(d)

Figure 3 

4. 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.

Acknowledgments

The authors are very grateful to the reviewers for their valuable comments and suggestions. This work was supported by the Youth Science Foundations of Education Department of Hebei Province (nos. 2010233, 2011236).