
Citation: Xinqi Dong, Manrui Zhang, Melissa A. Simon. Self-mastery among Chinese Older Adults in the Greater Chicago Area[J]. AIMS Medical Science, 2014, 1(1): 57-72. doi: 10.3934/medsci.2014.1.57
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Some human pathogens, such as hepatitis B virus (HBV), hepatitis C virus (HCV) and human immunodeficiency virus (HIV), possess the ability to suppress the immune response and damage the immune system, ultimately leading to the development of certain chronic diseases and even tumors [1]. In recent years, researchers have extensively studied numerous models pertaining to these infections [2,3,4]. In recent years, there has been notable progress in treating these infections. For example, the latest direct-acting antiviral drugs (DAA) for HCV can cure the majority of cases quickly, with cure rates reaching up to 95%[5,6]. For HBV, the optimal approach involves long-term medication to manage the condition, reduce viral transmission and stabilize host immune system[7,8]. Similarly, antiretroviral therapy (ART) is commonly used to suppress HIV replication[9]. Notably, a small group of elite controllers can naturally manage HIV without treatment[10]. However, challenges persist, including drug resistance and the risk of viral replication or recurrence[5,6,7,8,9]. In the past few years, immune checkpoint therapy, a method that shows significant efficacy in completely eliminating tumor cells for certain types of tumors, has been discovered, which activates the host's own immune system to inhibit tumor growth and spread [11,12,13]. Enhancing antiviral treatment by strengthening specific immune responses has become a hot topic among researchers.
In 2003, Komarova et al. [1] first established an immunosuppressive infection model to investigate the relationship between drug efficacy and treatment duration, the model is given as follows:
$ {dydt=ry(1−yK)−ay−pyz,dzdt=cyz1+ηy−bz−qyz:=zf(y), $
|
(1.1) |
where $ y $, $ z $ denote the population size of viruses and immune cells at time $ t $, respectively. Viral reproduction follows logistic growth, $ r $ represents the viral replication rate at a viral load $ K $. It is hypothesized that the virus population undergoes decay at a rate $ a $, and immune cells eliminate the viruses at a rate $ pyz $. Furthermore, immune cells are suppressed by the viruses at a rate $ qyz $ and die at a rate $ b $. In the context of antigenic stimulation, the proliferation of immune cells relies on both the viruses and immune cells, and it can be quantified by the expansion function
$ \Phi(y, z) = \frac{cyz}{1+\eta y}, $ |
where $ \eta $ represents the inhibitory effects of the viruses on the proliferation of immune cells.
In 2014, Shu et al. [14] introduced a time delay in the immune expansion function of the model described in [1] and studied the existence of Hopf bifurcation and periodic solutions. In 2017, Tian et al. [15] incorporated diffusion into the model presented in [14] and studied the direction and stability of Hopf bifurcations. However, viruses require a certain amount of time to effectively stimulate immune cells. We refer to this time period as the virus stimulation delay, denoted as $ \tau $. The immune cells at time $ t $ are essentially activated by the viruses at time $ t-\tau $. Therefore, in 2021, Li et al. [16] studied the function as follows:
$ Φ(yτ,z)=cy(t−τ)z1+ηy(t−τ). $
|
However, it has been noted by Wodarz (see pp 31, Chapter 2 in [17]) that the growth rate of immune cells, denoted as $ zf(y) $, in all of these models was assumed to be directly proportional to $ z $, but nonlinearity can better capture the realistic dynamics. Thus, a more realistic model, proposed by De Boer and Perelson [18,19] in 1995–1997, has been suggested to provide a better representation of the actual dynamics. In this model, the immune expansion function is described by the well-known Beddington-DeAngelis functional response [20,21], the form of which is as follows:
$ Φ(y,z)=cyz1+ηy+sz. $
|
It was initially used in predator-prey models [22,23,24]. The parameter $ s $ in the function represents the strength of competition between immune responses. This immune expansion function provides a more biologically realistic description of the interactions and competitive relationships between virus populations and immune cell populations within a host [25]. Thus the Beddington-DeAngelis functional response has also been widely discussed in immunosuppressive infection models [26,27,28].
In 2023, Chen et al. [29] introduced the Beddington-DeAngelis functional response into the immunosuppressive infection model established by Komarova et al. [1], the model is as follows:
$ {dydt=ry(1−yK)−ay−pyz,dzdt=cy(t−τ)z1+ηy(t−τ)+sz−bz−qyz. $
|
(1.2) |
In [29], Chen et al. discussed the local and global stability of equilibrium points in the absence of delay, as well as the stability and Hopf bifurcation of model (1.2) when delay is present. They designed antiviral drug treatment strategies from the perspectives of adaptivity and feasibility of immune competition strength and viral suppression intensity.
In practical problems, the spread of viruses within a host is possible. Thus we introduce the diffusion into the model proposed by Chen et al. in [29]. As far as we know, this task has not been attempted yet. Therefore, we study the reaction-diffusion system under the Neumann boundary condition as follows:
$ {∂y(x,t)∂t=d1Δy(x,t)+ry(x,t)(1−y(x,t)K)−ay(x,t)−py(x,t)z(x,t),∂z(x,t)∂t=d2Δz(x,t)+cy(x,t−τ)z(x,t)1+ηy(x,t−τ)+sz(x,t)−bz(x,t)−qy(x,t)z(x,t),∂νy(x,t)=∂νz(x,t)=0,x∈∂Ω,t>0,y(x,t)=ψ1(x,t),z(x,t)=ψ2(x,t),x∈Ω,t∈[−τ,0], $
|
(1.3) |
where $ \Omega = (0, l \pi) $ with $ l > 0 $, $ y = y(x, t) $ and $ z = z(x, t) $ denoting the population densities of viruses and immune cells at location $ x $ and time $ t $, respectively. The parameters $ s $ and $ \tau $ stand for the competition intensity between immune responses and the viral stimulation delay, respectively. The diffusion rates of viruses and immune cells are $ d_1 $ and $ d_2 $, respectively. The same parameters as in model (1.1) have the same biological meanings. Note that all parameters in (1.3) remain nonnegative.
The remaining sections of the paper are organized as follows: Section 2 discusses the local asymptotic stability of the positive constant steady state solution and the existence of Hopf bifurcation. In Section 3, we derive the normal form of the Hopf bifurcation by using the multiple time scales (MTS) method. Section 4 presents numerical simulations to illustrate the theoretical analysis, and the conclusion is given in Section 5.
In this section, we concentrate on studying the local asymptotic stability of the positive constant steady state solution and the existence of Hopf bifurcation for system (1.3). From a biological perspective, we only focus on the positive constant steady state solution $ (y_*, z_*) $. It is evident that $ (y_*, z_*) $ satisfies the following equations:
$ ry∗(1−y∗K)−ay∗−py∗z∗=0,cy∗z∗1+ηy∗+sz∗−bz∗−qy∗z∗=0. $
|
(2.1) |
Then we can write $ y_* $ in terms of $ z_* $ as
$ y∗=K(1−a+pz∗r), $
|
(2.2) |
substituting Eq (2.2) into Eq (2.1), we obtain
$ Az2+Bz+C=0, $
|
where
$ A=−qηK2p2+Kpqrs,B=2qη(r−a)pK2+[r(bη−c+q)p−(r−a)rqs]K−br2s,C=−qη(r−a)2K2+r(bη−c+q)(a−r)K−br2. $
|
(2.3) |
Denote
$ Δ1△=B2−4AC. $
|
(2.4) |
Lemma 1. For the positive constant steady state solution of system (1.3), we have the following results.
(i) If $ \Delta_1 > 0 $, $ AC > 0, \; AB < 0, \mathit{\text{and}} a+pz_* < r $ holds, then system (1.3) has two positive constant steady state solutions, $ E_{*_{1}} = (y_{*_{1}}, z_{*_{1}}) $ and $ E_{*_{2}} = (y_{*_{2}}, z_{*_{2}}) $. Here $ z_{*_{1}} = \frac{-B + \sqrt{\Delta_1}}{2A} $, $ z_{*_{2}} = \frac{-B - \sqrt{\Delta_1}}{2A} $, with $ y_{*_{1}} = K\left(1 - \frac{a + pz_{*_{1}}}{r}\right) $ and $ y_{*_{2}} = K\left(1 - \frac{a + pz_{*_{2}}}{r}\right) $;
(ii) If $ \Delta_1 > 0 $, $ AC < 0, \mathit{\text{and}} a+pz_* < r $ holds, then system (1.3) has one positive constant steady state solution $ E_{*_{3}} = (y_{*_{3}}, z_{*_{3}}) $. Here $ z_{*_{3}} = \frac{-B + \sqrt{\Delta_1}}{2A} $, with $ y_{*_{3}} = K\left(1 - \frac{a + pz_{*_{3}}}{r}\right) $;
(iii) If $ \Delta_1 = 0 $, $ AB < 0, \mathit{\text{and}} a+pz_* < r $ holds, then system (1.3) has one positive constant steady state solution $ E_{*_{4}} = (y_{*_{4}}, z_{*_{4}}) $. Here $ z_{*_{4}} = \frac{-B}{2A} $, with $ y_{*_{4}} = K\left(1 - \frac{a + pz_{*_{4}}}{r}\right) $.
Without loss of generality, we denote $ E_{*_{1}} $, $ E_{*_{2}} $, $ E_{*_{3}} $, and $ E_{*_{4}} $ as $ E_* $. Therefore, when the parameters of system (1.3) satisfy any one of the conditions (ⅰ)–(ⅱ) of Lemma 1, the positive constant steady state solution $ E_* = (y_*, z_*) $ exists.
Now, we analyze the stability of $ E_* = (y_*, z_*) $. Set $ U(x, t) = (y(x, t), z(x, t))^{\mathrm{T}} $, then the linearization of system (1.3) at $ E_* $ can be written as
$ ∂U(x,t)∂t=DΔU(x,t)+A∗U(x,t)+B∗U(x,t−τ), $
|
(2.5) |
where
$ D=(d100d2),A∗=(−ry∗K−py∗−qz∗−z∗s(b+qy∗)2cy∗),B∗=(00z∗(b+qy∗)[c−η(b+qy∗)]cy∗0). $
|
Therefore, the characteristic equation of the linearization of system (1.3) at $ E_* $ can be obtained as
$ λ2+Tnλ+Nn+C0(e−λτb0−q)=0,n∈N0={0,1,2,3,...}, $
|
(2.6) |
with
$ Tn=n2l2d1+n2l2d2+ry∗K+z∗s(b+qy∗)2cy∗,Nn=n4l4d1d2+n2l2d1⋅z∗s(b+qy∗)2cy∗+n2l2d2⋅ry∗K+rz∗s(b+qy∗)2Kc,C0=py∗z∗,b0=(b+qy∗)[c−η(b+qy∗)]cy∗. $
|
Lemma 2. When $ \tau = 0 $, the positive constant steady state solution $ E_* $ of system (1.3) is locally asymptotically stable if $ b_0-q > 0 $.
Proof. When $ \tau = 0 $, characteristic Eq (2.6) becomes
$ λ2+Tnλ+Nn+C0(b0−q)=0,n∈N0. $
|
(2.7) |
It is clear that $ T_n > 0 $, $ N_n > 0 $, thus if $ b_0-q > 0 $ holds, the real parts of the roots of Eq (2.6) are negative. Thus, $ E_* $ is locally asymptotically stable.
This completes the proof.
Suppose that $ \lambda = $i$ \omega\; (\omega > 0) $ is a root of Eq (2.6), then we have
$ {C0b0cos(ωτ)=ω2+C0q−Nn,C0b0sin(ωτ)=Tnω. $
|
(2.8) |
Square and add above equations, then denote $ g\mathrel{\overset{\triangle}{ = }}\omega^2. $ We obtain
$ φ(g)△=g2+(2C0q−2Nn+T2n)g+(C0q−Nn)2−C20b20=0. $
|
(2.9) |
Next, we will discuss the existence of $ \omega $. For the sake of convenience in our discussion, we present the following assumed conditions:
$ (H1)Δ2=0,2C0q−2Nn+T2n<0,(H2)Δ2>0,(C0q−Nn)2−C20b20<0,(H3)Δ2>0,2C0q−2Nn+T2n<0,(C0q−Nn)2−C20b20>0,(H4)Δ2<0,(H5)Δ2>0,2C0q−2Nn+T2n>0,(C0q−Nn)2−C20b20>0, $
|
where $ \Delta_2 = (2C_0 q - 2 N_n+T_n^2)^2-4\left[(C_0 q-N_n)^2-C_0^2 b_0^2\right] $, and $ N_n, T_n, b_0, C_0 $ are given in Eq (2.6).
Lemma 3. When $ \tau > 0 $, concerning the cases of positive roots of Eq (2.9), we have the following results:
(i) Equation (2.9) has a unique positive root if $ \mathrm{(H1)} $ or $ \mathrm{(H2)} $ is satisfied, then the positive constant steady state solution of system (1.3) will not exhibit stability switches,
(ii) Equation (2.9) has two positive roots if $ \mathrm{(H3)} $ is satisfied, then the positive constant steady state solution of system (1.3) will exhibit stability switches,
(iii) Equation (2.9) has no positive root if $ \mathrm{(H4)} $ or $ \mathrm{(H5)} $ is satisfied.
Proof. If the condition $ \mathit{\text{(H1)}} $ is satisfied, we have $ g_1 = g_2 $, $ g_1+g_2 = -(2C_0 q - 2 N_n+T_n^2) > 0 $. If the condition $ \mathit{\text{(H2)}} $ is satisfied, we have $ g_1\cdot g_2 = (C_0 q-N_n)^2-C_0^2 b_0^2 < 0 $. Both of them imply that there is a unique positive root $ g $ satisfying Eq (2.9). This proves (ⅰ).
Similarly, we conclude (ⅱ)–(ⅲ).
Considering the scenario with only a pair of purely imaginary roots, the unique positive $ \omega $ that satisfies Eq (2.9) can be described as follows:
$ ω+n=√−(2C0q−2Nn+T2n)+√Δ22,n∈N0. $
|
(2.10) |
Denote
$ Rn△=sin(ω+nτ)=Tnω+nC0b0,Qn△=cos(ω+nτ)=(ω+n)2+C0q−NnC0b0. $
|
(2.11) |
Now, define a set
$ I1={n∈N0|such that (H1) or (H2) holds}. $
|
(2.12) |
From Eq (2.11), we obtain
$ τ+n,j={1ω+n(arccos(Qn)+2jπ),forRn≥0,1ω+n(2π−arccos(Qn)+2jπ),forRn<0.n∈I1,j∈N0. $
|
(2.13) |
From Eq (2.13), we find that $ \{\tau_{n, j}^+\}_{j = 0}^{\infty} $ is increasing about $ j $ for some fixed $ n \in I_1 $. Thus, for fixed $ n $, one has $ \tau_{n, 0}^+ = \min _{j \in \mathbb{N}_0}\left\{\tau_{n, j}^ + \right\} $. To investigate the stability of $ E_* $, we define the smallest critical value of time delay
$ τc=min{τ+n,0|n∈I1}. $
|
(2.14) |
Theorem 1. If $ b_0-q > 0 $ and one of the two conditions $ {{\text{(H1)}}} $ or $ {{\text{(H2)}}} $ is satisfied, then the positive constant steady state solution $ E_* $ of system (1.3) is locally asymptotically stable when $ \tau \in [0, \tau_c) $, and unstable when $ \tau \in \left(\tau_c, \infty\right) $. Moreover, system (1.3) undergoes the Hopf bifurcation at $ E_* $ when $ \tau = \tau_c $, where $ \tau_c $ is given in Eq (2.14).
Proof. Differentiating the both sides of the characteristic Eq (2.6) with respect to time delay $ \tau $ gives rise to
$ (dλdτ)−1=2λ+Tnλ[C0q−(λ2+Tnλ+Nn)]−τλ. $
|
As a result, one has
$ (Redλdτ)−1|τ=τ+n,j=√Δ2T2n(ω+n,j)2+[Nn−C0q−(ω+n,j)2]2>0,n∈I1,j∈N0. $
|
(2.15) |
This means that the transversality condition is satisfied.
This completes the proof.
We have obtained that under certain conditions, the characteristic Eq (2.6) of system (1.3) has a unique pair of purely imaginary roots $ \lambda = \pm $i$ \omega_n^+ $, then system (1.3) undergoes the Hopf bifurcation at $ E_* $ when $ \tau = \tau_c $, with $ \tau_c $ as given at $ (2.14) $. In this section, we will utilize the method of MTS to derive the normal form of the Hopf bifurcation.
Denote that
$ f(y(x,t−τ),z(x,t))△=cy(x,t−τ)z(x,t)1+ηy(x,t−τ)+sz(x,t) $
|
(3.1) |
and
$ L=1+ηy∗+sz∗,α=1+sz∗,β=1+ηy∗. $
|
(3.2) |
Perform a Taylor expansion of $ f(y(x, t-\tau), z(x, t)) $ at $ E_* $, then we obtain
$ f(y(x,t−τ),z(x,t))=cy∗z∗L+cz∗αL2(y(x,t−τ)−y∗)+cy∗βL2(z(x,t)−z∗)−cηz∗αL3(y(x,t−τ)−y∗)2+c(2ηy∗z∗s+ηy∗+α)L3(y(x,t−τ)−y∗)(z(x,t)−z∗)−csy∗βL3(z(x,t)−z∗)2+cη2z∗αL4(y(x,t−τ)−y∗)3+cη[3sz∗αL4−1+2sz∗L3](y(x,t−τ)−y∗)2(z(x,t)−z∗)+cs[−1−z∗s(1+2ηy∗)+η2y2∗]L4(y(x,t−τ)−y∗)(z(x,t)−z∗)2+cs2y∗βL4(z(x,t)−z∗)3+⋯. $
|
(3.3) |
Now, we set $ \hat{y}(x, t) = y(x, \tau t)-y_* $, $ \hat{z}(x, t) = z(x, \tau t)-z_* $, and still denote $ \hat{y}(x, t) $ and $ \hat{z}(x, t) $ by $ y(x, t) $ and $ z(x, t) $, respectively. Then system (1.3) takes the form
$ {∂y(x,t)∂t=τ[d1Δy(x,t)+r(y(x,t)+y∗)(1−y(x,t)+y∗K)−a(y(x,t)+y∗)−p(y(x,t)+y∗)(z(x,t)+z∗)],∂z(x,t)∂t=τ[d2Δz(x,t)+cz∗αL2y(x,t−1)+cy∗βL2z(x,t)−cηz∗αL3y2(x,t−1)+c(2ηy∗z∗s+ηy∗+α)L3y(x,t−1)z(x,t)−csy∗βL3z2(x,t)+cη2z∗αL4y3(x,t−1)+cη[3sz∗αL4−1+2sz∗L3]y2(x,t−1)z(x,t)+cs[−1−z∗s(1+2ηy∗)+η2y2∗]L4y(x,t−1)z2(x,t)+cs2y∗βL4z3(x,t)−bz(x,t)−q[y(x,t)z(x,t)+y(x,t)z∗+z(x,t)y∗]],∂νy(x,t)=∂νz(x,t)=0,x∈∂Ω,t>0,y(x,t)=ψ1(x,t),z(x,t)=ψ2(x,t),x∈Ω=(0,lπ),t∈[−τ,0]. $
|
(3.4) |
Let $ h = \left(h_{11}, h_{12}\right)^{\mathrm{T}} $ be the eigenvector of the characteristic matrix of the linearization of system Eq (3.4) at $ E_* $ corresponding to the eigenvalue $ \mathrm{i} \omega \tau $ and $ h^* = d\left(h_{21}, h_{22}\right)^{\mathrm{T}} $ be the eigenvector of the adjoint matrix corresponding to the eigenvalue $ -\mathrm{i} \omega \tau $. They satisfy the inner product
$ \langle h^*, h\rangle = \bar{h}^*\mathrm{^T}\cdot h = 1. \quad $ |
By calculating, we have
$ h11=1,h12=−1py∗(iω+n2l2d1+rKy∗),h21=−1py∗[−iω+n2l2d2+z∗s(b+qy∗)2cy∗],h22=1,d=(h21ˉh11+h22ˉh12)−1. $
|
(3.5) |
Consider the delay $ \tau $ as a bifurcation parameter, let $ \tau = \tau_c+\varepsilon \mu $, where $ \tau_c $ is the critical point of the Hopf bifurcation, given as Eq (2.14). $ \mu $ is a small perturbation parameter, and $ \varepsilon $ is the characteristic scale used for non-dimensionalization. The solution of Eq (3.4) is assumed to take the form:
$ U(x,t)=U(x,T0,T1,T2,⋯)=+∞∑k=1εkUk(x,T0,T1,T2,⋯), $
|
(3.6) |
where
$ U(x,T0,T1,T2,⋯)=(y(x,T0,T1,T2,⋯),z(x,T0,T1,T2,⋯))T,Uk(x,T0,T1,T2,⋯)=(yk(x,T0,T1,T2,⋯),zk(x,T0,T1,T2,⋯))T. $
|
(3.7) |
The derivative with respect to $ t $ is now transformed into
$ ∂∂t=∂∂T0+ε∂∂T1+ε2∂∂T2+⋯=D0+εD1+ε2D2+⋯, $
|
(3.8) |
with the differential operator $ D_i = \frac{\partial}{\partial T_i}, \; i \in \mathbb{N}_0. $ Let
$ yj=yj(x,T0,T1,T2,⋯),yj,1=yj(x,T0−1,T1,T2,⋯),zj=zj(x,T0,T1,T2,⋯),zj,1=zj(x,T0−1,T1,T2,⋯), $
|
(3.9) |
where $ j\in \mathbb{N} = \{1, 2, 3, ...\} $. According to Eqs (3.7) and (3.8), we get
$ ∂U∂t=εD0U1+ε2D1U1+ε3D2U1+ε2D0U2+ε3D1U2+ε3D0U3+⋯. $
|
(3.10) |
To deal with the delayed terms, we expand $ y(x, t-1) $ at $ y(x, T_0-1, T_1, T_2, \cdots) $, then we have
$ y(x,t−1)=εy11+ε2y21+ε3y31−ε2D1y11−ε3D1y21−ε3D2y11+⋯, $
|
(3.11) |
where $ y_{j1} = y_j\left(x, T_0-1, T_1, T_2, \cdots\right), \; j\in \mathbb{N} $.
Substituting Eqs (3.6)–(3.11) into Eq (3.4) will lead to some perturbation equations. By collecting the coefficients with respect to different powers of $ \varepsilon $, we can obtain a set of ordered linear differential equations. First, for the $ \varepsilon $-order terms, we have
$ {D0y1−τc[ry1−ay1+d1Δy1−2ry∗y1K−py1z∗−py∗z1]=0,D0z1−τc[d2Δz1+cz∗αy11L2+cy∗βz1L2−bz1−q(y∗z1+y1z∗)]=0, $
|
(3.12) |
where $ L, \alpha, \beta $ are given by Eq (3.2). Thus the solution of Eq (3.12) can be expressed in the form as follows:
$ y1=G(T1,T2,⋯)eiωτcT0h11cos(nx)+c.c.,z1=G(T1,T2,⋯)eiωτcT0h12cos(nx)+c.c., $
|
(3.13) |
where $ h_{11}, h_{12} $ are given by Eq (3.5), and $ c. c. $ represents the complex conjugate of the previous term. Next, for the $ \varepsilon^2 $-order terms, we obtain
$ {D0y2−τc[ry2−ay2+d1Δy2−2ry∗y2K−py2z∗−py∗z2]=−D1y1−τcry21K−μay1+μry1+μd1Δy1−μpy1z∗−μpy∗z1−2μry∗y1K−τcpy1z1,D0z2−τc[d2Δz2+cz∗αy21L2+cy∗βz2L2−bz2−q(y∗z2+y2z∗)]=τc[−cz∗αD1y11L2−cηz∗αy211L3+c(2ηy∗z∗s+ηy∗+α)y11z1L3−cy∗sβz21L3−qy1z1]+μ[d2Δz1+cz∗αy11L2+cy∗βz1L2−bz1−q(y∗z1+y1z∗)]−D1z1, $
|
(3.14) |
where $ L, \alpha, \beta $ are given by Eq (3.2). We substitute solution Eq (3.13) into the right side of Eq (3.14), and denote the coefficient vector of $ \mathrm{e}^{\mathrm{i} \omega \tau_c T_0} $ as $ m_1 $, which satisfies the solvability condition
$ \langle h^*, (m_1,\cos(nx))\rangle = 0. $ |
Then one has
$ ∂G∂T1=MμG, $
|
(3.15) |
where
$ M=ˉh21M1+ˉh22M2h11ˉh21+h12ˉh22+τcz∗cαL2e−iωτch11ˉh22, $
|
(3.16) |
with
$ M1=−ah11+rh11−n2d1h11−pz∗h11−py∗h12−2ry∗Kh11,M2=−n2d2h12+cz∗αL2e−iωτch11+cy∗βL2h12−bh12−qy∗h12−qz∗h11. $
|
Suppose that the solution of Eq (3.14) is
$ {y2=+∞∑k=0(η0,kGˉG+η1,ke2iωτcT0G2+ˉη1,ke−2iωτcT0ˉG2)cos(kx),z2=+∞∑k=0(ζ0,kGˉG+ζ1,ke2iωτcT0G2+ˉζ1,ke−2iωτcT0ˉG2)cos(kx). $
|
(3.17) |
Let $ C_k = \langle\cos (n x) \cos (n x), \cos (k x)\rangle = \int_0^{l\pi} \cos (n x) \cos (n x) \cos (k x) \mathrm{d} x $. Besides, we have
$ ∫lπ0cos(nx)cos(kx)dx={0k≠n,lπ2k=n≠0,lπk=n=0. $
|
We substitute solution Eqs (3.13) and (3.17) into Eq (3.14), then make the inner product with $ \cos (kx) $ on both sides of the obtained equation, where $ k\in \mathbb{N}_0 $. By adding the obtained results, we can compare the coefficients in front of terms $ G^2 $ and $ G \bar{G} $, then we get
$ η1,k=Ck(W−(J1+2iω)J2py∗)∫lπ0cos(kx)cos(kx)dx[−(2iω+J1)(l1+2iω)py∗−cz∗αL2e−2iωτc−qz∗],η0,k=Ck(S−J1l2py∗)∫lπ0cos(kx)cos(kx)dx[−J1l1py∗−cz∗αL2−qz∗],ζ1,k=Ck∫lπ0cos(kx)cos(kx)dx⋅J2py∗−l1+2iωpy∗⋅η1,k,ζ0,k=Ck∫lπ0cos(kx)cos(kx)dx⋅l2py∗−l1py∗⋅η0,k, $
|
(3.18) |
where
$ W=−cηz∗αL3e−2iωτch211+c(2ηy∗z∗s+ηy∗+α)L3eiωτc−cy∗sβL3h212−qh11h12,S=−cηz∗αL32h11ˉh11+c(2ηy∗z∗s+ηy∗+α)L3(e−iωτch11ˉh12+eiωτcˉh11h12)−cy∗sβL32h12ˉh12−q(h11ˉh12+h12ˉh11),J1=d2k2−cy∗βL2+b+qy∗,J2=−rKh211−ph11h12,l1=a−r+2ry∗K+pz∗+d1k2,l2=−rK2h11ˉh11−p(h11ˉh12+h12ˉh11),L=1+ηy∗+sz∗,α=1+sz∗,β=1+ηy∗. $
|
Specifically, if $ n = 0 $, the condition for $ C_k \neq0 $ is $ k = 0 $. Thus we consider $ n = k = 0 $, then we have $ C_k = l \pi $, $ \int_0^{l\pi} \cos (n x) \cos (k x) \mathrm{d} x = l \pi $. Thus we can simply get the value of $ \eta_{1, 0} $, $ \eta_{0, 0} $, $ \zeta_{1, 0} $, $ \zeta_{0, 0} $. For the $ \varepsilon^3 $-order terms, we get
$ {D0y3+τc(a−r+2ry∗K+pz∗)y3−d1τcΔy3+τcpy∗z3=μd1Δy2−μ(a−r+2ry∗K+pz∗)y2−D2y1−D1y2−μpy1z1−μpy∗z2−μry21K−2τcry2y1K−τcpy1z2−τcpy2z1,D0z3−τcd2Δz3−τccz∗αL2y31−(τccy∗βL2−b−qy∗)z3=−D2z1−D1z2+τc[−cz∗αD1y21−cz∗αD2y11L2−−2cηz∗αD1y211+2cηz∗αy11y21L3+c(2ηy∗z∗s+ηy∗+α)y11z2L3+c(2ηy∗z∗s+ηy∗+α)(−D1y11z1+y21z1)L3−2cy∗sβz1z2L3+c(−1+z∗(−2ηy∗−1)s)sy11z21L4+cy∗s2βz31L4+cz∗η2αy311L4+cη[3sz∗αL4−2sz∗+1L3]y211z1−qy1z2−qy2z1]+μ[d2Δz2+−cz∗αD1y11+cz∗αy21L2+(cy∗βL2−b−qy∗)z2−cηz∗αy211L3+c(2ηy∗z∗s+ηy∗+α)y11z1L3−cy∗βsz21L3−qy1z1−qy2z∗], $
|
(3.19) |
where $ L, \alpha, \beta $ are given by Eq (3.2). We substitute solution Eqs (3.13) and (3.17) into Eq (3.19), then denote the coefficient vector of $ \mathrm{e}^{\mathrm{i} \omega \tau_c T_0} $ as $ m_2 $, which satisfies the solvability condition
$ \langle h^*, (m_2,\cos(nx))\rangle = 0. $ |
Since $ \mu $ is the disturbance parameter and $ \mu^2 $ has very little influence on the result, the items containing $ \mu^2 G $ can be ignored. Then we obtain
$ \frac{\partial G}{\partial T_2} = \chi G^2 \bar{G}, $ |
where
$ χ=RCk+QV, $
|
(3.20) |
with
$ R=ˉh21[−2τcrK(h11+∞∑k=0η0,k+ˉh11+∞∑k=0η1,k)−τcp(h11+∞∑k=0ζ0,k+ˉh11+∞∑k=0ζ1,k+h12+∞∑k=0η0,k+ˉh12+∞∑k=0η1,k)]+ˉh22τc[−2cηz∗αL3(e−iωτch11+∞∑k=0η0,k+e−iωτcˉh11+∞∑k=0η1,k)+c(2ηy∗z∗s+ηy∗+α)L3(e−iωτch11+∞∑k=0ζ0,k+eiωτcˉh11+∞∑k=0ζ1,k+h12+∞∑k=0η0,k+ˉh12+∞∑k=0η1,k)−2cy∗sβL3(h12+∞∑k=0ζ0,k+ˉh12+∞∑k=0ζ1,k)],Q=ˉh22τc[c(−1+z∗(−2ηy∗−1)s)sL4(2e−iωτch11h12ˉh12+eiωτcˉh11h212)+cy∗βs2L4(2h212ˉh12)+cz∗η2αL4(2e−iωτch211ˉh11)+cη[3sz∗αL4−2sz∗+1L3](e−2iωτch211ˉh12+2h11ˉh11h12)−q(h11+∞∑k=0ζ0,k+ˉh11+∞∑k=0ζ1,k+h12+∞∑k=0η0,k+ˉh12+∞∑k=0η1,k)]∫lπ0cos4(nx)dx,V=[h11ˉh21+h12ˉh22+τccz∗αL2e−iωτch11ˉh22]∫lπ0cos(nx)cos(nx)dx,L=1+ηy∗+sz∗,α=1+sz∗,β=1+ηy∗. $
|
Therefore, the normal form of the Hopf bifurcation for system Eq (1.3) reduced on the center manifold is
$ ˙G=MμG+χG2ˉG, $
|
(3.21) |
where $ M $ and $ \chi $ are given by Eqs (3.16) and (3.20), respectively. Now introduce $ G = r \mathrm{e}^{\mathrm{i} \theta } $ and put it into the normal form Eq (3.21), then one has
$ {˙r=Re(M)μr+Re(χ)r3,˙θ=Im(M)μ+Im(χ)r2. $
|
(3.22) |
For the stability of the periodic solutions of system (1.3), we have the following result.
Theorem 2. When $ \frac{\mathrm{Re}(M)\mu}{\mathrm{Re}(\chi)} < 0 $, system Eq (1.3) exists as periodic solutions near the positive constant steady state solution $ E_* $. Moreover,
(i) If $ \mathrm{Re}(M)\mu < 0 $, the bifurcating periodic solutions reduced on the center manifold are unstable, and when $ \mu > 0 $ ($ \mu < 0 $), the direction of bifurcation is forward (backward);
(ii) If $ \mathrm{Re}(M)\mu > 0 $, the bifurcating periodic solutions reduced on the center manifold are stable, and when $ \mu > 0 $ ($ \mu < 0 $), the direction of bifurcation is forward (backward).
In this section, we will perform numerical simulations for different values of $ \tau $. Based on the work of Komarova et al. [1], we fix
$ a=3.0day−1,p=1.0mm3cells−1day−1,q=1.0mm3virus−1day−1,r=6.0day−1. $
|
For the remaining parameters, we consider the actual meaning of the parameters and choose
$ d1=0.4mm−3day−1,K=10virusmm−3,b=2day−1,d2=0.1mm−3day−1,c=6.5mm3virus−1day−1,η=0.35mm−1virus,s=0.02mm3immunecells−1,l=2. $
|
It is easy to check that the condition (ⅱ) of Lemma 1 is satisfied, which implies that the system (1.3) has only one positive constant steady state solution,
$ E_* = (y_*,z_*) = (0.4600,2.7239). $ |
In addition, we can also find that this set of parameters satisfies the condition $ \text{(H2)} $ of Lemma 3. Furthermore, by Eqs (2.10), (2.12) and (2.14), we obtain
$ ω+0=2.1289,I1={n=0,1,2,3},τc=τ+0,0=0.0667. $
|
Therefore, by Theorem 2.1, $ E_* $ is locally asymptotically stable when $ \tau \in [0, \tau_c) = [0, 0.0667) $, and unstable when $ \tau \in \left(0.0667, +\infty\right) $. Take $ \tau = 0.055 \in [0, 0.0667) $ as an example, then $ E_* $ is locally asymptotically stable as is illustrated by the computer simulations (see Figure 1).
Biologically, when the virus stimulation delay does not exceed the threshold, immune cells can control the virus in the long-term at low levels. Although the virus is not completely eliminated, it does not cause serious disease deterioration, this is the ideal scenario that we hope for.
According to Eqs (3.16) and (3.20), we can get
$ M = 0.3874 -10.6020\mathrm{i},\; \; \; \; \chi = -0.4221 - 0.2652\mathrm{i}, $ |
which implies that $ \mathrm{Re}(M) > 0 $, and $ \mathrm{Re}(\chi) < 0 $. From Theorem 2, when $ \mu > 0 $, system (1.3) has stable bifurcating periodic solutions. In addition, the direction of the Hopf bifurcation is forward. Take $ \tau = 0.067\in \left(0.0667, +\infty\right) $ as an example, then system (1.3) generates a spatially homogeneous stable periodic solution (see Figure 2, where Figure 2(a), (c) depict simulations of the virus and immune cell dynamics, respectively, while Figure 2(b), (d) represent their respective local zoom-in plots).
Biologically, when the virus stimulation delay slightly exceeds the threshold $ \tau_c $, the densities of immune cells and virus exhibits periodic oscillations. As the number of immune cells increases, they effectively control the virus and reduce the viral load. After a while, an immunoevasion mechanism for the virus may emerge and cause the number of immune cells to drop, allowing the virus to remultiply. This scenario is not ideal, but it is still manageable.
Once $ \tau $ significantly exceeds the threshold, take $ \tau = 0.076\in \left(0.0667, +\infty\right) $ as an example, the virus proliferates rapidly (see Figure 3). This is the worst-case scenario, as the virus could spread rapidly and pose significant risks to the host.
In this paper, we incorporated diffusion into an immunosuppressive infection model with virus stimulation delay and Beddington-DeAngelis functional response. We studied the stability of the constant steady state solution of the model and provided conditions for the occurrence of Hopf bifurcation. Moreover, we analyzed the stability and direction of the Hopf bifurcation, and further derived the normal form. By analyzing the model, we obtained a threshold $ \tau_c $ for the virus stimulation delay $ \tau $ and discovered that when $ \tau $ was less than this threshold, the constant steady state solution was locally asymptotically stable. However, once $ \tau $ exceeded the threshold, the constant steady state solution lost stability, and the model exhibited stable periodic solutions. This implied that when the virus could quickly stimulate immune cells, the host possessed strong immunity, which could inhibit virus growth and spread. Conversely, rapid virus proliferation posed significant risks to the host. The discovery of the threshold could help biologists predict, judge, and control the growth of virus cells and select effective treatment strategies.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This study was funded by Fundamental Research Funds for the Central Universities of China (Grant No.2572022DJ07).
The authors declare there is no conflicts of interest.
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