Citation: Cuicui Jiang, Kaifa Wang, Lijuan Song. Global dynamics of a delay virus model with recruitment and saturation effects of immune responses[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1233-1246. doi: 10.3934/mbe.2017063
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In recent years, virus dynamics attracts more and more attentions of researchers and plays a crucial role in many diseases research, including AIDS, hepatitis and influenza. Many mathematical models have provided insights into virus infection and dynamics, as well as on how an infection can be managed, reduced or even eradicated ([3], [4], [7], [15], [17], [27], [38], [43], [44]). Since the basic three-dimensional viral infection model was proposed by Nowak et al. [21], Perelson et al. [26], Perelson and Nelson [25], Nowak and May [20], many people have established different within-host infection model, which help us to better understand virus infection and various drug therapy strategies by mathematical analysis, numerical simulations and clinical data ([13], [19], [22], [28], [29]). Note that immune responses play a critical part in the process of viral infections. Concretely, cytotoxic T lymphocyte (CTL) cells can attack infected cells, and antibody cells can neutralize viruses. To better understand the role of the immune function during virus infection, Wodarz proposed the following model with both CTL and antibody immune responses [41],
{˙T(t)=λ−d1T(t)−βT(t)V(t),˙I(t)=βT(t)V(t)−d2I(t)−pI(t)C(t),˙V(t)=rd2I(t)−d3V(t)−qA(t)V(t),˙C(t)=k1I(t)C(t)−d4C(t),˙A(t)=k2A(t)V(t)−d5A(t), | (1) |
where a dot denotes the differentiation with respect to time
After that, some researchers have taken into account the effect of immune responses including CTL responses or antibody responses ([24], [35], [36], [37], [39]). Some other researchers have incorporated the effect of CTL responses and intracellular delays ([11], [16], [18], [32], [45]). Concretely, the global dynamics of (1) with and without intracellular time delay is given in [24] and [42], respectively. Note that model (1) assumes that CTL and antibody responses are produced at bilinear rates. However, De Boer [5] pointed out that the bilinear rates cannot model several immune responses that are together controlling a chronic infection. In [5], De Boer has proposed an immune response function with the saturation. Incorporating the saturation effects of immune responses and the delay, [12] also obtained the global stability of the model, which is totally determined by the corresponding reproductive numbers. These results preclude the complicated behaviors such as the backward bifurcations and Hopf bifurcations which may be induced by saturation factors and time delay.
Note also that most of models assume CTL responses are activated by infected cells/antigenic stimulation, and antibody responses are activated by virus in these studies. However, as pointed out by Nowak and May [20], CTL responses have another function of self-regulating, i.e., the CTL responses are triggered by encountering foreign antigen and then adopts a constant level which is independent of the concentration of virions or infected cell. Bocharvor et al. have provided evidence the export of precursor CTL cells from the thymus [2]. Pang and Cui et al. have studied the export of specific precursor CTL cells from the thymus in [23], but they didn't considered intracellular time delay and antibody responses. Similarly, Wang and Wang have considered that neutralizing antibodies are produced at a constant rate after the injection [37], but they didn't take into account the effect of CTL responses and intracellular time delay.
Motivated by the above studies, we will formulate and analyze a virus dynamics model with the recruitment of immune responses, saturation effects of immune responses and an intracellular time delay, which can be described by the following functional differential equations:
{˙T(t)=λ−d1T(t)−βT(t)V(t),˙I(t)=βT(t−τ)V(t−τ)e−sτ−d2I(t)−pI(t)C(t),˙V(t)=rd2I(t)−d3V(t)−qA(t)V(t),˙C(t)=λ1+k1I(t)C(t)h1+C(t)−d4C(t),˙A(t)=λ2+k2A(t)V(t)h2+A(t)−d5A(t). | (2) |
Here, we use
The main aim of the present paper is to explore the effects of the recruitment of immune responses on virus infection. The organization of this paper is as follows. In the next section, some preliminary analyzes of the model (2) will be given. Stability of all equilibria are given in Section 3. In Section 4, some numerical simulations are given to explain the effects of
In this section, we will first prove the positivity and boundedness of solutions, and then derive the expression of the basic reproduction number for model (2).
Let
Proposition 1. Under the above initial conditions, all solutions of model (2) are nonnegative. In particular, the solution
Proof. We first verify that
˙V(t2)=rd2I(t2). |
By solving the second equation of model (2), we obtain
I(t2)=e∫t20−(d2+pC(ξ))dξ[I(0)+∫t20βT(θ−τ)V(θ−τ)e−sτe∫θ0(d2+pC(ξ))dξdθ]>0. |
It follows that
I(t)=e∫t0−(d2+pC(ξ))dξ[I(0)+∫t0βT(θ−τ)V(θ−τ)e−sτe∫θ0(d2+pC(ξ))dξdθ]. |
From the above expression of
It follows easily that
Proposition 2. All solutions of model (2) in
Proof. Set
L(t)=T(t)+I(t+τ)+13rV(t+τ)+d23k1C(t+τ)+d34k2rA(t+τ). |
Calculating the derivative of
˙L(t)=λ−d1T(t)−βT(t)V(t)+βT(t)V(t)−d2I(t+τ)−pI(t+τ)C(t+τ)+d23I(t+τ)−d33rV(t+τ)−q3rA(t+τ)V(t+τ)+d23k1λ1+d23I(t+τ)C(t+τ)h1+C(t+τ)−d23k1d4C(t+τ)+d34k2rλ2+d34rA(t+τ)V(t+τ)h2+A(t+τ)−d34k2rd5A(t+τ). | (3) |
Since
C(t+τ)h1+C(t+τ)≤1,A(t+τ)h2+A(t+τ)≤1, |
we obtain
˙L(t)≤λ−d1T(t)−d2I(t+τ)+d23I(t+τ)−d33rV(t+τ)+d23k1λ1+d23I(t+τ)−d23k1d4C(t+τ)+d34k2rλ2+d34rV(t+τ)−d34k2rd5A(t+τ)≤λ+d23k1λ1+d34k2rλ2−d1T(t)−d23I(t+τ)−d3413rV(t+τ)−d4d23k1C(t+τ)−d5d34k2rA(t+τ)≤λ+d23k1λ1+d34k2rλ2−mL(t), |
where
lim supt⟶∞L(t)≤λm+d2λ13k1m+d3λ24k2rm. |
From the first equation of model (2), we get
˙T(t)≤λ−d1T(t). |
It follows that
lim supt⟶∞T(t)≤λd1. |
Set
˙F(t)=˙T(t)+˙I(t+τ)≤λ−nF(t),n=min{d1,d2}, |
thus
lim supt⟶∞F(t)≤λn. |
Then,
lim supt⟶∞(T(t)+I(t+τ))≤λn. | (4) |
From the third equation of model (2) and (4), we have
lim supt⟶∞V(t)≤rd2λd3n. |
Further, let
M=max{λd1,λn,rd2λd3n,λm+d2λ13k1m+d3λ24k2rm}. |
The dynamics of model (2) can be analyzed in the following bounded feasible region
Γ={(T,I,V,C,A)∣0≤T≤M,0≤T+I≤M,0≤V≤M,0≤C≤M,0≤A≤M}. |
Based on the concept of the basic reproductive number for an epidemic disease presented in [6, 35], we know the basic reproductive number
From model (2), it is clear that healthy cells, CTL cells and antibody cells will stabilize to
R0=P1(φ1)P2(φ2)=βλd1rd2(d2+pλ1d4)1(d3+qλ2d5)e−sτ. |
Based on the above expression, we know that there are inverse proportional relationship between the basic reproduction number of virus (
In this section, we first discuss the existence of infection-free equilibrium, and then analyze its stability. Besides, using the uniform persistence theory, we obtain the existence of an endemic equilibrium. After that, the stability of an endemic equilibrium was proved by constructing Lyapunov functional.
Apparently, there is always an infection-free equilibrium in system (2):
T0=λd1,C0=λ1d4,A0=λ2d5. |
Next, we discuss the stability of the infection-free equilibrium
Theorem 3.1. When
Proof. First we define a Lyapunov functional
L0=∫T(t)T0(S−T0)SdS+esτI(t)+(1r+pC0rd2)esτV(t)+pesτk1∫C(t)C0(h1+S)(S−C0)SdS+(d2+pC0)rd2qesτk2∫A(t)A0(h2+S)(S−A0)SdS+∫0−τβT(t+θ)V(t+θ)dθ. |
Calculating the time derivative of
˙L0=λ−d1T(t)−βT(t)V(t)−T0T(t)(λ−d1T(t)−βT(t)V(t))+βT(t−τ)V(t−τ)−d2I(t)esτ−pI(t)C(t)esτ+1r(rd2I(t)esτ−d3V(t)esτ−qA(t)V(t)esτ)+pC0rd2(rd2I(t)esτ−d3V(t)esτ−qA(t)V(t)esτ)+pesτk1(h1+C(t)){λ1−d4C(t)+k1I(t)C(t)h1+C(t)−C0C(t)(λ1+k1I(t)C(t)h1+C(t)−d4C(t))}+(d2+pC0)rd2qesτk2(h2+A(t)){λ2+k2A(t)V(t)h2+A(t)−d5A(t)−A0A(t)(λ2+k2A(t)V(t)h2+A(t)−d5A(t))}+βT(t)V(t)−βT(t−τ)V(t−τ). |
Since
.L0=2d1T0−d1T(t)−T0T(t)d1T0+βT0V(t)−pI(t)C(t)esτ−d3rV(t)esτ−qrA(t)V(t)esτ+pI(t)C0esτ−pC0rd2d3V(t)esτ−pC0rd2qA(t)V(t)esτ+pesτk1λ1(h1+C(t))+pI(t)C(t)esτ−pesτk1d4C(t)(h1+C(t))−pesτk1λ1(h1+C(t))C0C(t)−pI(t)C0esτ+pesτk1d4C0(h1+C(t))+(d2+pC0)rd2qesτk2λ2(h2+A(t))+(d2+pC0)rd2esτqA(t)V(t)−(d2+pC0)rd2qesτk2d5A(t)(h2+A(t))−(d2+pC0)rd2qesτk2λ2A0A(t)(h2+A(t))−(d2+pC0)esτrd2qA0V(t)+(d2+pC0)rd2qk2esτd5A0(h2+A(t))=d1T0(2−T(t)T0−T0T(t))+(d2+pC0)(d3+qA0)esτrd2(R0−1)V(t)−pd4esτk1(C(t)−C0)2+pk1λ1h1esτ(2−C0C(t)−C(t)C0) |
−(d2+pC0)qesτrd2k2d5(A(t)−A0)2+(d2+pC0)qesτrd2k2λ2h2(2−A0A(t)−A(t)A0). |
Since the geometric mean is less than or equal to the arithmetical mean, it follows from
D0={(T(t),I(t),V(t),C(t),A(t))|˙L0=0}. |
It is easy to show that
In order to obtain the the existence of an endemic equilibrium, in this subsection, we investigate the uniform persistence of (2). We first introduce a preliminary theory. Let
Lemma 3.2. ([31], Theorem 3) Let
(H1)
(H2) There exists a finite sequence
(ⅰ)
(ⅱ) no subset of
(ⅲ)
(ⅳ)
Then there exists
By applying Lemma 3.2 to (2), we can obtain the following result for the uniform persistence of (2).
Theorem 3.3. If
Proof. Let
X0={˜ϕ∈X+:˜ϕ2(θ)≡0,˜ϕ3(θ)≡0 for θ∈[−τ,0]},X0=X+∖X0,M∂={ψ∈X+:Φt(ψ)∈X0,t≥0}. |
Basic analysis of (2) implies that
Let
{˙T(t)=λ−d1T(t),˙C(t)=λ1−d4C(t),˙A(t)=λ2−d5A(t). | (5) |
It then follows from the result in [14] that
Since
R0=βλd1rd2d2+pλ1d41d3+qλ2d5e−sτ>1, |
we have
(d2+pλ1d4)(d3+qλ2d5)<βλd1rd2e−sτ. | (6) |
Thus, there is sufficiently small
(d2+p(λ1d4+σ))(d3+q(λ2d5+σ))<β(λd1−σ)rd2e−sτ. |
Suppose
(T∗(t),I∗(t),V∗(t),C∗(t),A∗(t))→(λ/d1,0,0,λ1/d4,λ2/d5) as t→+∞. |
For sufficiently large
λd1−σ<T∗(t)<λd1+σ,λ1d4−σ<C∗(t)<λ1d4+σ,λ2d5−σ<A∗(t)<λ2d5+σ, |
if
{˙I∗(t)≥−d2I∗(t)+βV∗(t)(λd1−σ)e−sτ−pI∗(t)(λ1d4+σ),˙V∗(t)≥−d3V∗(t)+rd2I∗(t)−q(λ2d5+σ)V∗(t). | (7) |
Since
Aσ=(−d2−p(λ1d4+σ)β(λd1−σ)e−sτrd2−d3−q(λ2d5+σ)), | (8) |
the non-diagonal elements of (8) are positive, and from (6), we obtain
Now consider the following auxiliary system
{˙I∗(t)=−d2I∗(t)+βV∗(t)(λd1−σ)−pI∗(t)(λ1d4+σ),˙V∗(t)=−d3V∗(t)+rd2I∗(t)−q(λ2d5+σ)V∗(t). | (9) |
Note
Define a continuous function
p(¯ϕ)=min{¯ϕ2(0),¯ϕ3(0)},∀¯ϕ∈X+. |
It is clear that
Furthermore, from the first equation of (2), Proposition 1 and the above results, we have
˙T(t)=λ−d1T(t)−βT(t)V(t)>λ−d1T(t)−βMT(t)=λ−(d1+βM)T(t), |
Thus,
lim inft→+∞T(t)>λd1+βM. |
From the fourth equation of (2),
˙C(t)=λ1+k1I(t)C(t)h1+C(t)−d4C(t)≥λ1−d4C(t), |
we have
lim inft→+∞C(t)≥λ1d4. |
From the fifth equation of (2),
˙A(t)=λ2+k2A(t)V(t)h1+A(t)−d5A(t)≥λ2−d5A(t), |
Therefore, taking
lim inft→+∞T(t)≥ε,lim inft→+∞I(t)≥ε,lim inft→+∞V(t)≥ε,lim inft→+∞C(t)≥ε,lim inft→+∞A(t)≥ε |
are valid for any solution of system (2) with initial condition in
From the Theorem 3.1, we are easy to get that
Now, we discuss the stability of the endemic equilibrium
Theorem 3.4. When
Proof. Set
m1=βT1V1rd2I1. |
Define a Lyapunov functional
L1=∫T(t)T1(S−T1)SdS+esτ∫I(t)I1(S−I1)SdS+m1∫V(t)V1(S−V1)SdS+pesτk1∫C(t)C1(h1+C(t))(S−C1)SdS+m1qk2∫A(t)A1(h2+A(t))(S−A1)SdS+βT1V1∫0−τ(T(t+θ)V(t+θ)T1V1−1−lnT(t+θ)V(t+θ)T1V1)dθ. |
Calculating the time derivative of
.L1=λ−d1T(t)−βT(t)V(t)−T1T(t)(λ−d1T(t)−βT(t)V(t))−pI(t)C(t)esτ+βT(t−τ)V(t−τ)−d2I(t)esτ−I1I(t)(βT(t−τ)V(t−τ)−d2I(t)esτ−pI(t)C(t)esτ)+m1(rd2I(t)−d3V(t)−qA(t)V(t))−m1V1V(t)(rd2I(t)−d3V(t)−qA(t)V(t))+pesτk1(h1+C(t)){λ1+k1I(t)C(t)h1+C(t)−d4C(t)−C1C(t)(λ1+k1I(t)C(t)h1+C(t)−d4C(t))}+qm1k2(h2+A(t)){λ2+k2A(t)V(t)h2+A(t)−d5A(t)−A1A(t)(λ2+k2A(t)V(t)h2+A(t)−d5A(t))}+βT(t)V(t)−βT(t−τ)V(t−τ)+βT1V1lnT(t−τ)V(t−τ)T(t)V(t). |
Since
λ=d1T1+βT1V1, βT1V1=(d2I1+pI1C1)esτ, rd2I1=d3V1+qA1V1,λ1+k1I1C1h1+C1=d4C1, λ2+k2A1V1h2+A1=d5A1, |
we have
˙L1=d1T1(2−T(t)T1−T1T(t))+βT1V1−βT1V1T1T(t)+βT1V(t)−I1I(t)βT(t−τ)V(t−τ)+d2I1esτ+pI1C(t)esτ−m1d3V(t)−m1qA(t)V(t)−βT1V1V1I(t)V(t)I1+m1d3V1+m1qA(t)V1+pesτk1λ1(h1+C(t))−pesτk1d4C(t)(h1+C(t))−pesτk1C1C(t)λ1(h1+C(t))+pesτk1d4C1(h1+C(t))+m1qk2λ2(h2+A(t))+m1qA(t)V(t)−m1qk2d5A(t)(h2+A(t))−m1qk2A1A(t)λ2(h2+A(t))−m1qA1V(t)+m1qk2d5A1(h2+A(t))+βT1V1lnT(t−τ)V(t−τ)T(t)V(t)=d1T1(2−T(t)T1−T1T(t))+βT1V1(1−T1T(t)+lnT1T(t))+βT1V1(1 |
−I1T(t−τ)V(t−τ)I(t)T1V1+lnI1T(t−τ)V(t−τ)I(t)T1V1)+βT1V1(1−V1I(t)I1V(t)+lnV1I(t)I1V(t))+pk1λ1h1esτ(2−C(t)C1−C1C(t))−pk1d4esτ(C(t)−C1)2+m1qk2λ2h2(2−A1A(t)−A(t)A1)−m1qk2d5(A(t)−A1)2. |
Since the geometric mean is less than or equal to the arithmetical mean and
D1={(T(t),I(t),V(t),C(t),A(t))|˙L1=0}. |
It is easy to verify that
T1T(t)=I1T(t−τ)V(t−τ)I(t)T1V1=V1I(t)I1V(t)=1. |
Thus,
˙T(t)=λ−d1T1−βT1V(t)=0. |
As a result, we have
{˙I(t)=βT1V1e−sτ−d2I1−pI1C1=0,˙V(t)=rd2I1−d3V1−qA1V1=0, |
which implies
In this section, we implement numerical simulations to explore the effects of the recruitment of immune responses (
The all parameter values are shown in Table 1.
Par. | Value | Description | Ref. |
0-50 cells ml-day | Recruitment rate of healthy cells | [33,38] | |
| Death rate of healthy cells | [38] | |
| Infection rate of target cells by virus | [33,38] | |
| Death rate of infected cells | [41,46] | |
| Burst size of virus | [38] | |
| Clearance rate of free virus | [38] | |
Killing rate of CTL cells | [41,40] | ||
| Neutralizing rate of antibody | [41] | |
| Proliferation rate of CTL response | [2,41] | |
| Production rate of antibody response | [41] | |
| Mortality rate of CTL response | [2,40] | |
| Clearance rate of antibody | [41] | |
| 1/s is the average time | [32,47] | |
| Virus replication time | [38] | |
1200 | Saturation constant | Assumed | |
1500 | Saturation constant | Assumed | |
Varied | Rate of CTL export from thymus | [9] | |
Varied | Recruitment rate of antibody | [9] |
Figure 1 illustrates that
In this paper, the global dynamics of a within-host model with immune responses and intracellular time delay has been studied. By the method of Lyapunov functional and persistence theory, we obtain the global stability of the model (2) are completely determined by the values of the reproductive number. The results imply that the complicated behaviors such as backward bifurcations and Hopf bifurcations do not exist in the model with both immune responses and time delay.
Considering the basic reproductive number of virus
R0=R(τ)=λβrd2e−sτd1(d2+pλ1d4)(d3+qλ2d5) |
as a function of
The authors are very grateful to the anonymous referees for their valuable comments and suggestions. This research is supported by the National Natural Science Fund of P. R. China (No. 11271369).
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1. | Zhijun Liu, Lianwen Wang, Ronghua Tan, Spatiotemporal dynamics for a diffusive HIV-1 infection model with distributed delays and CTL immune response, 2022, 27, 1531-3492, 2767, 10.3934/dcdsb.2021159 |
Par. | Value | Description | Ref. |
0-50 cells ml-day | Recruitment rate of healthy cells | [33,38] | |
| Death rate of healthy cells | [38] | |
| Infection rate of target cells by virus | [33,38] | |
| Death rate of infected cells | [41,46] | |
| Burst size of virus | [38] | |
| Clearance rate of free virus | [38] | |
Killing rate of CTL cells | [41,40] | ||
| Neutralizing rate of antibody | [41] | |
| Proliferation rate of CTL response | [2,41] | |
| Production rate of antibody response | [41] | |
| Mortality rate of CTL response | [2,40] | |
| Clearance rate of antibody | [41] | |
| 1/s is the average time | [32,47] | |
| Virus replication time | [38] | |
1200 | Saturation constant | Assumed | |
1500 | Saturation constant | Assumed | |
Varied | Rate of CTL export from thymus | [9] | |
Varied | Recruitment rate of antibody | [9] |
Par. | Value | Description | Ref. |
0-50 cells ml-day | Recruitment rate of healthy cells | [33,38] | |
| Death rate of healthy cells | [38] | |
| Infection rate of target cells by virus | [33,38] | |
| Death rate of infected cells | [41,46] | |
| Burst size of virus | [38] | |
| Clearance rate of free virus | [38] | |
Killing rate of CTL cells | [41,40] | ||
| Neutralizing rate of antibody | [41] | |
| Proliferation rate of CTL response | [2,41] | |
| Production rate of antibody response | [41] | |
| Mortality rate of CTL response | [2,40] | |
| Clearance rate of antibody | [41] | |
| 1/s is the average time | [32,47] | |
| Virus replication time | [38] | |
1200 | Saturation constant | Assumed | |
1500 | Saturation constant | Assumed | |
Varied | Rate of CTL export from thymus | [9] | |
Varied | Recruitment rate of antibody | [9] |