一类具有细胞与细胞传染和病毒与细胞传染的时滞HIV-1传染病模型

CLC number: O175.1 Document code: A Article ID: 1001-7011(2018)02-0140-11

Received date: 2016-11-28

Foundation item: Supported by the National Natural Science Foundation of China (11401453); Natural Science Basic Research Plan of Shaanxi Province (2014JQ1018; 2014JQ1038); Shaanxi Provincial Education Department (16JK1331)

Biography: Ms.SHAO Mingyue (1993-), M. S. candidate, interested in: mathematical biology, E-mail: shaomingyue2015@126.com

不同程序中权利要求保护范围理解和解释的基本规则............................................................................................刘 鹏 12.41

Corresponding author: Mr. ZHANG Tailei (1980-), professor, Ph.D., interested in: mathematical biology, E-mail: t.l.zhang@126.com

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0 Introduction

Human immunodeficiency virus (HIV) is a retrovirus that targets cells with CD4+ receptors in the human body, including CD4+ T-cells, the main driver of the immune response. When the number of the CD4+ T-cells falls below a critical threshold, an HIV patient is diagnosed with Acquired Immune Deficiency Syndrome (AIDS). AIDS is a highly infectious disease with high mortality.

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Over the past two decades, many mathematical models have been developed to describe the infection with human immuno deficiency virus (HIV-1). Wang [1] developed a simple model with virus-to-cell that ignored the time delays. Hu [2] considered the dynamics with a discrete delay. Xu [3] developed an HIV-1 infection model with distributed intracellular delay. More realistic models include time delays such as papers [4-13]. In paper [4], the authors established the following mathematical model:

 

(1)

where T(t) ,I(t) ,V(t) and E(t) represent the concentration of susceptible CD4+ T-cells, productively infected T-cells, free virus particles and concentration of effector cells at time t. CD4+ T-cells are infected by free viral particles and infectious cells (productively infected cells) at rates β1T(t)V(t) and β2T(t)I(t), respectively. The term I(t-τ) accounts for the time needed to activate the CD8+ T-cells response, where τ is a constant.

In this paper, we consider a viral model with both virus-to-cell and cell-to-cell transmissions and two discrete delays, in which the first discrete delay describes the intracellular latency for the virus-to-cell infection, the second discrete delay describes the intracellular latency for the cell-to-cell infection. Without loss of generality, they all will be represented by τ. Next, we consider the following more general HIV-1 infection model:

 

(2)

Here e-μτ is the survival rate of cells that are infected by viruses or infected cells from time t-τ to t.

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The meaning of other parameters (s,d,,r,α,Tmax,d1,d2,d3,d4,N,p) are as follows (all the parameters are positive):

对比治疗前与治疗48 h后的RR（呼吸频率）、HR（心率）和动脉血气分析（PH、PaO2、PaCO2）结果。

 

Table 1 Meaning of the parameters

  

ParametersMeaning of the parameters sThe source of CD4+ T-cells from precursors dThe natural death rate of CD4+ T-cells rThe growth rate of CD4+ T-cells (d

As usual, initial functions of (2) are given as

T(θ)=φ1(θ), I(θ)=φ2(θ),V(θ)=φ3(θ) and E(θ)=φ4(θ) (-τ θ 0),

(3)

where the initial function and is the Banach space of continuous functions mapping the interval [-τ,0] into Y(t,φ)=(T(t,φ),I(t,φ),V(t,φ),E(t,φ)) is the unique solution of system (2) with initial function φ. In next section, we prove that every solution of system (2) with initial function (3) is positive and bounded. In section 2, we consider global stability of steady states and the uniformly persistent of the system (2). In section 3, we simulate the model numerically. Finally, we make a summary for this article.

1 Positivity and boundedness

Let Y(t,φ)=(T(t),I(t),V(t),E(t)) be a solution of system (2) with initial condition (3). Using similar arguments as the proof Theorem 2.1 in paper [4], we can obtain the following result.

We can easily know that X0 is positively invariant for p(t) (where p(t) is the family of solution operators corresponding to (2), the ω limit set ω(x) of x consists of y∈X). We define

Lemma 1 Solution Y(t,φ)=(T(t),I(t),V(t),E(t)) of system (2) with initial condition (3) are positive for all t≥0, and they are ultimately bounded.

Proof Firstly, we prove T(t) is positive for all t≥0. Otherwise, there exists a positive number t0, such that T(t)>0 for t∈[0,t0) and T(t0)=0, indicating that 0. From the first equation of system (2), we have a contradiction and hence we have T(t) is positive for all t≥0. From system (2), we have

 

 

 

From (3), it is easy to see that I(t-τ) and V(t-τ) are positive for t∈[0,τ), and T(t) is positive for all t≥0, therefore we have β1e-μτT(s-τ)V(s-τ)+β2e-μτT(s-τ)I(s-τ)>0 for t∈[0,τ). Furthermore, it can be shown that I(t) is positive for t∈[0,τ), and V(t) is positive for t∈[0,τ). Thus, for all t≥0 we have I(t) that is positive, which along with (3) implies that V(t) and E(t) are positive for all t≥0.

Next, we show solutions of system (2) are ultimately bounded. From the first equation of system (2), we obtain

Let From the comparison principle, sup T(t) T0, where Then T(t) is ultimately bounded.

Let H(t)=T(t)+I(t+τ), we have

 

where δ1=min(d,d1). From the comparison principle, we get sup H(t) implying that I(t) is ultimately bounded. Similarly, we can easily get that V(t) and E(t) are also ultimately bounded and sup V(t)+E(t) where δ2=min(d2,d4).

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In fact, we can see from Lemma 1 appearing later, that the set

a,V+E<b}.

2 Stability analysis of steady state

2.1 Asymptotical stability analysis of the infection-free steady state P0

System (2) has two steady states: the infection-free steady state P0=(T0,0,0,0) and the infection steady state P*=(T*,I*,V*,E*), where

 

I*

 

 

We define the basic reproduction number as follows Biologically, R0 represents the average number of secondary infections. In fact, is the basic reproduction number corresponding to virus-to-cell infection, while is the basic reproduction number corresponding to cell-to-cell infection.

Let us consider the following transformation: where is an steady state of system (2). Hence, we have the corresponding linearization system of system (2):

 

(4)

The characteristic equation is

 

Theorem 1 If R0<1, the infection-free steady state P0 is locally asymptotically stable. If R0>1, P0 is unstable.

Proof If R0<1, we can easily know that I* is negative. So system (2) has a unique infection-free steady state P0. Next we discuss the local asymptotic stability of P0. It is clear that characteristic equation (5) has two negative real roots and λ2=-d4. Now we consider the transcendental equation

λ2+(d1+d2)λ+d1d2-((λ+d2)β2e-μτ+Nd1β1e-μτ)T0e-λτ=0.

(6)

For τ=0, we have that d1+d2-T0β2>0 and d1d2(1-R0)>0. This shows that all roots of (6) have negative real parts for τ=0.

For τ>0, if (6) has pure imaginary roots λ=±iω (ω>0), then we have

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(7)

This implies that

 

(8)

where m=ω2, and We can easily know that equation (8) does not exist positive roots. Hence, it shows that any roots of (5) have negative real parts, P0 is locally asymptotically stable for all τ≥0 when R0<1. And P0 is unstable when R0>1.

Theorem 2 If R0<1, P0 is globally asymptotically stable.

Proof In papers [4-5], the authors used fluctuations lemma introduced in paper [14] to prove the global asymptotic stability of infection-free steady state. We also use this method to prove the global stability of P0. For acontinuous and bounded function f(t), we define

f sup f(t), f inf f(t).

So T(t),I(t),V(t),E(t) satisfy

0 T T <, 0 I I <, 0 V V <, 0 E E<.

(9)

We claim that T(t) T0 for t≥0 if T(0) T0. From the fluctuation lemma, for the second equation of system (2), we know that there is a sequence tn with tn→ such that I(tn)→I , 0. Thus

β1e-μτT(tn-τ)V(tn-τ)+β2e-μτT(tn-τ)I(tn-τ)-d1I(tn).

We have

d1I β1e-μτT(t0)V +β2e-μτT(t0)I .

(10)

From the third equation of system (2), we have

d2V Nd1I .

(11)

From (10) and (11), we get

d1I

(12)

and I ≥0. If I >0, we have d1 This is a contradiction with R0<1. So I =0 and

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Furthermore where and indicating Using similarly arguments to the fourth equation of system (2), we obtain From the first equation of system (2), we have The proof of Theorem 2 is completed.

2.2 Uniform persistence

Theorem 3 If R0>1, the solution of system (2) with the initial condition (3) is uniformly persistent with φ2(0)≠0 and φ3(0)≠0.

Proof Let

 

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∂X=XX0={φ∈X|φ2(0)=0 or φ3(0)=0}.

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We can easily know that

We claim that

M∂={(T,0,0,E)}.

(i) If 1<R0 P* does not exist.

For (a), from the third equation of (2) we have Hence there is a sufficiently small such that which contradicts to the condition of V(t) is non-negative. Similarly, we can obtain a contradiction for (b). This proves the claim M∂={(T,0,0,E)}.

Let A=∩X∈Ab ω(x), where Ab is global attractor of p(t) restricted to ∂X. We show that A={P0}. In fact, from A⊆M∂ and the second and first equations of (2), we have and Thus {P0} is the isolated invariant set in X.

Next we show that Ws(P0)∩X0=∅. If this is not true, then there exists a solution (T(t),I(t),V(t),E(t))∈X0 such that and For any sufficiently small constant ε>0, there exists a positive constant t1=t1(ε) such that T(t)>T0-ε>0 and E(t)<ε for all t≥t1.

We consider the following system:

 

(13)

M∂={φ∈X|Y(t)φ∈∂X0,∀t≥0}.

It follows that

 

(14)

If I,V→0, as t→, then by a standard comparison argument and the nonnegativity, the solution y1,y2 of (13) with the initial condition y1(t)=I(t), y2(t)=V(t), ∀t∈[t0,t0+τ] converges to (0,0). Thus where is defined by

Cite this: SHAO Mingyue, ZHANG Tailei, LIU Junli． A delayed model of HIV-1 with cell-to-cell and virus-to-cell transmissions [J]. Journal of Natural Science of Heilongjiang University, 2018, 35(2): 140-150.

 

thus we have

 

For a sufficiently small ε and R0>1, we have which contradicts to Thus we have Ws(P0)∩X0=∅.

Denote m:X→R+ and m(φ)=min{φ2(0),φ3(0)}, by paper [17] we have inf I(t)≥η and inf V(t)≥η. Moreover, from the fourth equation of system (2), we can know that inf E(t)≥ From the first equation of system (2), we can know that inf T(t)≥η1. This completes the proof of Theorem 3.

2.3 Stability of the infection steady state P*

Now we study the asymptotical stability of P*, when R0>1

Assuming Y(t)∈M∂ for all t≥0 is not true, then there exists t0>0 such that either (a) I(t0)>0 and V(t0)=0; or (b) I(t0)=0 and V(t0)>0.

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(ii) If P* exists. Under this condition, we discuss the asymptotically stability of P*.

From (5), we know the characteristic equation at the infected steady state is

λ4+a3λ3+a2λ2+a1λ+a0+(b3λ3+b2λ2+b1λ+b0)e-τλ=0,

(15)

where

 

 

 

 

b3=-β1e-μτT*,

 

 

b1=(β1e-μτV*+β2e-μτI*

 

 

For τ=0, we have and

By Routh-Hurwitz criterion [15], we know all solutions of (15) have negative real parts if and only if Hk>0 (k=1,2), where

 

(16)

At this time, P* is locally asymptotically stable.

For τ>0, if equation (15) has pure imaginary roots λ=±iω (ω>0), we have

 

(17)

This implies that

G(k)=k4+D1k3+D2k2+D3k+D4=0,

(18)

where k=ω2, .

Since the coefficients of equations (15) and (18) depend upon τ, this will increase the difficulty to deal with the characteristic equation (15). We know that if equation (18) has positive real roots, that is to say, the characteristic equation (15) have purely imaginary roots for certain value τ0 and the stability of P* may change. Similar to paper [10], we only study the sufficient condition to ensure the stability of P* does not change.

Lemma 2 If D4<0, then equation (18) has at least one positive root.

Proof We know G(0)=D4<0 and . Hence, equation (18) has at least one positive root.

Next we study From paper [16], we introduce Then we know that is equivalent to h3+q1h+q2=0, where and From Cardan formula, we know

 

where and so we get

Lemma 3 For D4>0,(i) if Δ>0, equation (18) has no positive roots when k1 0 or G(k1)>0;

(ii) if Δ 0, equation (18) has no positive roots when ki 0 or G(ki)>0, i=1,2,3.

Proof (i) For D4>0, if Δ>0, we know that h1 is unique real root, hence k1 is a unique real root of and We can easily know that G(k) is increasing when k>k1, G(k) is decreasing when k<k1. So equation (18) has no positive roots when k1 0 or G(k1)>0. The proof of (i) is completed.

(ii) For D4>0, if Δ 0, we know has three real roots k1,k2,k3. Let k*=max{k1,k2,k3}, G(k) is increasing when k>k*, hence equation (18) has no positive roots when k* 0. In addition, it is easy to prove that equation (18) has no positive roots when G(ki)>0, i=1,2,3.

The above analysis can be summarized into the following theorem.

Theorem 4 Suppose that Hk>0 (k=1,2), all roots of (15) have negative real roots, if any one of the following conditions holds:

(i) For D4>0, if Δ>0, equation (18) has no positive roots when k1 0 or G(k1)>0;

(ii) For D4>0, if Δ 0, equation (18) has no positive roots when ki 0 or G(ki)>0, i=1,2,3.

Then infection steady state P*=(T*,I*,V*,E*) is asymptotically stable, if (i) or (ii) holds.

Basing on Theorem 4，we construct a suitable Lyapunov functional to investigate the global stability of the infection steady state P* when r=0.

Theorem 5 The unique infection steady state P* is globally asymptotically stable when r=0.

Proof We define an Lyapunov functional as follows:

U=U1+U2+U3+U4+U5.

where

 

In combination with

s-dT*+rT*

β1e-μτT*V*+β2e-μτT*I*-d1I*-d3E*I*=0, pI*-d4E*=0.

So we have

 

where

 

Δ3=(β1T*V*+β2T*I*

Δ4=(β1T*V*+β2T*I*

It is easy to know that 0, and if and only if (T,I,V,E)=(T*,I*,V*,E*). Then the unique infection steady state P* is globally asymptotically stable.

3 Numerical simulation

In this section, we illustrate our results on stability by numerical simulations. For τ=1. From papers [2,4,9,13], we select a set of parameter values. s=10, d=0.1, β1=0.000 25, β2=0.000 65, μ=2, r=0.03, Tmax=1 500, α=1.2, d1=0.5, d2=3, d3=0.812, d4=0.618, N=50 and p=0.05. It is easy to know that R0=0.915 3<1 and P0=(137.458 6,0,0,0). Now we draw the corresponding image, see Fig.1.

  

  

 

Fig.1 P0 is globally asymptotically stable for τ=1

For τ=0, we select a set of parameter values. s=10, d=0.01, β1=0.000 25, β2=0.000 65, μ=3,r=0.1, Tmax=1 500, α=1.2, d1=0.4, d2=2.4, d3=0.812, d4=1.618, N=500 and p=0.05. It is easy to know that R0=78.050 0>1, P*=(37.718 5,16.352 2,1 362.7,0.505 3), H1=32.335 3>0 and H2=31.900 7>0. Now we draw the corresponding image, see Fig.2.

  

  

 

Fig.2 P* is globally asymptotically stable for τ=0

4 Summary

Basing on the model in papers [4,10], by considering both virus-to-cell and cell-to-cell transmissions, we derived a more realistic model with two delays.Two time delays will be represented by τ. In our analysis, the model has two steady states: the infection-free steady state and the infection steady state. Next we analyze the stability properties of the above two equilibria. First, we consider the stability of the infection-free equilibrium. Through rigorous analysis, we obtained the infection-free steady state is the unique equilibrium and globally asymptotically stable. Next, we consider the stability of the infection steady state. By analyzing the characteristic equation, we get a sufficient condition for the stability of the infection steady state. Since the coefficients of characteristic equation which is dependent on τ, this will increase the difficulty to deal with the stability of the infection steady state. Hopf bifurcation under certain conditions will be studied in the future.

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