Citation: Jinliang Wang, Xiu Dong. Analysis of an HIV infection model incorporating latency age and infection age[J]. Mathematical Biosciences and Engineering, 2018, 15(3): 569-594. doi: 10.3934/mbe.2018026
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Determining the threshold dynamics of infection-free and infection equilibrium in viral infection model has made great progress in the last decades [1,2,4,8,9,10,16,17,24,31,36,32,33,34,35,40,44]. A key insight in this progress is that if threshold value (named, the basic reproduction number) is less than one then the infection-free equilibrium is globally asymptotically stable otherwise the endemic equilibrium attracts all solutions (is globally asymptotically stable) whenever it exists. One method adopted here is due to the classical Volterra type Lyapunov function, which was discovered by Volterra [30]. These confirmed global stability properties of steady states for within host virus model establish our understanding the virus dynamical behaviors, that is, whether the viruses die out or not.
Even large discrete and continuous delay differential equations of viral infection models have been successfully treated by Volterra type Lyapunov function, (ⅰ) nonlinear incidence rate functions [8,24,31]; (ⅱ) discrete delays [10,16,35] and finite distributed delays [17,31,36], and infinite distributed delays [8,24,34]; (ⅲ) immune responses [24,35,44,39,41]; and (ⅳ) additional infection processes [31]. It is still a hot topic in in-host model to determine how these factors affect the virus dynamical behaviors. We also refer the reader to see these citations for more references.
Recently, age-structured viral infection model has attracted much attentions of researchers. HIV latency remains a major obstacle to viral elimination. Although HIV-1 replication can be controlled by antiretroviral therapy in suppress the plasma viral load to below the detection limit, the time spent in this progress may last half life of months or years [21]. Virus persisting in reservoirs, such as latency infected CD4+ T cells, may the reason that long-term low viral load persistence in patients on antiretroviral therapy and keeping the virus from being eliminated. These latency infected CD4+ T cells are not affected by immune responses but can produce virus once activated by relevant antigens.
Some recent studies reveals the decay dynamics of the latent reservoir. For example, a model has been developed by Muller et al. [11] to describe the heterogeneity of latent cell activation. An ordinary differential equations (ODEs) model has been studied by Kim and Perelson [7] to include decreasing activation of latently infected cells. Activation of these latently infected cells needs specificity antigen. A recent study by Strain et al. [22] reveals that the dynamics of latently infected CD4+ T cells are often heterogeneous. They argued that cells specific to frequently encountered antigens are activated soon while cells specific to rare antigens need more time to be activated. Thus, the activation rate depends on the time spent since the cell is latently infected (that is the time elapsed since the establishment of latency), which we refer as latency age for short. A recent paper by Alshorman et al. [1] introduced a latency age model to mathematically analyze the dynamics of the latent reservoir under combination therapy. They give an affirmed answers that the long-term activation rate of latently infected cells plays an important role in determining the dynamics.
Taking into account the picture that the mortality rate and viral production rate of infected cells may depend on the infection age of cells, Nelson et al. [15], Huang et al. [4] and Wang et al. [37] have studied age-structured model of HIV infection by considering age to be a continuous variable rather than be constant in ODEs models. These assumptions lead to a hybrid system of ODEs and partial differential equations (PDEs) formulation and allow us to have a good understanding on productively infected cells. Together with the infinite-dimensional nature of system, this formulation creates some mathematical difficulties in establishing the existence of a global compact attractor, even in other epidemic models (see some relevant references for our discussion on age-structured models, [6,43,42,25,27,26,14]).
Denote by
The following assumptions are a compromise between generality and simplicity.
Assumption 1.1 (ⅰ) There is a small fraction (
(ⅱ) When latently infected cells are activated to become productively infected cells, an age-dependent remove rate
(ⅲ) We assume that production rate of viral particles
Biologically, (ⅰ) of Assumption 1.1 comes from the evidences that a very small fraction of CD4+ T cell infection leads to HIV latency. They don't produce new virus unless activated by antigens, please see [18,19,20]. (ⅱ) of Assumption 1.1 based on the fact the latently infected cell population is very likely to be heterogeneous [22,23]. Cells specific to frequently encountered antigens may be preferentially activated and quickly removed from the reservoir. It may depends on the time elapsed since latent infection and affect the activation rate, that is the reason why we are interested in the latent infection age. (ⅲ) of Assumption 1.1 it is known that viral proteins and unspliced viral RNA accumulate within the cytoplasm of an infected cell, and thus, they actually ramps up [3,12,29]. Therefore, infection age should be incorporated into the model.
In this paper, we introduce the following HIV infection model with latency and infection age,
$ \left\{ \begin{array}{lll} \displaystyle \frac{ dT(t)}{ dt} =h -d T(t)-\beta T(t)V(t), \\ \displaystyle \left(\frac{\partial}{\partial t}+\frac{\partial}{\partial a}\right)e(a,t) =-\theta_1(a)e(a,t),\\ \displaystyle \left(\frac{\partial}{\partial t}+\frac{\partial}{\partial b}\right)i(b,t) =-\theta_2(b)i(b,t),\\ \displaystyle \frac{ dV(t)}{ dt} =\int_0^\infty p(b)i(b,t) db-cV(t), \end{array}\right. $ | (1) |
with boundary and initial conditions
$\begin{align*} \left\{ \begin{array}{lll} \displaystyle e(0,t) = f\beta T(t)V(t),\\ \displaystyle i(0,t) = (1-f)\beta T(t)V(t)+\int_0^\infty \xi(a)e(a,t) da,\\ \displaystyle T(0) =T_0 \ge 0 ,\ e(a,0) =e_0(a)\in L^1_{+}(0,\infty),\\ \displaystyle i(b,0) =i_0(b)\in L^1_{+}(0,\infty),\ V(0) =V_0 \ge 0, \end{array}\right. \end{align*}$ |
where
Mathematically, for the ease of simplicity, we make the following assumptions.
Assumption 1.2 (ⅰ)
(ⅱ) For
$ \bar{\theta_i}: = ess\sup\limits_{a\in[0,\infty)}\theta_i(a)<\infty,\ \ \bar{p}: = ess\sup\limits_{a\in[0,\infty)}p(a)<\infty,\ \ \bar{\xi}: = ess\sup\limits_{a\in[0,\infty)}\xi(a)<\infty, $ |
(ⅲ)
(ⅳ) There exists
(ⅴ) There exists a maximum age
Our goal of the present paper is to adopt previous model in [32,33] by incorporating the latency age for infected cells as discussed in [1], and to study the threshold dynamics of infection-free and infection equilibrium in viral infection model subject to latently age and infection age. We will show the existence of a compact attractor of all compact sets of nonnegative initial data and use the Lyapunov function to show that this attractor is the singleton set containing the endemic equilibrium. Roughly speaking, if the basic reproduction number is less than one then the infection-free equilibrium is globally asymptotically stable otherwise the endemic equilibrium attracts all solutions with active infection at some time.
The remaining part of this paper is organized as follows. In Section 2, we present some preliminary results including model formulation (equivalent integrated semigroup formulation and Volterra formulation), properties of solutions and existence of equilibria. Then we show the asymptotic smoothness of
For ease of notations, we introduce the following notations:
$ \Omega(a) = e^{-\int_0^a\theta_1(\tau)d\tau}\ \ \mbox{and}\ \ \Gamma(b) = e^{-\int_0^b \theta_2(\tau)d\tau}\ \ \mbox{for}\ \ a, b\ge 0. $ |
Biologically,
Then, by (ⅱ) and (ⅴ) of Assumption 1.2, we have that for all
$\begin{align}\label{E3} &\ 0\leq\Omega(a)\leq \textrm{e}^{-\mu_0a}\ \ \mbox{and}\ \ 0\leq\Gamma(b)\leq \textrm{e}^{-\mu_0b}, \notag\\ &\ \frac{d\Omega(a)}{da} = -\theta_1(a)\Omega(a)\ \ \mbox{and}\ \ \frac{d\Gamma(b)}{db} = -\theta_2(b)\Gamma(b). \end{align}$ | (2) |
Following the line of [27], we reformulate the model (1) as a semilinear Cauchy problem. Taking into account the boundary conditions, we consider the following state space,
$ \mathcal{X} = \mathbb{R}\times\mathbb{R}\times L^1((0,\infty), \mathbb{R})\times\mathbb{R}\times L^1((0,\infty), \mathbb{R})\times\mathbb{R}, $ |
$ \mathcal{X}_+ = \mathbb{R}_+\times\mathbb{R}_+\times L_+^1((0,\infty), \mathbb{R})\times\mathbb{R}_+\times L_+^1((0,\infty),\mathbb{R})\times\mathbb{R}_+, $ |
endowed with the usual product norm, and set
$ \mathcal{X}_0 = \mathbb{R} \times \{0\} \times L^1((0,\infty), \mathbb{R})\times \{0\} \times L^1((0,\infty), \mathbb{R})\times\mathbb{R}, $ |
$ \mathcal{X}_{0+} = \mathcal{X}_0\cap\mathcal{X}_+. $ |
We consider the linear operator
$\begin{equation*} \mathcal{A}\begin{pmatrix} T\\ \begin{pmatrix} 0\\ e\\ \end{pmatrix}\\ \begin{pmatrix} 0\\ i\\ \end{pmatrix}\\ V \end{pmatrix} = \begin{pmatrix} -dT\\ \begin{pmatrix} -e(0)\\ -e'-\theta_1(a)e\\ \end{pmatrix}\\ \begin{pmatrix} -i(0)\\ -i'-\theta_2(a)i\\ \end{pmatrix}\\ -cV \end{pmatrix} \end{equation*}$ |
with
$ \textrm{Dom}(\mathcal{A}) = \mathbb{R}\times \{0\} \times W^{1,1}((0,\infty), \mathbb{R})\times \{0\} \times W^{1,1}((0,\infty), \mathbb{R})\times\mathbb{R}, $ |
where
Define nonlinear operator
$ \mathcal{F}\begin{pmatrix} T\\ \begin{pmatrix} 0\\ e\\ \end{pmatrix}\\ \begin{pmatrix} 0\\ i\\ \end{pmatrix}\\ V \end{pmatrix} =\begin{pmatrix} h-\beta TV\\ \begin{pmatrix} f\beta TV\\ 0_{L_1}\\ \end{pmatrix}\\ \begin{pmatrix} (1-f)\beta TV+M\\ 0_{L_1}\\ \end{pmatrix}\\ N \end{pmatrix}. $ |
where
$ M(t) =\int_0^{\infty} \xi(a)e(a, t)da,\ \ N(t) =\int_0^{\infty} p(b)i(b, t)db. $ | (3) |
Then by setting
$ \frac{du(t)}{dt} = \mathcal{A}u(t)+\mathcal{F}(u(t))\ \ \mbox{for}\ \ t\geq 0\ \ \mbox{and}\ \ u(0)\in \mathcal{X}_{0+}. $ |
If any initial value
$ e(0,0) = f\beta T_0V_0 $ |
and
$ i(0,0) = (1-f)\beta T_0V_0+\int_0^{\infty}\xi(a)e_0(a)da, $ |
then (1) is well-posed under Assumption 1.2 due to Iannelli [6] and Magal [14]. Denote
$ \mathcal{Y} = \mathbb{R}_+\times L_+^1(0,\infty)\times L_+^1(0,\infty)\times \mathbb{R}_+ $ |
with the norm
$ \|(x, \varphi, \psi, y)\|_{\mathcal{Y}} = |x|+\|\varphi\|_{L^1}+\|\psi\|_{L^1}+|y|\ \ \mbox{for}\ (x, \varphi, \psi, y)\in \mathcal{Y}. $ |
In fact, for such solutions, it is not difficult to show that
Using the results presented in [14,27], thus we can get a continuous solution semi-flow
$ \Phi \left( t,X_0\right) = \Phi_t(X_0): =\left( T(t), e(\cdot, t), i(\cdot, t), V(t) \right), t\geq 0,\ \ X_0 \in \mathcal{Y}. $ |
The precise result is the following proposition.
Proposition 1. For system (1), there exists a unique strongly continuous semiflow
$ \int_0^{t}x(s)ds\in \textrm{Dom}(\mathcal{A}),\ \mbox{and}\ x(t) =x_0 +\mathcal{A}\int_0^tx(s)ds+ \int_0^t \mathcal{F}(x(s))ds, \forall t\geq0. $ |
According to the Volterra formulation (see Webb [42] and Iannelli [6]), integrating the second and third equations of (1) along the characteristic lines
$e(a,t) = \left\{ {\begin{array}{*{20}{l}} {f\beta T(t - a)V(t - a)\Omega (a) = e(0,t - a)\Omega (a),}&{{\rm{if}}\;\;t > a,}\\ {{e_0}(a - t)\frac{{\Omega (a)}}{{\Omega (a - t)}},}&{{\rm{if}}\;\;t \le a;} \end{array}} \right.$ | (4) |
and
$\begin{array}{*{20}{l}} {i(b,t) = }\\ {\left\{ {\begin{array}{*{20}{l}} {\left[ {(1 - f)\beta T(t - b)V(t - b) + M(t - b)} \right]\Gamma (b) = i(0,t - b)\Gamma (b),\;}&{{\rm{if}}\;t > b,}\\ {{i_0}(b - t)\frac{{\Gamma (b)}}{{\Gamma (b - t)}},}&{{\rm{if}}\;t \le b.} \end{array}} \right.} \end{array}$ | (5) |
Thus system (1) can be rewritten as the following Volterra-type equations,
$ \left\{ \begin{array}{ll} \displaystyle \frac{ dT(t)}{ dt} =h -d T(t)-\beta T(t)V(t), \\ \displaystyle \frac{ dV(t)}{ dt} =\int_0^t p(b)\Gamma(b)((1-f)\beta T(t-b)V(t-b)+M(t-b)) db\\ \displaystyle +\int_t^{\infty}p(b)i_0(b-t)\frac{\Gamma(b)}{\Gamma(b-t)}db-cV(t), \end{array}\right. $ |
where
Proposition 2. Define
$\begin{array}{l} \Xi : = \;\left\{ {{X_0} = ({T_0},{e_0},{i_0},{V_0}) \in {\cal Y}\;|\;{T_0} + \left\| {{e_0}(a)\left\| {_{{L^1}}} \right.} \right. \le \frac{h}{{{\mu _0}}},} \right.\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {T_0} + \left\| {{e_0}(a)} \right.\left\| {_{{L^1}}} \right. + \left\| {{i_0}} \right.(b)\left\| {_{{L^1}}} \right. \le \frac{h}{{{\mu _1}}},\;\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \left. {{V_0} \le \frac{{\bar ph}}{{c{\mu _0}}} + \frac{{h\bar p\bar \xi }}{{c\mu _0^2}},\;{{\left\| {{X_0}} \right\|}_{\cal Y}} \le \frac{h}{{{{\tilde \mu }_0}}}} \right\}, \end{array}$ |
where
$\Phi (t,{X_0}) \in \Xi \;\;for\;all\;\;t \ge 0\;\;and\;\;{X_0} \in \Xi .$ |
Moreover,
Proof. By (4) and changes of variables, we have
$\begin{align*} \|e(\cdot,t)\|_{L^1} = &\ \int_0^t e(0,t-a)\Omega(a) da+\int_t^\infty e_0(a-t)\frac{\Omega(a)}{\Omega(a-t)} da\\ =&\ \int_0^t e(0,\sigma)\Omega(t-\sigma) d\sigma+ \int_0^\infty e_0 (\tau)\frac{\Omega(t+\tau)}{\Omega(\tau)} d\tau. \end{align*}$ |
We derivative this equality,
$\frac{{d{{\left\| {e( \cdot ,t)} \right\|}_{{L^1}}}}}{{dt}} = e(0,t)\Omega (0) + \int_0^t e (0,\sigma )\frac{{d\Omega (t - \sigma )}}{{dt}}d\sigma + \int_0^\infty {\frac{{{e_0}(\tau )}}{{\Omega (\tau )}}} \frac{{d\Omega (t + \tau )}}{{dt}}d\tau .$ |
By (2) and changing of variables, we have
$\begin{align*} \frac{d\|e(\cdot,t)\|_{L^1}}{dt} =&\ e(0,t)\Omega(0)-\int_0^t e(0,\sigma)\theta_1(t-\sigma)\Omega(t-\sigma)d\sigma\\ &\ -\int_0^{\infty} \frac {e_0(\tau)}{\Omega(\tau)}\theta_1(t+\tau)\Omega(t+\tau)d\tau \nonumber\\ = &\ e(0,t)\Omega(0)-\int_0^\infty \theta_1(a)e(a,t) da. \end{align*}$ |
By the first equation in (1), (ⅳ) of Assumption 1.2, and use
$\begin{array}{*{20}{l}} {\frac{{d(T(t) + {{\left\| {e( \cdot ,t)} \right\|}_{{L^1}}})}}{{dt}} = h - dT(t) - \beta T(t)V(t) + f\beta T(t)V(t) - \int_0^\infty {{\theta _1}} (a)e(a,t)da}\\ {\quad \quad \quad \quad \quad \quad \quad \quad \quad \le h - {\mu _0}(T(t) + {{\left\| {e( \cdot ,t)} \right\|}_{{L^1}}}).} \end{array}$ |
We integrate this differential inequality and obtain the a priori estimate,
$ T(t)+ \|e(\cdot,t)\|_{L^1} \leq \frac{h}{\mu_0} -e^{-\mu_0 t}\left\{ \frac{h}{\mu_0} - ( T_0+ \|e(\cdot,t)\|_{L^1}) \right\}, t\geq 0. $ | (6) |
This implies that
$ T(t)+\|e(\cdot,t)\|_{L^1}\leq \frac{h}{\mu_0}. $ |
Similarly, we have
$ \frac{ d\|i(\cdot, t)\|_{L^1}}{ dt} = i(0, t)\Gamma(0)-\int_0^\infty \theta_2(b)i(b,t) db. $ |
We add the two equations,
$\begin{array}{l} \frac{{d(T(t) + {{\left\| {e( \cdot ,t)} \right\|}_{{L^1}}}) + {{\left\| {i( \cdot ,t)} \right\|}_{{L^1}}})}}{{dt}} = h - dT + \int_0^\infty \xi (a)e(a,t)da\\ \quad \;\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - \int_0^\infty {{\theta _1}} (a)e(a,t)da - \int_0^\infty {{\theta _2}} (b)i(b,t)db. \end{array}$ |
and since (ⅱ) and (ⅳ) of Assumption 1.2, obtain the estimates
$\frac{{d(T(t) + {{\left\| {e( \cdot ,t)} \right\|}_{{L^1}}}) + {{\left\| {i( \cdot ,t)} \right\|}_{{L^1}}})}}{{dt}} \le h + \bar \xi \frac{h}{{{\mu _0}}} - {\mu _0}(T(t) + {\left\| {e( \cdot ,t)} \right\|_{{L^1}}} + {\left. {i( \cdot ,t)} \right\|_{{L^1}}}).$ |
We integrate this differential inequality and obtain the a priori estimate,
$\begin{align}\label{Ii} &\ \|T(t)+\|e(\cdot, t)\|_{L^1})+ \|i(\cdot, t)\|_{L^1}\nonumber\\ \leq&\ \frac{h+\bar{\xi}\frac{h}{\mu_0}}{\mu_0}-e^{-\mu_0 t}\left\{\frac{h+\bar{\xi}\frac{h}{\mu_0}}{\mu_0}-(T(t)+ \| e(\cdot, t)\|_{L^1}+i(\cdot, t)\|_{L^1})\right\}, \ \ t\geq 0. \end{align}$ | (7) |
This implies
$ \frac{ dV(t)}{ dt} \leq \bar{p} \| i(\cdot,t)\|_{L^1}-c V(t)\leq \bar{p}\left(\frac{h}{\mu_0}+\frac{\bar{\xi}h}{\mu_0^2}\right) -c V(t). $ |
It follows (ⅳ) of Assumption 1.2, we have that
$\begin{align}\label{p} V(t) \leq&\ \frac{\bar{p}(\frac{h}{\mu_0}+\frac{\bar{\xi}h}{\mu_0^2})}{c}-e^{-c t}\left\{\frac{\bar{p}h}{c\mu_0}+ \frac{h\bar{p}\bar{\xi}}{c\mu_0^2} -V_0 \right\}\nonumber\\ \leq&\ \frac{\bar{p}h}{c\mu_0}+\frac{h\bar{p}\bar{\xi}}{c\mu_0^2}-e^{-\mu_0 t}\left\{ \frac{\bar{ph}}{c\mu_0}+\frac{h\bar{p}\bar{\xi}}{c\mu_0^2} -V_0 \right\} \end{align}$ | (8) |
Consequently, from (6), (7) and (8), we conclude that if
$\begin{array}{*{20}{c}} {{{\left\| {{\Phi _t}({X_0})} \right\|}_{\cal Y}}}&{ \le \;\left( {1 + \frac{{\bar \xi }}{{{\mu _0}}} + \frac{{\bar p}}{c} + \frac{{\bar p\bar \xi }}{{c{\mu _0}}}} \right)\frac{h}{{{\mu _0}}}} \end{array}$ | (9) |
$\begin{array}{*{20}{l}} {\quad \quad \quad \quad \quad \quad - {e^{ - {\mu _0}t}}\left\{ {\left( {1 + \frac{{\bar \xi }}{{{\mu _0}}} + \frac{{\bar p}}{c} + \frac{{\bar p\bar \xi }}{{c{\mu _0}}}} \right)\frac{h}{{{\mu _0}}} - {X_0}{_{\cal Y}}} \right\}}\\ {\;\quad \quad \quad \quad = \frac{h}{{{{\tilde \mu }_0}}} - {e^{ - {\mu _0}t}}\left\{ {\frac{h}{{{{\tilde \mu }_0}}} - {{\left\| {{X_0}} \right\|}_{\cal Y}}} \right\} \le \frac{h}{{{{\tilde \mu }_0}}}.} \end{array}$ | (10) |
In summary, we have shown that
As a consequence of Proposition 2, we have the following result.
Proposition 3. Let
(ⅰ)
(ⅱ)
(ⅲ)
System (1) always has an infection-free equilibrium
$ P^0 =(T^0, e^0(a), i^0(b), V^0): = (\frac{h}{d}, 0, 0, 0). $ |
The equations for an equilibrium are obtained from (1) by setting the time derivatives equal to
$ \left\{ \begin{array}{ll} \displaystyle h -d T^*-\beta T^*V^* = 0,\\ \displaystyle \frac{ d}{ da}e^*(a) = -\theta_1(a)e^*(a), \\ \displaystyle \frac{d}{da}i^*(b) = -\theta_2(b)i^*(b), \\ \displaystyle \int_0^\infty p(b)i^*(b) db = cV^*,\\ \displaystyle e^*(0) = f\beta T^*V^*,\\ \displaystyle i^*(0) = (1-f)\beta T^*V^*+\int_0^\infty\xi(a) e^*(a) da. \end{array} \right. $ | (11) |
Denote
$ K = \int_0^\infty \xi(a)\Omega(a) da,\ \ J = \int_0^\infty p(b)\Gamma(b) db. $ |
Biologically,
We define basic reproduction number,
$ \Re_0 =\frac{f\beta T^0 K J}{c} +\frac{(1-f)\beta T^0J}{c}. $ |
which accounts for the total number of virons resulted from a single viron through the virus-to-cell infection mod.
Direct calculation yields that if
$\begin{align}\label{*} &\ T^* = \frac{T^0}{\Re_0}, \ e^*(a) = fh\left(1-\frac{1}{\Re_0}\right)\Omega(a), \nonumber\\ &\ i^*(b) = (1-f+ fK)h\left(1-\frac{1}{\Re_0}\right)\Gamma(b), \ V^* = \frac{1}{c}\int_0^\infty p(b)i^*(b) db. \end{align}$ | (12) |
In summary, we have shown the following result.
Proposition 4. (ⅰ) System (1) always has an infection-free equilibrium
(ⅱ) If
By Proposition 2 and 3, the semiflow is point-dissipative and
Definition 3.1. [28] A set
Recall that
Proposition 5. For any solution of (1), the associated functions
Proof. Let
$\begin{align}\label{ML} &M(t+h)-M(t) =\ \int_0^{\infty} \xi(a)e(a, t+h)da-\int_0^{\infty} \xi(a)e(a, t)da\nonumber\\ \leq &\ \int_0^h \xi(a)e(a, t+h)da+\int_h^{\infty} \xi(a)e(a, t+h)da-\int_0^{\infty} \xi(a)e(a, t)da\nonumber\\ \leq &\ \int_0^h \xi(a)e(0, t+h-a)\Omega(a)da\nonumber\\ &\ +\int_h^{\infty} \xi(a)e(a, t+h)da-\int_0^{\infty} \xi(a)e(a, t)da. \end{align}$ | (13) |
By applying
$M(t + h) - M(t) \le f\beta {A^2}\bar \xi h + \int_0^\infty \xi (\sigma + h)e(\sigma + h,t + h)d\sigma - \int_0^\infty \xi (a)e (a,t)da$ |
It follows from (4) that
$ e(\sigma+h,t+h) = e(\sigma,t)\frac{\Omega(\sigma+h)}{\Omega(\sigma)}. $ |
Thus,
$\begin{align*} M(t+h)-M(t) \leq&\ f\beta A^2\bar{\xi} h+\int_0^{\infty} \left(\xi(a+h)\frac{\Omega(a+h)}{\Omega(a)}-\xi(a)\right)e(a,t)da\nonumber\\ =&\ f\beta A^2\bar{\xi} h+\int_0^{\infty}\left(\xi(a+h)e^{-\int_{a}^{a+h}\theta_1(s)ds}-\xi(a)\right)e(a,t)da\nonumber\\ =&\ f\beta A^2\bar{\xi} h+\int_0^{\infty} \xi(a+h)\left(e^{-\int_{a}^{a+h}\theta_1(s)ds}-1\right)e(a,t)da\nonumber\\ &\ + \int_0^{\infty} (\xi(a+h)-\xi(a))e(a,t)da. \end{align*}$ |
From (ⅱ) of Assumption 1.2, we obtain
$ 0\leq \xi(a+h)\left|e^{-\int_a^{a+h}\theta_1(s)ds}-1\right|\leq \bar{\xi}\bar{\theta_1}h. $ |
Recall that
$\begin{align*} M(t+h)-M(t)\leq f\beta A^2\bar{\xi}h+\bar{\xi}\bar{\theta_1}Ah+M_\xi Ah. \end{align*}$ |
Hence,
Next we divide
$\begin{eqnarray*} \Theta ( t,X_0 ) & := & (0,\tilde{\varphi}_e(\cdot,t),\tilde{\varphi}_i(\cdot,t),0 ), \\ \Psi ( t,X_0 ) & := & (T(t),\tilde{e}(\cdot,t),\tilde{i}(\cdot,t),V(t) ), \end{eqnarray*}$ |
where
$\begin{align*} \tilde{\varphi}_e ( a,t ) =\left\{ \begin{array}{ll} 0, \ \ &\ \mbox{if}\ \ t>a \geq 0,\\ e( a,t ),\ \ &\ \mbox{if}\ \ a \geq t \geq 0; \end{array} \right. &\ \tilde{\varphi}_i ( b,t ) =\left\{ \begin{array}{ll} 0,\ \ &\mbox{if}\ \ t>b \geq 0,\\ i( b,t ),\ \ &\mbox{if}\ \ b \geq t \geq 0; \end{array} \right. \end{align*}$ |
$\tilde e(a,t) = \left\{ {\begin{array}{*{20}{l}} {e(a,t),\;}&{{\rm{if}}\;\;t > a \ge 0,}\\ {0,\;}&{{\rm{if}}\;\;a \ge t \ge 0;} \end{array}} \right.\;\tilde i(b,t) = \left\{ {\begin{array}{*{20}{l}} {i(b,t),\;}&{{\rm{if}}\;\;t > b \ge 0,}\\ {0,}&{{\rm{if}}\;\;b \ge t \ge 0.} \end{array}} \right.$ |
Then
Theorem 3.2. For
(ⅰ) There exists a function
(ⅱ) For
proof Proof of (ⅰ) of Theorem 3.2. Let
${\tilde \varphi _e}(a,t) = \left\{ {\begin{array}{*{20}{l}} {0,\;\;}&{{\rm{if}}\;\;t > a \ge 0,}\\ {{e_0}(a - t)\frac{{\Omega (a)}}{{\Omega (a - t)}},\;}&{{\rm{if}}\;\;a \ge t \ge 0;} \end{array}} \right.$ |
and
${\tilde \varphi _i}(b,t) = \left\{ {\begin{array}{*{20}{l}} {0,\;\;}&{{\rm{if}}\;\;t > b \ge 0,}\\ {{i_0}(b - t)\frac{{\Gamma (b)}}{{\Gamma (b - t)}},\;}&{{\rm{if}}\;\;b \ge t \ge 0.} \end{array}} \right.$ |
Then, for
$\begin{align*} \left\| \Theta \left( t,X_0 \right) \right\|_{\mathcal{Y}} = &\ \left| 0\right| +\| \tilde{\varphi}_e(\cdot,t)\|_{L^1} +\| \tilde{\varphi}_i ( \cdot,t ) \|_{L^1} + \left| 0\right| \\ = &\ \int_t^\infty\left| e_0 (a-t)\frac{\Omega(a)}{\Omega(a-t)}\right| da +\int_t^\infty\left| i_0 (b-t)\frac{\Gamma(b)}{\Gamma(b-t)}\right| db \\ =&\ \int_0^\infty\left| e_0 (\sigma)\frac{\Omega(\sigma+t)}{\Omega(\sigma)}\right| d\sigma + \int_0^\infty\left| i_0 (\sigma)\frac{\Gamma(\sigma+t)}{\Gamma(\sigma)}\right| d\sigma \\ = &\ \int_0^\infty\left| e_0 (\sigma)e^{-\int_\sigma^{\sigma+t}\theta_1(\tau) d\tau}\right| d\sigma + \int_0^\infty\left| i_0 (\sigma)e^{-\int_\sigma^{\sigma+t}\theta_2(\tau) d\tau}\right| d\sigma \\ \leq &\ e^{-\mu_0t}\| e_0\|_{L^1} + e^{-\mu_0t}\| i_0\|_{L^1} \\ \leq &\ e^{-\mu_0t}\left\| X_0 \right\|_{\mathcal{Y}}. \end{align*}$ |
Proof of (ⅱ) of Theorem 3.2. It is sufficient to show that
(ⅰ) The supremum of
(ⅱ)
(ⅲ)
(ⅳ)
It follows from (4), (5), Proposition 3 and (2) that
Next we verify condition (ⅲ). For sufficiently small
$\begin{array}{l} \int_0^\infty {\left| {\tilde e(a + h,t) - \tilde e(a,t)} \right|} da = \int_0^{t - h} | e(a + h,t) - e(a,t)|da + \int_{t - h}^t | 0 - e(a,t)|da\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad = \int_0^{t - h} | e(0,t - a - h)\Omega (a + h) - e(0,t - a)\Omega (a)|da\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + \int_{t - h}^t | e(0,t - a)\Omega (a)|da\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \le \;{\Delta _1} + {\Delta _2} + f\beta {A^2}h, \end{array}$ |
where
$ \Delta_1 = \int_0^{t-h} e(0,t-a-h)\Big|\Omega(a+h)-\Omega(a)\Big| da $ |
and
$ \Delta_2 =\int_0^{t-h} \Big| e(0,t-a-h)-e(0,t-a)\Big|\Omega(a)da. $ |
We first get an estimate of
$\begin{align*} \int_0^{t-h}|\Omega(a+h)-\Omega(a)| da =&\ \int_0^{t-h}\Big(\Omega(a)-\Omega(a+h)\Big) da\\ = &\ \int_0^{t-h}\Omega(a) da-\int_h^t\Omega(a) da\\ = &\ \int_0^{t-h}\Omega(a) da-\int_h^{t-h}\Omega(a) da -\int_{t-h}^t\Omega(a) da\\ = &\ \int_0^h\Omega(a) da-\int_{t-h}^t\Omega(a) da \\ \leq &\ h, \end{align*}$ |
it follows from Proposition 3 that
$ \Delta_1\le f\beta A^2h. $ |
Next we estimate
$\begin{align*} \Delta_2 =&\ \int_0^{t-h} \Big|f\beta T(t-a-h)V(t-a-h)-f\beta T(t-a)V(t-a)\Big|\Omega(a) da. \end{align*}$ |
It is easy to see that
$ \Delta_2\le Gh \int_0^{t-h} e^{-\mu_0a}da \le \frac{Gh}{\mu_0}. $ |
Hence
$ \int_0^\infty \left| \tilde{e}(a+h,t)-\tilde{e}(a,t)\right| da\leq\ \left(2f\beta A^2+\frac{G}{\mu_0}\right) h, $ |
and condition (ⅲ) directly follows.
As to
$\begin{array}{l} \int_0^\infty {\left| {\tilde i(b + h,t) - \tilde i(b,t)} \right|} db = \;\int_0^{t - h} | i(b + h,t) - i(b,t)|db + \int_{t - h}^t | 0 - i(b,t)|db\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\; = \;\int_0^{t - h} | i(0,t - b - h)\Gamma (b + h) - i(0,t - b)\Gamma (b)|db\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;\quad \; + \int_{t - h}^t | i(0,t - b)\Gamma (b)|db\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\; \le {\Upsilon _1} + {\Upsilon _2} + \left[ {(1 - f)\beta {A^2} + \bar \xi A} \right]h, \end{array}$ |
where
$ \Upsilon_1 = \int_0^{t-h} i(0,t-b-h)\Big|\Gamma(b+h)-\Gamma(b)\Big| db $ |
and
$ \Upsilon_2 =\int_0^{t-h} \Big| i(0,t-b-h)-i(0,t-b)\Big|\Gamma(b)db. $ |
Similarly, we have
$ \Upsilon_1\le \left[(1-f)\beta A^2+\bar{\xi}A\right] h. $ |
Next we estimate
$\begin{array}{l} {\Upsilon _2} = \;\int_0^{t - h} | (1 - f)\beta T(t - a - h)V(t - a - h) + M(t - a - h)\\ \quad \quad \quad \quad \quad - (1 - f)\beta T(t - a)V(t - a) - M(t - a)|\Gamma (b)da\\ \quad \quad \le (1 - f)\beta \int_0^{t - h} | T(t - a - h)V(t - a - h) - T(t - a)V(t - a)|\Gamma (b)da\\ \quad \quad \quad \quad + \int_0^{t - h} | M(t - a - h) - M(t - a)|\Gamma (b)da. \end{array}$ |
As before,
$ \Upsilon_2\leq M_1h\int_0^{t-h}e^{-\mu_0b}db\leq \frac{Hh}{\mu_0}. $ |
Finally, we have
$ \int_0^\infty \left| \tilde{i}(b+h,t)-\tilde{i}(b,t)\right| db \leq \left\{2\left[(1-f)\beta A^2+\bar{\xi}A\right] +\frac{H}{\mu_0}\right\}h, $ |
thus condition (ⅲ) directly follows. This completes the proof.
Consequently, we have the following theorem for the semi-flow
Theorem 3.3. The semi-flow
This section is spent on proving that (1) is uniformly persistent under the condition
Let
$\left\{ {\begin{array}{*{20}{l}} {\frac{{dT(t)}}{{dt}} = h - dT(t) - \frac{1}{f}\hat e(t),}\\ {e(a,t) = \left\{ {\begin{array}{*{20}{l}} {\hat e(t - a)\Omega (a),\;\;}&{{\rm{if}}\;\;t \ge a \ge 0,}\\ {{e_0}(a - t)\frac{{\Omega (a)}}{{\Omega (a - t)}},\;}&{{\rm{if}}\;\;a \ge t \ge 0;} \end{array}} \right.}\\ {i(b,t) = \left\{ {\begin{array}{*{20}{l}} {\hat i(t - b)\Gamma (b),\;}&{{\rm{if}}\;\;t \ge b \ge 0,}\\ {{i_0}(b - t)\frac{{\Gamma (b)}}{{\Gamma (b - t)}},\;}&{{\rm{if}}\;\;b \ge t \ge 0,} \end{array}} \right.} \end{array}} \right.$ | (14) |
where
$\begin{align}\label{e_int_1} \hat{e}(t) =&\ f\beta T(t) V(t) \end{align}$ | (15) |
and
$\begin{array}{l} \begin{array}{*{20}{l}} {\hat i(t) = }&{\;(1 - f)\beta T(t)V(t) + \int_0^t \xi (a)\Omega (a)\hat e(t - a)da} \end{array}\\ \quad \quad \quad \quad \quad \quad + \int_t^\infty \xi (a)\frac{{\Omega (a)}}{{\Omega (a - t)}}{e_0}(a - t)da. \end{array}$ | (16) |
Lemma 4.1. If
$ \limsup\limits_{t \to \infty} \hat{e}(t) > \epsilon_0. $ | (17) |
Proof. We first get an estimate on
$ \hat{i}(t) \geq (1-f)\beta T(t) V(t)+\int_{0}^{t} \xi(a) \Omega(a) \hat{e}(t-a) da. $ | (18) |
Solving the fourth equation of (1) with initial condition
$ V(t) = V_0e^{-ct}+\int_0^t\int_0^\infty p(b)i(b,\tau)db\cdot e^{-c(t-\tau)}d\tau $ |
Then
$ V(t) \geq \int_0^t e^{-c(t-\tau)}\int_0^\tau p(b)i(b,\tau) db d\tau =\int_0^t e^{-c(t-\tau)}\int_0^\tau p(b)\Gamma(b)\hat{i}(\tau-b) db d\tau. $ |
This, combined with (18), gives us
$\begin{array}{*{20}{l}} {\hat i(t) \ge }&{\;(1 - f)\beta T(t)\int_0^t {{e^{ - c(t - \tau )}}} \int_0^\tau p (b)\Gamma (b)\hat i(\tau - b)dbd\tau }\\ {}&{ + f\beta \int_0^t \xi (a)\Omega (a)T(t - a)\int_0^{t - a} {{e^{ - c(t - a - \tau )}}} \int_0^\tau p (b)\Gamma (b)\hat i(\tau - b)dbd\tau da.} \end{array}$ | (19) |
Since
$ \frac{(1-f)\beta}{c} \frac{h-\epsilon_1}{d} \int_{0}^{\infty} p(b) \Gamma(b) db +\frac{f\beta}{c} \frac{h-\epsilon_1}{d} \int_{0}^{\infty} p(b) \Gamma(b) db \int_0^\infty \xi(a)\Omega(a) da> \ 1. \label{R0_e} $ |
We claim that (17) holds for this
$ \hat{e}(t) \leq \epsilon_0\ \ \mbox{for all}\ \ t\geq T. $ |
Then it follows from (14) that
$\begin{align}\label{i_int_3} \hat{i}(t) \geq &\ (1-f)\beta \frac{h-\epsilon_1}{d} \int_{0}^{t} e^{-c(t-\tau)} \int_{0}^{\tau} p(b) \Gamma(b) \hat{i}(\tau-b) db d\tau \nonumber \\ &\ + f\beta \frac{h-\epsilon_1}{d}\int_{0}^{t} \xi(a) \Omega(a)\int_{0}^{t-a}e^{-c(t-a-\tau)}\int_0^\tau p(b)\Gamma(b)\hat{i}(\tau-b) db d\tau da \end{align}$ | (20) |
for all
$\begin{array}{l} {\cal L}[\hat i] \ge \;(1 - f)\beta \frac{{h - {_1}}}{d}\int_0^\infty {{e^{ - \lambda t}}} \int_0^t {{e^{ - c(t - \tau )}}} \int_0^\tau p (b)\Gamma (b)\hat i(\tau - b)dbd\tau dt + \\ \quad \quad f\beta \frac{{h - {_1}}}{d}\int_0^\infty {{e^{ - \lambda t}}} \int_0^t \xi (a)\Omega (a)\int_0^{t - a} {{e^{ - c(t - a - \tau )}}} \int_0^\tau p (b)\Gamma (b)\hat i(\tau - b)dbd\tau dadt \end{array}$ |
$\begin{align*} = &\ (1-f)\beta \frac{h-\epsilon_1}{d}\frac{1}{c+\lambda} \int_{0}^{\infty} e^{-\lambda b} p(b) \Gamma(b) db\mathcal{L}[\hat{i}] \\ &\ +f\beta \frac{h-\epsilon_1}{d}\frac{1}{c+\lambda} \int_{0}^{\infty} e^{-\lambda b} p(b) \Gamma(b) db \int_0^\infty e^{-\lambda a} \xi(a) \Omega(a) da \mathcal{L}[\hat{i}]. \end{align*}$ |
Here
$1\ge \frac{(1-f)\beta }{c}\frac{h-{{\epsilon }_{1}}}{d}\int_{0}^{\infty }{p}(b)\Gamma (b)db+\frac{f\beta }{c}\frac{h-{{\epsilon }_{1}}}{d}\int_{0}^{\infty }{p}(b)\Gamma (b)db\int_{0}^{\infty }{\xi }(a)\Omega (a)da,$ |
which yields a contradiction.
In order to apply a technique used by Smith and Thieme [25,Chapter 9] (see also McCluskey [13,Section 8]), we consider a total
Let
$\begin{align*} \phi(r+t) =&\ \Phi(t,\phi(r))\ \ \mbox{for}\ \ t\geq 0\ \ \mbox{and}\ \ r\in \mathbb{R}, \\ e(a,r) =&\ e\left(0,r-a \right) \Omega (a) =\hat{e}(r-a) \Omega (a)\ \ \mbox{for}\ \ r\in \mathbb{R} \ \mbox {and}\ \ a \geq 0, \\ i(b,r) =&\ i\left(0,r-b \right) \Gamma (b) =\hat{i}(r-b) \Gamma (b)\ \ \mbox{for} \ \ r\in \mathbb{R}\ \ \mbox {and}\ \ b \geq 0. \end{align*}$ |
So it follows from (14)-(15) that
$ \left\{ \begin {array}{lll} \displaystyle \frac{ dT(r)}{ dr} =h -d T(r)-\frac{1}{f}\hat{e}(r), \\ \displaystyle \hat{e}(r) =f\beta T(r) V(r), \\ \displaystyle \hat{i}(r) =(1-f)\beta T(r) V(r)+\int_{0}^{\infty} \xi(a) \Omega(a) \hat{e}(r-a) da, \\ \displaystyle \frac{ dV(r)}{ dr} =\int_{0}^{\infty} p(b) \Gamma(b) \hat{i}(r-b) db-c V(r),\ \ \mbox{for}\ \ r\in \mathbb{R}. \end {array}\right. $ |
By the similar arguments as in McCluskey [13,Section 8] and Wang et al. [38,Section 5] and a slight modification of the proof in [17, Lemma 4.1], actually, a total
Thus Lemma 4.1 tells us that if
Theorem 4.2. If
When
$ \liminf\limits_{t\to \infty} \left\| e(\cdot ,t) \right\|_{L^1} \geq \hat{e}^\infty \int_{0}^{\infty} \Omega (a) da, $ |
where
Theorem 4.3. If
$ \liminf\limits_{t\to \infty} T(t)\geq \epsilon, \liminf\limits_{t\to \infty}\|e(\cdot,t)\|_{L^1} \geq \epsilon, \liminf\limits_{t\to \infty}\|i(\cdot,t)\|_{L^1} \geq \epsilon, \liminf\limits_{t\to \infty} V(t)\geq \epsilon. $ |
This section is devoted to investigate the local stability of equilibria of (1).
Theorem 5.1. (ⅰ) If
(ⅱ) If
Proof. Proof. of (ⅰ) of Theorem 5.1. Linearizing (1) around the infection-free equilibrium
$\begin{align*} x_1(t) = T(t)-\frac{h}{d},\ x_2(a,t) = e(a, t),\ x_3(b, t) = i(b,t),\ x_4(t) = V(t), \end{align*}$ |
we get
$ \left\{ \begin{array}{lll} \displaystyle \frac{ dx_1(t)}{ dt} =-d x_1(t)-\frac{\beta h}{d}x_4(t), \\ \displaystyle \left(\frac{\partial}{\partial t}+\frac{\partial}{\partial a}\right)x_2(a,t) =-\theta_1(a)x_2(a,t),\\ \displaystyle \left(\frac{\partial}{\partial t}+\frac{\partial}{\partial b}\right)x_3(b,t) =-\theta_2(b)x_3(b,t),\\ \displaystyle \frac{ dx_4(t)}{ dt} =\int_0^\infty p(b)x_3(b,t) db-cx_4(t),\\ \displaystyle x_2(0,t) = f\beta\frac{h}{d}x_4(t),\\ \displaystyle x_3(0,t) = (1-f)\beta\frac{h}{d}x_4(t)+\int_0^\infty \xi(a)x_2(a,t)da. \end{array}\right. $ | (21) |
Set
$\begin{align}\label{z1} x_1(t) = x_1^0e^{\lambda t}, ~~x_2(a, t) = x_2^0(a)e^{\lambda t}, ~~x_3(b, t) = x_3^0(b)e^{\lambda t}, ~~x_4(t) = x_4^0 e^{\lambda t}, \end{align}$ | (22) |
where
$\begin{align*} \lambda x_1^0 = -dx_1^0-\frac{h\beta}{d}x_4^0, \end{align*}$ |
$ \left\{ \begin{array}{ll} \displaystyle \lambda x_2^0(a)+\frac{dx_2^0(a)}{d a} = -\theta_1(a)x_2^0(a),\\ \displaystyle x_2^0(0) = f \beta\frac{ h}{d}x_4^0, \end{array}\right. $ | (23) |
$ \label{Lsx_3} \left\{ \begin{array}{ll} \displaystyle \lambda x_3^0(b)+\frac{dx_3^0(b)}{d b} = -\theta_2(b)x_3^0(b),\\ \displaystyle x_3^0(0) = (1-f)\frac{h \beta}{d}x_4^0+\int_0^\infty \xi(a)x_2^0(a)da, \end{array}\right. $ | (24) |
$\begin{align}\label{Lsx_4} \lambda x_4^0 = \int_0^\infty p(b)x_3^0(b)db-cx_4^0. \end{align}$ | (25) |
We integrate the first equation of Equ (23) and Equ (24) from
$\begin{align} \label{Ls2} x_2^0(a) =&\ x_2^0(0)e^{-\lambda a-\int_0^a\theta_1(s)ds}, \end{align}$ | (26) |
and
$\begin{align} \label{Ls3} x_3^0(b) =&\ x_3^0(0)e^{-\lambda b-\int_0^b\theta_2(s)ds} \notag\\ = &\ \left[(1-f)\frac{h \beta}{d}x_4^0+\int_0^\infty \xi(a)x_2^0(a)da\right]e^{-\lambda b-\int_0^b\theta_2(s)ds}. \end{align}$ | (27) |
From (25), (26) and (27),
$\begin{align} \label{Ls4} x_4^0 =&\ \frac{\int_0^\infty p(b)x_3^0(b)db}{\lambda+c} \notag\\ = &\ \frac{1-f}{\lambda+c}\frac{h \beta}{d}x_4^0\int_0^\infty p(b)e^{-\lambda b-\int_0^b\theta_2(s)ds}db\notag\\ &\ +\frac{x_2^0(0)}{\lambda+c}\frac{h}{d}\int_0^\infty \xi(a)e^{-\lambda a-\int_0^a\theta_1(s)ds}da\int_0^\infty p(b)e^{-\lambda b-\int_0^b\theta_2(s)ds} db. \end{align}$ | (28) |
Combining (23) into (28), it follows the equation that
$ \mathcal{W}(\lambda) = 1,\label{CE} $ | (29) |
where
$\begin{align*} \mathcal{W}(\lambda) = &\ \frac{f\beta}{\lambda+c}\frac{h}{d}\int_0^\infty \xi(a) e^{-\lambda a-\int_0^a\theta_1(s) ds} da \int_0^\infty p(b)e^{-\lambda b-\int_0^b\theta_2(s) ds} db \\ &\ +\frac{1-f}{\lambda+c}\frac{h\beta}{d} \int_0^\infty p(b)e^{-\lambda b-\int_0^b\theta_2(s) ds} db. \end{align*}$ |
Since
$ \lim\limits_{\lambda\to\infty}\mathcal{W}(\lambda) = 0,\ \lim\limits_{\lambda\to-\infty}\mathcal{W}(\lambda) = \infty,\ \mathcal{W}'(\lambda)<0, $ |
Thus (29) admits a unique real root,
$ 1 = |\mathcal{W}(\lambda)| = |\mathcal{W}(\mu+\nu i)|\leq \mathcal{W}(\mu), $ |
which implies that
Proof of (ⅱ) of Theorem 5.1. Linearizing the system (1) at
$\begin{align*} &\ y_1(t) = T(t)-T^*,\ y_2(a,t) = e(a, t)-e^*(a),\\ &\ y_3(b, t) = i(b,t)-i^*(b),\ y_4(t) = V(t)-V^*, \end{align*}$ |
we get
$ \left\{ \begin{array}{lll} \displaystyle \frac{ dy_1(t)}{ dt} =-d \Re_0 y_1(t)-\beta T^*y_4(t), \\ \displaystyle \left(\frac{\partial}{\partial t}+\frac{\partial}{\partial a}\right)y_2(a,t) =-\theta_1(a)y_2(a,t),\\ \displaystyle \left(\frac{\partial}{\partial t}+\frac{\partial}{\partial b}\right)y_3(b,t) =-\theta_2(b)y_3(b,t),\\ \displaystyle \frac{ dy_4(t)}{ dt} =\int_0^\infty p(b)y_3(b,t) db-cy_4(t),\\ \displaystyle y_2(0,t) = fd(\Re_0-1)y_1(t)+f\beta T^*y_4(t),\\ \displaystyle y_3(0,t) = (1-f)d(\Re_0-1)y_1(t)+(1-f)\beta T^*y_4(t)+\int_0^\infty \xi(a)y_2(a,t)da, \end{array}\right. $ | (30) |
Set
$\begin{align}\label{z2} y_1(t) = y_1^0e^{\lambda t}, ~~y_2(a, t) = y_2^0(a)e^{\lambda t}, ~~y_3(b, t) = y_3^0(b)e^{\lambda t}, ~~y_4(t) = y_4^0 e^{\lambda t}, \end{align}$ | (31) |
where
$\begin{align}\label{Lsx_5} \lambda y_1^0 = -d\Re_0y_1^0-\beta T^*y_4^0, \end{align}$ | (32) |
$ \left\{ \begin{array}{ll} \displaystyle \lambda y_2^0(a)+\frac{dy_2^0(a)}{d a} = -\theta_1(a)y_2^0(a),\\ \displaystyle y_2^0(0) = fd(\Re_0-1)y_1^0+f\beta T^*y_4^0, \end{array}\right. $ | (33) |
$\left\{ {\begin{array}{*{20}{l}} {\lambda y_3^0(b) + \frac{{dy_3^0(b)}}{{db}} = - {\theta _2}(b)y_3^0(b),}\\ {y_3^0(0) = (1 - f)d({\Re _0} - 1)y_1^0 + (1 - f)\beta {T^*}y_4^0 + y_2^0(0)\int\limits_0^\infty \xi (a){e^{ - \lambda a - \int\limits_0^a {{\theta _1}} (s)ds}}da,} \end{array}} \right.$ | (34) |
and
$\begin{align}\label{Lsx_8} \lambda y_4^0 = \int_0^\infty p(b)y_3^0(b)db-cy_4^0. \end{align}$ | (35) |
We integrate the first equation of (33), (34) from
$\begin{align*} y_2^0(a) =&\ y_2^0(0)e^{-\lambda a-\int_0^a \theta_1(s)ds}, \end{align*}$ |
and
$\begin{align*} y_3^0(b) =&\ y_3^0(0)e^{-\lambda b-\int_0^b\theta_2(s)ds} \notag\\ = &\ \left[(1-f)d(\Re_0-1)y_1^0+(1-f)\beta T^*y_4^0\right]e^{-\lambda b-\int_0^b\theta_2(s)ds}\\ &\ +y_2^0(0)\int_0^\infty \xi(a)e^{-\lambda a-\int_0^a \theta_1(s)ds}da\cdot e^{-\lambda b-\int_0^b\theta_2(s)ds}. \end{align*}$ |
and from (35), we have
$\begin{align} \label{Ls8} y_4^0 =&\ \frac{\int_0^\infty p(b)y_3^0(b)db}{\lambda+c} \notag\\ = &\ \frac{1-f}{\lambda+c}\left(d(\Re_0-1)y_1^0+\beta T^*y_4^0\right)\int_0^\infty p(b)e^{-\lambda b-\int_0^b\theta_2(s)ds}db\nonumber\\ & +\frac{y_2^0(0)}{\lambda+c}\int_0^\infty \xi(a)e^{-\lambda a-\int_0^a\theta_1(s)ds}da\int_0^\infty p(b)e^{-\lambda b-\int_0^b\theta_2(s)ds} db. \end{align}$ | (36) |
Combining (32), (33) into (36), yields the characteristic equation at
$ G(\lambda) = (\lambda+d)\mathcal{W}_1(\lambda)-\lambda-d\Re_0 = 0, $ | (37) |
where
$\begin {eqnarray*} \mathcal{W}_1(\lambda) & = &\ \frac{(1-f)\beta T^*}{\lambda+c} \int_0^\infty p(b)e^{-\lambda b-\int_0^b\theta_2(s) ds} db \\ && +\frac{f\beta T^*}{\lambda+c}\int_0^\infty \xi(a)e^{-\lambda a-\int_0^a\theta_1(s) ds} da \int_0^\infty p(b)e^{-\lambda b-\int_0^b\theta_2(s) ds} db. \end{eqnarray*} $ |
It is sufficient to show that (37) has no roots with non-negative real parts. Suppose that it has a root
$ (\mu+\nu i+d)\mathcal{W}_1(\mu+\nu i)-\mu -\nu i -d\Re_0 = 0. $ |
Separating the real part of the above equality gives
$ \mathrm{Re}\ \mathcal{W}_1(\mu+\nu i) = \frac{(\mu+d\Re_0)(\mu+d)+\nu^2}{(\mu+d)^2+\nu^2}>1. $ | (38) |
Noticing that
$ \mathrm{Re}\ \mathcal{W}_1(\mu+\nu i)\leq |\mathcal{W}_1(\mu)| = \mathcal{W}_1(\mu)\leq \mathcal{W}_1(0) = 1, $ |
which yields a contradiction. This completes the proof.
This section is devoted to investigate the global stability of the equilibria by using Lyapunov functionals under the threshold value. In what follows, we introduce an important function
Theorem 6.1. The infection-free equilibrium
Proof. Considering the candidate Lyapunov function as follows,
$ \mathcal{L}_{IFE}(t) = \mathcal{L}_1(t)+\mathcal{L}_2(t)+\mathcal{L}_3(t)+\mathcal{L}_4(t), $ |
where
$\begin{align*} \frac{ d\mathcal{L}_1(t)}{ dt} = - dT_0\bigg(\frac{T^0}{T}+\frac{T}{T_0}-2\bigg)-\beta TV+\beta T^0V. \end{align*}$ |
By integration by parts, we calculate the derivative of
$\begin{align*} \frac{ d\mathcal{L}_2(t)}{ dt} = &\ \int_0^\infty \phi(a)\frac{\partial e(a,t)}{\partial t} da =-\int_0^\infty \phi(a) \bigg[\theta_1(a)e(a,t)+\frac{\partial e(a,t)}{\partial a}\bigg] da \\ = &\ -\phi(a)e(a,t) \bigg|_0^\infty+\int_0^\infty \phi'(a)e(a,t) da-\int_0^\infty \phi(a)\theta_1(a)e(a,t) da \\ = &\ \phi(0)e(0,t)+\int_0^\infty \bigg(\phi'(a)-\phi(a)\theta_1(a)\bigg)e(a,t) da. \end{align*}$ |
An argument similar to the one used in calculating the derivative of
$ \frac{ d\mathcal{L}_3(t)}{ dt} = \psi(0)i(0,t)+\int_0^\infty \bigg(\psi'(b)-\psi(b)\theta_2(b)\bigg)i(b,t) db. $ |
We calculate the derivative of
$ \frac{ d\mathcal{L}_4(t)}{ dt} =\frac{\beta T^0}{c}\int_0^\infty p(b)i(b,t) db-\beta T^0V. $ |
Secondly, we have
$\begin{array}{l} \frac{{d{{\cal L}_{IFE}}(t)}}{{dt}} = \; - d{T^0}(\frac{{{T^0}}}{T} + \frac{T}{{{T_0}}} - 2) - \beta TV + \phi (0)f\beta TV + \psi (0)(1 - f)\beta TV\\ \quad \quad \quad \quad \;\; + \int_0^\infty ( \phi '(a) - \phi (a){\theta _1}(a) + \psi (0)\xi (a))e(a,t)da\\ \quad \quad \quad \quad \;\; + \int_0^\infty ( \psi '(b) - \psi (b){\theta _2}(b) + \frac{{\beta {T^0}}}{c}p(b))i(b,t)db. \end{array}$ |
Choosing
$ \left\{ \begin{array}{lll} \displaystyle \psi(b) = \int_b^\infty \frac{\beta T^0}{c}p(u) e^{-\int_b^u \theta_2(\omega) d\omega} du,\\ \displaystyle \phi(a) = \int_a^\infty \psi(0)\xi(u) e^{-\int_a^u \theta_1(\omega) d\omega} du.\\ \end{array}\right. $ |
Then it is easy to see that
$ \left\{ \begin{array}{lll} \displaystyle \psi(0) = \frac {\beta T^0J}{c},\ \phi(0) = \frac {\beta T^0JK}{c},\\ \displaystyle \psi'(b)-\psi(b)\theta_2(b)+\frac{\beta T^0}{c}p(b) = 0,\\ \displaystyle \phi'(a)-\phi(a)\theta_1(a)+\psi(0)\xi(a) = 0. \end{array}\right. $ |
Consequently,
$ \frac{ d\mathcal{L}_{IFE}(t)}{ dt} =- dT_0\bigg(\frac{T^0}{T}+\frac{T}{T_0}-2\bigg)+\big(\Re_0-1\big)\beta TV. $ |
Notice that
To establish the global stability of the infection equilibrium, we introduce the following Lemma.
Lemma 6.2. Suppose that
$\begin {eqnarray} \label{L1} (1-f)\beta T^*V^*\bigg[1-\frac{e(0, t)i^*(0)}{e^*(0) i(0, t)}\bigg]+\int_0^\infty \xi(a) e^*(a)\bigg[1-\frac{e(a, t)i^*(0)}{e^*(a) i(0, t)}\bigg] d a & =& 0, \end{eqnarray}$ | (39) |
Proof. We give the proof for (39). In fact,
$\begin{align*} &\ (1-f)\beta T^*V^*+\int_0^\infty \xi(a) e^*(a)da\\ &\ -(1-f)\beta T^*V^*\frac{e(0,t)i^*(0)}{e^*(0) i(0,t)}-\int_0^\infty \xi(a) e^*(a)\frac{e(a,t)i^*(0)}{e^*(a) i(0,t)} d a \end{align*}$ |
$\begin{align*} = &\ i^*(0)- \left((1-f)\beta TV+\int_0^\infty \xi(a) e(a,t)da\right)\frac{i^*(0)}{i(0,t)}\\ = &\ 0 \end{align*}$ |
This immediately gives (39).
Theorem 6.3. If
Proof. Let
$ G[x,y] = x-y-y \ln \frac{x}{y},\ \mbox{for}\ x, y>0. $ |
It is easy to see that
Considering the following candidate Lyapunov function,
$ \mathcal{L}_{EE}(t) = \mathcal{H}_1(t)+\mathcal{H}_2(t)+\mathcal{H}_3(t)+\mathcal{H}_4(t), $ |
where
$\begin{align*} &\ \mathcal{H}_1(t) =G[T, T^*],\ \mathcal{H}_2(t) =\int_0^\infty \phi_1(a)G \big[ e(a,t),e^*(a)\big]da,\\ &\ \mathcal{H}_3(t) =\int_0^\infty \psi_1(b) G \big[i(b,t), i^*(b) \big] db,\ \mathcal{H}_4(t) =\frac{\beta T^*}{c}G [V,V^*]. \end{align*}$ |
We define
$ \psi_1 (b) = \int_b^\infty \frac{\beta T^*}{c}p(u) e^{-\int_b^u \theta_2(\omega) d\omega} du, $ |
and
$ \phi_1 (a) = \int_a^\infty \psi_1(0)\xi(u) e^{-\int_a^u \theta_1(\omega) d\omega} du, $ |
it follows that
$\begin{align*} \psi_1'(b)-\psi_1(b)\theta_2(b) =&\ -\frac{\beta T^*}{c}p(b). \end{align*}$ |
$\begin{align*} \phi_1'(a)-\phi_1(a)\theta_1(a) =&\ -\psi_1(0)\xi(a). \end{align*}$ |
Firstly, we calculate the derivative of
$ \frac{ d\mathcal{H}_1(t)}{ dt} =-dT^*\bigg(\frac{T}{T^*}+\frac{T^*}{T}-2\bigg)+\frac{1}{f}\left(1-\frac{T^*}{T}\right)\left(e^*(0)-e(0,t)\right) $ |
By using (4),
$\begin{align*} \mathcal{H}_2(t) =&\ \int_0^t \phi_1(a) G[e(0,t-a)\Omega(a), e^*(a)] da\\ &\ +\int_t^\infty \phi_1(a) G[e_0(a-t)e^{-\int_{a-t}^a \theta_1(\omega) d\omega}, e^*(a)] da\\ = &\ \int_0^t \phi_1(t-r) G[e(0,r)\Omega(t-r), e^*(t-r)] dr\\ &\ +\int_0^\infty \phi_1(t+r) G[e_0(r)e^{-\int_r^{t+r} \theta_1(\omega) d\omega}, e^*(t+r)] dr\\ = &\ \mathcal{B_1}(t)+\mathcal{B_2}(t). \end{align*}$ |
The derivative of
$\begin{align*} &\ \frac{ d\mathcal{B_1}(t)}{ dt}\\& =\phi_1(0) G[e(0,t),e^*(0)] +\int_0^t \phi_1'(t-r) G\left[e(0,r)e^{-\int_0^{t-r} \theta_1(\omega) d\omega},e^*(t-r)\right] dr\\ &\ -\int_0^t \phi_1(t-r) \theta_1(t-r)\bigg[e(0,r)e^{-\int_0^{t-r} \theta_1(\omega) d\omega} G_x\left[e(0,r)e^{-\int_0^{t-r} \theta_1(\omega) d\omega},e^*(t-r)\right]\\ &\ + e^*(t-r) G_y\left[e(0,r)e^{-\int_0^{t-r} \theta_1(\omega) d\omega},e^*(t-r)\right] \bigg] dr, \end{align*}$ |
and
$\begin{align*} &\ \frac{ d\mathcal{B_2}(t)}{ dt} =\int_0^\infty \phi_1'(t+r) G\left[e_0(r)e^{-\int_r^{t+r} \theta_1(\omega) d\omega},e^*(t+r)\right] dr\\ &\ -\int_0^\infty \phi_1(t+r) \theta_1(t+r) \bigg[e_0(r)e^{-\int_r^{t+r} \theta_1(\omega) d\omega} G_x\left[e_0(r)e^{-\int_r^{t+r} \theta_1(\omega) d\omega}, e^*(t+r)\right]\\ &\ +e^*(t+r) G_y\left[e_0(r)e^{-\int_r^{t+r} \theta_1(\omega) d\omega}, e^*(t+r)\right]\bigg] dr. \end{align*}$ |
We obtain the derivative of
$\begin{array}{l} \frac{{d{{\cal H}_2}(t)}}{{dt}} = \;{\phi _1}(0)G[e(0,t),{e^*}(0)] + \int_0^\infty [ {\phi _{1'}}(a) - {\phi _1}(a){\theta _1}(a)]G[e(a,t),{e^*}(a)]da\\ \quad \quad \quad \; = {\phi _1}(0)G[e(0,t),{e^*}(0)] - \int_0^\infty {{\psi _1}} (0)\xi (a)G[e(a,t),{e^*}(a)]da. \end{array}$ |
A similar argument as in the derivative of
$\begin{align*} \frac{ d\mathcal{H}_3(t)}{ dt} =&\ \psi_1(0) G[i(0,t),i^*(0)]+\int_0^\infty \big[\psi'(b)-\psi(b)\theta_2(b) \big] G[i(b,t),i^*(b)] db \\ =&\ \psi_1(0) G[i(0,t),i^*(0)]-\int_0^\infty \frac{\beta T^*}{c}p(b) G[i(b,t),i^*(b)] db. \end{align*}$ |
We calculate the derivative of
$\frac{{d{{\cal H}_4}(t)}}{{dt}} = \frac{{\beta {T^*}}}{c}\int_0^\infty p (b)i(b,t)db - \beta {T^*}V + \beta {T^*}{V^*} - \frac{{\beta {T^*}{V^*}}}{{cV}}\int_0^\infty p (b)i(b,t)db.$ |
If follows from
$\begin{array}{*{20}{l}} {\frac{{d{{\cal L}_{EE}}}}{{dt}} = }&{\; - d{T^*}(\frac{T}{{{T^*}}} + \frac{{{T^*}}}{T} - 2) + \frac{1}{f}\left( {1 - \frac{{{T^*}}}{T}} \right)\left( {{e^*}(0) - e(0,t)} \right)}\\ {}&{ + {\phi _1}(0)G[e(0,t),{e^*}(0)] - \int_0^\infty {{\psi _1}} (0)\xi (a)G[e(a,t),{e^*}(a)]da}\\ {}&{ + {\psi _1}(0)G[i(0,t),{i^*}(0)] - \int_0^\infty {\frac{{\beta {T^*}}}{c}} p(b)G[i(b,t),{i^*}(b)]db}\\ {}&{ + \int_0^\infty {\frac{{\beta {T^*}}}{c}} p(b)i(b,t)db + \beta {T^*}{V^*} - \beta {T^*}V - \frac{{{V^*}}}{V}\int_0^\infty {\frac{{\beta {T^*}}}{c}} p(b)i(b,t)db.{\rm{ }}} \end{array}$ | (40) |
Recall that
$ (1-f)(\beta T^*V^*-\beta TV)+\int_0^\infty \xi(a)\bigg(e^*(a)-e(a,t)\bigg) da = i^*(0)-i(0,t), $ |
and
$ \frac{f\beta T^* K J}{c}+\frac{(1-f)\beta T^* J}{c} = \left(\frac{f\beta K J}{c}+\frac{(1-f)\beta J}{c}\right)\frac{T^0}{\Re_0} = 1. $ |
Thus (40) becomes
$\begin{array}{l} \frac{{d{{\cal L}_{EE}}(t)}}{{dt}} = \; - d{T^*}(\frac{T}{{{T^*}}} + \frac{{{T^*}}}{T} - 2) + \frac{1}{f}\left( {1 - \frac{{{T^*}}}{T}} \right)\left( {{e^*}(0) - e(0,t)} \right)\\ \quad \quad \quad \quad + \frac{1}{f}G[e(0,t),{e^*}(0)] - \int_0^\infty {\frac{{\beta {T^*}}}{c}} p(b)G[i(b,t),{i^*}(b)]db\\ \quad \quad \quad \quad + \frac{{\beta {T^*}J}}{c}\left[ {(1 - f)\beta {T^*}{V^*}\ln \frac{{e(0,t){i^*}(0)}}{{{e^*}(0)i(0,t)}} + \int_0^\infty \xi (a){e^*}(a)\ln \frac{{e(a,t){i^*}(0)}}{{{e^*}(a)i(0,t)}}da} \right]\;\\ \quad \quad \quad \quad + \int_0^\infty {\frac{{\beta {T^*}}}{c}} p(b)i(b,t)db + \beta {T^*}{V^*} - \beta {T^*}V - \frac{{{V^*}}}{V}\int_0^\infty {\frac{{\beta {T^*}}}{c}} p(b)i(b,t)db. \end{array}$ |
It follows that,
$\begin{array}{*{20}{l}} {\frac{{d{{\cal L}_{EE}}(t)}}{{dt}}}&{ = \; - d{T^*}(\frac{T}{{{T^*}}} + \frac{{{T^*}}}{T} - 2) - \frac{1}{f}{e^*}(0)\left( {\frac{{{T^*}}}{T} - \ln \frac{{e(0,t)}}{{{e^*}(0)}}} \right)}\\ {}&{ - \int_0^\infty {\frac{{\beta {T^*}}}{c}} p(b)G[i(b,t),{i^*}(b)]db}\\ {}&{ + \frac{{\beta {T^*}J}}{c}\left[ {(1 - f)\beta {T^*}{V^*}\ln \frac{{e(0,t){i^*}(0)}}{{{e^*}(0)i(0,t)}} + \int_0^\infty \xi (a){e^*}(a)\ln \frac{{e(a,t){i^*}(0)}}{{{e^*}(a)i(0,t)}}da} \right]}\\ {}&{ + \int_0^\infty {\frac{{\beta {T^*}}}{c}} p(b)i(b,t)db + \beta {T^*}{V^*} - \frac{{{V^*}}}{V}\int_0^\infty {\frac{{\beta {T^*}}}{c}} p(b)i(b,t)db.} \end{array}$ | (41) |
Recall that
$\begin{align*} &\ \frac{d\mathcal{L}_{EE}(t)}{dt} =-dT^*\bigg(\frac{T}{T^*}+\frac{T^*}{T}-2\bigg)\\ &\ +\frac{\beta T^* J}{c}\left[(1-f)\beta T^*V^*\ln\frac{e(0,t)i^*(0)}{e^*(0) i(0,t)}+\int_0^\infty \xi(a) e^*(a)\ln\frac{e(a,t)i^*(0)}{e^*(a) i(0,t)} d a\right]\\ &\ +\int_0^\infty\frac{\beta T^*}{c} p(b)i^*(b)\bigg(2+\ln\frac{i(b,t)}{i^*(b)}-\frac{T^*}{T}-\ln \frac{e(0,t)}{e^*(0)}-\frac{V^* i(b,t)}{V i^*(b)}\bigg) db. \end{align*}$ |
Further, we have
$\begin{array}{*{20}{l}} {\frac{{d{{\cal L}_{EE}}(t)}}{{dt}} = }&{\; - d{T^*}(\frac{T}{{{T^*}}} + \frac{{{T^*}}}{T} - 2)}\\ {}&{ + \frac{{\beta {T^*}J}}{c}(1 - f)\beta {T^*}{V^*}\left( {1 - \frac{{e(0,t){i^*}(0)}}{{{e^*}(0)i(0,t)}} + \ln \frac{{e(0,t){i^*}(0)}}{{{e^*}(0)i(0,t)}}} \right)}\\ {}&{ + \frac{{\beta {T^*}J}}{c}\int_0^\infty \xi (a){e^*}(a)\left( {1 - \frac{{e(a,t){i^*}(0)}}{{{e^*}(a)i(0,t)}} + \ln \frac{{e(a,t){i^*}(0)}}{{{e^*}(a)i(0,t)}}} \right)da}\\ {}&{ + \int_0^\infty {\frac{{\beta {T^*}p(b)}}{c}} {i^*}(b)(2 - \frac{{{T^*}}}{T} + \ln \frac{{{T^*}}}{T} - \frac{{{V^*}i(b,t)}}{{V{i^*}(b)}} + \ln \frac{{{V^*}i(b,t)}}{{V{i^*}(b)}})db} \end{array}$ | (42) |
$\begin{align*} &\ -\frac{\beta T^* J}{c}\bigg\{(1-f)\beta T^*V^*\bigg[1-\frac{e(0,t)i^*(0)}{e^*(0) i(0,t)}\bigg]\nonumber\\ &\ +\int_0^\infty \xi(a) e^*(a)\bigg[1-\frac{e(a,t)i^*(0)}{e^*(a) i(0,t)}\bigg] d a \bigg\}.\nonumber \end{align*}$ |
Recall that Lemma 6.2 holds. Putting (39) into (42), we have
$\begin{align*} \frac{d\mathcal{L}_{EE}(t)}{dt} =&\ -dT^*\bigg(\frac{T}{T^*}+\frac{T^*}{T}-2\bigg)\\ &\ -\frac{\beta T^*J}{c}\bigg[\int_0^\infty \xi(a) e^*(a)g\bigg(\frac{e(a,t)i^*(0)}{e^*(a) i(0,t)}\bigg)da\\ &\ +(1-f)\int_0^\infty\frac{\beta T^* }{c} p(b)i^*(b)g\bigg(\frac{e(0,t)i^*(0)}{e^*(0) i(0,t)}\bigg)db\bigg]\\ &\ -\int_0^\infty\frac{\beta T^* }{c} p(b)i^*(b)\bigg[g\bigg(\frac{T^*}{T}\bigg) +g\bigg(\frac{V^* i(b,t)}{Vi^*(b)}\bigg)\bigg] db\\ \leq &\ 0 \end{align*}$ |
and
$ \frac{i(b,t)}{i^*(b)}=\frac{i(0,t)}{i^*(0)}=\frac{V}{V^*}=\frac{e(0,t)}{e^*(0)}=\frac{e(a,t)}{e^*(a)}, \ \ \mbox{for all $a$, $b\geq 0$.} $ |
It is not difficult to check that the largest invariant subset
This paper is devoted to the global dynamics of an HIV infection model subject to latency age and infection age, where the model formulation, basic reproduction number computation, and rigorous mathematical analysis, such as relative compactness and persistence of the solution semi-flows, and existence of a global attractor are involved. We have shown that the existence of a compact attractor of all compact sets of nonnegative initial data and used the Lyapunov functional to show that this attractor is the singleton set containing the equilibrium. Given that the model is so complex, the proof does require some rigorous calculation. The dynamics (at least the long-term dynamics) of the model do not appear to have been altered by adding the
The authors would like to thank the anonymous referees and editor for very helpful suggestions and comments which led to improvements of our original manuscript. J. Wang is supported by National Natural Science Foundation of China (No. 11226255 and No. 11201128), the Science and Technology Innovation Team in Higher Education Institutions of Heilongjiang Province (No. 2014TD005). X. Dong is supported by Graduate Students Innovation Research Program of Heilongjiang University (No. YJSCX2017-177HLJU).
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