Qualitative analysis of hepatitis B virus infection model with impulsive vaccination and time delay

Acta Mathematica Scientia 2011,31B(3):1020–1034 http://actams.wipm.ac.cn

QUALITATIVE ANALYSIS OF HEPATITIS B VIRUS INFECTION MODEL WITH IMPULSIVE VACCINATION AND TIME DELAY∗

)1,†

Qiao Meihong (

)2

Qi Huan (

Chen Yingchun (


)3

1. School of Mathematics and Physics, China University of Geoscience, Wuhan 430074, China 2. Department of Control Science & Engineering, Huazhong University of Science & Technology, Wuhan 430074, China 3. National Key Lab of Defence Science and Technology on Acoustic Countermeasure, Zhanjiang Branch, Zhanjiang 524022, China E-mail: [email protected]

Abstract According to the epidemic state, propagation mode and transformation between HBV infection states, the Hepatitis B Virus (HBV) infection with impulsive vaccination and time delay are modelled and analyzed. The control methods of impulsive vaccination and active therapy are adopted. By using comparative theorem of impulsive differential equation, the sufficient conditions that Hepatitis B Virus will be eliminated eventually or be persistent are derived. Key words impulsive vaccination; time delay; globally asymptotical stability; persistence 2000 MR Subject Classification

1

37C75

Introduction

Being a common disease, Hepatitis B Virus (HBV) is distributed all over the world. The prevention and treatment of HBV are one of the most concerning problems in the world [1]. According to the report of WHO, about two billion persons have been infected with HBV, among which 350 million are chronic HBV infectors [2]. In each year about one million infectors die of HBV-induced hypohepatia, hepatocirrhosis and liver cancer [3, 4]. Natural course of the HBV infection is as follows. Through effective contact with HBV patients or HBV carriers, the susceptible become HBV infectors. HBV carriers are infective at the later stage of latent period, which is about 45–150 days, with an average of 90 days. Then the HBV carriers become acute HBV infector after the latent period. The period of acute HBV ∗ Received

February 23, 2009; revised December 7, 2009. This work was supported by the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) CUGL100238 and 2011CB710600 (973 Program). † Corresponding author: Qiao Meihong.

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is about four months. More than 90% of acute HBV infectors could be cured after six months of treatment while the rest of them turn into chronic HBV infectors. From present statistical data in China, existing HBV cases are almost chronic HBV infectors and the majority don’t have obvious acute period. As few chronic HBV infectors could be cured completely, chronic HBV infectors usually become carriers. So the majority of acute and chronic HBV infectors tend to be HBV carriers and only a few of them obtain HBV immunity after treatment [5]. 50–75% of chronic HBV infectors have active activities of virus replication and liver inflammation. The natural course of disease infection can last as long as 30–50 years. Continuous chronic HBV infection exhibits various kinds of clinical symptoms, such as carriers without symptom, acute or chronic hepatitis, hepatocirrhosis and even hepatocellular carcinoma. In China, Southeast Asia and Africa, 50% of hepatocirrhosis and 70–90% of liver cancers are caused by chronic HBV infection and 7–30% of HBV carriers are infected by variants of HBV [6]. Presently, some literatures [7–11] researched epidemic mathematical models. And most of all, [7] described a mathematical model developed to predict the dynamics of HBV transmission and to evaluate the long term effectiveness of the vaccination programme. Based on the mechanism of HBV infection, we think it is more interesting and significative, if impulsive vaccination can be considered in the HBV model. Therefore, HBV infection model with impulsive vaccination and time delay be researched in this paper. The rest of the paper is organized as follows. The HBV control model with impulsive vaccination and time delay is established according to the epidemic state, propagation mode and transformation between infection states. The control methods of impulsive vaccination and active therapy are adopted. In Section 3, global attractability of disease-free periodic solution is proved. Section 4 gives out uniform persistence of the system. Finally, a brief discussion and biological sense are given in Section 5. By using comparative theorem of impulsive differential equation, the sufficient conditions that HBV will be eliminated eventually or be persistent are derived.

2

Modelling

There was much interest in modeling epidemic dynamics (e.g., [12–17] and the references cited therein) as epidemic models are very useful in the control of epidemic diseases. For simplicity, some assumptions are made as follows before establishing the control model of HBV. (a) The infection rate is directly proportional to the product of the susceptible number and infector number. (b) The cure rate is directly proportional to the number of infectors. (c) The natural death rate is μ and the death rates of acute patients, chronic sufferers with e antigen-positive and chronic sufferers with e antigen-negative are λ1 , λ2 , λ3 , respectively, where λi > 0, i = 1, 2, 3, and λ2 > λ3 . (d) Supposing constant input rate A > 0, input population (new births and immigrants) are all susceptible. Hepatitis B virus (HBV), a member of the hepadnaviruses (hepatotropic DNA viruses) family, causes acute and chronic infection of the liver [18]. Most persons who are infected by

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the acute HBV could be cured. However, persons who are infected by the chronic HBV can hardly be cured. The medicine can only suppress HBV replication. Usually, chronic HBV can be controlled by changing e antigen-positive to e antigen-negative. Because the death rate and infectivity of chronic sufferers with e antigen-negative are lower than those of chronic sufferers with e antigen-positive. The relationships between parameters in the HBV model are shown in Fig. 1.

Fig. 1

S (Susceptible) people can become I1 (acute HBV infectors), or I2 (chronic e

antigen-positive sufferers) through HBV infection. Most I1 (acute HBV infectors) could be cured to be R (recuperators), Usually, Chronic HBV can be controlled by changing I2 (chronic e antigen-positive sufferers) to I3 (chronic e antigen-negative sufferers). μ is the natural death rate, λ1 , λ2 , λ3 are the death rates of acute patients, chronic sufferers with e antigen-positive and chronic sufferers with e antigen-negative respectively. β1 is acute HBV infection rate, β2 is chronic HBV infection rate. γ2 is the cure rate of acute HBV, γ3 is the negative conversion rate.

Based on the above hypotheses, the HBV model with impulsive vaccination and time delay is established as follows ⎧ ⎫ ˙ ⎪ ⎪ S(t) = A − μS(t) − β S(t)I (t) − β S(t)I (t) ⎪ ⎪ 1 1 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −(μ+λ +γ )ω 1 2 ˙ ⎪ ⎪ I (t) = β S(t)I (t) − (μ + λ + γ )I (t) − β e S(t − ω)I (t − ω) 1 1 1 1 2 1 1 1 ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ −(μ+λ +γ )ω ⎪ 1 2 ˙ ⎪ I2 (t) = β2 S(t)I2 (t) − (μ + λ2 + γ3 )I2 (t) + β1 e S(t − ω)I1 (t − ω) t = kτ, ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ (1) I˙3 (t) = γ3 I2 (t) − (λ3 + μ)I3 (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ˙ ⎪ R(t) = γ2 I1 (t) − μR(t) ⎪ ⎪ ⎫ ⎪ ⎪ ⎪ S(t+ ) = (1 − p)S(t) ⎬ ⎪ ⎪ ⎪ ⎪ t = kτ. ⎪ ⎩ R(t+ ) = R(t) + pS(t) ⎭ The population size is N (t) = S(t) + I1 (t) + I2 (t) + I3 (t) + R(t), where S(t) is the number of the susceptible, I1 (t) is the number of acute HBV infectors, I2 (t) is the number of chronic e antigen-positive sufferers, I3 (t) is the number of chronic e antigennegative sufferers and R(t) is the number of the recuperators. A denotes constant input rate,

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β1 denotes acute HBV infection rate, β2 denotes chronic HBV infection rate, γ2 denotes cure rate of acute HBV, γ3 denotes the negative conversion rate, ω denotes the latent period from acute HBV to chronic HBV, p denotes vaccination rate, τ denotes the interval between two vaccinations. All coefficients in the model are positive. By system (1) and N˙ (t) = A − μN (t) − λ1 I1 (t) − λ2 I2 (t) − λ3 I3 (t),

(2)

we know the population size changes along with time t. When I1 (t) ≡ I2 (t) ≡ I3 (t) ≡ 0, we have A lim N (t) = . t→∞ μ From system (2), N (t) ≤ A μ holds true when t is sufficiently large. Substitute R(t) for N (t) − S(t) − I1 (t) − I2 (t) − I3 (t), and then system (1) becomes ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎫ ˙ ⎪ S(t) = A − μS(t) − β1 S(t)I1 (t) − β2 S(t)I2 (t) ⎪ ⎪ ⎪ ⎪ −(μ+λ +γ )ω 1 2 ⎪ S(t − ω)I1 (t − ω) ⎪ I˙1 (t) = β1 S(t)I1 (t) − (μ + λ1 + γ2 )I1 (t) − β1 e ⎪ ⎬ −(μ+λ1 +γ2 )ω ˙ I2 (t) = β2 S(t)I2 (t) − (μ + λ2 + γ3 )I2 (t) + β1 e S(t − ω)I1 (t − ω) t = kτ, ⎪ ⎪ ⎪ ⎪ I˙3 (t) = γ3 I2 (t) − (λ3 + μ)I3 (t) ⎪ ⎪ ⎪ ⎪ ⎭ N˙ (t) = A − μN (t) − λ1 I1 (t) − λ2 I2 (t) − λ3 I3 (t) ⎫ (3) S(t+ ) = (1 − p)S(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ I1 (t+ ) = I1 (t) ⎪ ⎬ + t = kτ. I2 (t ) = I2 (t) ⎪ ⎪ ⎪ ⎪ I3 (t+ ) = I3 (t) ⎪ ⎪ ⎪ ⎪ ⎭ + N (t ) = N (t)

Now we discuss the dynamic characteristic of system (3). Suppose it satisfies the initial value, 5 (φ1 (θ), φ2 (θ), φ3 (θ), φ4 (θ), φ5 (θ)) ∈ C+ = C([−ω, 0], R+ ),

φi (0) > 0,

i = 1, 2, 3, 4, 5.

(4)

In biological sense, we will discuss (3) on the closed set

A A 5 . Ω = (S, I1 , I2 , I3 , N ) ∈ R |S ≥ 0, I1 ≥ 0, I2 ≥ 0, I3 ≥ 0, S + I1 + I2 + I3 ≤ , 0 ≤ N ≤ μ μ Obviously, Ω is the positive invariable set of (3). Because (3) satisfies the condition that the solution is unique on R+ × Ω and the maximum existing region of the solution is [t0 , ∞), every solution of (3) is piecewise continuous function with numerable discontinuity points. Now, we give some key definitions. Suppose x(t) = (S(t), I1 (t), I2 (t), I3 (t), N (t)) and x∗ (t) = (S ∗ (t), I1∗ (t), I2∗ (t), I3∗ (t), N ∗ (t)) are respectively arbitrary solution and fixed solution of (3). Definition 1 For arbitrary initial value x0 , if there exists a positive number M ≥ m > 0 (independent of x0 ), such that m ≤ x(t) ≤ M holds true when t is sufficiently large, then we call (3) is uniformly persistent. That is, HBV will become endemic.

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Definition 2

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If lim |x(t) − x∗ (t) | = 0 holds true for arbitrary initial value x0 , then we t→∞

call x∗ (t) is globally attractive. Definition 3 If there exists a δ = δ(t0 , ε, η) > 0 for arbitrary ε > 0, η > 0 such that |x(t) − x∗ (t) | < ε holds true when |x0 − x∗0 | < δ, t > t0 and |t − t0 | > η, then we call x∗ (t) is stable. Definition 4 If x∗ (t) is stable and globally attractive, we call x∗ (t) is of globally asymptotical stability. When the disease-free periodic solution of (3) exists and is globally asymptotical stable, HBV will eventually disappear. Definition 5 (Equicontinuity) For arbitrary ε > 0 if there exists δ > 0, such that |x(t1 ) − x(t2 ) | < ε holds true when t1 , t2 ∈ [0, T ] and |t1 − t2 | < δ, then we call x(t) is equicontinuous on [0, T ].

3

Global Attractivity of Disease-Free Periodic Solution

First, we will show the existence of disease-free periodic solution. If it does exist, HBV will disappear forever, i.e., I1 (t) = I2 (t) = I3 (t) = 0 for t ≥ 0. In this situation, the increase of the susceptible and total population size satisfies the following equations ⎫ ⎧ ⎪ ˙ ⎪ S(t) = A − μS(t) ⎬ ⎪ ⎪ t = kτ, ⎪ ⎪ ⎨ N(t) ˙ = A − μN (t) ⎭ ⎫ ⎪ ⎪ S(t+ ) = (1 − p)S(t) ⎬ ⎪ ⎪ t = kτ. ⎪ ⎪ ⎩ N (t+ ) = N (t) ⎭

(5)

By the second and fourth equation of (5), we have lim N (t) = A μ. t→∞ In the following, we will prove that there exists a periodic solution with period τ for the susceptible. Consider the limit system, ⎧ ⎨ S(t) ˙ = A − μS(t), t = kτ, (6) ⎩ S(t+ ) = (1 − p)S(t), t = kτ. Obviously, by Lemma 1 in [19], there exists a unique periodic solution of globally asymptotical stability in (6),  pe−μ(t−kτ ) A S (t) = 1− , μ 1 − (1 − p)e−μτ ∗

kτ < t ≤ (k + 1)τ.

(7)

Hence, there exists a disease-free periodic solution (S ∗ (t), 0, 0, 0, A μ ) in (3). Then we will prove this periodic solution is globally attractive. Theorem 1 If R1∗ < 1 and R2∗ < 1, the disease-free periodic solution (S ∗ (t), 0, 0, 0, A μ ) of (3) is globally attractive, where R1∗ =

β1 A(eμτ − 1) , μ(μ + λ1 + γ2 )(eμτ + p − 1)

R2∗ =

β2 A(eμτ − 1) . μ(μ + λ2 + γ3 )(eμτ + p − 1)

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Proof Because R1∗ < 1, we can choose a sufficiently small ε1 > 0 such that  A(eμτ − 1) + ε1 < μ + λ1 + γ2 . β1 μ(eμτ + p − 1)

(8)

By the fifth equation of (3) and N˙ (t) ≤ A − μN (t), we know there exists an integer k1 > 0 such that A N (t) ≤ , t > k1 τ. (9) μ By the first equation of (3), we derive that  A ˙ − S(t) . S(t) < μ μ When t > k1 τ and k > k1 , we consider the following impulsive differential comparative system ⎧  A ⎪ ⎨ x(t) ˙ =μ − x(t) , t = kτ, μ (10) ⎪ ⎩ x(t+ ) = (1 − p)x(t), t = kτ. By Lemma 1 in [19], we know x∗ (t) = S ∗ (t) is the periodic solution of (10) with globally asymptotical stability and there exists k2 > k1 such that when k > k2 , S(t) < S ∗ (t) + ε1 ≤

A(eμτ − 1) + ε1 = δ, μ(eμτ + p − 1)

kτ < t ≤ (k + 1)τ.

(11)

The second equation of (3) implies I˙1 (t) ≤ (β1 δ − μ − λ1 − γ2 )I1 (t),

t > kτ,

k > k2 .

Thus lim I1 (t) = 0.

t→∞

(12)

Therefore, for arbitrary sufficiently small ε2 > 0, there exists an integer k3 > k2 such that I1 (t) < ε2 for all t > k3 τ. Then, from the third equation of (3) we derive that, when t > kτ + ω, k > k3 , I˙2 (t) ≤ β2 δI2 (t) − (μ + λ2 + γ3 )I2 (t) + β1 e−(μ+λ1 +γ2 )ω δε2 = (β2 δ − μ − λ2 − γ3 )I2 (t) + β1 e−(μ+λ1 +γ2 )ω δε2 , i.e.,  I2 (t) ≤

I2 (kτ ) +

β1 e−(μ+λ1 +γ2 )ω δε2 [(β2 δ−μ−λ2 −γ3 )(t−kτ )] β1 e−(μ+λ1 +γ2 )ω δε2 e − . β2 δ − μ − λ2 − γ3 β2 δ − μ − λ2 − γ3

Hence lim I2 (t) = 0.

t→∞

(13)

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Similarly, for arbitrary sufficiently small ε3 > 0, there exists an integer k4 > k3 such that I2 (t) < ε3 for all t > k4 τ. Then, according to the fourth equation of (3), we get I˙3 (t) ≤ γ3 ε3 − (μ + λ3 )I3 (t) for all t > kτ, k > k4 . Obviously, lim I3 (t) = 0.

t→∞

(14)

Therefore, for arbitrary sufficiently small ε4 > 0, there exists an integer k5 > k4 such that I3 (t) < ε4 for all t > k5 τ. Finally, according to the fifth equation of (3), we have N˙ (t) > A − μN (t) − λ1 ε2 − λ2 ε3 − λ3 ε4 , t > k5 τ. When t > k5 τ, consider the differential comparative system ˙ Z(t) = A − μZ(t) − λ1 ε2 − λ2 ε3 − λ3 ε4 . Obviously, lim Z(t) =

t→∞

(15)

A − λ1 ε2 − λ2 ε3 − λ3 ε4 . μ

According to the differential comparative lemma in [20], we prove that, when t > k6 τ, there exists k6 > k5 such that N (t) ≥

A − λ1 ε2 − λ2 ε3 − λ3 ε4 − ε5 . μ

(16)

Because ε2 , ε3 , ε4 , ε5 are arbitrary sufficiently small positive numbers, from (9) and (16), we have A lim N (t) = . (17) t→∞ μ By (12) and (13), we know that, for arbitrary ε6 > 0, there exists k7 > k6 such that I1 (t) < ε6 ,

I2 (t) < ε6 ,

when t > k7 τ. When t > k7 τ, the first equation of (3) implies ˙ S(t) ≥ A − (μ + β1 ε6 + β2 ε6 )S(t). When t > kτ and k > k7 , consider the following impulsive differential comparative system, ⎧ ⎨ u(t) ˙ = A − (μ + β1 ε6 + β2 ε6 )u(t), t = kτ, (18) ⎩ u(t+ ) = (1 − p)u(t), t = kτ.

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Based on Lemma 1 in [19], we know that there exists a unique periodic solution of system (18): u ¯e (t) =

 A A e−(μ+β1 ε6 +β2 ε6 )(t−kτ ) , + u∗ − μ + β1 ε6 + β2 ε6 μ + β1 ε6 + β2 ε6 kτ < t ≤ (k + 1)τ.

It is globally asymptotical stable, where u∗ =

(1 − p)(1 − e−(μ+β1 ε6 +β2 ε6 )τ ) A . μ + β1 ε6 + β2 ε6 1 − (1 − p)e−(μ+β1 ε6 +β2 ε6 )τ

By the impulsive differential comparative lemma in [20], there exists an integer k8 > k7 such that S(t) > u ¯e (t) − ε6 , kτ < t ≤ (k + 1)τ, k > k8 . (19) Because ε1 and ε6 are both arbitrary small, (11) and (19) imply  pe−μ(t−kτ ) ) A ∗ 1− , kτ < t ≤ (k + 1)τ. lim S(t) = S (t) = t→∞ μ 1 − (1 − p)e−μτ

(20)

By (12), (13), (14), (17) and (20), we derive that the disease-free periodic solution (S ∗ (t), 0, 0, 0, A μ ) of (3) is globally attractive. According to Theorem 1, we can easily derive the following conclusion. Proposition 1 1) When Aβ1 < μ(μ+ λ1 + γ2 ) and Aβ2 < μ(μ+ λ2 + γ3 ), the disease-free periodic solution ∗ (S (t), 0, 0, 0, A μ ) of (3) is globally attractive. 2) When Aβ1 > μ(μ + λ1 + γ2 ), Aβ2 > μ(μ + λ2 + γ3 ) and the impulsive vaccination period τ < min{τ1∗ , τ2∗ }, the disease-free periodic solution (S ∗ (t), 0, 0, 0, A μ ) of (3) is globally attractive, where

 pμ(μ + λ1 + γ2 ) 1 ∗ τ1 = ln 1 + , μ Aβ1 − μ(μ + λ1 + γ2 )

 pμ(μ + λ2 + γ3 ) 1 . τ2∗ = ln 1 + μ Aβ2 − μ(μ + λ2 + γ3 ) 3) When Aβ1 > μ(μ + λ1 + γ2 ), Aβ2 < μ(μ + λ2 + γ3 ) and the impulsive vaccination period τ < τ1∗ , the disease-free periodic solution (S ∗ (t), 0, 0, 0, A μ ) of (3) is globally attractive. 4) When Aβ1 < μ(μ + λ1 + γ2 ), Aβ2 > μ(μ + λ2 + γ3 ) and the impulsive vaccination period τ < τ2∗ , the disease-free periodic solution (S ∗ (t), 0, 0, 0, A μ ) of (3) is globally attractive. Proposition 2 If the impulsive vaccination rate p > max{p∗1 , p∗2 }, disease-free periodic solution (S ∗ (t), 0, 0, 0, A μ ) of (3) is globally attractive, where p∗1 =

[Aβ1 − μ(μ + λ1 + γ2 )](eμτ − 1) , μ(μ + λ1 + γ2 )

p∗2 =

[Aβ2 − μ(μ + λ2 + γ3 )](eμτ − 1) . μ(μ + λ2 + γ3 )

Fig. 2 shows the time series and the orbits of system (3), where the parameters satisfy the conditions of Theorem 1, where the red line represents susceptible people; the black line

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represents the healthy people resumed from HBV infection; the blue line represents I1 (acute HBV infectors); the green line represents I2 (chronic e antigen-positive sufferers); the yellow line represents I3 (chronic e antigen-negative sufferers). From Fig. 2, it is obvious that HBV disappears in the condition of R1∗ < 1 and R2∗ < 1. The definitions of R1∗ and R2∗ are as above-mentioned.

Fig. 2

The constant input rate A = 1; the natural death rate μ = 0.1; the acute HBV

infection rate β1 = 0.1; the chronic HBV infection rate β2 = 0.1; the death rate of acute patients λ1 = 0.8; the death rate of chronic sufferers with e antigen-positive λ2 = 0.99; the death rate of chronic sufferers with e antigen-negative λ3 = 0.9; the cure rate of acute HBV γ2 = 0.1; the negative coversion rate γ3 = 0.8; the threshold quantity of acute HBV transforms to chronic HBV ω = 50; the interval of immunization τ = 5; the immunization rate p = 0.9.

4

Uniform Persistence of the System Before proving system (3) to be uniformly persistent, we first prove a lemma. Let Aβ1 (1 − e−(μ+λ1 +γ2 )ω )(1 − p)(eμτ − 1) R1∗ = , μ(μ + λ1 + γ2 )(eμτ + p − 1) μ A(1 − e−(μ+λ1 +γ2 )ω )(1 − p)(eμτ − 1) − , I¯1 = μτ (μ + λ1 + γ2 )(e + p − 1) β1 R2∗ = I¯2 =

Aβ2 (1 − p)(eμτ − 1) , μ(μ + λ2 + γ3 )(eμτ + p − 1)

μ A(1 − p)(eμτ − 1) − . (μ + λ2 + γ3 )(eμτ + p − 1) β2

If R1∗ > 1, we have I¯1 > 0. Lemma 1 If R1∗ > 1 and R2∗ > 1, then for arbitrary t0 > 0, I1 (t) < I¯1 and I2 (t) < I¯2 don’t hold true for all t > t0 .

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Proof Suppose Lemma 1 doesn’t hold true, there exist t0 > 0 and t1 > t0 + ω, such that if t ≥ t1 − ω, I1 (t) < I¯1 , I2 (t) < I¯2 . (21) Noting that the infection period of acute infector is ω, the second equation of system (3) implies that, when t ≥ ω,  t I1 (t) = β1 S(u)I1 (u)e−(μ+λ1 +γ2 )(t−u) du. (22) t−ω

By the first equation of (3), we have, when t ≥ t1 − ω, ˙ S(t) ≥ A − (μ + β1 I¯1 + β2 I¯2 )S(t). When t ≥ t1 − ω, consider the following impulsive differential comparative system ⎧ ⎨ v(t) ˙ = A − (μ + β1 I¯1 + β2 I¯2 )v(t), t = kτ, ⎩ v(t+ ) = (1 − p)v(t), t = kτ.

(23)

According to Lemma 1 in [19], there exists a unique periodic solution in system (23):  A A ¯ ¯ ∗ + v e−(μ+β1 I1 +β2 I)(t−kτ ) , v¯e (t) = − μ + β1 I¯1 + β2 I¯ μ + β1 I¯1 + β2 I¯ kτ < t ≤ (k + 1)τ. It is globally asymptotical stable, where v∗ =

¯

¯

(1 − p)(e(μ+β1 I1 +β2 I)τ − 1) A . ¯ μ + β1 I¯1 + β2 I¯ e(μ+β1 I¯1 +β2 I)τ +p−1

By the impulsive differential comparative theorem in [20], there exists t2 (≥ t1 − ω) such that, when t ≥ t2 , we have . S(t) ≥ v¯e (t) − ε > v ∗ − ε = S L , (24) where ε is a sufficiently small positive number. Note that R1∗ > 1, thus I¯1 > 0. Hence, when ε > 0 is sufficiently small,we get β1 S L (1 − e−(μ+λ1 +γ2 )ω ) > μ + λ1 + γ2 . According to (22) and (24), we have  t S L I1 (u)e−(μ+λ1 +γ2 )(t−u) du I1 (t) ≥ β1

(25)

t−ω

for all t ≥ t2 . Supposing I1l =

min

t∈[t2 ,t2 +ω)

I1 (t), we can predicate I1 (t) ≥ I1l for all t ≥ t2 . Otherwise,

there exists a t3 ≥ t2 + ω such that I1 (t3 ) = I1l and I1 (t) ≥ I1 (t3 ) for t3 ≥ t ≥ t2 . Thus by (25), we have 1 − e−(μ+λ1 +γ2 )ω I1 (t3 ) > β1 S L I1 (t3 ) . μ + λ1 + γ2

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Therefore, β1 S L (1 − e−(μ+λ1 +γ2 )ω ) < μ + λ1 + γ2 . This is contradict with β1 S L (1 − e−(μ+λ1 +γ2 )ω ) > μ + λ1 + γ2 . The above predication is proved. Choose a constant R1 such that 1 < R1 < R1∗ . We can predicate that when t ≥ t2 + ω, I1 (t) > I1l R1 . Notice that I1 (t2 + ω) ≥ β1 S L



t2 +ω

t2

I1l e−(μ+λ1 +γ2 )(t2 +ω−u) du > I1l R1 .

If the prediction is not true, there exists a t4 > t2 + ω such that, when t2 + ω ≤ t ≤ t4 , I1 (t) ≥ I1l R1

and I1 (t4 ) = I1l R1 .

On the other hand, if t2 + ω ≤ t ≤ t4 , we have I1 (t) ≥ β1 S L



t

t−ω

I1l e−(μ+λ1 +γ2 )(t2 −u) du = β1 S L I1l

1 − e−(μ+λ1 +γ2 )ω > I1l R1 . μ + λ1 + γ2

Hence I1 (t4 ) > I1l R1 . It is a contradiction. When t > t2 + kω, I1 (t) ≥ I1l R1k . When t is sufficiently large, I1 (t) ≥ I¯1 is incompatible with I1 (t) < I¯1 for all t ≥ t0 . Theorem 2 If R1∗ > 1, there exists an integer q1 such that any positive solution of (3) satisfies lim inf I1 (t) > q1 .

t→∞ ¯

Proof Define q1 = min{ I21 , q0 }, where q0 is defined in (26).

According to Lemma 1, the positive solution (S(t), I1 (t), I2 (t), I3 (t), N (t)) of (3) is discussed under two cases. First, when t is sufficiently large, I1 (t) ≥ I¯1 holds. Second, when t is sufficiently large, I1 (t) vibrates about I¯1 . In the first case, lim inf I1 (t) > q. Obviously, Theorem 2 holds true. t→∞ In the second case, suppose I1 (t∗ ) = I1 (t∗ + ξ) = I¯1 . If t∗ ≤ t ≤ t∗ + ξ, then we have I1 (t) ≤ I¯1 such that β1 S L (1 − e−(μ+λ1 +γ2 )ω ) > μ + λ1 + γ2 for t∗ ≤ t ≤ t∗ + ξ and sufficiently large t∗ . Because the positive solution of (3) is eventually bounded and I1 (t) has no impulse, we know I1 (t) is equicontinuous. Thus, there exists T (0 < T < ω and T is independent of the selection of t∗ ) such that I¯1 I1 (t) > 2 for all t∗ < t ≤ t∗ + T. When ξ ≤ T, the result is obvious.

No.3

M.H. Qiao et al: QUALITATIVE ANALYSIS OF HEPATITIS B VIRUS INFECTION MODEL 1031

Now we consider T < ξ ≤ ω for t∗ < t ≤ t∗ + ξ. We derive I1 (t) ≥ β1 S

L

≥ β1 S L >



t

I1 (u)e t−ω  t∗ +T t∗

−(μ+λ1 +γ2 )(t−u)

du

I1 (u)e−(μ+λ1 +γ2 )(t−u) du

β1 I¯1 L −(μ+λ1 +γ2 )ω . S Te = q0 , 2

(26)

i.e., when T < ξ ≤ ω, I1 (t) ≥ q1 , for t∗ < t ≤ t∗ + ξ; when ξ > ω, by the second equation of (3), we derive I1 (t) ≥ q1 for t∗ < t ≤ t∗ + ω. Similar to the proof of Lemma 1, we know I1 (t) ≥ q1 for t∗ + ω < t ≤ t∗ + ξ. Because ∗ ∗ [t , t + ξ] is arbitrary (as long as t∗ is sufficiently large), in the second case we can derive lim inf I1 (t) > q1 . t→∞

Rewrite the third equation of (3) as I˙2 (t) = β2 S(t)I2 (t) + β1 e−(μ+λ1 +γ2 )ω S(t)I1 (t) − (μ + λ2 + γ3 )I2 (t)  t −(μ+λ1 +γ2 )ω d S(θ)I1 (θ)dθ. −β1 e dt t−ω

(27)

Consider any positive solution of (3). Define V (t) = I2 (t) + β1 e−(μ+λ1 +γ2 )ω



t

t−ω

S(θ)I1 (θ)dθ.

By (27), we make derivative of V for the positive solution of (3), V˙ (t) = β2 S(t)I2 (t) − (μ + λ2 + γ3 )I2 (t) + β1 e−(μ+λ1 +γ2 )ω S(t)I1 (t) > (β2 S(t) − (μ + λ2 + γ3 ))I2 (t).

(28)

Because R2∗ > 1 and I2∗ (t) > 0, there exists a sufficiently small ε > 0 such that

 ¯ ¯ (1 − p)(e(β1 I1 +β2 I2 +μ)τ − 1) A β2 − ε > 1. μ + λ2 + γ3 β1 I¯1 + β2 I¯2 + μ e(β1 I¯1 +β2 I¯2 +μ)τ + p − 1 We have

β2 S L μ+λ2 +γ3

> 1 from (29). By (24) and (28), we have

 V˙ (t) > (β2 S(t) − (μ + λ2 + γ3 ))I2 (t) = (μ + λ2 + γ3 ) for t ≥ t2 . Let I2l =

(29)

min

t∈[t2 ,t2 +ω]

β2 S L − 1 I2 (t) μ + λ2 + γ3

I2 (t).

In the following we will prove I2 (t) ≥ I2l for all t ≥ t2 . Otherwise, there exists T0 ≥ 0 such that I2 (t) ≥ I2l

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for t2 ≤ t ≤ t2 + ω + T0 , I2l = I2 (t2 + ω + T0 ), I˙2 (t2 + ω + T0 ) ≤ 0. Whereas, by the third equation of (3) and (24), we know I˙2 (t2 + ω + T0 ) ≥ β2 S(t2 + ω + T0 )I2 (t2 + ω + T0 ) +β1 e−(μ+λ1 +γ2 )ω S(t2 + T0 )I1 (t2 + T0 ) − (μ + λ2 + γ3 )I2l . It is a contradiction. Therefore, I2 (t) ≥ I2l for all t ≥ t2 . By (28), we know  ˙ V (t) > (μ + λ2 + γ3 )

β2 S L − 1 I2l μ + λ2 + γ3

for t ≥ t2 . When t → ∞, V (t) → ∞. This is contradict with V (t) ≤ A + A2 β1 ωe−(μ+λ1 +γ2 )ω . So for arbitrary t0 > 0, I2 (t) isn’t always smaller than I¯2 for t ≥ t0 . We will consider two cases in the following. First, I2 (t) ≥ I¯2 for sufficiently large t. Second, I2 (t) vibrates about I¯2 for sufficiently large t. ¯ Define q2 = min{ I22 , q}, where q will be described later. We need to prove I2 (t) ≥ q2 for sufficiently large t. In the first case, the result is obvious. In the second case, suppose t∗ > 0 and ζ > 0 satisfing I2 (t∗ ) = I2 (t∗ + ζ) = I¯2 and I2 (t) < I¯2 for t∗ < t < t∗ + ζ such that S(t) > S L for t∗ < t < t∗ + ζ. Because the positive solution of (3) is eventually bounded and there is no impulse in I2 (t), I2 (t) is uniformly continuous, i.e., there exists T1 (0 < T1 < ω, T1 is independent of t∗ ) ¯ such that I2 (t) > I22 . When t∗ ≤ t ≤ t∗ + T1 and ζ ≤ T1 , we have the result. Consider the case of T1 < ζ ≤ ω. Because I˙2 (t) > −(μ + λ2 + γ3 )I2 (t) and I2 (t∗ ) = I¯2 , obviously we have I2 (t) > q = I¯2 e−(μ+λ2 +γ3 )ω , t∗ < t < t∗ + ζ. When ζ > ω, by the third equation of (3), we derive I2 (t) ≥ q2 , t ∈ [t∗ , t∗ + ω]. Similar to the above proof, we have I2 (t) ≥ q2 , t ∈ [t∗ + ω, t∗ + ζ]. Because [t∗ , t∗ + ζ] is arbitrary (t∗ is sufficiently large), in the second case, I2 (t) ≥ q2 holds true for sufficiently large t.

No.3

M.H. Qiao et al: QUALITATIVE ANALYSIS OF HEPATITIS B VIRUS INFECTION MODEL 1033

From the above discussion, we know q2 is independent of the positive solution of (3). That is, arbitrary positive solution of (3) satisfies lim I2 (t) ≥ q2 . t→∞

Theorem 3 When R1∗ > 1 and R2∗ > 1, system (3) is persistent. Proof By the fourth equation of (3), we have I2 (t) ≥ q2 for sufficiently large t. When t is sufficiently large, we derive I˙3 (t) ≥ γ3 q2 − (μ + λ3 )I3 (t). Therefore, lim I3 (t) ≥

t→∞

γ3 q2 = I¯3 = q3 . μ + λ3

When t is sufficiently large, according to the first equation of (3), we have  A A A A ˙ S(t). S(t) ≥ A − μS(t) − β1 S(t) − β2 S(t) = A − μ + β1 + β2 μ μ μ μ Similar to the proof of Lemma 1, we have lim S(t) ≥ q4 ,

t→∞

where q4 =

A

A μ+β1 A μ +β2 μ

Let

(1−p)(e

(μ+β1 A +β2 A )τ μ μ −1)

(μ+β1 A +β2 A )τ μ μ e +p−1

− ε0 (ε0 is sufficiently small).

A Ω0 = (S, I1 , I2 , I3 , N ) : q4 ≤ S, q3 ≤ I3 , q2 ≤ I2 , q1 ≤ I1 , S + I1 + I2 + I3 ≤ N ≤ . μ We know Ω0 is a globally attractive region in Ω from the above discussion and Theorem 2. The positive solution of (3), which satisfies the initial value condition (4), will ultimately enter and remain in Ω0 . That means system (3) is persistent. From Theorem 2 and Theorem 3, we can easily derive the following conclusion. Proposition 3 When p < min{p1 , p2 }, HBV will become endemic and system (3) is persistent, where p1 =

Aβ1 (1 − e−(μ+λ1 +γ2 )ω ) − μ(μ + λ1 + γ2 ) Aβ1 (1 − e−(μ+λ1 +γ2 )ω ) +

μ(μ+λ1 +γ2 ) eμτ −1

(Aβ1 (1 − e−(μ+λ1 +γ2 )ω ) > μ(μ + λ1 + γ2 ),

,

p2 =

Aβ2 − μ(μ + λ2 + γ3 ) Aβ2 +

μ(μ+λ2 +γ3 ) eμτ −1

Aβ2 > μ(μ + λ2 + γ3 )).

When τ > max{τ1 , τ2 }, HBV will become endemic and the system is persistent, where 

1 pμ(μ + λ1 + γ2 ) +1 , τ1 = ln μ Aβ1 (1 − p)(1 − e−(μ+λ1 +γ2 )ω ) − μ(μ + λ1 + γ2 )

 pμ(μ + λ2 + γ3 ) 1 +1 . τ2 = ln μ Aβ2 (1 − p) − μ(μ + λ2 + γ3 ) From Theorem 2 and Theorem 3, we know, when R1∗ < 1 and R2∗ < 1, HBV will disappear, and when R1∗ > 1 and R2∗ > 1, HBV will uniformly persistent.

5

Conclusions

Based on the mechanism of HBV infection, we have established a mathematical HBV model with impulsive vaccination and time delay, according to the epidemic state, propagation mode

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and transformation between infection states. The control methods of impulsive vaccination and active therapy are adopted. By using comparative theorem of impulsive differential equation, the sufficient conditions that HBV will be eliminated eventually or persistent are derived. We prove that the infection-free periodic solution is globally asymptotically stable when R1∗ < 1 and R2∗ < 1. The numerical simulations show that HBV declines sharply and eventually is eliminated. Moreover, we derive two interesting deductions of HBV decrease. When R1∗ > 1 and R2∗ > 1, the infection-free equilibrium is unstable and the system is uniformly persistent, i.e., HBV will exist persistently. The dynamic characteristic of (3) for R1∗ < 1 < R1∗ and R2∗ < 1 < R2∗ will be studied in future research. References [1] Maddrey W C. Hepatitis B-an Important Public Health Issue. Clin Lab, 2001: 47–51 [2] Lee W M. Hepatitis B virus infection. N Engl J Med, 1997, 24: 1733–1745 [3] Beasley R P, Hwang L Y, Lin C C, Chien C S. Hepatocellular carcinoma and Hepatitis B virus: A prospective study of 22 707 men in Taiwan. Lancet, 1981, 2(8256): 1129–1133 [4] Weissberg J I, Andres L L, Smith C I, et al. Survival in chronic hepatitis B: An analysis of 379 patients. Ann Intern Med, 1984, 101(5): 613–616 [5] Liu S J, Zhu Q. HBV Disabusing and Answering Doubt. Beijing: Chinese Medicine and Scientific Technology Publishing Company, 2003 [6] Ezquieta B, Cueva E, Oliver A, et al. SHOX intragenic micro-satellite analysis in patients with short stature. J Pediatr Endecrinol Metab, 2002, 15(2): 139–148 [7] Zhao S J, Xu Z Y, Lu Y. Mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China. International Journal of Epidemiology, 2000, 29(4): 744–752 [8] Li J Q, Ma Z E, Zhou Y C. Global of SIS epidemic model with a simple vaccination and multiple endemic equilibria. Acta Mathematica Scientia, 2006, 26B(1): 83–93 [9] Matveev A S, Savkin A V. Application of optimal control theory to analysis of cancer chemotherapy regimens. Systems and Control Letters, 2002, 46(5): 311–321 [10] Alfonseca M, Bravo M T M, Torrea J L. Mathematical model for the analysis of hepatitis B and AIDS epidemics. Simulation, 2000, 74(4): 219–226 [11] Ogren P, Martin C F. Vaccination strategies for epidemics in highly mobile populations. Applied Mathematics and Computation, 2002, 127(2): 261–272 [12] Anderson R M, May R M. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University, 1991 [13] Capasso V. Mathematical Structures of Epidemic Systems. Lecture Notes in Biomath, Vol 97. Heidelberg: Springer, 1993 [14] Hethcote H W. The mathematics of infectious diseases. SIAM Rev, 2000, 42: 599–653 [15] Kamo M, Sasaki A. The effect of cross-immunity and seasonal forcing in a multi-strain epidemic model. Physica D, 2002, 165: 228–241 [16] Wang W, Ruan S. Simulating the SARS outbreak in Beijing with limited data. J Theoret Biol, 2004, 227: 369–379 [17] Wang W, Zhao X Q. An age-structured epidemic model in a patchy environment. SIAM J Appl Math, 2005, 65: 1597–1614 [18] Park N H, Song I H, Chung Y H. Chronic hepatitis B in hepatocarcinogercesis. Postgrad Med J, 2006, 82: 507–515 [19] Gao S J, Chen L S, Teng Z D. Pulse vaccination of an SEIR epidemic model with time delay. Nonlinear Analysis: Real World Applications, 2008: 599–607 [20] Lakshmikantham V, Bainov D D, Simeonov P S. Theory of Impulsive Differential Equations. Singapore: World Scientific, 1989