Dynamic Behavior for an SIRS Model with Nonlinear Incidence Rate and Treatment

Abstract

This paper considers an SIRS model with nonlinear incidence rate and treatment. It is assumed that susceptible and infectious individuals have constant immigration rates. We investigate the existence of equilibrium and prove the global asymptotical stable results of the endemic equilibrium. We then obtained that the model undergoes a Hopf bifurcation and existences a limit cycle. Some numerical simulations are given to illustrate the analytical results.

1. Introduction

Treatment is an important and effective method to prevent and control the spread of various infectious diseases, such as measles, tuberculosis, and flu [1–4]. In classical epidemic models, the treatment rate of the infective is assumed to be proportional to the number of the infective individuals [5]. This is unsatisfactory because the resources for treatment should be quite large. In fact, every community should have a suitable capacity for treatment. If it is too large, the community pays for unnecessary cost. If it is small, the community has the risk of the outbreak of a disease. Thus, it is realistic to maintain a suitable capacity of disease treatment. Wang and Ruan [6] considered an SIR model in which the capacity for the treatment of a disease in a community is a constant. Namely, they used the following function: ℎ(𝐼)=𝑘,if𝐼>0,0,if𝐼=0, (1) which was used by [7]. This seems more reasonable when we consider the limitation of the treatment resource of a community.

There are many reasons for using nonlinear incidence rate, and various forms of nonlinear incidence rates have been proposed recently. Liu et al. [8] proposed a nonlinear saturated incidence function 𝑔(𝐼)𝑆=𝛽𝑆𝐼^𝑝/(1+𝛼𝐼^𝑞) to model the effect of behavioral changes to certain communicable disease, where 𝛽𝑆𝐼^𝑝 describes the infection force of the disease and 1/(1+𝛼𝐼^𝑞) measures the inhibition effect from the behavioral change of the susceptible individuals when the number of infectious individuals increases. The case when 𝑝=1, 𝑞=2 was used by [9]. We assume the population can be partitioned into three compartments: susceptible, infective, and recovered. Let 𝑆, 𝐼, and 𝑅 denote the numbers of susceptible, infective, and recovered individuals, respectively. Motivated by the works [6, 7, 9], we will formulate an SIRS model with nonlinear incidence rate and constant immigration rates for susceptible and infectious individuals [10]. Namely, we consider the following SIRS model: 𝑆^′=1−𝑝𝐴−𝛽𝐼𝑆1+𝛼𝐼²𝑅−𝑑𝑆+𝛾𝑅,_′(𝐼),𝐼=𝑚𝐼−𝑑+𝛾𝑅+ℎ_′=𝑝𝐴+𝛽𝐼𝑆1+𝛼𝐼²−(𝑑+𝑚)𝐼−ℎ(𝐼), (2) where 𝑑 is the rate of natural death, 𝑚 is the rate for recovery, 𝛽 is the proportionality constant, 𝛾 is the rate at which recovered individuals lose immunity and return to susceptible class, 𝛼 is the parameter measures of the psychological or inhibitory effect, and (1−𝑝)𝐴, 𝑝𝐴 are constant recruitments of susceptible and infective individuals, respectively. It is assumed that all the parameters are positive constants. It is easy to see that the total population size 𝑁 implies 𝑁^′=𝐴−𝑑𝑁. Since 𝑁 tends to a constant as 𝑡 tends to infinity, we assume that the population is in equilibrium and investigate the behavior of (2) on the plane 𝑆+𝐼+𝑅=𝐴/𝑑=𝑁₀>0. Let (𝑆(𝑡),𝐼(𝑡),𝑅(𝑡)) be a solution of (2) with initial conditions 𝑆(0)≥0, 𝐼(0)≥0, and 𝑅(0)≥0. This solution will satisfy 𝑆(𝑡)≥0, 𝐼(𝑡)≥0, and 𝑅(𝑡)≥0 for all 𝑡≥0 since 𝑆^′=(1−𝑝)𝐴+𝛾𝑅>0 if  (0,𝐼(𝑡),𝑅(𝑡))∈𝑅³₊, 𝐼^′=𝑝𝐴>0 if (𝑆(𝑡),0,𝑅(𝑡))∈𝑅³₊, and 𝑅^′=𝑚𝐼+ℎ(𝐼)≥0 if (𝑆(𝑡),𝐼(𝑡),0)∈𝑅³₊. Consequently, 𝑅³₊ is positively invariant for (2). Thus, we restrict our attention to the following reduced model: 𝐼^′𝑁=𝑝𝐴−𝑘+𝛽𝐼₀−𝐼−𝑅1+𝛼𝐼²−()𝑅𝑑+𝑚𝐼,_′=𝑚𝐼−𝑑+𝛾𝑅+𝑘, (3) where 𝑘 is the treatment constant.

From the epidemiological interpretation, our discussion on (3) will be restricted in the following bounded domain: 𝐷=(𝐼,𝑅)∶𝐼>0,𝑅>0,0<𝐼+𝑅<𝑁₀ (4) which is a positively invariant set for (3).

The paper is organized as follows. In the next section, we investigate the existence and stability of equilibrium for (3). In Section 3, we study the Hopf bifurcation and limit cycle. Some numerical simulations are given to illustrate the analytical results. Section 4 is a brief discussion.

2. Existence and Stability of Equilibrium

In this section, we first consider the existence of equilibrium of (3) and their global stability. In order to find endemic equilibrium of (3), we substitute 𝑅=𝑚𝐼+𝑘𝑑+𝛾 (5) into 𝑁𝑝𝐴−𝑘+𝛽𝐼₀−𝐼−𝑅1+𝛼𝐼²=(𝑑+𝑚)𝐼 (6) to obtain the cubic equation 𝑓(𝐼)=𝛼(𝑑+𝑚)𝐼³+𝛽𝑑+𝛾+𝑚+𝛼𝐼𝑑+𝛾𝑘−𝑝𝐴²+𝑑+𝑚+𝛽𝑘𝑑+𝛾−𝛽𝑁₀𝐼+𝑘−𝑝𝐴=0. (7)

Let 𝐼₁, 𝐼₂, and 𝐼₃ be three roots of (7). Then, we get 𝛽𝛼()𝐴𝑑+𝑚𝑑−𝑘=1𝑑+𝛾𝛼−𝐼₁𝐼₂+𝐼₁𝐼₃+𝐼₃𝐼₂,𝛽𝛼()𝑚𝑑+𝑚1+𝑑+𝛾=𝐼₁𝐼₂𝐼₃𝛼−𝐼₁+𝐼₂+𝐼₃. (8)

When 𝑅₀=𝑘/𝑝𝐴<1, we can see that 𝐼₁+𝐼₂+𝐼₃𝐼₁𝐼₂𝐼₃1<𝛼<𝐼₁𝐼₂+𝐼₂𝐼₃+𝐼₁𝐼₃; (9) then there is a unique positive root of (7). Direct calculations show that 𝑓𝑁₀=1−𝑝𝐵+𝛾𝑁₀+𝑁²₀𝛽𝛾𝑁₀(𝑑+𝜀)+𝛾𝛼𝑁₀+1−𝑝𝑑𝛼𝑁₀𝑓>0,𝑝𝐴−𝑘=𝑑+𝑚𝛽𝑝𝐴1−𝑅₀(𝑑+𝑚)²×𝑑+𝛾−𝑘𝛾+𝑁₀𝑝−1𝑑𝑑+𝛾+𝑚−𝑁₀𝛾𝑚<0if𝑅₀<1. (10) From biological considerations, it is easy to see the positive root 𝐼∈𝑝𝐴−𝑘𝑑+𝑚,𝑁₀. (11)

Based on the above analysis, we obtain the following theorem.

Theorem 1. There is a unique endemic equilibrium 𝐸₀(𝐼_𝑒,𝑅_𝑒) of (3) if 𝑅₀<1.

Theorem 2. The endemic equilibrium 𝐸₀ of (3) is locally asymptotically stable if 𝑅₀<1.

Proof. The Jacobian matrix of (3) at 𝐸₀ takes the form 𝐽𝐸₀=𝑝𝐴𝑅₀−1𝐼_𝑒−2𝛼𝛽𝐼²_𝑒𝑁₀−𝐼_𝑒−𝑅_𝑒1+𝛼𝐼²_𝑒²−𝛽𝐼_𝑒1+𝛼𝐼²_𝑒−𝛽𝐼_𝑒1+𝛼𝐼²_𝑒.𝑚−𝑑−𝛾 (12)It is easy to obtain tr(𝐽(𝐸₀))<0 and det(𝐽(𝐸₀))>0 when 𝑅₀<1. This completes the proof.

Theorem 3. The endemic equilibrium 𝐸₀ of (3) is globally asymptotically stable if 𝑅₀<1.

Proof. Taking Dulac function 𝐷=1+𝛼𝐼²𝐼, (13) we obtain 𝜕(𝑃𝐷)+𝜕()𝜕𝐼𝑄𝐷()𝜕𝑅=−𝛽−2𝛼𝑑+𝑚𝐼−𝑝𝐴1−𝑅₀𝐼²+𝛼𝑝𝐴1−𝑅₀−𝑑+𝛾𝐼−𝛼𝐼𝑑+𝛾≤−𝛽−𝛼𝑝𝐴1−𝑅₀𝐼−𝑝𝐴1−𝑅₀𝐼²−𝑑+𝛾𝐼−𝛼𝐼𝑑+𝛾, (14) where (𝑃,𝑄) is the vector field of (3). Obviously, 𝜕(𝑃𝐷)+𝜕()𝜕𝐼𝑄𝐷𝜕𝑅<0if𝑅₀<1. (15) Then by Dulac’s criteria, the system (3) admits no limit cycles or separatrix cycles. The global stability of 𝐸₀ follows from the Poincare-Bendixson Theorem. This completes the proof.

Theorem 4. There will be one or two endemic equilibria 𝐸_∗(𝐼_∗,𝑅_∗) of the system (3) if 𝑅₀>1.

Proof. Based on the analysis of Theorem 1, when 𝑅₀>1, we obtain the roots of (7) satisfying 𝐼₁+𝐼₂+𝐼₃𝛽𝑅=−𝑑+𝛾+𝑚+𝛼𝑝𝐴₀−1𝑑+𝛾𝛼()𝐼𝑑+𝑚𝑑+𝛾<0,₁𝐼₂𝐼₃𝑅=−𝑝𝐴₀−1𝛼(𝑑+𝑚)<0; (16) then there exist one positive real root and two conjugate complex roots with negative real parts or two positive real roots and one negative root. Thus, there will be one or two endemic equilibria in (3). This completes the proof.

3. Hopf Bifurcation

In this section, we study the Hopf bifurcation and limit cycle of the system (3). For simplicity of computation, we consider the following system which is equivalent to (3): 𝐼^′=𝛽𝐼𝑁₀−𝐼−𝑅−(𝑑+𝑚)𝐼1+𝛼𝐼²+1−𝑅₀𝑝𝐴1+𝛼𝐼²,𝑅^′=𝑚𝐼−𝑑+𝛾𝑅+𝑝𝐴𝑅₀1+𝛼𝐼². (17)

Let 𝑥=𝐼−𝐼_∗, 𝑦=𝑅−𝑅_∗ to translate 𝐸_∗ to (0,0). Then, (17) becomes 𝑥^′=𝑎₁₁𝑥+𝑎₁₂𝑦+𝑓₁,𝑦𝑥,𝑦_′=𝑎₂₁𝑥+𝑎₂₂𝑦+𝑓₂𝑥,𝑦, (18) where 𝑓₁(𝑥,𝑦) and 𝑓₂(𝑥,𝑦) represent the higher order terms and 𝑎₁₁=𝛽𝑁₀−2𝐼_∗−𝑅_∗−(𝑑+𝑚)1+3𝛼𝐼²_∗−2𝑝𝐴𝛼𝑅₀−1𝐼_∗,𝑎₁₂=−𝛽𝐼_∗,𝑎₂₁=𝑚1+𝛼𝐼²_∗,𝑎₂₂=−𝑑+𝛾1+𝛼𝐼²_∗. (19)

To obtain the Hopf bifurcation, we fix parameters such that tr(𝐽(𝐸_∗))=0, which is equivalent to 𝑎₁₁=(𝑑+𝛾)(1+𝛼𝐼²_∗). Let 𝑋=𝑥,𝑌=𝑎₁₁𝑥+𝑎₁₂𝑦; (20) then (18) is reduced to 𝑋^′=𝑌+𝑓₁𝑋,𝑌−𝑎₁₁𝑋𝑎₁₂,𝑌^′=−𝛿𝑋+𝑎₁₁𝑓₁𝑋,𝑌−𝑎₁₁𝑋𝑎₁₂+𝑎₁₂𝑓₂𝑋,𝑌−𝑎₁₁𝑋𝑎₁₂, (21) where 𝛿=(𝑑+𝑟)²1+𝛼𝐼²_∗²−𝑚𝛽𝐼_∗1+𝛼𝐼²_∗()>0if𝑑+𝑟²1+𝛼𝐼²_∗>𝑚𝛽𝐼_∗. (22)

Let 𝑌𝑢=−𝑋,𝑣=√𝛿; (23) we obtain the normal form of the Hopf bifurcation: 𝑢^′=−√𝛿𝑣+𝐹₁(),𝑣𝑢,𝑣_′=√𝛿𝑢+𝐹₂(𝑢,𝑣), (24) where 𝐹₁()=√𝑢,𝑣𝛿𝑢𝐼_∗−(𝑑+𝑚)𝛼𝑢³−𝛽+3𝑑𝛼𝐼_∗+3𝑚𝛼𝐼_∗−𝑝𝐴𝛼𝑅₀−()−1𝑑+𝛼1+𝛼𝐼²_∗𝐼_∗𝑢²,𝐹₂()=1𝑢,𝑣√𝛿𝛼−𝑚𝛼+𝑑+𝛾²1+𝛼𝐼²_∗𝛽𝐼_∗𝑢³+1√𝛿2𝑚𝛼𝐼_∗−2𝛼𝑑+𝛾²1+𝛼𝐼²_∗𝛽+𝛼𝑑+𝛾𝑢²𝑣𝛽𝐼_∗−𝑚1+𝛼𝐼²_∗√𝛿−2𝛼𝑑+𝛾𝑢𝑣²𝛽+𝑑+𝛾1+𝛼𝐼²_∗𝑣√𝛿𝛽𝐼_∗. (25)(26)

Set the Lyapunov number by 1𝜎=𝜕16³𝐹₁𝜕𝑢³+𝜕³𝐹₁𝜕𝑢𝜕𝑣²+𝜕³𝐹₂𝜕𝑢²+𝜕𝜕𝑣³𝐹₂𝜕𝑣³+𝜕²𝐹₁𝜕𝜕𝑢𝜕𝑣²𝐹₁𝜕𝑢²+𝜕²𝐹₁𝜕𝑣²−𝜕²𝐹₂𝜕𝜕𝑢𝜕𝑣²𝐹₂𝜕𝑢²+𝜕²𝐹₂𝜕𝑣²−𝜕²𝐹₁𝜕𝑢²𝜕²𝐹₂𝜕𝑢²+𝜕²𝐹₁𝜕𝑣²𝜕²𝐹₂𝜕𝑣², (27) which can be reduced to 1𝜎=162𝛼𝑑+𝛾𝛽𝐼_∗−6𝛼(𝑑+𝑚). (28)

So we have the following Hopf bifurcation results.

Theorem 5. There exist Hopf bifurcation and limit cycle in the system (17), when 𝑎₁₁=𝑑+𝛾1+𝛼𝐼²_∗,(𝑑+𝑟)²1+𝛼𝐼²_∗>𝑚𝛽𝐼_∗. (29)

To illustrate the theorem, let us consider the following parameters.

𝛽=0.01 (see [11]), 𝑚=0.1 (see [12]), 𝐴=4.4236, 𝑑=𝑘=𝛾=0.1, 𝛼=0.3995, and 𝑝=0.01.

We have 𝑅₀=2.2606>1, and the equilibria 𝐸₁(0.25,0.625), 𝐸₂(1.3505,1.1752) exist (see Theorem 4). The equilibrium 𝐸₂ is unstable saddle. The parameter values satisfy conditions (29) of Theorem 5 and 𝜎=3.965. Therefore, (17) has an unstable periodic orbit which encircles 𝐸₁. Its phase portrait is illustrated in Figure 1. The time series of the infective and recovered individuals are given in Figures 2 and 3, respectively.

Figure 1 

The phase portraits of (17).

Figure 2 

Time series of infective individuals.

Figure 3 

Time series of recovered individuals.

4. Conclusion

In this paper, we discuss an SIRS epidemic model with nonlinear incidence rate and treatment. It is assumed that susceptible and infectious individuals have constant immigration rates. We investigate the existence and stability of equilibria of (3) and study the Hopf bifurcation and limit cycle. Some numerical simulations are given to illustrate the analytical results. Without the treatment and recruitment of infectious, (2) becomes the SIRS model (see [9]).

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 and 2011236).