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 by means of the bifurcation theory. The stability of epidemic periodic solutions is discussed. Furthermore, some numerical simulations are given to illustrate our results.
The SIR epidemic models have attracted much attention in recent years. In most cases, ordinary differential equations are used to build SIR epidemic models [1–5]. 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 [8–13].
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  introduced a constant treatment in an SIR modelwhich simulated a limited capacity for treatment. Further, Wang  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 . Recently, Zhou and Fan  introduced the following continually differentiable treatment function:where represents the maximal medical resources supplied per unit time and 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 is of saturated type and reflects the “psychological” effect or the inhibition effect .
In , 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 : 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. is the proportion of input population without immunity.
In , 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.
2. The Existence and Stability of the Disease-Free Periodic Solution
In this section, we investigate the existence of the disease-free periodic solution of system (8). In this case, infectious individuals are entirely absent from the population permanently, that is, System (8) yields
Lemma 1 (see ). System (9) has a unique globally asymptotically stable periodic solution , where
According to Lemma 1, we obtain the following result.
Theorem 2. System (8) has a disease-free periodic solution .
Next, we will discuss the stability of the periodic solution . The associated eigenvalues of the periodic solution are
According to Floquet theory of impulsive differential equation, the periodic solution is locally asymptotically stable if , that is to say, where is defined in (11). Denote
Theorem 3. If , then the disease-free periodic solution of system (8) is locally asymptotically stable.
Remark 4. In , 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 Hence, the pulse vaccination strategy is beneficial.
Next, we will prove the global attractivity of the disease-free periodic solution of system (8).
Theorem 5. If , then the disease-free periodic solution of system (8) is globally attractive.
Proof. If , then . Since , one can choose small enough such that Let . By system (7), we have Consider the following impulsive comparison system: By the comparison theorem of impulsive differential equation, we have and as . Hence, for given , we havefor all large enough. For simplification, we may assume (23) holds for all . It is obvious that, for all ,From the first and third equations of system (8), we obtain Consider the following impulsive comparison system: According to Lemma 1 and the comparison theorem of impulsive differential equation, we get and as . Hence, for and all large enoughFor simplification, we may suppose (27) holds for all .
By the second equation of system (8), we getwhich leads toTherefore, and as . From (28), we obtainSince is a periodic solution, there exists a constant such that, for all So Hence, as . Without loss of generality, we may assume that for
From the first and third equations of system (8), we have Consider the following impulsive comparison system: By the comparison theorem of impulsive differential equation and Lemma 1, we getwhereBy (34), it is easy to see that as . From (27) and (33), we obtain that, for any , there exists such that for .
Letting , we have , for large enough, which implies as . So the disease-free periodic solution of system (8) is globally attractive. The proof is completed.
Synthesizing Theorems 3 and 5, we obtain the following result.
Theorem 6. The disease-free periodic solution of system (8) is globally asymptotically stable if .
3. The Persistence of the Disease
Theorem 7. If , then there exists a positive constant , such that for any positive solution of system (8), ; that is, the disease is uniformly strongly persistent.
Proof. We consider a solution of system (8) and a constant Suppose there exists , such that . Let This time interval of length can be finite or infinite. Firstly, we prove that is finite when is appropriately chosen. Assume that We see that holds in the interval . Then, from the first and third equations of system (8), we have Consider the following impulsive comparison system:According to Lemma 1, we obtain that system (37) has a globally asymptotically stable positive periodic solution , where By the comparison principle, there exists such that, for any and Meanwhile, as .
From the second equation of system (8), we obtain for Let and integrating between pulses for yieldsThen we have Define Since as , then , as and If and are sufficiently small, then from , we obtain that Thus, as , which contradicts the boundedness of . Therefore, if is sufficiently small.
Let us fix these previous and , for which Then we discuss the following two possibilities: (1) for all large .(2) oscillates about for all large .If possibility () holds, then we complete our proof. Next, we will consider possibility (). From the oscillatory property, we can define an unbounded, monotone increasing sequence such that and Choose an arbitrary . Let , and similarly . Let We see that , and from the continuity of , we obtain that There exists such , because We claim that Otherwise, we assume that
From the second equation of system (8), we have , which givesFrom the assumption , we obtain Thus, However, from (40), we getBy (42) and (44), we get which is a contradiction.
Hence, and for Thus, in case (2), we can setFor any sufficiently large for which , we have , since we can choose such that for some .
Finally, if there exists , such that for all , 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.
4. 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.
4.1. The Poincaré Map
Suppose the disease-free periodic solution with the initial point and period passes through the points and at time and then jumps to the point due to vaccination pulse. Thus,
Consider another solution of system (8) with the initial point . This disturbed trajectory starting from the point reaches the point at time and then jumps to the point . Thus,
Denote and then . Let Then system (8) may be written asBy the Taylor expansion, we havewhereFor , letwhereFrom systems (48), (49), and (51), we getIt follows from system (51) thatUsing system (54), we obtainwhere
From system (55), the following Poincaré map is obtained:
4.2. Transcritical Bifurcation
In this subsection, we discuss the existence of a transcritical bifurcation by means of map (57).
The fixed point of map (57) corresponds to the disease-free periodic solution of system (8). The associated eigenvalues of the fixed point are given bywhere is defined in (11).
DenotewhereIf and , then
By the above analysis, we find that one of the eigenvalues of the fixed point is . An eigenvalue with is associated with a transcritical bifurcation in map (57). Hence, is a candidate for a transcritical bifurcation point in map (57).
Theorem 8. Assume that , where are shown in (59) and (60), respectively. (1)If , then a supercritical bifurcation occurs at in system (8). For some , system (8) has a positive periodic solution for (2)If , then a subcritical bifurcation occurs at in system (8). For some , system (8) has a positive periodic solution for
Proof. Let ; then map (57) can be rewritten aswhereAccording to map (62), we may let and use the translation ; then map (62) becomeswhereNow the center manifold theorem is used to determine the nature of the bifurcations of the fixed point at . There exists a center manifold for (65) which can be locally represented as follows: Letting , and substituting into (65) yields . Equating term of like powers to zero gives , , Then . Hence, map (65) restricted to the center manifold is given by where Then we consider the following equation:We find
Firstly, we consider case (1), that is, If , then Thus, by the implicit function theorem, there exists and continuously differentiable function , such thatwhere and
Let , where ; then (70) can be written asIt is easy to see that According to Remark 4, we have So Therefore, Hence, (71) has a positive root if is small enough. However, So, Thus, system (8) undergoes a supercritical bifurcation at if
Similar to the above analysis, we may prove that system (8) undergoes a subcritical bifurcation at if This completes the proof.
5. The Stability of Epidemic Periodic Solutions
In the following, we discuss the stability of positive periodic solutions of system (8). According to Theorem 8, system (8) has a positive periodic solution with the initial point and period .
Next, we discuss the local stability of the periodic solution . Suppose that is any solution of system (8). LetSubstituting (73) into (8), we obtain the linearization of system (8) as follows:whereLet . Calculating the upper-right derivative of along the solution of system (74), we have
By (10), for all , we haveThus, from (27), we obtain that, for any and all large enough,According to (24), we obtain that, for all large enough,Since is a positive periodic solution of system (8), it is obvious that, by (78) and (79), for all , we get . In addition, by Theorem 7, there exists such that, for all
SoIf then we may let