On the Stability Property of the Infection-Free Equilibrium of a Viral Infection Model

Abstract

The dynamics of a viral infection model with nonautonomous lytic immune response is studied from the perspective of dying out of the disease. With the help of the theory of exponential dichotomy of linear systems, we give a new proof about the global asymptotic stability of the infection-free equilibrium for the case 𝑅₀=1. The result improves and complements one of the results of Wang et al. (2006).

1. Introduction

Throughout this paper, given a bounded continuous function 𝑝 defined on 𝑅, let 𝑝^𝑙 and 𝑝^𝑢be defined as 𝑝^𝑙=inf_𝑡∈𝑅𝑝(𝑡),𝑝^𝑢=sup_𝑡∈𝑅𝑝(𝑡). (1.1)

The aim of this paper is to investigate the stability property of the infection-free equilibrium of the following nonautonomous viral infection model: 𝑥̇(𝑡)𝑦̇(𝑡)(𝑡)𝑧̇(𝑡)=𝜆−𝑑𝑥−𝛽𝑥𝑦,=𝛽𝑥𝑦−𝑎𝑦−𝑝𝑦𝑧,=𝑐𝑦−𝑏𝑧, (1.2) where 𝑥(𝑡), 𝑦(𝑡), and 𝑧(𝑡), represents susceptible host cells, a virus population, and a CTL response, respectively. Susceptible host cells are generated at a rate 𝜆, die at a rate 𝑑𝑥, and become infected by virus at a rate 𝛽𝑥𝑦. Infected cells die at a rate 𝑎𝑦 and are killed by the CTL response at a rate 𝑝(𝑡)𝑦𝑧. The CTL response expands in response to viral antigen derived from infected cells at a rate 𝑐𝑦 and decay in the absence of antigenic stimulation at a rate 𝑏𝑧. We assume that 𝜆, 𝑑, 𝛽, 𝑎, 𝑐, and 𝑏 are all positive constants and 𝑝(𝑡) is a continuous, real-valued functions which is bounded above and below by positive constants. For more detail deduction and background of the above model, see Wang et al. [1] and Fan and Wang [2].

We consider (1.2) together with the following initial conditions 𝑥(0)>0,𝑦(0)>0,𝑧(0)>0. (1.3) It is not difficult to see that solutions of (1.2)-(1.3) are well defined and positive for all 𝑡≥0.

Recently, Wang et al. [1] proposed and studied the dynamic behaviors of the system (1.2). Obviously, system (1.2) admits one and only one steady state 𝐸₀=(𝑥^∗,0,0)=(𝜆/𝑑,0,0), which represents the infection-free equilibrium. The basic reproductive ratio of the virus is given by 𝑅₀=𝜆𝛽/𝑎𝑑. It can be expected that disease dies out if 𝑅₀<1 and becomes endemic if 𝑅₀>1. However, it seems not an easy thing to deal with the critical case 𝑅₀=1. In [1], the authors obtained the following interesting result.

Theorem A. The infection-free equilibrium 𝐸₀ is globally asymptotically stable if 𝑅₀=1.

Indeed, in their proof of Theorem A, by using the variation of constants formula for inhomogeneous linear ordinary differential equations, the solution to the third equality of system (1.2) takes the form 𝑧(𝑡)=𝑧(0)𝑒^−𝑏𝑡+∫^𝑡₀𝑐𝑦(𝑠)𝑒^𝑏𝑠𝑑𝑠𝑒^𝑏𝑡. (1.4) From this equality, they immediately declared that 𝑧(𝑡)−𝑐𝑦(𝑡)𝑏⟶0as𝑡⟶+∞. (1.5) Maybe it is obviously to some scholars that equality (1.4) implies (1.5), however, we found it is not an easy thing for us to understand this deduction. Since (1.4) and (1.5) play crucial role in their prove of Theorem A, it motivated us to propose the following interesting issue.

Is It Possible for Us to Give a Different Proof of Theorem A?
On the other hand, we argue that it is more suitable to consider a general nonautonomous 𝑝(𝑡) than that of a periodic function. Thus, it is natural to propose the following question.

Whether the Conclusion of Theorem A Still Holds Under the Assumption That 𝑝(𝑡) is a General Positive Nonautonomous Continuous Function?
The aim of this paper is, by applying the theory of exponential dichotomies of linear system [3, 4] and adapting some analysis technique recently developed by Chen et al. [5–8], to give an affirmed answer to above two issues, more precisely, we obtain the following theorem.

Theorem B. Let 𝑝(𝑡) be a positive continuous function bounded above and below by positive constants. Then the infection-free equilibrium 𝐸₀ of system (1.2) is globally asymptotically stable if 𝑅₀=1.

We will prove Theorem B in the next section and give a numeric simulation in Section 3. We end this paper by a briefly discussion. For more works on viral infection model, one could refer to [1–6, 9–17] and the references cited therein. For the works about the stability of differential equations, one could refer to [7, 8, 18, 19] and the references cited therein.

2. Proof of Theorem B

Now we state several lemmas which will be useful in proving of our main result.

Lemma 2.1 (see [13]). If 𝑎>0, 𝑏>0 and 𝑥̇≥𝑥(𝑏−𝑎𝑥), when 𝑡≥𝑡₀ and 𝑥(𝑡₀)>0, one has liminf_𝑡→+∞𝑥(𝑡)≥𝑏𝑎. (2.1) If 𝑎>0, 𝑏>0 and 𝑥̇≤𝑥(𝑏−𝑎𝑥), when 𝑡≥𝑡₀ and 𝑥(𝑡₀)>0, one has limsup_𝑡→+∞𝑥(𝑡)≤𝑏𝑎. (2.2)

Lemma 2.2. Let 𝑏 be a positive constant and let 𝑐(𝑡) is a nonnegative continuous bounded function, then system 𝑥̇(𝑡)=−𝑏𝑥+𝑐(𝑡) (2.3) admits a unique bounded solution 𝑥^∗(𝑡), which is globally attractive.

Proof. Since 𝑏 is a positive constant, it follows that system 𝑥̇(𝑡)=−𝑏𝑥 (2.4) admits the exponential dichotomies. From He [3, page 59] or Lin [4, page 55] we know that (2.3) admits a unique bounded solution 𝑥^∗(𝑡)=^𝑡_−∞𝑒^{−𝑏(𝑡−𝑠)}𝑐(𝑠)𝑑𝑠. (2.5) Let 𝑥(𝑡) be any solution of system (2.3), and 𝑢(𝑡)=𝑥(𝑡)−𝑥^∗(𝑡). Then 𝑥̇(𝑡)(𝑡)(𝑡),𝑥̇=−𝑏𝑥+𝑐_∗(𝑡)=−𝑏𝑥^∗(𝑡)+𝑐(𝑡). (2.6) It follows that 𝑢 satisfies 𝑢̇(𝑡)=−𝑏𝑢(𝑡). (2.7) Thus, 𝑢(𝑡)=𝑢(0)𝑒^−𝑏𝑡⟶0as𝑡⟶+∞, (2.8) that is, lim_𝑡→+∞𝑥(𝑡)−𝑥^∗(𝑡)=0. (2.9) This ends the proof of Lemma 2.3.

Lemma 2.3 (see [1]). All solutions of system (1.2) are positive for 𝑡>0 and there exists 𝑀>0, such that all the solutions satisfy 𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)<𝑀 for all large 𝑡.

Lemma 2.4 (see [1]). Let 𝑥^∞=limsup_𝑡→+∞𝑥(𝑡). Then 𝑥^∞≤𝑥^∗∶=𝜆/𝑑.

Proof of Theorem B. Let (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡))^𝑇 be any positive solution of system (1.2), from Lemma 2.4 it follows that 𝑦(𝑡) is bounded for all 𝑡>0. From the third equation of system (1.2) we have 𝑧̇(𝑡)=𝑐𝑦(𝑡)−𝑏𝑧(𝑡). (2.10) It follows from Lemma 2.3 that system (2.10) admits a unique bounded solution 𝑧^∗(𝑡)=^𝑡_−∞𝑒^{−𝑏(𝑡−𝑠)}𝑐𝑦(𝑠)𝑑𝑠, (2.11) also, 𝑧(𝑡)−𝑧^∗(𝑡)⟶0as𝑡⟶+∞, (2.12) For arbitrarily small positive constant 𝜀 (without loss of generality, we may assume that 2(𝑏+(𝛽+𝑝^𝑢)𝜀)𝜀<1), it follows from (2.12) and Lemma 2.2 that there exists a 𝑇₁>0 such that for all 𝑡>𝑇₁𝑧(𝑡)>𝑧^∗(𝑡)𝑥(𝑡)−𝜀,<𝑥^∗+𝜀. (2.13) Substituting (2.13) to the second equation of system (1.2) leads to 𝑦̇=𝛽𝑥𝑦−𝑎𝑦−𝑝(𝑡)𝑦𝑧≤𝛽𝑥^∗+𝜀−𝑎−𝑝(𝑡)𝑧^∗−𝜀𝑦. (2.14) Noting that 𝑅₀=1, which implies that 𝛽𝑥^∗−𝑎=0, thus, above inequality leads to 𝑦̇(𝑡)≤𝛽+𝑝(𝑡)𝜀−𝑐𝑝(𝑡)^𝑡_−∞𝑒^{−𝑏(𝑡−𝑠)}𝑦(𝑠)𝑦(𝑡)≤𝑑𝑠𝛽+𝑝^𝑢𝜀−𝑐𝑝^𝑙∞₀𝑒^−𝑏𝑠𝑦()𝑦(𝑡)≤𝑡−𝑠𝑑𝑠𝛽+𝑝^𝑢𝜀−𝑐𝑝^𝑙𝑡₀𝑒^−𝑏𝑠𝑦(𝑡−𝑠)𝑑𝑠𝑦(𝑡). (2.15) It follows from (2.15) that 𝑦̇(𝑡)≤𝛽+𝑝^𝑢𝜀𝑦(𝑡). (2.16) For 𝑡−𝑠≥0, integrating (2.16) on [𝑡−𝑠,𝑡], we derive 𝑦(𝑡−𝑠)≥𝑦(𝑡)𝑒^{−(𝛽+𝑝𝑢)𝜀𝑠}. (2.17) Substituting (2.17) into (2.15) leads to 𝑦̇(𝑡)≤𝛽+𝑝^𝑢𝜀−𝑐𝑝^𝑙𝑡₀𝑒^−𝑏𝑠𝑦(𝑡)𝑒^{−(𝛽+𝑝𝑢)𝜀𝑠}𝑦(𝑡)=𝑑𝑠𝛽+𝑝^𝑢𝜀−𝑐𝑝^𝑙𝑡₀𝑒^{−(𝑏+(𝛽+𝑝𝑢)𝜀)𝑠}𝑑𝑠𝑦(𝑡)𝑦(𝑡). (2.18) Noting that ^𝑡₀𝑒^{−(𝑏+(𝛽+𝑝𝑢)𝜀)𝑠}𝑑𝑠⟶^∞₀𝑒^{−(𝑏+(𝛽+𝑝𝑢)𝜀)𝑠}1𝑑𝑠=𝑏+𝛽+𝑝^𝑢𝜀as𝑡⟶+∞. (2.19) It follows from (2.19) that for above 𝜀>0, there exists an enough large 𝑇₁>0 such that for all 𝑡>𝑇₁, ^𝑡₀𝑒^{−(𝑏+(𝛽+𝑝𝑢)𝜀)𝑠}1𝑑𝑠≥𝑏+𝛽+𝑝^𝑢𝜀1−𝜀≥2𝑏+𝛽+𝑝^𝑢𝜀. (2.20) Substituting (2.20) into (2.18), for 𝑡≥𝑇₁, one has 𝑦̇(𝑡)≤𝛽+𝑝^𝑢𝜀−𝑐𝑝^𝑙2𝑏+𝛽+𝑝^𝑢𝜀𝑦(𝑡)𝑦(𝑡). (2.21)
Applying Lemma 2.1 to (2.18), it immediately follows thatlimsup_𝑡→+∞𝑦(𝑡)≤2𝑏+𝛽+𝑝^𝑢𝜀𝛽+𝑝^𝑢𝜀𝑐𝑝^𝑙. (2.22) Since 𝑦(𝑡)>0 for all 𝑡>0, it follows that 0≤liminf_𝑡→+∞𝑦(𝑡)≤limsup_𝑡→+∞𝑦(𝑡)≤2𝑏+𝛽+𝑝^𝑢𝜀𝛽+𝑝^𝑢𝜀𝑐𝑝^𝑙. (2.23) Since 𝜀 is arbitrarily small positive constant, setting 𝜀→0 in (2.23) leads to lim_𝑡→+∞𝑦(𝑡)=0. (2.24) The rest of the proof is similarly to the proof of Theorem 2.2 in [1] and we omit the detail here.

3. An Example

Consider the following viral infection model: ̇̇3𝑥=1−𝑥−𝑥𝑦,𝑦=𝑥𝑦−𝑦−^−𝑡√+1+cos3𝑡3+sin(𝑡)3̇1𝑦𝑧,𝑧=21𝑦−3𝑧. (3.1)

In this case, corresponding to system (1.2), 𝜆=𝑎=𝑑=𝛽=1, 𝑐=1/2, 𝑏=1/3, 𝑝(𝑡)=1+cos(√3𝑡)/3+sin(𝑡)/3. Obviously, 𝑅₀=𝜆𝛽/𝑎𝑑=1. Thus, as a consequence of Theorem B, the infection-free equilibrium 𝐸₀ is globally asymptotically stable. Numeric simulations (Figures 1, 2, and 3) support this conclusion. We mention here that since 𝑝(𝑡) is general nonautonomous continuous function, Theorem A could not be applied to system (3.1).

Figure 1 

Dynamic behaviors of the first component 𝑥(𝑡) of the solution (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡))𝑇 to system (3.1) with the initial condition (0.1,0.1,0.1)𝑇, (1,1,1)𝑇, (2,3,4)𝑇, and (0.6,1,2)𝑇, respectively.

Figure 2 

Dynamic behaviors of the second component 𝑦(𝑡) of the solution (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡))𝑇 to system (3.1) with the initial condition (0.1,0.1,0.1)𝑇, (1,1,1)𝑇, (2,3,4)𝑇, and (0.6,1,2)𝑇, respectively.

Figure 3 

Dynamic behaviors of the third component 𝑧(𝑡) of the solution (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡))𝑇 to system (3.1) with the initial condition (0.1,0.1,0.1)𝑇, (1,1,1)𝑇, (2,3,4)𝑇, and (0.6,1,2)𝑇, respectively.

4. Conclusion

In this paper, we revisit the model proposed by Wang et al. [1]. By applying the theory of exponential dichotomy of linear systems and the differential inequality theory, we show that for general nonautonomous positive continuous coefficient 𝑝(𝑡), 𝑅₀=1 is enough to ensure the global asymptotic stability of the infection-free equilibrium.

Acknowledgments

The author is grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. Also, this work was supported by the Technology Innovation Platform project of Fujian Province (2009J1007).