Qualitative and Bifurcation Analysis of an SIR Epidemic Model with Saturated Treatment Function and Nonlinear Pulse Vaccination

An SIR epidemic model with saturated treatment function and nonlinear pulse vaccination is studied. The existence and stability of the disease-free periodic solution are investigated. The sufficient conditions for the persistence of the disease are obtained. The existence of the transcritical and flip bifurcations is considered bymeans of the bifurcation theory.The stability of epidemic periodic solutions is discussed. Furthermore, some numerical simulations are given to illustrate our results.


Introduction
The SIR epidemic models have attracted much attention in recent years.In most cases, ordinary differential equations are used to build SIR epidemic models [1][2][3][4][5].However, impulsive differential equations [6,7] are also suitable for the mathematical simulation of evolutionary processes in which the parameters state variables undergo relatively long periods of smooth variation followed by a short-term rapid change in their values.Many results have been obtained for SIR epidemic models described by impulsive differential equations [8][9][10][11][12][13].
In the classical epidemic models, it is usually assumed that the removal rate of the infective individuals is proportional to the number of the infective individuals, which implies that the medical resources such as drugs, vaccines, hospital beds, and isolation places are very sufficient for the infectious disease.However, in reality, every community or country has an appropriate or limited capacity for treatment and vaccination.
In order to investigate the effect of the limited capacity for treatment on the spread of infectious disease, Wang and Ruan [14] introduced a constant treatment in an SIR model which simulated a limited capacity for treatment.Further, Wang [15] modified the constant treatment to which meant that the treatment rate was proportional to the number of the infective individuals before the capacity of treatment was reached and then took its maximum value  0 .Recently, Zhou and Fan [16] introduced the following continually differentiable treatment function: where  ≥ 0 represents the maximal medical resources supplied per unit time and  > 0 is half-saturation constant, which measures the efficiency of the medical resource supply in the sense that if  is smaller, then the efficiency is higher.They investigated the following SIR model: where (), (), and () denote the numbers of susceptible, infective, and recovered individuals at time , respectively. is the recruitment rate of the population,  is the natural death rate of the population,  is the natural recovery rate, and  is the disease-related mortality.The incidence rate /(1 + ) is of saturated type and reflects the "psychological" effect or the inhibition effect [17].
In [16], the authors addressed some problems on system (4) such as the existence of endemic equilibria and backward bifurcation, the locally and globally asymptotic stability of the disease-free equilibrium and endemic equilibrium, and the existence of the Hopf bifurcation.
In addition to the treatment, vaccination is often restricted by limited medical resources.The vaccination success rate always has some saturation effect.That is, vaccination rate can be expressed as a saturation function as follows [18]: Here,  is the maximum pulse immunization rate. is the half-saturation constant, that is, the number of susceptible individuals when the vaccination rate is half the largest vaccination rate.They established the following SIR epidemic model: where  represents the total number of input population. (0 <  ≤ 1) is the proportion of input population without immunity.
In [18], the authors addressed some problems on system (6) such as the existence and stability of the disease-free periodic solution of system (6) and the existence of the transcritical bifurcations.
Motivated by [16,18], the following SIR epidemic model with saturated treatment function and nonlinear pulse vaccination is considered: Noticing that variable  just appears in the third and sixth equations of model (7), we only need to consider the following subsystem of model (7): The remaining part of this paper is organized as follows.In the next section, we discuss the existence and stability of the disease-free periodic solution of system (8).In Section 3, the persistence of the disease is considered.In Section 4, the existence of positive periodic solutions is discussed by using the bifurcation theory.We study the stability of the positive periodic solution of system (8) in Section 5.In Section 6, we consider the existence of flip bifurcations by means of the bifurcation theory.In Section 7, some numerical simulations are given to illustrate our results.Finally, some concluding remarks are given.

The Existence and Stability of the Disease-Free Periodic Solution
In this section, we investigate the existence of the diseasefree periodic solution of system (8).In this case, infectious individuals are entirely absent from the population permanently, that is, () = 0,  ≥ 0. System (8) yields Lemma 1 (see [18]).System ( 9) has a unique globally asymptotically stable periodic solution  * (), where According to Lemma 1, we obtain the following result.
Next, we will discuss the stability of the periodic solution ( * (), 0).The associated eigenvalues of the periodic solution ( * (), 0) are According to Floquet theory of impulsive differential equation, the periodic solution where  is defined in (11).Denote Theorem 3. If  0 < 1, then the disease-free periodic solution ( * (), 0) of system ( 8) is locally asymptotically stable.
Remark 4. In [16], the basic reproduction number of system (8) without the pulse vaccination is In this paper, the corresponding basic reproduction number of system (8) is We claim that  < /.Otherwise, we assume that  > /.By system (9), we find that  * () <  if  > /.In addition, However,  * () is a periodic solution with period .It is a contradiction.Thus,  < /.So  0 <  * 0 .Hence, the pulse vaccination strategy is beneficial.
Synthesizing Theorems 3 and 5, we obtain the following result.

The Persistence of the Disease
Theorem 7. If  0 > 1, then there exists a positive constant , such that for any positive solution () of system (8), lim inf →∞ () ≥ ; that is, the disease is uniformly strongly persistent.
From the second equation of system (8) Finally, if there exists  3 >  0 + 1 , such that () >  * for all  >  3 , then the same  <  * works as well as a lower estimate.
Note that  depends only on the fixed constants  * and ; thus we get strong uniform persistence.

The Existence of Transcritical Bifurcations
In this section, we will discuss the existence of transcritical bifurcations by means of the bifurcation theory.We let the half-saturation constant  be the bifurcation parameter.

Transcritical Bifurcation.
In this subsection, we discuss the existence of a transcritical bifurcation by means of map (57).
By the above analysis, we find that one of the eigenvalues of the fixed point (0, 0) is 1.An eigenvalue with 1 is associated with a transcritical bifurcation in map (57).Hence, (0, 0,  0 ) is a candidate for a transcritical bifurcation point in map (57).
Similar to the above analysis, we may prove that system (8) undergoes a subcritical bifurcation at  =  0 if  2 > 0. This completes the proof.
Lemma 10 (see [19]).Let   :  →  be a one-parameter family of map such that   0 has a fixed point  0 with eigenvalue −1.Assume the following conditions: Then there is a smooth curve of fixed points of   passing through ( 0 ,  0 ), the stability of which changes at ( 0 ,  0 ).
There is also a smooth curve  passing through ( 0 ,  0 ) so that  \ ( 0 ,  0 ) is a union of hyperbolic period-two orbits.
In (F2), the sign of  determines the stability and direction of bifurcation of the orbits of period two.If  is positive, the orbits are stable; if  is negative, they are unstable.

Numerical Simulation
In this section, we will give bifurcation diagrams and phase portraits of system (8) to illustrate the above theoretical analyses and find new interesting complex dynamical behaviors by using numerical simulations.The bifurcation parameters are considered in the following two cases.
The solution of system (8) from the initial point (15.5, 5.2) with  = 3 tends to the stable disease-free periodic solution when  increases (see Figure 1 The bifurcation diagram of system (8) with respect to  is presented in Figure 2. It is seen from the bifurcation diagram that the disease-free periodic solution is stable for  ∈ (0, 3.866) and unstable for  ∈ (3.866, +∞).A positive -periodic solution bifurcates from the disease-free periodic solution at  ≈ 3.866 through transcritical bifurcation.This positive -periodic solution is stable for  ∈ (3.866, 14.45) and unstable for  ∈ (14.45, +∞).A positive 2-periodic solution bifurcates from the positive -periodic solution at  ≈ 14.45 through flip bifurcation.
The phase space plots for different values of  are drawn in Figure 3. Figure 3(a) shows a -periodic solution in system (8) with  = 10.
The bifurcation diagram of system (8) with respect to  is presented in Figure 4. System (8) presents complicated dynamics in this case.From Figure 4, we can see that there exist the chaotic regions and period orbits as the parameter    varies.Figure 4 depicts that there are , 2, 3, 4 periodic windows.From Figure 5, we show the following behaviors: the period-windows within the chaotic regions, the inverse period-doubling bifurcation from 4-periodic orbits to chaos, and the inverse bifurcation from the 3periodic orbits to chaos.If we consider  as a parameter, then the bifurcation diagram of system (8) with  = 6 is presented in Figure 6.It is seen from the bifurcation diagram that there is a route from chaos to stable periodic solutions via a cascade of reverse period-doubling bifurcation.

Discussion
It is well known that one important strategy to control epidemic disease is vaccination.In this paper, an SIR epidemic model with saturated treatment function and nonlinear pulse vaccination is studied.By Remark 4, we may see that the basic reproduction number  * 0 of system (8) without the pulse vaccination in [16] is greater than the corresponding basic reproduction number  0 of system (8).Hence, pulse vaccination may reduce the basic reproduction number.
To control the disease, a strategy should reduce the basic reproduction number to below unity.Thus, the introduction of pulse vaccination is helpful in controlling the epidemic diseases.For the control of the epidemic diseases, chaos may cause the diseases to run a higher risk of outbreak due to the unpredictability.Thus, it is necessary to delay or eliminate chaos.It is well known that the flip bifurcations may lead to chaos.So the flip bifurcations should be controlled.We enrich the medical resources (i.e., decrease ) to prevent the flip bifurcations.According to Theorem 11, we may reduce  below  1 .Then, the flip bifurcations can be eliminated.In addition, the flip bifurcation will not occur if the stability of the epidemic periodic solutions is not changed.Thus, we may also eliminate the flip bifurcation using Theorem 9.This prevents disease outbreaks.In the following, we eradicate the disease by means of preventing the transcritical bifurcations.

( 1 )
Consider the following set of parameters:

Figure 5 :
Figure 5: Bifurcation diagrams of system (8) with respect to  in case (2) of Section 7.