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Effect of cellular reservoirs and delays on the global dynamics of HIV

Advances in Difference Equations20182018:85

https://doi.org/10.1186/s13662-018-1523-0

  • Received: 13 December 2017
  • Accepted: 13 February 2018
  • Published:

Abstract

We investigate general HIV infection models with three types of infected cells: latently infected cells, long-lived productively infected cells, and short-lived productively infected cells. We incorporate three discrete or distributed time delays into the models. Moreover, we consider the effect of humoral immunity on the dynamical behavior of the HIV. The HIV-target incidence rate, production/proliferation, and removal rates of the cells and HIV are represented by general nonlinear functions. We show that the solutions of the proposed model are nonnegative and ultimately bounded. We derive two threshold parameters which fully determine the existence and stability of the three steady states of the model. Using Lyapunov functionals, we establish the global stability of the steady states of the model. The theoretical results are confirmed by numerical simulations.

Keywords

  • HIV infection
  • HAART
  • Global stability
  • Humoral immune response
  • Latency
  • Viral reservoirs

MSC

  • 34D20
  • 34D23
  • 37N25
  • 92B05

1 Introduction

Mathematical modeling and analysis of within-host human immunodeficiency virus (HIV) dynamics have become one of the hot topics during the last decades [131]. These works can help researchers better understand the HIV dynamical behavior and provide new suggestions for clinical treatment. Most of the mathematical models presented in the literature have focused on modeling the interaction between three main compartments: uninfected CD4+ T cells (s), infected cells (y), and free HIV particles (p). Other models have differentiated between latent and active infected cells by introducing a new variable (w) for the latently infected cells [3237]. In [38], an HIV mathematical model has been presented by considering three types of infected cells: latently infected cells (w), short-lived productively infected cells (y), and long-lived productively infected cells (u) as follows:
$$\begin{aligned}& \dot{s}(t) =\rho-\beta_{1}s(t)+\omega s(t) \biggl( 1- \frac{s(t)}{s _{\max}} \biggr) -(1-\varepsilon_{1}) (\overline{\lambda} _{1}+\overline{\lambda}_{2}+\overline{\lambda} _{3})s(t)p(t), \end{aligned}$$
(1)
$$\begin{aligned}& \dot{w}(t) =(1-\varepsilon_{1})\overline{\lambda}_{1} s(t)p(t)-(a_{1}+\beta_{2})w(t), \end{aligned}$$
(2)
$$\begin{aligned}& \dot{y}(t) =(1-\varepsilon_{1})\overline{\lambda}_{2} s(t)p(t)+a_{1}w(t)-\beta_{3}y(t), \end{aligned}$$
(3)
$$\begin{aligned}& \dot{u}(t) =(1-\varepsilon_{1})\overline{\lambda}_{3} s(t)p(t)-\beta_{4}u(t), \end{aligned}$$
(4)
$$\begin{aligned}& \dot{p}(t) =(1-\varepsilon_{2})\overline{N}\beta_{3}y(t)+(1- \varepsilon_{2})\overline{M}\beta_{4}u(t)- \beta_{5}p(t), \end{aligned}$$
(5)
where ρ is the creation rate of the uninfected CD4+ T cells, \(\beta_{i}\), \(i=1,\ldots,5\), are the death rate constants of the five compartments s, w, y, u, and p, respectively. The model incorporates reverse transcriptase inhibitor (RTI) with efficacy \(\varepsilon_{1}\) and protease inhibitor (PI) with efficacy \(\varepsilon_{2}\), where \(\varepsilon_{1},\varepsilon_{2}\in{}[0,1]\). The parameters ω and \(s_{\max}\) are the maximum proliferation rate and the maximum level of concentration of the uninfected CD4+ T cells, respectively. The latently infected cells are activated at rate \(a_{1}w\). The parameters and are the average numbers of HIV particles generated in the lifetime of the short-lived and long-lived infected cells, respectively. The uninfected CD4+ T cells become infected with infectivity \(\overline{\lambda }_{1}+\overline{\lambda}_{2}+\overline{\lambda}_{3}\). Elaiw et al. [27] have generalized the above model by incorporating the humoral immune response and considering general nonlinear functions for the generation and removal rates of all compartments:
$$\begin{aligned}& \dot{s}(t) =\pi\bigl(s ( t ) \bigr)-(1-\varepsilon_{1}) (\overline{ \lambda }_{1}+\overline{\lambda}_{2}+\overline{ \lambda}_{3})\chi\bigl(s ( t ) ,p ( t ) \bigr), \end{aligned}$$
(6)
$$\begin{aligned}& \dot{w}(t) =(1-\varepsilon_{1})\overline{\lambda}_{1} \chi\bigl(s ( t ) ,p ( t ) \bigr)-(a_{1}+\beta_{2})g_{1} \bigl(w ( t ) \bigr), \end{aligned}$$
(7)
$$\begin{aligned}& \dot{y}(t) =(1-\varepsilon_{1})\overline{\lambda}_{2} \chi\bigl(s ( t ) ,p ( t ) \bigr)+a_{1}g_{1}\bigl(w ( t ) \bigr)-\beta_{3}g_{2}\bigl(y ( t ) \bigr), \end{aligned}$$
(8)
$$\begin{aligned}& \dot{u}(t) =(1-\varepsilon_{1})\overline{\lambda}_{3} \chi\bigl(s ( t ) ,p ( t ) \bigr)-\beta_{4}g_{3}\bigl(u ( t ) \bigr), \end{aligned}$$
(9)
$$\begin{aligned}& \begin{aligned}[b] \dot{p}(t) &=(1-\varepsilon_{2})\overline{N}\beta_{3}g_{2} \bigl(y ( t ) \bigr)+(1-\varepsilon_{2})\overline{M} \beta_{4}g_{3}\bigl(u ( t ) \bigr) \\ &\quad{} -\beta_{5}g_{4} \bigl(p ( t ) \bigr)-qg_{4}\bigl(p ( t ) \bigr)g _{5}\bigl(x ( t ) \bigr), \end{aligned} \end{aligned}$$
(10)
$$\begin{aligned}& \dot{x}(t) =rg_{4}\bigl(p ( t ) \bigr)g_{5}\bigl(x ( t ) \bigr)-\beta _{6}g_{5}\bigl(x ( t ) \bigr), \end{aligned}$$
(11)
where x represents the concentration of the B cells. π, χ, \(g_{i}\), \(i=1,\ldots,5\), are general nonlinear functions. Model (6)–(11) assumes that, once the HIV contacts a CD4+ T cell, it becomes infected in the same time. Neglecting the time delays is an unrealistic assumption (see, e.g., [29, 30]).

The aim of this paper is to propose HIV infection models which improve model (6)–(11) by taking into account three time delays, discrete or distributed. We derive two threshold parameters and present some mild sufficient conditions for the existence and global stability of the steady states of the models.

2 HIV dynamics model with discrete delays

We formulate a nonlinear HIV dynamics model with latent reservoirs, humoral immunity, and discrete time delays:
$$\begin{aligned}& \dot{s} =\pi\bigl(s ( t ) \bigr)-(1-\varepsilon_{1}) (\overline{ \lambda }_{1}+\overline{\lambda}_{2}+\overline{ \lambda}_{3})\chi\bigl(s ( t ) ,p ( t ) \bigr), \end{aligned}$$
(12)
$$\begin{aligned}& \dot{w} =(1-\varepsilon_{1})\overline{\lambda}_{1}e^{-\mu _{1}\tau_{1}} \chi\bigl(s ( t-\tau_{1} ) ,p ( t-\tau_{1} ) \bigr)-(a_{1}+\beta_{2})g_{1}\bigl(w ( t ) \bigr), \end{aligned}$$
(13)
$$\begin{aligned}& \dot{y} =(1-\varepsilon_{1})\overline{\lambda}_{2}e^{-\mu _{2}\tau_{2}} \chi\bigl(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) \bigr)+a_{1}g_{1}\bigl(w ( t ) \bigr)-\beta_{3}g_{2} \bigl(y ( t ) \bigr), \end{aligned}$$
(14)
$$\begin{aligned}& \dot{u} =(1-\varepsilon_{1})\overline{\lambda}_{3}e^{-\mu _{3}\tau_{3}} \chi\bigl(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) \bigr)- \beta_{4}g_{3}\bigl(u ( t ) \bigr), \end{aligned}$$
(15)
$$\begin{aligned}& \dot{p} =(1-\varepsilon_{2})\overline{N}\beta_{3}g_{2} \bigl(y ( t ) \bigr)+(1-\varepsilon_{2})\overline{M} \beta_{4}g_{3}\bigl(u ( t ) \bigr)-\beta_{5}g_{4} \bigl(p ( t ) \bigr)-qg_{4}\bigl(p ( t ) \bigr)g _{5}\bigl(x ( t ) \bigr), \end{aligned}$$
(16)
$$\begin{aligned}& \dot{x} =rg_{4}\bigl(p ( t ) \bigr)g_{5}\bigl(x ( t ) \bigr)-\beta_{6}g _{5}\bigl(x ( t ) \bigr). \end{aligned}$$
(17)
All the parameters are positive. Here, \(\tau_{1}\) is the time between viral entry and latent infection (i.e., the integration of viral DNA into cell’s DNA has finished), while \(\tau_{2}\) and \(\tau_{3}\) are the times between viral entry and viral production form short-lived productively infected and long-lived productively infected cells, respectively. The factor \(e^{-\mu_{i}\tau_{i}}\) accounts for the loss of target cells during the delay period of length \(\tau_{i}\), where \(\mu_{i}\) is a constant. Let us define \(\lambda_{i}=(1-\varepsilon _{1})\overline{\lambda}_{i}\), \(i=1,2,3\), \(N=(1-\varepsilon _{2})\overline{N}\), \(M=(1-\varepsilon_{2})\overline{M}\), and \(\lambda=\lambda_{1}+\lambda_{2}+\lambda_{3}\). Functions χ, π, \(g_{i}\), \(i=1,\ldots,5\), are continuously differentiable and satisfy the following hypotheses:
(H1).
  1. (i)

    There exists \(s_{0}\) such that \(\pi(s_{0})=0\), \(\pi(s)>0\) for \(s\in{}[0,s_{0})\);

     
  2. (ii)

    \(\pi^{\prime}(s)<0\) for \(s\in(0,\infty)\);

     
  3. (iii)

    There are \(b>0\) and \(\overline{b}>0\) such that \(\pi(s)\leq b- \overline{b}s\) for \(s\in{}[0,\infty)\).

     
(H2).
  1. (i)

    \(\chi(s,p)>0\) and \(\chi(0,p)=\chi(s,0)=0\) for \(s,p\in(0,\infty)\);

     
  2. (ii)

    \(\frac{\partial\chi(s,p)}{\partial s}>0\), \(\frac{\partial \chi(s,p)}{\partial p}>0\), and \(\frac{\partial\chi(s,0)}{\partial p}>0\) for all \(s,p\in(0,\infty)\);

     
  3. (iii)

    \(( \frac{\partial\chi(s,0)}{\partial p} ) ^{\prime }>0\) for \(s\in(0,\infty)\).

     
(H3).
  1. (i)

    \(g_{j}(\eta)>0\) for \(\eta\in(0,\infty)\), \(g_{j}(0)=0\), \(j=1,\ldots,5\);

     
  2. (ii)

    \(g_{j}^{\prime}(\eta)>0\) for \(\eta\in(0,\infty)\), \(j=1,2,3,5\), \(g_{4}^{\prime}(\eta)>0\), for \(\eta\in{}[0, \infty)\);

     
  3. (iii)

    there are \(\alpha_{j}>0\), \(j=1,\ldots,5\), such that \(g_{j}(\eta)\geq\alpha_{j}\eta\) for \(\eta\in{}[0,\infty)\).

     
(H4).
  1. (i)

    \(\frac{\partial}{\partial p} ( \frac{\chi(s,p)}{g _{4}(p)} ) \leq0\) for \(p\in(0,\infty)\).

     
We consider system (12)–(17) with the initial conditions:
$$\begin{aligned} \begin{gathered} s ( \theta ) =\varphi_{1} ( \theta ) ,\quad\quad w ( \theta ) = \varphi_{2} ( \theta ) ,\quad\quad y ( \theta ) =\varphi_{3} ( \theta ) , \\ u ( \theta ) =\varphi_{4} ( \theta ) ,\quad\quad p ( \theta ) = \varphi_{5} ( \theta ) ,\quad\quad x ( \theta ) =\varphi_{6} ( \theta ) , \\ \varphi_{j} ( \theta ) \geq0,\quad \theta\in{}[-\varsigma ,0], \\ \varphi_{j} ( \theta ) \in C \bigl( [ -\varsigma,0 ] , \mathbb{R}_{\geq0}^{6} \bigr), \quad j=1,\ldots,6, \end{gathered} \end{aligned}$$
(18)
where \(\varsigma=\max \{ \tau_{1},\tau_{2},\tau_{3} \} \) and C is the Banach space of continuous functions mapping the interval \([ -\varsigma,0 ] \) into \(\mathbb{R}_{\geq0}^{6}\). Then the uniqueness of the solution for \(t>0\) is guaranteed [39].

2.1 Preliminaries

Lemma 1

Let hypotheses (H1)(H3) be valid, then the solutions of system (12)(17) are nonnegative and ultimately bounded.

Proof

Let us write system (12)–(17) in the matrix form \(\dot{k} ( t ) =L ( k ( t ) ) \), where \(k=(s,w,y, u,p,x)^{T}\), \(L=(L_{1},L_{2},L_{3},L_{4},L_{5},L_{6})^{T}\), and
L ( k ( t ) ) = ( L 1 ( k ( t ) ) L 2 ( k ( t ) ) L 3 ( k ( t ) ) L 4 ( k ( t ) ) L 5 ( k ( t ) ) L 6 ( k ( t ) ) ) , L = ( π ( s ( t ) ) λ χ ( s ( t ) , p ( t ) ) λ 1 e μ 1 τ 1 χ ( s ( t τ 1 ) , p ( t τ 1 ) ) ( a 1 + β 2 ) g 1 ( w ( t ) ) λ 2 e μ 2 τ 2 χ ( s ( t τ 2 ) , p ( t τ 2 ) ) + a 1 g 1 ( w ( t ) ) β 3 g 2 ( y ( t ) ) λ 3 e μ 3 τ 3 χ ( s ( t τ 3 ) , p ( t τ 3 ) ) β 4 g 3 ( u ( t ) ) N β 3 g 2 ( y ( t ) ) + M β 4 g 3 ( u ( t ) ) β 5 g 4 ( p ( t ) ) q g 4 ( p ( t ) ) g 5 ( x ( t ) ) r g 4 ( p ( t ) ) g 5 ( x ( t ) ) β 6 g 5 ( x ( t ) ) ) .
We have
$$ L_{i} \bigl( k ( t ) \bigr) \big| _{k_{i} ( t ) \in\mathbb{R}_{>0}^{6}}\geq0, \quad i=1,\ldots,6. $$
Using Lemma 2 in [40], the solutions of system (12)–(17) with the initial states (18) satisfy \(X(t)\in\mathbb{R}_{\geq0}^{6}\) for all \(t\geq0\). The nonnegativity of the model’s solution implies that \(\lim\sup_{t\rightarrow\infty }s ( t ) \leq M_{1}\), where \(M_{1}=\frac{b}{\bar{b} }\). Let
$$\begin{aligned} T(t) & =Ne^{-\mu_{1}\tau_{1}}s(t-\tau_{1})+Ne^{-\mu_{2}\tau_{2}}s(t- \tau_{2})+Me^{-\mu_{3}\tau_{3}}s(t-\tau_{3}) \\ &\quad{} +Nw(t)+Ny(t)+Mu(t)+\frac{1}{2}p(t)+\frac{q}{2r}x(t), \end{aligned}$$
then
$$\begin{aligned} \dot{T}(t) & =Ne^{-\mu_{1}\tau_{1}} \bigl[ \pi\bigl(s ( t-\tau_{1} ) \bigr)-\lambda\chi\bigl(s ( t-\tau_{1} ) ,p ( t-\tau_{1} ) \bigr) \bigr] \\ & \quad{} +Ne^{-\mu_{2}\tau_{2}} \bigl[ \pi\bigl(s ( t-\tau_{2} ) \bigr)- \lambda \chi\bigl(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) \bigr) \bigr] \\ &\quad{} +Me^{-\mu_{3}\tau_{3}} \bigl[ \pi\bigl(s ( t-\tau_{3} ) \bigr)- \lambda \chi\bigl(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) \bigr) \bigr] \\ &\quad{} +N \bigl[ \lambda_{1}e^{-\mu_{1}\tau_{1}}\chi\bigl(s ( t- \tau_{1} ) ,p ( t-\tau_{1} ) \bigr)- ( a_{1}+ \beta_{2} ) g_{1}\bigl(w(t)\bigr) \bigr] \\ &\quad{} +N \bigl[ \lambda_{2}e^{-\mu_{2}\tau_{2}}\chi\bigl(s ( t- \tau_{2} ) ,p ( t-\tau_{2} ) \bigr)+a_{1}g_{1} \bigl(w(t)\bigr)-\beta _{3}g_{2}\bigl(y(t)\bigr) \bigr] \\ & \quad{}+M \bigl[ \lambda_{3}e^{-\mu_{3}\tau_{3}}\chi\bigl(s ( t- \tau_{3} ) ,p ( t-\tau_{3} ) \bigr)-\beta_{4}g_{3} \bigl(u ( t ) \bigr) \bigr] \\ & \quad{}+\frac{1}{2} \bigl( N\beta_{3}g_{2}\bigl(y ( t ) \bigr)+M\beta_{4}g _{3}\bigl(u ( t ) \bigr)- \beta_{5}g_{4}\bigl(p ( t ) \bigr)-qg_{4}\bigl(p ( t ) \bigr)g_{5}\bigl(x ( t ) \bigr) \bigr) \\ &\quad{} +\frac{q}{2r} \bigl( rg_{4}\bigl(p ( t ) \bigr)g_{5}\bigl(x ( t ) \bigr)-\beta_{6}g_{5} \bigl(x ( t ) \bigr) \bigr) \\ & \leq Ne^{-\mu_{1}\tau_{1}} \bigl[ b-\overline{b}s ( t-\tau_{1} ) \bigr] +Ne^{-\mu_{2}\tau_{2}} \bigl[ b-\overline{b}s ( t-\tau_{2} ) \bigr] +Me^{-\mu_{3}\tau_{3}} \bigl[ b- \overline{b}s ( t-\tau_{3} ) \bigr] \\ & \quad{}-N\beta_{2}\alpha_{1}w(t)-\frac{1}{2}N \beta_{3}\alpha_{2}y(t)- \frac{1}{2}M \beta_{4}\alpha_{3}u(t)-\frac{1}{2} \beta_{5}\alpha_{4}p(t)- \frac{q}{2r} \beta_{6}\alpha_{5}p(t) \\ & \leq b \bigl( Ne^{-\mu_{1}\tau_{1}}+Ne^{-\mu_{2}\tau_{2}}+Me^{-\mu _{3}\tau_{3}} \bigr) \\ & \quad{}-\sigma \biggl[ Ne^{-\mu_{1}\tau_{1}}s(t-\tau_{1})+Ne^{-\mu_{2}\tau _{2}}s(t- \tau_{2})+Me^{-\mu_{3}\tau_{3}}s(t-\tau_{3}) \\ & \quad{} +Nw(t)+Ny(t)+Mu(t)+\frac{1}{2}p(t)+\frac{q}{2r}x(t) \biggr] \\ & \leq b ( 2N+M ) -\sigma T(t), \end{aligned}$$
where \(\sigma=\min\{ \overline{b},\beta_{2}\alpha_{1},\frac{1}{2} \beta_{3}\alpha_{2},\frac{1}{2}\beta_{4}\alpha_{3},\beta_{5}\alpha _{4},\beta_{6}\alpha_{5}\}\). Then \(\lim\sup_{t\rightarrow\infty}T(t) \leq\frac{b ( 2N+M ) }{\sigma}\). The nonnegativity of the system’s variables implies that
$$\begin{aligned}& \lim\sup_{t\rightarrow\infty}w(t) \leq\frac{b ( 2N+M ) }{N\sigma}=M_{2}, \\& \lim\sup_{t\rightarrow\infty}y(t) \leq\frac{b ( 2N+M ) }{N\sigma}=M_{2}, \\& \lim\sup_{t\rightarrow\infty}u(t) \leq\frac{b ( 2N+M ) }{M\sigma}=M_{3}, \\& \lim\sup_{t\rightarrow\infty}p(t) \leq\frac{2b ( 2N+M ) }{\sigma}=M_{4}, \\& \lim\sup_{t\rightarrow\infty}x(t) \leq\frac{2rb ( 2N+M ) }{q\sigma}=M_{5}. \end{aligned}$$
Therefore, \(s ( t ) \), \(w ( t ) \), \(y ( t ) \), \(u ( t ) \), \(p ( t ) \), and \(x ( t ) \) are ultimately bounded. □
According to Lemma 1, we can show that the region
$$ \begin{aligned}[b] \Omega={}& \bigl\{ (s,w,y,u,p,x)\in C^{6}: \Vert s \Vert \leq M_{1}, \Vert w \Vert \leq M_{2}, \Vert y \Vert \leq M_{2}, \\ &\Vert u \Vert \leq M_{3}, \Vert p \Vert \leq M_{4}, \Vert x \Vert \leq M_{5} \bigr\} \end{aligned} $$
(19)
is positively invariant with respect to system (12)–(17), where \(\Vert \phi \Vert =\sup_{t\rightarrow\infty}\phi(t)\).

Lemma 2

Suppose that hypotheses (H1)(H4) are valid (12)(17), then there exist two bifurcation parameters \(R_{0}\) and \(R_{1}\) with \(R_{0}>R_{1}>0\) such that
  1. (i)

    if \(R_{0}\leq1\), then there exists only one steady state \(\Pi_{0}\);

     
  2. (ii)

    if \(R_{1}\leq1< R_{0}\), then there exist two steady states \(\Pi_{0}\) and \(\Pi_{1}\);

     
  3. (iii)

    if \(R_{1}>1\), then there exist three steady states \(\Pi_{0}\), \(\Pi_{1}\), and \(\Pi_{2}\).

     

Proof

Let \(\Pi(s,w,y,u,p,x)\) be any steady state of (12)–(17) satisfying the following equations:
$$\begin{aligned}& 0 =\pi(s)-\lambda\chi(s,p), \end{aligned}$$
(20)
$$\begin{aligned}& 0 =\lambda_{1}e^{-\mu_{1}\tau_{1}}\chi(s,p)-(a_{1}+ \beta_{2})g _{1}(w), \end{aligned}$$
(21)
$$\begin{aligned}& 0 =\lambda_{2}e^{-\mu_{2}\tau_{2}}\chi(s,p)+a_{1}g_{1}(w)- \beta_{3}g_{2}(y), \end{aligned}$$
(22)
$$\begin{aligned}& 0 =\lambda_{3}e^{-\mu_{3}\tau_{3}}\chi(s,p)-\beta_{4}g _{3}(u), \end{aligned}$$
(23)
$$\begin{aligned}& 0 =N\beta_{3}g_{2}(y)+M\beta_{4}g_{3}(u)- \beta_{5}g_{4}(p)-qg_{4}(p)g _{5}(x), \end{aligned}$$
(24)
$$\begin{aligned}& 0 =rg_{4}(p)g_{5}(x)-\beta_{6}g_{5}(x). \end{aligned}$$
(25)
From Eq. (25) we have two possible solutions: \(g_{5}(x)=0\) and \(g_{4}(p)=\beta_{6}/r\). The first possibility \(g_{5}(x)=0\) implies that \(x=0\). Hypothesis (H3) implies that \(g_{i}^{-1}\), \(i=1,\ldots,5\), exist, strictly increasing and \(g_{i}^{-1}(0)=0\). Let us define
$$\begin{aligned} \begin{gathered} \psi_{1}(s) =g_{1}^{-1} \biggl( \frac{\lambda_{1}e^{-\mu _{1}\tau_{1}}}{\lambda(a_{1}+\beta_{2})}\pi(s) \biggr) , \\ \psi_{2}(s)=g_{2}^{-1} \biggl( \frac{a_{1}\lambda _{1}e^{-\mu_{1}\tau_{1}}+(a_{1}+\beta_{2})\lambda _{2}e^{-\mu_{2}\tau_{2}}}{\beta_{3}\lambda(a_{1}+\beta_{2})}\pi (s) \biggr) , \\ \psi_{3}(s) =g_{3}^{-1} \biggl( \frac{\lambda_{3}e^{-\mu _{3}\tau_{3}}}{\beta_{4}\lambda}\pi(s) \biggr) , \quad\quad\psi_{4}(s)=g_{4}^{-1} \biggl( \frac{ \gamma}{\lambda}\pi(s) \biggr) , \end{gathered} \end{aligned}$$
(26)
where
$$ \gamma=\frac{N(a_{1}\lambda_{1}e^{-\mu_{1}\tau_{1}}+(a_{1}+ \beta_{2})\lambda_{2}e^{-\mu_{2}\tau_{2}})+M\lambda_{3}(a_{1}+\beta _{2})e^{-\mu_{3}\tau_{3}}}{\beta_{5}(a_{1}+\beta _{2})}. $$
It follows from Eqs. (20)–(24) that
$$ w=\psi_{1}(s),\quad\quad y=\psi_{2}(s),\quad\quad u= \psi_{3}(s),\quad \quad p=\psi_{4}(s). $$
(27)
Obviously, \(\psi_{i}(s)>0\) for \(s\in{}[0,s_{0})\) and \(\psi_{i}(s _{0})=0\), \(i=1,\ldots,4\). From Eqs. (20), (26), and (27) we obtain
$$ \gamma\chi\bigl(s,\psi_{4}(s)\bigr)-g_{4}\bigl( \psi_{4}(s)\bigr)=0. $$
(28)
Equation (28) admits a solution \(s=s_{0}\) which gives the infection-free steady state \(\Pi_{0}(s_{0},0, 0,0,0,0)\). Let
$$ \Phi_{1}(s)=\gamma\chi\bigl(s,\psi_{4}(s) \bigr)-g_{4}\bigl(\psi_{4}(s)\bigr)=0. $$
It is clear from hypotheses (H1) and (H2) that \(\Phi_{1}(0)=-g_{4}( \psi_{4}(0))<0\) and \(\Phi_{1}(s_{0})=0\). Moreover,
$$ \Phi_{1}^{\prime} ( s_{0} ) =\gamma \biggl[ \frac{\partial \chi(s_{0},0)}{\partial s}+\psi_{4}^{\prime}(s_{0}) \frac{\partial \chi(s_{0},0)}{\partial p} \biggr] -g_{4}^{\prime}(0)\psi_{4}^{ \prime}(s_{0}). $$
We note from hypothesis (H2) that \(\frac{\partial\chi(s_{0},0)}{ \partial s}=0\). Then
$$ \Phi_{1}^{\prime} ( s_{0} ) =\psi_{4}^{\prime}(s_{0})g_{4} ^{\prime}(0) \biggl( \frac{\gamma}{g_{4}^{\prime}(0)}\frac{\partial \chi(s_{0},0)}{\partial p}-1 \biggr) . $$
From Eq. (26), we get
$$ \Phi_{1}^{\prime} ( s_{0} ) =\frac{\gamma}{\lambda} \pi^{\prime}(s_{0}) \biggl( \frac{\gamma}{g_{4}^{\prime}(0)}\frac{ \partial\chi(s_{0},0)}{\partial p}-1 \biggr) . $$
Hence, from hypothesis (H1), we have \(\pi^{\prime}(s_{0})<0\). Therefore, if \(\frac{\gamma}{g_{4}^{\prime}(0)}\frac{\partial \chi(s_{0},0)}{\partial p}>1\), then \(\Phi_{1}^{\prime}(s_{0})<0\), and there exists \(s_{1}\in(0,s_{0})\) such that \(\Phi_{1}(s_{1})=0\). Hypotheses (H1)–(H3) imply that
$$ w_{1}=\psi_{1}(s_{1})>0, \quad\quad y_{1}=\psi_{2}(s_{1})>0, \quad\quad u _{1}=\psi_{3}(s_{1})>0, \quad\quad p_{1}=\psi_{4}(s_{1})>0. $$
(29)
It means that a humoral-inactivated infection steady state \(\Pi_{1}(s _{1},w_{1},y_{1},u_{1},p_{1},0)\) exists when \(\frac{\gamma}{g_{4} ^{\prime}(0)}\frac{\partial\chi(s_{0},0)}{\partial p}>1\). Let us define the basic infection reproduction number as follows:
$$ R_{0}=\frac{\gamma}{g_{4}^{\prime}(0)}\frac{\partial\chi(s_{0},0)}{ \partial p}. $$
The other solution of Eq. (25) is \(g_{4}(p_{2})=\frac{\beta _{6}}{r}\), which yields \(p_{2}=g_{4}^{-1} ( \frac{\beta_{6}}{r} ) >0\). Substitute \(p=p_{2}\) in Eq. (20) and let \(\Phi_{2}(s)=\pi(s)-\lambda\chi(s,p_{2})=0\). According to hypotheses (H1) and (H2), \(\Phi_{2}\) is strictly decreasing, \(\Phi_{2}(0)=\pi(0)>0\) and \(\Phi_{2}(s_{0})=-\lambda\chi(s_{0},p _{2})<0\). Thus, there exists unique \(s_{2}\in(0,s_{0})\) such that \(\Phi_{2}(s_{2})=0\). It follows from Eqs. (24) and (27) that
$$\begin{aligned}& w_{2} =\psi_{1}(s_{2})>0,\quad\quad y_{2}=\psi_{2}(s_{2})>0, \quad\quad u_{2}=\psi_{3}(s_{2})>0, \quad\quad p_{2}=g_{4}^{-1} \biggl( \frac{ \beta_{6}}{r} \biggr) >0, \\& x_{2} =g_{5}^{-1} \biggl( \frac{\beta_{5}}{q} \biggl( \gamma\frac{ \chi(s_{2},p_{2})}{g_{4}(p_{2})}-1 \biggr) \biggr) . \end{aligned}$$
Thus, \(x_{2}>0\) when \(\gamma\frac{\chi(s_{2},p_{2})}{g_{4}(p_{2})}>1\). Now we define the humoral immune response activation number as follows:
$$ R_{1}=\gamma\frac{\chi(s_{2},p_{2})}{g_{4}(p_{2})}. $$
If \(R_{1}>1\), then \(x_{2}=g_{5}^{-1} ( \frac{\beta_{5}}{q}(R_{1}-1) ) >0\), and there exists a humoral-activated infection steady state \(\Pi_{2}(s_{2},w_{2},y_{2},u _{2},p_{2},x_{2})\).
Clearly, from hypotheses (H2) and (H4), we have
$$ R_{1}=\gamma\frac{\chi(s_{2},p_{2})}{g_{4}(p_{2})}\leq\gamma \lim_{p\rightarrow0^{+}} \frac{\chi(s_{2},p)}{g_{4}(p)}=\frac{ \gamma}{g_{4}^{\prime}(0)}\frac{\partial\chi(s_{2},0)}{\partial p}< \frac{\gamma}{g_{4}^{\prime}(0)} \frac{\partial\chi(s_{0},0)}{ \partial p}=R_{0}. $$
 □
We will use the following equalities throughout the paper:
$$\begin{aligned}& \begin{gathered} \ln \biggl( \frac{\chi(s ( t-\tau_{1} ) ,p ( t-\tau_{1} ) )}{\chi(s,p)} \biggr) \\ \quad =\ln \biggl( \frac{g_{1}( \hat{w})\chi(s ( t-\tau_{1} ) ,p ( t-\tau_{1} ) )}{g _{1}(w)\chi(\hat{s},\hat{p})} \biggr) +\ln \biggl( \frac{\chi(\hat{s}, \hat{p})}{\chi(s,\hat{p})} \biggr) \\ \quad \quad{} +\ln \biggl( \frac{g_{4}(p)\chi(s,\hat{p})}{g_{4}(\hat{p})\chi(s,p)} \biggr) + \ln \biggl( \frac{g_{4}(\hat{p})g_{2}(y)}{g_{4}(p)g_{2}(\hat{y})} \biggr) +\ln \biggl( \frac{g_{2}(\hat{y})g_{1}(w)}{g_{2}(y)g_{1}(\hat{w})} \biggr) , \\ \ln \biggl( \frac{\chi(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) )}{\chi(s,p)} \biggr) \\ \quad =\ln \biggl( \frac{g_{2}( \hat{y})\chi(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) )}{g _{2}(y)\chi(\hat{s},\hat{p})} \biggr) +\ln \biggl( \frac{\chi(\hat{s}, \hat{p})}{\chi(s,\hat{p})} \biggr) \\ \quad \quad{} +\ln \biggl( \frac{g_{4}(\hat{p})g_{2}(y)}{g_{4}(p)g_{2}(\hat{y})} \biggr) +\ln \biggl( \frac{g_{4}(p)\chi(s,\hat{p})}{g_{4}(\hat{p})\chi(s,p)} \biggr) , \\ \ln \biggl( \frac{\chi(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) )}{\chi(s,p)} \biggr) \\ \quad =\ln \biggl( \frac{g_{3}( \hat{u})\chi(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) )}{g _{3}(u)\chi(\hat{s},\hat{p})} \biggr) +\ln \biggl( \frac{\chi(\hat{s}, \hat{p})}{\chi(s,\hat{p})} \biggr) \\ \quad \quad{} +\ln \biggl( \frac{g_{3}(u)g_{4}(\hat{p})}{g_{3}(\hat{u})g_{4}(p)} \biggr) +\ln \biggl( \frac{g_{4}(p)\chi(s,\hat{p})}{g_{4}(\hat{p})\chi(s,p)} \biggr) . \end{gathered} \end{aligned}$$
(30)

2.2 Global properties

The following theorems investigate the global stability of the steady states of system (12)–(17).

Let us define the function \(F:(0,\infty)\rightarrow [0,\infty)\) as \(F(z)=z-1-\ln z\). Denote \((s,w,y,u, p,x)=(s(t),w(t),y(t),u(t),p(t),x(t))\).

Theorem 1

If \(R_{0}\leq1\) and hypotheses (H1)(H4) are valid, then \(\Pi_{0}\) is globally asymptotically stable (GAS).

Proof

Define a Lyapunov functional \(V_{0}\) as follows:
$$\begin{aligned} V_{0} & =s-s_{0}- \int_{s_{0}}^{s}\lim_{p\rightarrow0^{+}} \frac{ \chi(s_{0},p)}{\chi(\eta,p)}\,d\eta+\ell_{1}w+\ell_{2}y+ \ell_{3}u \\ &\quad{} +\ell_{1} \int_{0}^{\tau_{1}}\lambda_{1}e^{-\mu_{1}\tau_{1}} \chi\bigl(s ( t-\theta ) ,p ( t-\theta ) \bigr)\,d\theta \\ & \quad{} +\ell_{2} \int_{0}^{\tau_{2}}\lambda_{2}e^{-\mu_{2}\tau_{2}} \chi\bigl(s ( t-\theta ) ,p ( t-\theta ) \bigr)\,d\theta \\ &\quad{}+ \ell_{3} \int_{0}^{\tau_{3}}\lambda_{3}e^{-\mu_{3}\tau_{3}} \chi\bigl(s ( t-\theta ) ,p ( t-\theta ) \bigr)\,d\theta+ \ell_{4}p+ \ell_{5}x, \end{aligned}$$
where
$$\begin{aligned} \begin{gathered} \lambda =\lambda_{1}\ell_{1}e^{-\mu_{1}\tau_{1}}+ \lambda_{2}\ell_{2}e^{-\mu _{2}\tau_{2}}+\lambda_{3} \ell _{3}e^{-\mu_{3}\tau_{3}}, \quad \quad (a_{1}+ \beta_{2})\ell_{1}=a _{1}\ell_{2}, \\ \ell_{2} =N\ell_{4}, \quad \quad\ell_{3}=M \ell_{4}, \quad \quad q\ell_{4}=r\ell_{5}. \end{gathered} \end{aligned}$$
(31)
The solution of Eqs. (31) is given by
$$ \ell_{1}=\frac{a_{1}N\lambda}{\gamma\beta_{5}(a_{1}+\beta _{2})}, \quad\quad \ell_{2}= \frac{N\lambda}{\gamma\beta_{5}}, \quad\quad \ell_{3}=\frac{M\lambda}{\gamma\beta_{5}}, \quad\quad \ell_{4}=\frac{ \lambda}{\gamma\beta_{5}}, \quad\quad \ell_{5}= \frac{q\lambda}{r\gamma\beta_{5}}. $$
(32)
We calculate \(\frac{dV_{0}}{dt}\) along the trajectories of (12)–(17) as follows:
$$\begin{aligned} \frac{dV_{0}}{dt} & = \biggl( 1-\lim_{p\rightarrow0^{+}}\frac{\chi(s _{0},p)}{\chi(s,p)} \biggr) \bigl( \pi(s)-\lambda\chi(s,p) \bigr) \\ & \quad{} +\ell_{1} \bigl( \lambda_{1}e^{-\mu _{1}\tau_{1}}\chi\bigl(s ( t-\tau_{1} ) ,p ( t- \tau_{1} ) \bigr)-(a_{1}+\beta_{2})g_{1}(w) \bigr) \\ & \quad{} +\ell_{2} \bigl( \lambda_{2} e^{-\mu_{2}\tau_{2}}\chi\bigl(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) \bigr)+a_{1}g_{1}(w)- \beta_{3}g_{2}(y) \bigr) \\ & \quad{}+\ell_{3} \bigl( \lambda_{3}e^{-\mu _{3}\tau_{3}}\chi\bigl(s ( t-\tau_{3} ) ,p ( t- \tau_{3} ) \bigr)- \beta_{4}g_{3}(u) \bigr) \\ & \quad{} +\ell_{1} \lambda_{1}e ^{-\mu_{1}\tau_{1}} \bigl( \chi(s,p)-\chi\bigl(s ( t- \tau_{1} ) ,p ( t-\tau_{1} ) \bigr) \bigr) \\ & \quad{}+ \ell_{2}\lambda_{2}e^{- \mu_{2}\tau_{2}} \bigl( \chi(s,p)-\chi \bigl(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) \bigr) \bigr) \\ & \quad{}+\ell_{3}\lambda_{3}e^{-\mu_{3}\tau_{3}} \bigl( \chi(s,p)- \chi\bigl(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) \bigr) \bigr) \\ & \quad{} + \ell_{4} \bigl( N\beta_{3}g_{2}(y)+M \beta_{4}g_{3}(u)-\beta_{5}g_{4}(p)-qg _{4}(p)g_{5}(x) \bigr) \\ & \quad{}+\ell_{5}\bigl(rg_{4}(p)g_{5}(x)- \beta_{6}g_{5}(x)\bigr). \end{aligned}$$
(33)
Collecting terms of Eq. (33) and using \(\pi(s_{0})=0\), we obtain
$$\begin{aligned} \frac{dV_{0}}{dt} & = \bigl( \pi(s)-\pi(s_{0}) \bigr) \biggl( 1- \lim_{p\rightarrow0^{+}}\frac{\chi(s_{0},p)}{\chi(s,p)} \biggr) \\ &\quad{} + \biggl( \frac{\lambda\chi(s,p)}{g_{4}(p)} \lim_{p\rightarrow0^{+}}\frac{\chi(s_{0},p)}{\chi(s,p)}- \ell_{4} \beta_{5} \biggr) g_{4}(p)- \ell_{5}\beta_{6}g_{5}(x) \\ & \leq \bigl( \pi(s)-\pi(s_{0}) \bigr) \biggl( 1- \lim _{p\rightarrow0^{+}}\frac{\chi(s_{0},p)}{\chi(s,p)} \biggr) \\ &\quad{}+ \biggl( \lim _{p\rightarrow0^{+}}\frac{\lambda\chi(s,p)}{g _{4}(p)}\lim_{p\rightarrow0^{+}} \frac{\chi(s_{0},p)}{\chi(s,p)}- \ell_{4}\beta_{5} \biggr) g_{4}(p)-\ell_{5}\beta_{6}g_{5}(x) \\ & = \bigl( \pi(s)-\pi(s_{0}) \bigr) \biggl( 1-\frac{\partial\chi(s _{0},0)/\partial p}{\partial\chi(s,0)/\partial p} \biggr) \\ &\quad{}+\ell_{4} \beta_{5} \biggl( \frac{\lambda}{\ell_{4}\beta_{5}g_{4}^{ \prime}(0)} \frac{\partial\chi(s_{0},0)}{\partial p}-1 \biggr) g_{4}(p)- \ell_{5} \beta_{6}g_{5}(x) \\ & = \bigl( \pi(s)-\pi(s_{0}) \bigr) \biggl( 1-\frac{\partial\chi(s _{0},0)/\partial p}{\partial\chi(s,0)/\partial p} \biggr) +\ell_{4} \beta_{5}(R_{0}-1)g_{4}(p)- \ell_{5}\beta_{6}g_{5}(x). \end{aligned}$$
By hypotheses (H1) and (H2), we obtain
$$ \bigl( \pi(s)-\pi(s_{0}) \bigr) \biggl( 1-\frac{\partial\chi(s_{0},0)/ \partial p}{\partial\chi(s,0)/\partial p} \biggr) \leq0. $$
Therefore, if \(R_{0}\leq1\), then \(\frac{dV_{0}}{dt}\leq0\) for \(s,p,x>0\). Clearly, \(\frac{dV_{0}}{dt}=0\) at \(\Pi_{0}\). Applying LaSalle’s invariance principle (LIP), we get that \(\Pi_{0}\) is GAS. □

Lemma 3

If \(R_{0}>1\) and hypotheses (H1)(H4) are valid, then
$$ \operatorname {sgn}(R_{1}-1)=\operatorname {sgn}(p_{1}-p_{2})=\operatorname {sgn}(s_{2}-s_{1}). $$

Proof

Using hypotheses (H1) and (H2), for \(s_{1},s_{2},p _{1},p_{2}>0\), we get
$$\begin{aligned}& ( s_{1}-s_{2} ) \bigl( \pi(s_{2})- \pi(s_{1}) \bigr) >0, \end{aligned}$$
(34)
$$\begin{aligned}& (s_{2}-s_{1}) \bigl(\chi(s_{2},p_{2})- \chi(s_{1},p_{2})\bigr)>0, \end{aligned}$$
(35)
$$\begin{aligned}& ( p_{2}-p_{1} ) \bigl( \chi(s_{1},p_{2})- \chi(s_{1},p_{1}) \bigr) >0, \end{aligned}$$
(36)
and from hypothesis (H4), we obtain
$$ (p_{1}-p_{2}) \biggl( \frac{\chi(s_{1},p_{2})}{g_{2}(p_{2})}- \frac{ \chi(s_{1},p_{1})}{g_{2}(p_{1})} \biggr) >0. $$
(37)
First, we show that \(\operatorname {sgn}(p_{1}-p_{2})=\operatorname {sgn}(s_{2}-s_{1})\). Suppose that \(\operatorname {sgn}(p_{2}-p_{1})=\operatorname {sgn}(s_{2}-s_{1})\). Using the steady state conditions of \(\Pi_{1}\) and \(\Pi_{2}\), we obtain
$$ \begin{aligned} \pi(s_{2})-\pi(s_{1})&=\lambda \bigl[ \chi(s_{2},p_{2})- \chi(s_{1},p _{1}) \bigr] \\ &=\lambda \bigl[ \bigl( \chi(s_{2},p_{2})-\chi(s_{1},p_{2}) \bigr)+\bigl( \chi(s_{1},p_{2})-\chi(s_{1},p_{1}) \bigr) \bigr] . \end{aligned} $$
Therefore, from inequalities (34)–(36) we obtain \(\operatorname {sgn}( s_{2}-s_{1} ) =\operatorname {sgn}( s_{1}-s_{2} ) \), which is a contradiction; hence, \(\operatorname {sgn}( p_{1}-p_{2} ) =\operatorname {sgn}( s _{2}-s_{1} ) \). Using Eq. (29) and the definition of \(R_{1}\), we get
$$ \begin{aligned} R_{1}-1&=\gamma \biggl( \frac{\chi(s_{2},p_{2})}{g_{4}(p_{2})}-\frac{ \chi(s_{1},p_{1})}{g_{4}(p_{1})} \biggr) \\ &=\gamma \biggl[ \frac{1}{g _{4}(p_{2})} \bigl( \chi(s_{2},p_{2})- \chi(s_{1},p_{2}) \bigr) +\frac{ \chi(s_{1},p_{2})}{g_{4}(p_{2})}- \frac{\chi(s_{1},p_{1})}{g_{4}(p _{1})} \biggr] . \end{aligned} $$
Thus, from Eqs. (35) and (37) we obtain \(\operatorname {sgn}(R_{1}-1)=\operatorname {sgn}(p _{1}-p_{2})\). □

Theorem 2

If \(R_{1}\leq1< R_{0}\) and hypotheses (H1)(H4) are valid, then \(\Pi_{1}\) is GAS.

Proof

Let
$$\begin{aligned} V_{1} & =s-s_{1}- \int_{s_{1}}^{s}\frac{\chi(s_{1},p_{1})}{ \chi(\eta,p_{1})}\,d\eta+ \ell_{1} \biggl( w-w_{1}- \int_{w_{1}} ^{w}\frac{g_{1}(w_{1})}{g_{1}(\eta)}\,d\eta \biggr) \\ &\quad{} + \ell_{2} \biggl( y-y _{1}- \int_{y_{1}}^{y}\frac{g_{2}(y_{1})}{g_{2}(\eta)}\,d\eta \biggr) +\ell_{3} \biggl( u-u_{1}- \int_{u_{1}}^{u}\frac{g_{3}(u_{1})}{g _{3}(\eta)}\,d\eta \biggr) \\ &\quad{} + \ell_{1}\lambda_{1}\chi(s_{1},p_{1})e^{- \mu_{1}\tau_{1}} \int_{0}^{\tau_{1}}F \biggl( \frac{\chi(s ( t-\theta ) ,p ( t-\theta ) )}{\chi(s_{1},p_{1})} \biggr) \,d\theta \\ &\quad{} +\ell_{2}\lambda_{2}\chi(s_{1},p_{1})e^{-\mu_{2}\tau_{2}} \int_{0} ^{\tau_{2}}F \biggl( \frac{\chi(s ( t-\theta ) ,p ( t- \theta ) )}{\chi(s_{1},p_{1})} \biggr) \,d \theta \\ & \quad{}+\ell_{3}\lambda_{3}\chi(s_{1},p_{1})e^{-\mu_{3}\tau_{3}} \int_{0} ^{\tau_{3}}F \biggl( \frac{\chi(s ( t-\theta ) ,p ( t- \theta ) )}{\chi(s_{1},p_{1})} \biggr) \,d \theta \\ &\quad{}+\ell_{4} \biggl( p-p _{1}- \int_{p_{1}}^{p}\frac{g_{4}(p_{1})}{g_{4}(\eta)}\,d\eta \biggr) + \ell_{5}x. \end{aligned}$$
Calculating \(\frac{dV_{1}}{dt}\) along the solutions of (12)–(17), we obtain
$$\begin{aligned} \frac{dV_{1}}{dt} & = \biggl( 1- \frac{\chi(s_{1},p_{1})}{\chi(s,p_{1})} \biggr) \bigl( \pi(s)- \lambda \chi(s,p) \bigr) \\ & \quad{}+\ell_{1} \biggl( 1-\frac{g_{1}(w_{1})}{g_{1}(w)} \biggr) \bigl( \lambda_{1}e^{-\mu_{1}\tau_{1}}\chi\bigl(s ( t-\tau_{1} ) ,p ( t-\tau_{1} ) \bigr)-(a_{1}+\beta_{2})g_{1}(w) \bigr) \\ & \quad{}+\ell_{2} \biggl( 1-\frac{g_{2}(y_{1})}{g_{2}(y)} \biggr) \bigl( \lambda_{2}e^{-\mu_{2}\tau_{2}}\chi\bigl(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) \bigr)+a_{1}g_{1}(w)- \beta_{3}g_{2}(y) \bigr) \\ & \quad{}+\ell_{3} \biggl( 1-\frac{g_{3}(u_{1})}{g_{3}(u)} \biggr) \bigl( \lambda_{3}e^{-\mu_{3}\tau_{3}}\chi\bigl(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) \bigr)-\beta_{4}g_{3}(u) \bigr) \\ &\quad{} +\ell_{1}\lambda_{1}e^{-\mu_{1}\tau_{1}} \bigl( \chi(s,p)- \chi\bigl(s ( t-\tau_{1} ) ,p ( t-\tau_{1} ) \bigr) \bigr) \\ & \quad{}+\ell_{1}\lambda_{1}\chi(s_{1},p_{1})e^{-\mu_{1}\tau _{1}} \ln \biggl( \frac{\chi(s ( t-\tau_{1} ) ,p ( t-\tau _{1} ) )}{\chi(s,p)} \biggr) \\ &\quad{} +\ell_{2}\lambda_{2}e^{-\mu_{2}\tau_{2}} \bigl( \chi(s,p)- \chi\bigl(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) \bigr) \bigr) \\ &\quad{} +\ell_{2}\lambda_{2}\chi(s_{1},p_{1})e^{-\mu_{2}\tau _{2}} \ln \biggl( \frac{\chi(s ( t-\tau_{2} ) ,p ( t-\tau _{2} ) )}{\chi(s,p)} \biggr) \\ &\quad{} +\ell_{3}\lambda_{3}e^{-\mu_{3}\tau_{3}} \bigl( \chi(s,p)- \chi\bigl(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) \bigr) \bigr) \\ & \quad{}+\ell_{3}\lambda_{3}\chi(s_{1},p_{1})e^{-\mu_{3}\tau _{3}} \ln \biggl( \frac{\chi(s ( t-\tau_{3} ) ,p ( t-\tau _{3} ) )}{\chi(s,p)} \biggr) \\ & \quad{}+\ell_{4} \biggl( 1-\frac{g_{4}(p_{1})}{g_{4}(p)} \biggr) \bigl(N \beta_{3}g _{2}(y)+M\beta_{4}g_{3}(u)- \beta_{5}g_{4}(p)-qg_{4}(p)g_{5}(x) \bigr) \\ &\quad{}+\ell_{5}\bigl(rg _{4}(p)g_{5}(x)- \beta_{6}g_{5}(x)\bigr). \end{aligned}$$
(38)
Collecting terms of Eq. (38) and applying the conditions of the steady state \(\Pi_{1}\)
$$\begin{aligned}& \pi(s_{1}) =\lambda\chi(s_{1},p_{1}), \\& (a_{1}+\beta_{2})g_{1}(w_{1}) = \lambda_{1} e^{-\mu_{1}\tau _{1}}\chi(s_{1},p_{1}), \\& \ell_{2}\beta_{3}g_{2}(y_{1})= \bigl( \ell_{1}\lambda_{1}e^{-\mu_{1}\tau_{1}}+ \ell_{2}\lambda_{2}e^{-\mu _{2}\tau_{2}}\bigr) \chi(s_{1},p_{1}), \\& \beta_{4}g_{3}(u_{1}) =\lambda_{3}e^{-\mu_{3}\tau_{3}} \chi(s_{1},p_{1}), \quad\quad\ell_{4} \beta_{5}g_{4}(p_{1})=\lambda \chi(s_{1},p_{1}), \end{aligned}$$
we get
$$\begin{aligned} \frac{dV_{1}}{dt} & = \bigl( \pi(s)-\pi(s_{1}) \bigr) \biggl( 1- \frac{ \chi(s_{1},p_{1})}{\chi(s,p_{1})} \biggr) +\lambda\chi(s _{1},p_{1}) \biggl( 1-\frac{\chi(s_{1},p_{1})}{\chi(s,p_{1})} \biggr) \\ & \quad{} +\lambda\chi(s_{1},p_{1}) \biggl( \frac{\chi(s,p)}{\chi (s,p_{1})}-\frac{g _{4}(p)}{g_{4}(p_{1})} \biggr) \\ & \quad{} -\ell_{1}\lambda_{1} e^{-\mu_{1}\tau_{1}} \chi(s_{1},p_{1})\frac{g _{1}(w_{1})\chi(s ( t-\tau_{1} ) ,p ( t-\tau_{1} ) )}{g_{1}(w)\chi(s_{1},p_{1})}+\ell_{1} \lambda_{1} e^{-\mu _{1}\tau_{1}}\chi(s_{1},p_{1}) \\ & \quad{}-\ell_{2}\lambda_{2}e^{-\mu_{2}\tau_{2}} \chi(s_{1},p_{1})\frac{g _{2}(y_{1})\chi(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) )}{g_{2}(y)\chi(s_{1},p_{1})} \\ & \quad{} -\ell_{1} \lambda_{1} e^{-\mu _{1}\tau_{1}}\chi(s_{1},p_{1}) \frac{g_{2}(y_{1})g_{1}(w)}{g_{2}(y)g _{1}(w_{1})} \\ &\quad{} +\bigl(\ell_{1}\lambda_{1}e^{-\mu_{1}\tau_{1}}+ \ell_{2}\lambda_{2}e^{-\mu _{2}\tau_{2}}\bigr) \chi(s_{1},p_{1}) \\ & \quad{}-\ell_{3}\lambda_{3}e^{-\mu_{3}\tau _{3}} \chi(s_{1},p_{1})\frac{g_{3}(u_{1}) \chi(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) )}{\chi(s _{1},p_{1})g_{3}(u)} \\ & \quad{}+\ell_{3}\lambda_{3}e^{-\mu_{3}\tau_{3}} \chi(s_{1},p _{1})+\ell_{1}\lambda_{1}e^{-\mu_{1}\tau_{1}} \chi(s_{1},p _{1})\ln \biggl( \frac{\chi(s ( t-\tau_{1} ) ,p ( t-\tau _{1} ) )}{\chi(s,p)} \biggr) \\ &\quad{} +\ell_{2}\lambda_{2}e^{-\mu_{2}\tau_{2}} \chi(s_{1},p _{1})\ln \biggl( \frac{\chi(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) )}{\chi(s,p)} \biggr) \\ & \quad{}+ \ell_{3}\lambda _{3}e^{-\mu_{3}\tau_{3}}\chi(s_{1},p_{1}) \ln \biggl( \frac{\chi(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) )}{\chi(s,p)} \biggr) \\ & \quad{}-\bigl(\ell_{1}\lambda_{1}e^{-\mu_{1}\tau_{1}}+ \ell_{2}\lambda_{2}e^{-\mu _{2}\tau_{2}}\bigr) \chi(s_{1},p_{1})\frac{g_{2}(y)g _{4}(p_{1})}{g_{2}(y_{1})g_{4}(p)} \\ & \quad{}-\ell_{3} \lambda_{3}e^{- \mu_{3}\tau_{3}}\chi(s_{1},p_{1}) \frac{g_{3}(u)g_{4}(p_{1})}{g_{3}(u _{1})g_{4}(p)} \\ & \quad{}+\lambda\chi(s_{1},p_{1})+r\ell_{5} \biggl( g_{4}(p_{1})-\frac{ \beta_{6}}{r} \biggr) g_{5}(x). \end{aligned}$$
(39)
Using inequalities (30) with \(\hat{s}=s_{1}\), \(\hat{w}=w_{1}\), \(\hat{y}=y_{1}\), and \(\hat{p}=p_{1}\), we can obtain
$$\begin{aligned} \frac{dV_{1}}{dt} & = \bigl( \pi(s)-\pi(s_{1}) \bigr) \biggl( 1- \frac{ \chi(s_{1},p_{1})}{\chi(s,p_{1})} \biggr) +\lambda\chi(s _{1},p_{1}) \biggl( \frac{\chi(s,p)}{\chi(s,p_{1})}-\frac{g_{4}(p)}{g _{4}(p_{1})} \biggr) \biggl( 1- \frac{\chi(s,p_{1})}{\chi(s,p)} \biggr) \\ & \quad -\lambda\chi(s_{1},p_{1}) \biggl[ F \biggl( \frac{\chi(s _{1},p_{1})}{\chi(s,p_{1})} \biggr) +F \biggl( \frac{g_{4}(p)\chi(s,p _{1})}{g_{4}(p_{1})\chi(s,p)} \biggr) \biggr] \\ & \quad -\ell_{1}\lambda_{1} e^{-\mu_{1}\tau_{1}} \chi(s_{1},p_{1}) \biggl[ F \biggl( \frac{g_{1}(w_{1})\chi(s ( t-\tau_{1} ) ,p ( t-\tau_{1} ) )}{g_{1}(w)\chi(s_{1},p_{1})} \biggr) +F \biggl( \frac{g_{2}(y_{1})g_{1}(w)}{g_{2}(y)g_{1}(w_{1})} \biggr) \biggr] \\ & \quad -\ell_{2}\lambda_{2}e^{-\mu_{2}\tau_{2}} \chi(s_{1},p_{1})F \biggl( \frac{g _{2}(y_{1})\chi(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) )}{g_{2}(y)\chi(s_{1},p_{1})} \biggr) \\ &\quad -\ell_{3}\lambda_{3}e^{-\mu_{3}\tau_{3}} \chi(s_{1},p _{1}) \biggl[ F \biggl( \frac{g_{3}(u_{1})\chi(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) )}{g_{3}(u)\chi(s_{1},p_{1})} \biggr) +F \biggl( \frac{g_{3}(u)g_{4}(p_{1})}{g_{3}(u_{1})g_{4}(p)} \biggr) \biggr] \\ & \quad - \bigl( \ell_{1}\lambda_{1}e^{-\mu_{1}\tau_{1}}+ \ell_{2}\lambda _{2}e^{-\mu_{2}\tau_{2}} \bigr) \chi(s_{1},p_{1})F \biggl( \frac{g _{2}(y)g_{4}(p_{1})}{g_{2}(y_{1})g_{4}(p)} \biggr) \\ &\quad +r\ell_{5} \bigl( g_{4}(p_{1})-g_{4}(p_{2}) \bigr) g_{5}(x). \end{aligned}$$
Hypotheses (H1), (H2), (H4), Lemma 3, and the condition \(R_{1}\leq1\) imply that
$$\begin{aligned}& \bigl( \pi(s)-\pi(s_{1}) \bigr) \biggl( 1-\frac{\chi(s_{1},p_{1})}{ \chi(s,p_{1})} \biggr) \leq0, \\& \biggl( \frac{\chi(s,p)}{\chi(s,p_{1})}- \frac{g_{4}(p)}{g_{4}(p_{1})} \biggr) \biggl( 1- \frac{\chi(s,p_{1})}{ \chi(s,p)} \biggr) \leq0, \\& g_{4}(p_{1})-g_{4}(p_{2}) \leq0. \end{aligned}$$
It follows that, for all \(s,y,p,x>0\), we have \(\frac{dV_{1}}{dt} \leq0\) and \(\frac{dV_{1}}{dt}=0\) at \(\Pi_{1}\). By LIP \(\Pi_{1}\) is GAS. □

Theorem 3

If \(R_{1}>1\) and hypotheses (H1)(H4) are valid, then \(\Pi_{2}\) is GAS.

Proof

Define
$$\begin{aligned} V_{2} & =s-s_{2}- \int_{s_{2}}^{s}\frac{\chi(s_{2},p_{2})}{ \chi(\eta,p_{2})}\,d\eta+ \ell_{1} \biggl( w-w_{2}- \int_{w_{2}} ^{w}\frac{g_{1}(w_{2})}{g_{1}(\eta)}\,d\eta \biggr) \\ & \quad{}+ \ell_{2} \biggl( y-y _{2}- \int_{y_{2}}^{y}\frac{g_{2}(y_{2})}{g_{2}(\eta)}\,d\eta \biggr) +\ell_{3} \biggl( u-u_{2}- \int_{u_{2}}^{u}\frac{g_{3}(u_{2})}{g _{3}(\eta)}\,d\eta \biggr) \\ & \quad{} + \ell_{1}\lambda_{1}\chi(s_{2},p_{2})e^{- \mu_{1}\tau_{1}} \int_{0}^{\tau_{1}}F \biggl( \frac{\chi(s ( t-\theta ) ,p ( t-\theta ) )}{\chi(s_{2},p_{2})} \biggr) \,d \theta \\ & \quad{}+\ell_{2}\lambda_{2}\chi(s_{2},p_{2})e^{-\mu_{2}\tau_{2}} \int_{0} ^{\tau_{2}}F \biggl( \frac{\chi(s ( t-\theta ) ,p ( t- \theta ) )}{\chi(s_{2},p_{2})} \biggr) \,d\theta \\ & \quad{}+\ell_{3}\lambda_{3}\chi(s_{2},p_{2})e^{-\mu_{3}\tau_{3}} \int_{0} ^{\tau_{3}}F \biggl( \frac{\chi(s ( t-\theta ) ,p ( t- \theta ) )}{\chi(s_{2},p_{2})} \biggr) \,d\theta \\ & \quad{}+\ell_{4} \biggl( p-p_{2}- \int_{p_{2}}^{p}\frac{g_{4}(p_{2})}{g _{4}(\eta)}\,d\eta \biggr) + \ell_{5} \biggl( x-x_{2}- \int_{x_{2}} ^{x}\frac{g_{5}(x_{2})}{g_{5}(\eta)}\,d\eta \biggr) . \end{aligned}$$
Calculating \(\frac{dV_{2}}{dt}\) along the solutions of model (12)-(17), we get
$$\begin{aligned} \frac{dV_{2}}{dt} & = \biggl( 1- \frac{\chi(s_{2},p_{2})}{\chi(s,p_{2})} \biggr) \bigl( \pi(s)- \lambda \chi(s,p) \bigr) \\ & \quad{}+\ell_{1} \biggl( 1-\frac{g_{1}(w_{2})}{g_{1}(w)} \biggr) \bigl( \lambda_{1}e^{-\mu_{1}\tau_{1}}\chi\bigl(s ( t-\tau_{1} ) ,p ( t-\tau_{1} ) \bigr)-(a_{1}+\beta_{2})g_{1}(w) \bigr) \\ &\quad{} +\ell_{2} \biggl( 1-\frac{g_{2}(y_{2})}{g_{2}(y)} \biggr) \bigl( \lambda_{2}e^{-\mu_{2}\tau_{2}}\chi\bigl(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) \bigr)+a_{1}g_{1}(w)- \beta_{3}g_{2}(y) \bigr) \\ &\quad{} +\ell_{3} \biggl( 1-\frac{g_{3}(u_{2})}{g_{3}(u)} \biggr) \bigl( \lambda_{3}e^{-\mu_{3}\tau_{3}}\chi\bigl(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) \bigr)-\beta_{4}g_{3}(u) \bigr) \\ &\quad{} +\ell_{1}\lambda_{1}e^{-\mu_{1}\tau_{1}} \bigl( \chi(s,p)- \chi\bigl(s ( t-\tau_{1} ) ,p ( t-\tau_{1} ) \bigr) \bigr) \\ &\quad{} +\ell_{1}\lambda_{1}e^{-\mu_{1}\tau_{1}} \chi(s_{2},p _{2})\ln \biggl( \frac{\chi(s ( t-\tau_{1} ) ,p ( t-\tau_{1} ) )}{\chi(s,p)} \biggr) \\ &\quad{} +\ell_{2}\lambda_{2}e^{-\mu_{2}\tau_{2}} \bigl( \chi(s,p)- \chi\bigl(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) \bigr) \bigr) \\ &\quad{} +\ell_{2}\lambda_{2}e^{-\mu_{2}\tau_{2}} \chi(s_{2},p _{2})\ln \biggl( \frac{\chi(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) )}{\chi(s,p)} \biggr) \\ &\quad{} +\ell_{3}\lambda_{3}e^{-\mu_{3}\tau_{3}} \bigl( \chi(s,p)- \chi\bigl(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) \bigr) \bigr) \\ &\quad{} +\ell_{3}\lambda_{3}e^{-\mu_{3}\tau_{3}} \chi(s_{2},p _{2})\ln \biggl( \frac{\chi(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) )}{\chi(s,p)} \biggr) \\ & \quad{}+\ell_{4} \biggl( 1-\frac{g_{4}(p_{2})}{g_{4}(p)} \biggr) \bigl(N \beta_{3}g _{2}(y)+M\beta_{4}g_{3}(u)- \beta_{5}g_{4}(p)-qg_{4}(p)g_{5}(x) \bigr) \\ &\quad{} +\ell_{5} \biggl( 1-\frac{g_{5}(x_{2})}{g_{5}(x)} \biggr) \bigl(rg_{4}(p)g _{5}(x)-\beta_{6}g_{5}(x) \bigr). \end{aligned}$$
(40)
Collecting terms of Eq. (40) and using the steady state conditions for \(\Pi_{2}\):
$$\begin{aligned}& \pi(s_{2}) =\lambda\chi(s_{2},p_{2}), \\& (a_{1}+\beta_{2})g_{1}(w_{2}) = \lambda_{1} e^{-\mu_{1}\tau _{1}}\chi(s_{2},p_{2}), \\& \ell_{2}\beta_{3}g_{2}(y_{2})= \bigl( \ell_{1}\lambda_{1}e^{-\mu_{1}\tau_{1}}+ \ell_{2}\lambda_{2}e^{-\mu _{2}\tau_{2}}\bigr) \chi(s_{2},p_{2}), \\& \beta_{4}g_{3}(u_{2}) =\lambda_{3}e^{-\mu_{3}\tau_{3}} \chi(s_{2},p_{2}), \quad\quad \ell_{4} \beta_{5}g_{4}(p_{2})=\lambda \chi(s_{2},p_{2})-q\ell _{4}g_{4}(p_{2})g_{5}(x_{2}), \\& \ell_{4}\beta_{5}g_{4}(p) =\lambda \chi(s_{2},p_{2})\frac{g _{4}(p)}{g_{4}(p_{2})}-q\ell_{4}g_{4}(p)g_{5}(x_{2}), \end{aligned}$$
we obtain
$$\begin{aligned} \frac{dV_{2}}{dt} & = \bigl( \pi(s)-\pi(s_{2}) \bigr) \biggl( 1- \frac{ \chi(s_{2},p_{2})}{\chi(s,p_{2})} \biggr) +\lambda\chi(s _{2},p_{2}) \biggl( 1-\frac{\chi(s_{2},p_{2})}{\chi(s,p_{2})} \biggr) \\ & \quad{}+\lambda\chi(s_{2},p_{2}) \biggl( \frac{\chi(s,p)}{\chi (s,p_{2})}-\frac{g _{4}(p)}{g_{4}(p_{2})} \biggr) \\ & \quad{}-\ell_{1}\lambda_{1} e^{-\mu_{1}\tau_{1}} \chi(s_{2},p_{2})\frac{g _{1}(w_{2})\chi(s ( t-\tau_{1} ) ,p ( t-\tau_{1} ) )}{g_{1}(w)\chi(s_{2},p_{2})}+\ell_{1} \lambda_{1} e^{-\mu _{1}\tau_{1}}\chi(s_{2},p_{2}) \\ & \quad{}-\ell_{2}\lambda_{2}e^{-\mu_{2}\tau_{2}} \chi(s_{2},p_{2})\frac{g _{2}(y_{2})\chi(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) )}{g_{2}(y)\chi(s_{2},p_{2})} \\ & \quad{}-\ell_{1} \lambda_{1} e^{-\mu _{1}\tau_{1}}\chi(s_{2},p_{2}) \frac{g_{2}(y_{2})g_{1}(w)}{g_{2}(y)g _{1}(w_{2})} \\ & \quad{}+\bigl(\ell_{1}\lambda_{1}e^{-\mu_{1}\tau_{1}}+ \ell_{2}\lambda_{2}e^{-\mu _{2}\tau_{2}}\bigr) \chi(s_{2},p_{2}) \\ & \quad{}-\ell_{3}\lambda_{3}e^{-\mu_{3}\tau _{3}} \chi(s_{2},p_{2})\frac{g_{3}(u_{2}) \chi(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) )}{g_{3}(u) \chi(s_{2},p_{2})} \\ & \quad{}+\ell_{3}\lambda_{3}e^{-\mu_{3}\tau_{3}} \chi(s_{2},p _{2})+\ell_{1}\lambda_{1}e^{-\mu_{1}\tau_{1}} \chi(s_{2},p _{2})\ln \biggl( \frac{\chi(s ( t-\tau_{1} ) ,p ( t-\tau _{1} ) )}{\chi(s,p)} \biggr) \\ & \quad{}+\ell_{2}\lambda_{2}e^{-\mu_{2}\tau_{2}} \chi(s_{2},p _{2})\ln \biggl( \frac{\chi(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) )}{\chi(s,p)} \biggr) \\ & \quad{}+ \ell_{3}\lambda _{3}e^{-\mu_{3}\tau_{3}}\chi(s_{2},p_{2}) \ln \biggl( \frac{\chi(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) )}{\chi(s,p)} \biggr) \\ & \quad{}-\bigl(\ell_{1}\lambda_{1}e^{-\mu_{1}\tau_{1}}+ \ell_{2}\lambda_{2}e^{-\mu _{2}\tau_{2}}\bigr) \chi(s_{2},p_{2})\frac{g_{2}(y)g _{4}(p_{2})}{g_{2}(y_{2})g_{4}(p)} \\ & \quad{}-\ell_{3} \lambda_{3}e^{- \mu_{3}\tau_{3}}\chi(s_{2},p_{2}) \frac{g_{3}(u)g_{4}(p_{2})}{g_{3}(u _{2})g_{4}(p)}+\lambda\chi(s_{2},p_{2}). \end{aligned}$$
(41)
By inequalities (30) with \(\hat{s}=s_{2}\), \(\hat{w}=w_{2}\), \(\hat{y}=y _{2}\), and \(\hat{p}=p_{2}\), we can get
$$\begin{aligned} \frac{dV_{2}}{dt} & = \bigl( \pi(s)-\pi(s_{2}) \bigr) \biggl( 1- \frac{ \chi(s_{2},p_{2})}{\chi(s,p_{2})} \biggr) +\lambda\chi(s _{2},p_{2}) \biggl( \frac{\chi(s,p)}{\chi(s,p_{2})}-\frac{g_{4}(p)}{g _{4}(p_{2})} \biggr) \biggl( 1- \frac{\chi(s,p_{2})}{\chi(s,p)} \biggr) \\ & \quad -\lambda\chi(s_{2},p_{2}) \biggl[ F \biggl( \frac{\chi(s _{2},p_{2})}{\chi(s,p_{2})} \biggr) +F \biggl( \frac{g_{4}(p)\chi(s,p _{2})}{g_{4}(p_{2})\chi(s,p)} \biggr) \biggr] \\ & \quad -\ell_{1}\lambda_{1} e^{-\mu_{1}\tau_{1}} \chi(s_{2},p_{2}) \biggl[ F \biggl( \frac{g_{1}(w_{2})\chi(s ( t-\tau_{1} ) ,p ( t-\tau_{1} ) )}{g_{1}(w)\chi(s_{2},p_{2})} \biggr) +F \biggl( \frac{g_{2}(y_{2})g_{1}(w)}{g_{2}(y)g_{1}(w_{2})} \biggr) \biggr] \\ & \quad -\ell_{2}\lambda_{2}e^{-\mu_{2}\tau_{2}} \chi(s_{2},p_{2})F \biggl( \frac{g _{2}(y_{2})\chi(s ( t-\tau_{2} ) ,p ( t-\tau_{2} ) )}{g_{2}(y)\chi(s_{2},p_{2})} \biggr) \\ & \quad -\ell_{3}\lambda_{3}e^{-\mu_{3}\tau_{3}} \chi(s_{2},p _{2}) \biggl[ F \biggl( \frac{g_{3}(u_{2})\chi(s ( t-\tau_{3} ) ,p ( t-\tau_{3} ) )}{g_{3}(u)\chi(s_{2},p_{2})} \biggr) +F \biggl( \frac{g_{3}(u)g_{4}(p_{2})}{g_{3}(u_{2})g_{4}(p)} \biggr) \biggr] \\ & \quad - \bigl( \ell_{1}\lambda_{1}e^{-\mu_{1}\tau_{1}}+ \ell_{2}\lambda _{2}e^{-\mu_{2}\tau_{2}} \bigr) \chi(s_{2},p_{2})F \biggl( \frac{g _{2}(y)g_{4}(p_{2})}{g_{2}(y_{2})g_{4}(p)} \biggr) . \end{aligned}$$
According to hypotheses (H1), (H2), and (H4), we get \(\frac{dV_{2}}{dt}\leq0\) and \(\frac{dV_{2}}{dt}=0\) at \(\Pi_{2}\). LIP implies that \(\Pi_{2}\) is GAS. □

3 Model with delay-distributed

In the next model, we consider a general delay-distributed HIV infection model with humoral immunity as follows:
$$\begin{aligned}& \dot{s} =\pi\bigl(s ( t ) \bigr)-\lambda\chi\bigl(s ( t ) ,p ( t ) \bigr), \end{aligned}$$
(42)
$$\begin{aligned}& \dot{w} =\lambda_{1} \int_{0}^{h_{1}}f_{1} ( \tau ) e^{-\mu_{1}\tau}\chi\bigl(s ( t-\tau ) ,p ( t-\tau ) \bigr)\,d\tau-(a_{1}+ \beta_{2})g_{1}\bigl(w ( t ) \bigr), \end{aligned}$$
(43)
$$\begin{aligned}& \dot{y} =\lambda_{2} \int_{0}^{h_{2}}f_{2} ( \tau ) e^{-\mu_{2}\tau}\chi\bigl(s ( t-\tau ) ,p ( t-\tau ) \bigr)\,d\tau+a_{1}g_{1} \bigl(w ( t ) \bigr)-\beta_{3}g_{2}\bigl(y ( t ) \bigr), \end{aligned}$$
(44)
$$\begin{aligned}& \dot{u} =\lambda_{3} \int_{0}^{h_{3}}f_{3} ( \tau ) e^{-\mu_{3}\tau}\chi\bigl(s ( t-\tau ) ,p ( t-\tau ) \bigr)\,d\tau- \beta_{4}g_{3}\bigl(u ( t ) \bigr), \end{aligned}$$
(45)
$$\begin{aligned}& \dot{p} =N\beta_{3}g_{2}\bigl(y ( t ) \bigr)+M \beta_{4}g_{3}\bigl(u ( t ) \bigr)-\beta_{5}g_{4} \bigl(p ( t ) \bigr)-qg_{4}\bigl(p ( t ) \bigr)g _{5}\bigl(x ( t ) \bigr), \end{aligned}$$
(46)
$$\begin{aligned}& \dot{x} =rg_{4}\bigl(p ( t ) \bigr)g_{5}\bigl(x ( t ) \bigr)-\beta_{6}g _{5}\bigl(x ( t ) \bigr), \end{aligned}$$
(47)
where \(f_{i} ( \tau ) e^{-\mu_{i}\tau}\) over the time interval \([ 0,h_{i} ] \), \(i=1,2,3\), represent the probabilities that uninfected cells contacted by HIV at time \(t-\tau\) survived τ time units and became infected at time t.
The probability distribution function \(f_{i} ( \tau ) \) is assumed to satisfy \(f_{i} ( \tau ) >0\) and
$$ \int_{0}^{h_{i}}f_{i} ( \tau ) \,d\tau=1,\quad\quad \int _{0}^{h_{i}}f_{i} ( \eta ) e^{\upsilon\eta}\,d \eta< \infty, \quad i=1,2,3, $$
where υ is a positive constant. Let us denote \(\Theta_{i}( \tau)=f_{i}(\tau)e^{-\mu_{i}\tau}\) and \(F_{i}=\int_{0}^{h _{i}}\Theta_{i}(\tau)\,d\tau\); thus \(0< F_{i}\leq1\), \(i=1,2,3\).

Lemma 4

Let hypotheses (H1)(H3) be valid, then the solutions of system (42)(47) are nonnegative and ultimately bounded.

Proof

The nonnegativity of solutions of system (42)–(47) can easily be shown as given in Lemma 1.

From Eq. (42) we have that \(\lim\sup_{t\rightarrow\infty}s ( t ) \leq\frac{b}{\bar{b} }=M_{1}\). Let
$$\begin{aligned} G(t) & =N \int_{0}^{h_{1}}\Theta_{1}(\tau)s(t-\tau)\,d \tau+N \int_{0}^{h_{2}}\Theta_{2}(\tau)s(t-\tau)\,d \tau+M \int _{0}^{h_{3}}\Theta_{3}(\tau)s(t-\tau)\,d \tau \\ & \quad{} +Nw(t)+Ny(t)+Mu(t)+\frac{1}{2}p(t)+\frac{q}{2r}x(t), \end{aligned}$$
then
$$\begin{aligned} \dot{G}(t) & =N \int_{0}^{h_{1}}\Theta_{1}(\tau) \bigl[ \pi \bigl(s ( t-\tau ) \bigr)-\lambda\chi\bigl(s ( t-\tau ) ,p ( t-\tau ) \bigr) \bigr] \,d\tau \\ & \quad{} +N \int_{0}^{h_{2}}\Theta_{2}(\tau) \bigl[ \pi \bigl(s ( t- \tau ) \bigr)-\lambda\chi\bigl(s ( t-\tau ) ,p ( t-\tau ) \bigr) \bigr] \,d\tau \\ & \quad{} +M \int_{0}^{h_{3}}\Theta_{3}(\tau) \bigl[ \pi \bigl(s ( t- \tau ) \bigr)-\lambda\chi\bigl(s ( t-\tau ) ,p ( t-\tau ) \bigr) \bigr] \,d\tau \\ & \quad{} +N \biggl[ \lambda_{1} \int_{0}^{h_{1}}\Theta_{1}(\tau)\chi\bigl(s ( t-\tau ) ,p ( t-\tau ) \bigr)\,d\tau- ( a_{1}+ \beta_{2} ) g_{1}\bigl(w(t)\bigr) \biggr] \\ &\quad{} +N \biggl[ \lambda_{2} \int_{0}^{h_{2}}\Theta_{2}(\tau)\chi\bigl(s ( t-\tau ) ,p ( t-\tau ) \bigr)\,d\tau+a_{1}g_{1}\bigl(w(t) \bigr)- \beta_{3}g_{2}\bigl(y(t)\bigr) \biggr] \\ &\quad{} +M \biggl[ \lambda_{3} \int_{0}^{h_{3}}\Theta_{3}(\tau)\chi\bigl(s ( t-\tau ) ,p ( t-\tau ) \bigr)\,d\tau-\beta_{4}g_{3}\bigl(u ( t ) \bigr) \biggr] \\ &\quad{} +\frac{1}{2} \bigl[ N\beta_{3}g_{2}\bigl(y ( t ) \bigr)+M\beta_{4}g _{3}\bigl(u ( t ) \bigr)- \beta_{5}g_{4}\bigl(p ( t ) \bigr)-qg_{4}\bigl(p ( t ) \bigr)g_{5}\bigl(x ( t ) \bigr) \bigr] \\ &\quad{} +\frac{q}{2r} \bigl[ rg_{4}\bigl(p ( t ) \bigr)g_{5}\bigl(x ( t ) \bigr)-\beta_{6}g_{5} \bigl(x ( t ) \bigr) \bigr] \\ & \leq N \int_{0}^{h_{1}}\Theta_{1}(\tau) \bigl[ b- \overline{b}s ( t-\tau ) \bigr] \,d\tau+N \int_{0} ^{h_{2}}\Theta_{2}(\tau) \bigl[ b- \overline{b}s ( t-\tau ) \bigr] \,d\tau \\ &\quad{} +M \int_{0}^{h_{3}}\Theta_{3}(\tau) \bigl[ b- \overline{b}s ( t-\tau ) \bigr] \,d\tau \\ &\quad{} -N\beta_{2}\alpha_{1}w(t)-\frac{1}{2}N \beta_{3}\alpha_{2}y(t)- \frac{1}{2}M \beta_{4}\alpha_{3}u(t)-\frac{1}{2} \beta_{5}\alpha_{4}p(t)- \frac{q}{2r} \beta_{6}\alpha_{5}x(t) \\ & \leq b ( NF_{1}+NF_{2}+MF_{3} ) -\sigma \biggl[ N \int_{0} ^{h_{1}}\Theta_{1}(\tau)s(t-\tau)\,d \tau+N \int_{0}^{h_{2}} \Theta_{2}(\tau)s(t-\tau)\,d \tau \\ &\quad{} +M \int_{0}^{h_{3}}\Theta_{3}(\tau)s(t-\tau)\,d \tau+Nw(t)+Ny(t)+Mu(t)+ \frac{1}{2}p(t)+\frac{q}{2r}x(t) \biggr] \\ & \leq b ( 2N+M ) -\sigma G(t). \end{aligned}$$
Hence, \(\lim\sup_{t\rightarrow\infty}G(t)\leq\frac{b ( 2N+M ) }{\sigma}\) and
$$\begin{aligned}& \lim\sup_{t\rightarrow\infty}w(t) \leq M_{2},\quad\quad \lim \sup_{t\rightarrow\infty}y(t)\leq M_{2}, \quad\quad \lim \sup _{t\rightarrow\infty}u(t)\leq M_{3}, \\& \lim\sup_{t\rightarrow\infty}p(t) \leq M_{4}, \quad\quad \lim \sup_{t\rightarrow\infty}x(t)\leq M_{5}, \end{aligned}$$
where σ, \(M_{1},\ldots,M_{5}\) are given in the previous section. Therefore, \(s ( t ) \), \(w ( t ) \), \(y ( t ) \), \(u ( t ) \), \(p ( t ) \), and \(x ( t ) \) are ultimately bounded. Moreover, the set Ω defined by (19) is also positively invariant with respect to system (42)–(47). □

3.1 Steady states

Lemma 5

Suppose that hypotheses (H1)(H4) are valid (42)(47), then there exist two bifurcation parameters \(R_{0}\) and \(R_{1}\) with \(R_{0}>R_{1}>0\) such that
  1. (i)

    if \(R_{0}\leq1\), then there exists only one steady state \(\Pi_{0}\);

     
  2. (ii)

    if \(R_{1}\leq1< R_{0}\), then there exist two steady states \(\Pi_{0}\) and \(\Pi_{1}\);

     
  3. (iii)

    if \(R_{1}>1\), then there exist three steady states \(\Pi_{0}\), \(\Pi_{1}\), and \(\Pi_{2}\).

     

Proof

Let \(\Pi(s,w,y,u,p,x)\) be any steady state of (42)–(47) satisfying the following equations:
$$\begin{aligned}& 0 =\pi(s)-\lambda\chi(s,p), \end{aligned}$$
(48)
$$\begin{aligned}& 0 =\lambda_{1}F_{1}\chi(s,p)-(a_{1}+ \beta_{2})g_{1}(w), \end{aligned}$$
(49)
$$\begin{aligned}& 0 =\lambda_{2}F_{2}\chi(s,p)+a_{1}g_{1}(w)- \beta_{3}g _{2}(y), \end{aligned}$$
(50)
$$\begin{aligned}& 0 =\lambda_{3}F_{3}\chi(s,p)-\beta_{4}g_{3}(u), \end{aligned}$$
(51)
$$\begin{aligned}& 0 =N\beta_{3}g_{2}(y)+M\beta_{4}g_{3}(u)- \beta_{5}g_{4}(p)-qg_{4}(p)g _{5}(x), \end{aligned}$$
(52)
$$\begin{aligned}& 0 =rg_{4}(p)g_{5}(x)-\beta_{6}g_{5}(x). \end{aligned}$$
(53)
From Eq. (53) we obtain two possible solutions: \(g_{5}(x)=0\) and \(g_{4}(p)=\beta_{6}/r\). First, we consider the case \(g_{5}(x)=0\), then from hypothesis (H3) we have \(x=0\). Hypothesis (H3) implies that \(g_{i}^{-1}\), \(i=1,\ldots,5\), exist, strictly increasing and \(g_{i}^{-1}(0)=0\). Let us define
$$\begin{aligned}& \begin{gathered} \alpha_{1}(s) =g_{1}^{-1} \biggl( \frac{\lambda_{1}F_{1}}{ \lambda(a_{1}+\beta_{2})}\pi(s) \biggr) , \quad\quad\alpha_{2}(s)=g _{2}^{-1} \biggl( \frac{a_{1}\lambda_{1}F_{1}+(a_{1}+\beta _{2})\lambda_{2}F_{2}}{\beta_{3}\lambda(a_{1}+\beta_{2})} \pi(s) \biggr) , \\ \alpha_{3}(s) =g_{3}^{-1} \biggl( \frac{\lambda_{3}F_{3}}{ \beta_{4}\lambda}\pi(s) \biggr) , \quad\quad\alpha_{4}(s)=g_{4}^{-1} \biggl( \frac{\zeta}{\lambda}\pi(s) \biggr) , \end{gathered} \end{aligned}$$
(54)
where
$$ \zeta=\frac{N(a_{1}\lambda_{1}F_{1}+(a_{1}+\beta_{2})\lambda _{2}F_{2})+M\lambda_{3}(a_{1}+\beta_{2})F_{3}}{ \beta_{5}(a_{1}+\beta_{2})}. $$
From Eqs. (48)–(53) we can get
$$ w=\alpha_{1}(s),\quad\quad y=\alpha_{2}(s), \quad\quad u= \alpha_{3}(s),\quad\quad p=\alpha_{4}(s). $$
(55)
Obviously, \(\alpha_{i}(s)>0\) for \(s\in{}[0,s_{0})\) and \(\alpha_{i}(s_{0})=0\), \(i=1,\ldots,4\). From Eqs. (48), (54), and (55) we obtain
$$ \zeta\chi\bigl(s,\alpha_{4}(s)\bigr)-g_{4}\bigl( \alpha_{4}(s)\bigr)=0. $$
(56)
Equation (56) admits a solution \(s=s_{0}\) which gives the infection-free steady state \(\Pi_{0}(s_{0},0, 0,0,0,0)\). Let
$$ \Psi_{1}(s)=\zeta\chi\bigl(s,\alpha_{4}(s) \bigr)-g_{4}\bigl(\alpha_{4}(s)\bigr)=0. $$
By hypotheses (H1) and (H2) we have \(\Psi_{1}(0)=-g_{4}(\alpha_{4}(0))<0\) and \(\Psi_{1}(s_{0})=0\). Moreover,
$$ \Psi_{1}^{\prime} ( s_{0} ) =\zeta \biggl[ \frac{\partial \chi(s_{0},0)}{\partial s}+\alpha_{4}^{\prime}(s_{0}) \frac{\partial \chi(s_{0},0)}{\partial p} \biggr] -g_{4}^{\prime}(0)\alpha_{4}^{ \prime}(s_{0}). $$
From hypothesis (H2) we note that \(\frac{\partial\chi(s_{0},0)}{ \partial s}=0\). Then
$$ \Psi_{1}^{\prime} ( s_{0} ) =\alpha_{4}^{\prime}(s_{0})g _{4}^{\prime}(0) \biggl( \frac{\zeta}{g_{4}^{\prime}(0)}\frac{\partial \chi(s_{0},0)}{\partial p}-1 \biggr) . $$
From Eq. (54), we get
$$ \Psi_{1}^{\prime} ( s_{0} ) =\frac{\zeta}{\lambda} \pi^{\prime}(s_{0}) \biggl( \frac{\zeta}{g_{4}^{\prime}(0)} \frac{ \partial\chi(s_{0},0)}{\partial p}-1 \biggr) . $$
Therefore, using hypothesis (H1), we get \(\pi^{\prime}(s_{0})<0\). So that, if \(\frac{\zeta}{g_{4}^{\prime}(0)}\frac{\partial\chi(s_{0},0)}{ \partial p}>1\), then \(\Psi_{1}^{\prime}(s_{0})<0\) and there exists \(s_{1}\in(0,s_{0})\) such that \(\Psi_{1}(s_{1})=0\). Hypotheses (H1)–(H3) imply that
$$ w_{1}=\alpha_{1}(s_{1})>0,\quad\quad y_{1}=\alpha_{2}(s_{1})>0, \quad\quad u_{1}=\alpha_{3}(s_{1})>0, \quad\quad p_{1}=\alpha_{4}(s_{1})>0. $$
(57)
It means that a humoral-inactivated infection steady state \(\Pi_{1}(s _{1},w_{1},y_{1},u_{1},p_{1},0)\) exists when \(\frac{\zeta}{g_{4}^{ \prime}(0)}\frac{\partial\chi(s_{0},0)}{\partial p}>1\). Now we can define
$$ R_{0}=\frac{\zeta}{g_{4}^{\prime}(0)}\frac{\partial\chi(s_{0},0)}{ \partial p}. $$
The other solution of Eq. (53) is \(g_{4}(p_{2})=\frac{\beta _{6}}{r}\), which yields \(p_{2}=g_{4}^{-1} ( \frac{\beta_{6}}{r} ) >0\). Substitute \(p=p_{2}\) in Eq. (48) and let \(\Psi_{2}(s)=\pi(s)-\lambda\zeta(s,p_{2})=0\). According to hypotheses (H1) and (H2), \(\Psi_{2}\) is strictly decreasing, \(\Psi_{2}(0)=\pi(0)>0\), and \(\Psi_{2}(s_{0})=-\lambda\zeta(s_{0},p _{2})<0\). Thus, there exists a unique \(s_{2}\in(0,s_{0})\) such that \(\Psi_{2}(s_{2})=0\). It follows from Eqs. (52) and (55) that
$$\begin{aligned}& w_{2} =\alpha_{1}(s_{2})>0,\quad\quad y_{2}=\alpha_{2}(s_{2})>0, \quad\quad u_{2}=\alpha_{3}(s_{2})>0,\quad\quad p_{2}=g_{4}^{-1} \biggl( \frac{ \beta_{6}}{r} \biggr) >0, \\& x_{2} =g_{5}^{-1} \biggl( \frac{\beta_{5}}{q} \biggl( \zeta\frac{ \chi(s_{2},p_{2})}{g_{4}(p_{2})}-1 \biggr) \biggr) . \end{aligned}$$
Thus, \(x_{2}>0\) when \(\zeta\frac{\chi(s_{2},p_{2})}{g_{4}(p_{2})}>1\). Let us define the parameter \(R_{1}\) as follows:
$$ R_{1}=\zeta\frac{\chi(s_{2},p_{2})}{g_{4}(p_{2})}. $$
If \(R_{1}>1\), then \(x_{2}=g_{5}^{-1} ( \frac{\beta_{5}}{q}(R_{1}-1) ) >0\), and there exists a humoral-activated infection steady state \(\Pi_{2}(s_{2},w_{2},y_{2},u _{2},p_{2},x_{2})\).
Clearly, from hypotheses (H2) and (H4), we have
$$ R_{1}=\zeta\frac{\chi(s_{2},p_{2})}{g_{4}(p_{2})}\leq\zeta\lim_{p\rightarrow0^{+}} \frac{\chi(s_{2},p)}{g_{4}(p)}=\frac{\zeta}{g _{4}^{\prime}(0)}\frac{\partial\chi(s_{2},0)}{\partial p}< \frac{ \zeta}{g_{4}^{\prime}(0)} \frac{\partial\chi(s_{0},0)}{\partial p}=R _{0}. $$
 □

3.2 Global properties

Theorem 4

If \(R_{0}\leq1\) and hypotheses (H1)(H4) hold true for system (42)(47), then \(\Pi_{0}\) is GAS.

Proof

Define
$$\begin{aligned} W_{0} & =s-s_{0}- \int_{s_{0}}^{s}\lim_{p\rightarrow0^{+}} \frac{ \chi(s_{0},p)}{\chi(\eta,p)}\,d\eta+\kappa_{1}w+\kappa_{2}y+\kappa _{3}u \\ &\quad{} +\kappa_{1}\lambda_{1} \int_{0}^{h_{1}}\Theta_{1}(\tau) \int_{0}^{\tau}\chi\bigl(s ( t-\theta ) ,p ( t- \theta ) \bigr)\,d\theta \,d\tau \\ & \quad{} +\kappa_{2}\lambda_{2} \int_{0}^{h_{2}}\Theta_{2}(\tau) \int_{0}^{\tau}\chi\bigl(s ( t-\theta ) ,p ( t- \theta ) \bigr)\,d\theta \,d\tau \\ &\quad{} +\kappa_{3}\lambda_{3} \int_{0} ^{h_{3}}\Theta_{3}(\tau) \int_{0}^{\tau}\chi\bigl(s ( t- \theta ) ,p ( t-\theta ) \bigr)\,d\theta \,d\tau \\ & \quad{} +\kappa_{4}p+\kappa_{5}x, \end{aligned}$$
(58)
where
$$\begin{aligned}& \lambda =\lambda_{1}\kappa_{1}F_{1}+ \lambda_{2}\kappa_{2}F_{2}+\lambda _{3}\kappa_{3}F_{3},\quad \quad(a_{1}+ \beta_{2})\kappa_{1}=a_{1}\kappa_{2}, \\& \kappa_{2} =N\kappa_{4},\quad\quad\kappa_{3}=M \kappa_{4},\quad\quad q\kappa_{4}=r \kappa_{5}. \end{aligned}$$
(59)
The solution of Eqs. (59) is given by
$$ \kappa_{1}=\frac{a_{1}N\lambda}{\zeta\beta_{5}(a_{1}+\beta _{2})},\quad\quad \kappa_{2}= \frac{N\lambda}{\zeta\beta_{5}},\quad\quad \kappa _{3}=\frac{M\lambda}{\zeta\beta_{5}},\quad \quad \kappa_{4}=\frac{\lambda}{\zeta\beta_{5}},\quad\quad \kappa_{5}= \frac{q\lambda}{r\zeta\beta_{5}}. $$
(60)
We evaluate \(\frac{dW_{0}}{dt}\) along the solutions of (42)-(47) as follows:
$$\begin{aligned} \frac{dW_{0}}{dt} & = \biggl( 1-\lim_{p\rightarrow0^{+}}\frac{\chi(s _{0},p)}{\chi(s,p)} \biggr) \bigl( \pi(s)-\lambda\chi(s,p) \bigr) +\kappa_{1} \lambda_{1} \int_{0}^{h_{1}}\Theta_{1}(\tau)\chi\bigl(s ( t-\tau ) ,p ( t-\tau ) \bigr)\,d\tau \\ & \quad{} -\kappa_{1}(a_{1}+\beta_{2})g_{1}(w)+ \kappa_{2} \lambda_{2} \int_{0}^{h_{2}}\Theta_{2}(\tau)\chi\bigl(s ( t-\tau ) ,p ( t-\tau ) \bigr)\,d\tau+a_{1}\kappa_{2}g _{1}(w) \\ & \quad{} -\kappa_{2}\beta_{3}g_{2}(y)+ \kappa_{3}\lambda_{3} \int_{0}^{h_{3}}\Theta_{3}(\tau)\chi\bigl(s ( t-\tau ) ,p ( t-\tau ) \bigr)\,d\tau-\kappa_{3}\beta_{4}g_{3}(u) \\ & \quad{} +\kappa_{1}\lambda_{1} \int_{0}^{h_{1}}\Theta_{1}( \tau) \bigl( \chi(s,p)-\chi\bigl(s ( t-\tau ) ,p ( t- \tau ) \bigr) \bigr) \,d\tau \\ &\quad{} +\kappa_{2}\lambda_{2} \int_{0}^{h_{2}}\Theta_{2}( \tau) \bigl( \chi(s,p)-\chi\bigl(s ( t-\tau ) ,p ( t- \tau ) \bigr) \bigr) \,d\tau \\ &\quad{} +\kappa_{3}\lambda_{3} \int_{0}^{h_{3}}\Theta_{3}( \tau) \bigl( \chi(s,p)-\chi\bigl(s ( t-\tau ) ,p ( t- \tau ) \bigr) \bigr) \,d\tau \\ & \quad{} +\kappa_{4} \bigl( N\beta_{3}g_{2}(y)+M \beta_{4}g_{3}(u)-\beta_{5}g _{4}(p)-qg_{4}(p)g_{5}(x) \bigr) \\ & \quad{} +\kappa_{5}\bigl(rg_{4}(p)g_{5}(x)- \beta_{6}g_{5}(x)\bigr). \end{aligned}$$
(61)
Collecting terms of Eq. (61) and using \(\pi(s_{0})=0\), we obtain
$$\begin{aligned} \frac{dW_{0}}{dt} & = \bigl( \pi(s)-\pi(s_{0}) \bigr) \biggl( 1- \lim_{p\rightarrow0^{+}}\frac{\chi(s_{0},p)}{\chi(s,p)} \biggr) \\ &\quad{} + \biggl( \frac{\lambda\chi(s,p)}{g_{4}(p)} \lim_{p\rightarrow0^{+}}\frac{\chi(s_{0},p)}{\chi(s,p)}- \kappa_{4} \beta_{5} \biggr) g_{4}(p)- \kappa_{5}\beta_{6}g_{5}(x) \\ & \leq \bigl( \pi(s)-\pi(s_{0}) \bigr) \biggl( 1- \lim _{p\rightarrow0^{+}}\frac{\chi(s_{0},p)}{\chi(s,p)} \biggr) \\ &\quad{} + \biggl( \lim _{p\rightarrow0^{+}}\frac{\lambda\chi(s,p)}{g _{4}(p)}\lim_{p\rightarrow0^{+}} \frac{\chi(s_{0},p)}{\chi(s,p)}- \kappa_{4}\beta_{5} \biggr) g_{4}(p)-\kappa_{5}\beta_{6}g_{5}(x) \\ & = \bigl( \pi(s)-\pi(s_{0}) \bigr) \biggl( 1-\frac{\partial\chi(s _{0},0)/\partial p}{\partial\chi(s,0)/\partial p} \biggr) \\ &\quad{} +\kappa _{4}\beta_{5} \biggl( \frac{\lambda}{\kappa_{4}\beta_{5}g_{4} ^{\prime}(0)} \frac{\partial\chi(s_{0},0)}{\partial p}-1 \biggr) g _{4}(p)-\kappa_{5} \beta_{6}g_{5}(x) \\ & = \bigl( \pi(s)-\pi(s_{0}) \bigr) \biggl( 1-\frac{\partial\chi(s _{0},0)/\partial p}{\partial\chi(s,0)/\partial p} \biggr) +\kappa _{4}\beta_{5}(R_{0}-1)g_{4}(p)- \kappa_{5}\beta_{6}g_{5}(x). \end{aligned}$$
(62)
From hypotheses (H1) and (H2), we have
$$ \bigl( \pi(s)-\pi(s_{0}) \bigr) \biggl( 1-\frac{\partial\chi(s_{0},0)/ \partial p}{\partial\chi(s,0)/\partial p} \biggr) \leq0. $$
Therefore, if \(R_{0}\leq1\), then \(\frac{dW_{0}}{dt}\leq0\). Thus, \(\Pi_{0}\) is GAS. □

Theorem 5

If \(R_{1}\leq1< R_{0}\) and hypotheses (H1)(H4) are valid for (42)(47), then \(\Pi_{1}\) is GAS.

Proof

We introduce
$$\begin{aligned} W_{1} & =s-s_{1}- \int_{s_{1}}^{s}\frac{\chi(s_{1},p_{1})}{ \chi(\eta,p_{1})}\,d\eta+ \kappa_{1} \biggl( w-w_{1}- \int_{w_{1}} ^{w}\frac{g_{1}(w_{1})}{g_{1}(\eta)}\,d\eta \biggr) \\ &\quad{} + \kappa_{2} \biggl( y-y _{1}- \int_{y_{1}}^{y}\frac{g_{2}(y_{1})}{g_{2}(\eta)}\,d\eta \biggr) +\kappa_{3} \biggl( u-u_{1}- \int_{u_{1}}^{u}\frac{g_{3}(u_{1})}{g _{3}(\eta)}\,d\eta \biggr) \\ &\quad{}+ \kappa_{1}\lambda_{1}\chi(s_{1},p_{1}) \int_{0}^{h_{1}}\Theta_{1}(\tau) \int_{0}^{\tau}F \biggl( \frac{\chi(s ( t-\theta ) ,p ( t-\theta ) )}{ \chi(s_{1},p_{1})} \biggr) \,d \theta \,d\tau \\ & \quad{} +\kappa_{2}\lambda_{2}\chi(s_{1},p_{1}) \int_{0}^{h_{2}} \Theta_{2}(\tau) \int_{0}^{\tau}F \biggl( \frac{\chi(s ( t- \theta ) ,p ( t-\theta ) )}{\chi(s_{1},p_{1})} \biggr) \,d \theta \,d\tau \\ &\quad{} +\kappa_{3}\lambda_{3}\chi(s_{1},p_{1}) \int_{0}^{h_{3}} \Theta_{3}(\tau) \int_{0}^{\tau}F \biggl( \frac{\chi(s ( t- \theta ) ,p ( t-\theta ) )}{\chi(s_{1},p_{1})} \biggr) \,d \theta \,d\tau \\ &\quad{} +\kappa_{4} \biggl( p-p_{1}- \int_{p_{1}}^{p}\frac{g _{4}(p_{1})}{g_{4}(\eta)}\,d\eta \biggr) + \kappa_{5}x. \end{aligned}$$
Evaluating \(\frac{dW_{1}}{dt}\) along the trajectories of (42)–(47), we have
$$\begin{aligned} \frac{dW_{1}}{dt} & = \biggl( 1- \frac{\chi(s_{1},p_{1})}{\chi(s,p_{1})} \biggr) \bigl( \pi(s)- \lambda \chi(s,p) \bigr) \\ & \quad{} +\kappa_{1} \biggl( 1-\frac{g_{1}(w_{1})}{g_{1}(w)} \biggr) \biggl( \lambda_{1} \int_{0}^{h_{1}}\Theta_{1}(\tau)\chi\bigl(s ( t- \tau ) ,p ( t-\tau ) \bigr)\,d\tau-(a_{1}+\beta_{2})g_{1}(w) \biggr) \\ & \quad{} +\kappa_{2} \biggl( 1-\frac{g_{2}(y_{1})}{g_{2}(y)} \biggr) \biggl( \lambda_{2} \int_{0}^{h_{2}}\Theta_{2}(\tau)\chi\bigl(s ( t- \tau ) ,p ( t-\tau ) \bigr)\,d\tau+a_{1}g_{1}(w)- \beta_{3}g _{2}(y) \biggr) \\ & \quad{} +\kappa_{3} \biggl( 1-\frac{g_{3}(u_{1})}{g_{3}(u)} \biggr) \biggl( \lambda_{3} \int_{0}^{h_{3}}\Theta_{3}(\tau)\chi\bigl(s ( t- \tau ) ,p ( t-\tau ) \bigr)\,d\tau-\beta_{4}g_{3}(u) \biggr) \\ & \quad{} +\kappa_{1}\lambda_{1} \int_{0}^{h_{1}}\Theta_{1}( \tau) \biggl( \chi(s,p)-\chi\bigl(s ( t-\tau ) ,p ( t- \tau ) \bigr)+\chi(s_{1},p_{1}) \\ &\quad{}\times \ln \biggl( \frac{\chi(s ( t- \tau ) ,p ( t-\tau ) )}{\chi(s,p)} \biggr) \biggr) \,d \tau \\ & \quad{} +\kappa_{2}\lambda_{2} \int_{0}^{h_{2}}\Theta_{2}( \tau) \biggl( \chi(s,p)-\chi\bigl(s ( t-\tau ) ,p ( t- \tau ) \bigr)+\chi(s_{1},p_{1}) \\ &\quad{}\times\ln \biggl( \frac{\chi(s ( t- \tau ) ,p ( t-\tau ) )}{\chi(s,p)} \biggr) \biggr) \,d \tau \\ & \quad{} +\kappa_{3}\lambda_{3} \int_{0}^{h_{3}}\Theta_{3}( \tau) \biggl( \chi(s,p)-\chi\bigl(s ( t-\tau ) ,p ( t- \tau ) \bigr)+\chi(s_{1},p_{1}) \\ &\quad{}\times\ln \biggl( \frac{\chi(s ( t- \tau ) ,p ( t-\tau ) )}{\chi(s,p)} \biggr) \biggr) \,d \tau \\ & \quad{} +\kappa_{4} \biggl( 1-\frac{g_{4}(p_{1})}{g_{4}(p)} \biggr) \bigl(N\beta _{3}g_{2}(y)+M\beta_{4}g_{3}(u)- \beta_{5}g_{4}(p)-qg_{4}(p)g_{5}(x) \bigr) \\ & \quad{} +\kappa_{5}\bigl(rg_{4}(p)g_{5}(x)- \beta_{6}g_{5}(x)\bigr). \end{aligned}$$
(63)
Collecting terms of Eq. (63) and applying the conditions of the steady state \(\Pi_{1}\):
$$\begin{aligned}& \pi(s_{1}) =\lambda\chi(s_{1},p_{1}), \\& (a_{1}+\beta_{2})g_{1}(w_{1}) = \lambda_{1} F_{1}\chi(s _{1},p_{1}), \quad\quad \kappa_{2}\beta_{3}g_{2}(y_{1})=( \kappa_{1}\lambda_{1}F_{1}+\kappa _{2} \lambda_{2}F_{2})\chi(s_{1},p _{1}), \\& \beta_{4}g_{3}(u_{1}) =\lambda_{3}F_{3} \chi(s_{1},p_{1}),\quad\quad \kappa_{4} \beta_{5}g_{4}(p_{1})=\lambda \chi(s_{1},p_{1}), \end{aligned}$$
we get
$$\begin{aligned} \frac{dW_{1}}{dt} & = \bigl( \pi(s)-\pi(s_{1}) \bigr) \biggl( 1- \frac{ \chi(s_{1},p_{1})}{\chi(s,p_{1})} \biggr) \\ & \quad{} +\lambda\chi(s_{1},p_{1}) \biggl( 1-\frac{\chi(s_{1},p _{1})}{\chi(s,p_{1})} \biggr) +\lambda\chi(s_{1},p_{1}) \biggl( \frac{\chi(s,p)}{\chi(s,p_{1})}- \frac{g_{4}(p)}{g_{4}(p_{1})} \biggr) \\ & \quad{} -\kappa_{1}\lambda_{1} \chi(s_{1},p_{1}) \int_{0} ^{h_{1}}\Theta_{1}(\tau) \frac{\chi(s ( t-\tau ) ,p ( t- \tau ) )g_{1}(w_{1})}{\chi(s_{1},p_{1})g_{1}(w)}\,d\tau+\kappa _{1}\lambda_{1} F_{1}\chi(s_{1},p_{1}) \\ &\quad{} -\kappa_{2}\lambda_{2}\chi(s_{1},p_{1}) \int_{0}^{h_{2}} \Theta_{2}(\tau) \frac{\chi(s ( t-\tau ) ,p ( t-\tau ) )g_{2}(y_{1})}{\chi(s_{1},p_{1})g_{2}(y)}\,d\tau \\ & \quad{} -\kappa_{1}\lambda_{1} F_{1} \chi(s_{1},p_{1})\frac{g_{2}(y _{1})g_{1}(w)}{g_{2}(y)g_{1}(w_{1})}+(\kappa_{1} \lambda_{1}F _{1}+\kappa_{2}\lambda_{2}F_{2}) \chi(s_{1},p_{1}) \\ & \quad{} -\kappa_{3}\lambda_{3}\chi(s_{1},p_{1}) \int_{0} ^{h_{3}}\Theta_{3}(\tau) \frac{\chi(s ( t-\tau ) ,p ( t- \tau ) g_{3}(u_{1}))}{\chi(s_{1},p_{1})g_{3}(u)}\,d\tau+\kappa _{3}\lambda_{3}F_{3} \chi(s_{1},p_{1}) \\ & \quad{} +\kappa_{1}\lambda_{1}\chi(s_{1},p_{1}) \int_{0} ^{h_{1}}\Theta_{1}(\tau)\ln \biggl( \frac{\chi(s ( t-\tau ) ,p ( t-\tau ) )}{\chi(s,p)} \biggr) \,d\tau \\ & \quad{} +\kappa_{2}\lambda_{2}\chi(s_{1},p_{1}) \int_{0} ^{h_{2}}\Theta_{2}(\tau)\ln \biggl( \frac{\chi(s ( t-\tau ) ,p ( t-\tau ) )}{\chi(s,p)} \biggr) \,d\tau \\ & \quad{} +\kappa_{3}\lambda_{3}\chi(s_{1},p_{1}) \int_{0} ^{h_{3}}\Theta_{3}(\tau)\ln \biggl( \frac{\chi(s ( t-\tau ) ,p ( t-\tau ) )}{\chi(s,p)} \biggr) \,d\tau \\ & \quad{} -(\kappa_{1}\lambda_{1}F_{1}+ \kappa_{2}\lambda _{2}F_{2})\chi(s_{1},p_{1}) \frac{g_{2}(y)g_{4}(p_{1})}{g_{2}(y_{1})g _{4}(p)}-\kappa_{3}\lambda_{3}F_{3} \chi(s_{1},p_{1})\frac{g _{3}(u)g_{4}(p_{1})}{g_{3}(u_{1})g_{4}(p)} \\ &\quad{} +\lambda\chi(s_{1},p_{1})+r\kappa_{5} \biggl( g_{4}(p_{1})-\frac{ \beta_{6}}{r} \biggr) g_{5}(x). \end{aligned}$$
Using inequalities (30) with \(\hat{s}=s_{1}\), \(\hat{w}=w_{1}\), \(\hat{y}=y_{1}\), \(\hat{p}=p_{1}\), and \(\tau_{1}=\tau_{2}=\tau_{3}= \tau\), we can obtain
$$\begin{aligned} \frac{dW_{1}}{dt} & = \bigl( \pi(s)-\pi(s_{1}) \bigr) \biggl( 1- \frac{ \chi(s_{1},p_{1})}{\chi(s,p_{1})} \biggr) \\ & \quad{} +\lambda\chi(s_{1},p_{1}) \biggl( \frac{\chi(s,p)}{ \chi(s,p_{1})}- \frac{g_{4}(p)}{g_{4}(p_{1})} \biggr) \biggl( 1-\frac{ \chi(s,p_{1})}{\chi(s,p)} \biggr) \\ & \quad{} -\lambda\chi(s_{1},p_{1}) \biggl[ F \biggl( \frac{\chi(s _{1},p_{1})}{\chi(s,p_{1})} \biggr) +F \biggl( \frac{g_{4}(p)\chi(s,p _{1})}{g_{4}(p_{1})\chi(s,p)} \biggr) \biggr] \\ & \quad{} -\kappa_{1}\lambda_{1} \chi(s_{1},p_{1}) \int_{0} ^{h_{1}}\Theta_{1}(\tau)F \biggl( \frac{g_{1}(w_{1})\chi(s ( t- \tau ) ,p ( t-\tau ) )}{g_{1}(w)\chi(s_{1},p_{1})} \biggr) \,d\tau \\ & \quad{} -\kappa_{2}\lambda_{2}\chi(s_{1},p_{1}) \int_{0}^{h_{2}} \Theta_{2}(\tau)F \biggl( \frac{g_{2}(y_{1})\chi(s ( t-\tau ) ,p ( t-\tau ) )}{g_{2}(y)\chi(s_{1},p_{1})} \biggr) \,d\tau \\ & \quad{} -\kappa_{3}\lambda_{3}\chi(s_{1},p_{1}) \int_{0} ^{h_{3}}\Theta_{3}(\tau)F \biggl( \frac{g_{3}(u_{1})\chi(s ( t- \tau ) ,p ( t-\tau ) )}{g_{3}(u)\chi(s_{1},p_{1})} \biggr) \\ & \quad{} -\kappa_{1}\lambda_{1}F_{1} \chi(s_{1},p_{1})F \biggl( \frac{g_{2}(y _{1})g_{1}(w)}{g_{2}(y)g_{1}(w_{1})} \biggr) \\ &\quad{} - ( \kappa_{1}\lambda _{1}F_{1}+\kappa_{2} \lambda_{2}F_{2} ) \chi(s_{1},p_{1})F \biggl( \frac{g_{2}(y)g_{4}(p_{1})}{g_{2}(y_{1})g _{4}(p)} \biggr) \\ & \quad{} -\kappa_{3}\lambda_{3}F_{3} \chi(s_{1},p_{1})F \biggl( \frac{g _{3}(u)g_{4}(p_{1})}{g_{3}(u_{1})g_{4}(p)} \biggr) +r \kappa_{5} \bigl( g _{4}(p_{1})-g_{4}(p_{2}) \bigr) g_{5}(x). \end{aligned}$$
Using hypotheses (H1), (H2), (H4), and Lemma 3, we get \(\frac{dW_{1}}{dt}\leq0\), where the equality occurs at \(\Pi_{1}\). By LIP, \(\Pi_{1}\) is GAS. □

Theorem 6

If \(R_{1}>1\) and hypotheses (H1)(H4) are valid for (42)(47), then \(\Pi_{2}\) is GAS.

Proof

Define
$$\begin{aligned} W_{2} & =s-s_{2}- \int_{s_{2}}^{s}\frac{\chi(s_{2},p_{2})}{ \chi(\eta,p_{2})}\,d\eta+ \kappa_{1} \biggl( w-w_{2}- \int_{w_{2}} ^{w}\frac{g_{1}(w_{2})}{g_{1}(\eta)}\,d\eta \biggr) \\ & \quad{} + \kappa_{2} \biggl( y-y _{2}- \int_{y_{2}}^{y}\frac{g_{2}(y_{2})}{g_{2}(\eta)}\,d\eta \biggr) +\kappa_{3} \biggl( u-u_{2}- \int_{u_{2}}^{u}\frac{g_{3}(u_{2})}{g _{3}(\eta)}\,d\eta \biggr) \\ & \quad{}+ \kappa_{1}\lambda_{1}\chi(s_{2},p_{2}) \int_{0}^{h_{1}}\Theta_{1}(\tau) \int_{0}^{\tau}F \biggl( \frac{\chi(s ( t-\theta ) ,p ( t-\theta ) )}{ \chi(s_{2},p_{2})} \biggr) \,d \theta \,d\tau \\ & \quad{} +\kappa_{2}\lambda_{2}\chi(s_{2},p_{2}) \int_{0}^{h_{2}} \Theta_{2}(\tau) \int_{0}^{\tau}F \biggl( \frac{\chi(s ( t- \theta ) ,p ( t-\theta ) )}{\chi(s_{2},p_{2})} \biggr) \,d \theta \,d\tau \\ &\quad{} +\kappa_{3}\lambda_{3}\chi(s_{2},p_{2}) \int_{0}^{h_{3}} \Theta_{3}(\tau) \int_{0}^{\tau}F \biggl( \frac{\chi(s ( t- \theta ) ,p ( t-\theta ) )}{\chi(s_{2},p_{2})} \biggr) \,d \theta \,d\tau \\ & \quad{} +\kappa_{4} \biggl( p-p_{2}- \int_{p_{2}}^{p}\frac{g_{4}(p_{2})}{g _{4}(\eta)}\,d\eta \biggr) + \kappa_{5} \biggl( x-x_{2}- \int_{x _{2}}^{x}\frac{g_{5}(x_{2})}{g_{5}(\eta)}\,d\eta \biggr) . \end{aligned}$$
Calculating \(\frac{dW_{2}}{dt}\) along the solutions of model (42)–(47), we get
$$\begin{aligned} \frac{dW_{2}}{dt} & = \biggl( 1- \frac{\chi(s_{2},p_{2})}{\chi(s,p_{2})} \biggr) \bigl( \pi(s)- \lambda \chi(s,p) \bigr) \\ & \quad{} +\kappa_{1} \biggl( 1-\frac{g_{1}(w_{2})}{g_{1}(w)} \biggr) \biggl( \lambda_{1} \int_{0}^{h_{1}}\Theta_{1}(\tau)\chi\bigl(s ( t- \tau ) ,p ( t-\tau ) \bigr)\,d\tau-(a_{1}+\beta_{2})g_{1}(w) \biggr) \\ & \quad{} +\kappa_{2} \biggl( 1-\frac{g_{2}(y_{2})}{g_{2}(y)} \biggr) \biggl( \lambda_{2} \int_{0}^{h_{2}}\Theta_{2}(\tau)\chi\bigl(s ( t- \tau ) ,p ( t-\tau ) \bigr)\,d\tau+a_{1}g_{1}(w)- \beta_{3}g _{2}(y) \biggr) \\ & \quad{} +\kappa_{3} \biggl( 1-\frac{g_{3}(u_{2})}{g_{3}(u)} \biggr) \biggl( \lambda_{3} \int_{0}^{h_{3}}\Theta_{3}(\tau)\chi\bigl(s ( t- \tau ) ,p ( t-\tau ) \bigr)\,d\tau-\beta_{4}g_{3}(u) \biggr) \\ &\quad{} +\kappa_{1}\lambda_{1} \int_{0}^{h_{1}}\Theta_{1}( \tau) \biggl( \chi(s,p)-\chi\bigl(s ( t-\tau ) ,p ( t- \tau ) \bigr)+\chi(s_{2},p_{2}) \\ &\quad{}\times \ln \biggl( \frac{\chi(s ( t- \tau ) ,p ( t-\tau ) )}{\chi(s,p)} \biggr) \biggr) \,d \tau \\ & \quad{} +\kappa_{2}\lambda_{2} \int_{0}^{h_{2}}\Theta_{2}( \tau) \biggl( \chi(s,p)-\chi\bigl(s ( t-\tau ) ,p ( t- \tau ) \bigr)+\chi(s_{2},p_{2}) \\ &\quad{}\times\ln \biggl( \frac{\chi(s ( t- \tau ) ,p ( t-\tau ) )}{\chi(s,p)} \biggr) \biggr) \,d \tau \\ & \quad{} +\kappa_{3}\lambda_{3} \int_{0}^{h_{3}}\Theta_{3}( \tau) \biggl( \chi(s,p)-\chi\bigl(s ( t-\tau ) ,p ( t- \tau ) \bigr)+\chi(s_{2},p_{2}) \\ &\quad{}\times\ln \biggl( \frac{\chi(s ( t- \tau ) ,p ( t-\tau ) )}{\chi(s,p)} \biggr) \biggr) \,d \tau \\ & \quad{} +\kappa_{4} \biggl( 1-\frac{g_{4}(p_{2})}{g_{4}(p)} \biggr) \bigl(N\beta _{3}g_{2}(y)+M\beta_{4}g_{3}(u)- \beta_{5}g_{4}(p)-qg_{4}(p)g_{5}(x) \bigr) \\ & \quad{} +\kappa_{5} \biggl( 1-\frac{g_{5}(x_{2})}{g_{5}(x)} \biggr) \bigl(rg_{4}(p)g _{5}(x)-\beta_{6}g_{5}(x) \bigr). \end{aligned}$$
(64)
Collecting terms of Eq. (64) and applying the steady state conditions for \(\Pi_{2}\):
$$\begin{aligned}& \pi(s_{2}) =\lambda\chi(s_{2},p_{2}), \\& (a_{1}+\beta_{2})g_{1}(w_{2}) = \lambda_{1} F_{1}\chi(s _{2},p_{2}), \quad\quad \kappa_{2}\beta_{3}g_{2}(y_{2})=( \kappa_{1}\lambda_{1}F_{1}+\kappa _{2} \lambda_{2}F_{2})\chi(s_{2},p _{2}), \\& \beta_{4}g_{3}(u_{2}) =\lambda_{3}F_{3} \chi(s_{2},p_{2}), \quad\quad \kappa _{4} \beta_{5}g_{4}(p_{2})=\lambda \chi(s_{2},p_{2})-q\kappa _{4}g_{4}(p_{2})g_{5}(x_{2}), \\& \kappa_{4}\beta_{5}g_{4}(p) =\lambda \chi(s_{2},p_{2})\frac{g _{4}(p)}{g_{4}(p_{2})}-q\kappa_{4}g_{4}(p)g_{5}(x_{2}) \end{aligned}$$
we obtain
$$\begin{aligned} \frac{dW_{2}}{dt} & = \bigl( \pi(s)-\pi(s_{2}) \bigr) \biggl( 1- \frac{ \chi(s_{2},p_{2})}{\chi(s,p_{2})} \biggr) +\lambda\chi(s _{2},p_{2}) \biggl( 1-\frac{\chi(s_{2},p_{2})}{\chi(s,p_{2})} \biggr) \\ & \quad{} +\lambda\chi(s_{2},p_{2}) \biggl( \frac{\chi(s,p)}{ \chi(s,p_{2})}- \frac{g_{4}(p)}{g_{4}(p_{2})} \biggr) \\ & \quad{}-\kappa_{1}\lambda _{1} \chi(s_{2},p_{2}) \int_{0}^{h_{1}}\Theta_{1}( \tau) \frac{g_{1}(w_{2})\chi(s ( t-\tau ) ,p ( t- \tau ) )}{g_{1}(w)\chi(s_{2},p_{2})}\,d\tau \\ & \quad{} +\kappa_{1}\lambda_{1} F_{1} \chi(s_{2},p_{2})-\kappa_{2} \lambda_{2}\chi(s_{2},p_{2}) \int_{0}^{h_{2}}\Theta_{2}(\tau) \frac{g _{2}(y_{2})\chi(s ( t-\tau ) ,p ( t-\tau ) )}{g _{2}(y)\chi(s_{2},p_{2})}\,d\tau \\ &\quad{} -\kappa_{1}\lambda_{1} F_{1} \chi(s_{2},p_{2})\frac{g_{2}(y _{2})g_{1}(w)}{g_{2}(y)g_{1}(w_{2})}+(\kappa_{1} \lambda_{1}F _{1}+\kappa_{2}\lambda_{2}F_{2}) \chi(s_{2},p_{2}) \\ &\quad{} -\kappa_{3}\lambda_{3}\chi(s_{2},p_{2}) \int_{0} ^{h_{3}}\Theta_{3}(\tau) \frac{g_{3}(u_{2})\chi(s ( t-\tau ) ,p ( t-\tau ) )}{g_{3}(u)\chi(s_{2},p_{2})}\,d\tau+\kappa _{3}\lambda_{3}F_{3} \chi(s_{2},p_{2}) \\ &\quad{} +\kappa_{1}\lambda_{1}\chi(s_{2},p_{2}) \int_{0} ^{h_{1}}\Theta_{1}(\tau)\ln \biggl( \frac{\chi(s ( t-\tau ) ,p ( t-\tau ) )}{\chi(s,p)} \biggr) \,d\tau \\ &\quad{} +\kappa_{2}\lambda_{2}\chi(s_{2},p_{2}) \int_{0} ^{h_{2}}\Theta_{2}(\tau)\ln \biggl( \frac{\chi(s ( t-\tau ) ,p ( t-\tau ) )}{\chi(s,p)} \biggr) \,d\tau \\ &\quad{} +\kappa_{3}\lambda_{3}\chi(s_{2},p_{2}) \int_{0} ^{h_{3}}\Theta_{3}(\tau)\ln \biggl( \frac{\chi(s ( t-\tau ) ,p ( t-\tau ) )}{\chi(s,p)} \biggr) \,d\tau \\ &\quad{} -(\kappa_{1}\lambda_{1}F_{1}+ \kappa_{2}\lambda _{2}F_{2})\chi(s_{2},p_{2}) \frac{g_{2}(y)g_{4}(p_{2})}{g_{2}(y_{2})g _{4}(p)} \\ &\quad{} -\kappa_{3}\lambda_{3}F_{3} \chi(s_{2},p_{2})\frac{g_{3}(u)g _{4}(p_{2})}{g_{3}(u_{2})g_{4}(p)}+\lambda \chi(s_{2},p_{2}). \end{aligned}$$
(65)
Using inequalities (30) with \(\hat{s}=s_{2}\), \(\hat{w}=w_{2}\), \(\hat{y}=y_{2}\), \(\hat{p}=p_{2}\), and \(\tau_{1}=\tau_{2}=\tau_{3}= \tau\), we can obtain
$$\begin{aligned} \frac{dW_{2}}{dt} & = \bigl( \pi(s)-\pi(s_{2}) \bigr) \biggl( 1- \frac{ \chi(s_{2},p_{2})}{\chi(s,p_{2})} \biggr) \\ & \quad{} +\lambda\chi(s_{2},p_{2}) \biggl( \frac{\chi(s,p)}{ \chi(s,p_{2})}- \frac{g_{4}(p)}{g_{4}(p_{2})} \biggr) \biggl( 1-\frac{ \chi(s,p_{2})}{\chi(s,p)} \biggr) \\ &\quad{} -\lambda\chi(s_{2},p_{2}) \biggl[ F \biggl( \frac{\chi(s _{2},p_{2})}{\chi(s,p_{2})} \biggr) +F \biggl( \frac{g_{4}(p)\chi(s,p _{2})}{g_{4}(p_{2})\chi(s,p)} \biggr) \biggr] \\ &\quad{} -\kappa_{1}\lambda_{1} \chi(s_{2},p_{2}) \int_{0} ^{h_{1}}\Theta_{1}(\tau)F \biggl( \frac{g_{1}(w_{2})\chi(s ( t- \tau ) ,p ( t-\tau ) )}{g_{1}(w)\chi(s_{2},p_{2})} \biggr) \,d\tau \\ & \quad{} -\kappa_{2}\lambda_{2}\chi(s_{2},p_{2}) \int_{0}^{h_{2}} \Theta_{2}(\tau)F \biggl( \frac{g_{2}(y_{2})\chi(s ( t-\tau ) ,p ( t-\tau ) )}{g_{2}(y)\chi(s_{2},p_{2})} \biggr) \,d\tau \\ &\quad{} -\kappa_{3}\lambda_{3}\chi(s_{2},p_{2}) \int_{0} ^{h_{3}}\Theta_{3}(\tau)F \biggl( \frac{g_{3}(u_{2})\chi(s ( t- \tau ) ,p ( t-\tau ) )}{g_{3}(u)\chi(s_{2},p_{2})} \biggr) \,d\tau \\ & \quad{} -\kappa_{1}\lambda_{1}F_{1} \chi(s_{2},p_{2})F \biggl( \frac{g _{2}(y_{2})g_{1}(w)}{g_{2}(y)g_{1}(w_{2})} \biggr) \\ & \quad{}- ( \kappa _{1}\lambda_{1}F_{1}+\kappa_{2} \lambda_{2}F_{2} ) \chi(s_{2},p_{2})F \biggl( \frac{g_{2}(y)g_{4}(p_{2})}{g_{2}(y_{2})g _{4}(p)} \biggr) \\ & \quad{} -\kappa_{3}\lambda_{3}F_{3} \chi(s_{2},p_{2})F \biggl( \frac{g _{3}(u)g_{4}(p_{2})}{g_{3}(u_{2})g_{4}(p)} \biggr) . \end{aligned}$$
According to hypotheses (H1), (H2), and (H4), we get \(\frac{dW_{2}}{dt}\leq0\). Applying LIP, one can show that \(\Pi_{2}\) is GAS. □

4 Numerical simulations

We now perform some computer simulations on the following application:
$$\begin{aligned}& \dot{s} ( t ) =\rho-\beta_{1}s ( t ) +\omega s ( t ) \biggl( 1- \frac{s ( t ) }{s_{\max}} \biggr) -\frac{(1- \varepsilon_{1})\overline{\lambda}s ( t ) p ( t ) }{1+\eta p ( t ) }, \end{aligned}$$
(66)
$$\begin{aligned}& \dot{w} ( t ) =\frac{(1-\varepsilon_{1})\overline{\lambda}_{1}e^{-\mu_{1}\tau_{1}} s ( t-\tau_{1} ) p ( t-\tau_{1} ) }{1+\eta p ( t-\tau_{1} ) }-(a_{1}+ \beta_{2})w ( t ) , \end{aligned}$$
(67)
$$\begin{aligned}& \dot{y} ( t ) =\frac{(1-\varepsilon_{1})\overline{\lambda}_{2} e^{-\mu_{2}\tau _{2}}s ( t-\tau_{2} ) p ( t-\tau_{2} ) }{1+\eta p ( t-\tau_{2} ) }+a_{1}w ( t ) - \beta_{3}y ( t ) , \end{aligned}$$
(68)
$$\begin{aligned}& \dot{u} ( t ) =\frac{(1-\varepsilon_{1})\overline{\lambda}_{3} e^{-\mu_{3}\tau _{3}}s ( t-\tau_{3} ) p ( t-\tau_{3} ) }{1+\eta p ( t-\tau_{3} ) }-\beta_{4}u ( t ) , \end{aligned}$$
(69)
$$\begin{aligned}& \dot{p} ( t ) =(1-\varepsilon_{2})\overline{N}\beta_{3}y ( t ) +(1-\varepsilon_{2})\overline{M}\beta_{4}u ( t ) - \beta_{5}p ( t ) -qp ( t ) x ( t ) , \end{aligned}$$
(70)
$$\begin{aligned}& \dot{x} ( t ) =rp ( t ) x ( t ) -\beta _{6}x ( t ) . \end{aligned}$$
(71)
We assume that \(\omega<\beta_{1}\). In this application, we consider the following specific forms of the general functions:
$$ \begin{gathered} \pi\bigl(s ( t ) \bigr)=\rho-\beta_{1}s ( t ) +\omega s ( t ) \biggl( 1-\frac{s ( t ) }{s_{\max}} \biggr) ,\quad\quad\chi\bigl(s ( t ) ,p ( t ) \bigr)= \frac{s ( t ) p ( t ) }{1+\eta p ( t ) }, \\ g _{i}(\theta)=\theta,\quad i=1,\ldots,5. \end{gathered} $$
First we verify hypotheses (H1)–(H4) for the chosen forms, then we solve the system using MATLAB. Clearly, \(\pi(0)=\rho>0\) and \(\pi(s_{0})=0\), where
$$ s_{0}=\frac{s_{\max}}{2\omega} \biggl( \omega-\beta_{1}+\sqrt{( \omega-\beta_{1})^{2}+\frac{4\rho\omega}{s_{\max}}} \biggr) . $$
We have
$$ \pi^{\prime}(s)=-\beta_{1}+\omega-\frac{2\omega s}{s_{\max}}< 0. $$
(72)
Clearly, \(\pi(s)>0\) for \(s\in{}[0,s_{0})\) and
$$ \pi(s)=\rho-(\beta_{1}-\omega)s-\omega\frac{s^{2}}{s_{\max}} \leq\rho-( \beta_{1}-\omega)s. $$
Then hypothesis (H1) is satisfied. We also have \(\chi(s,p)>0\), \(\chi(0,p)=\chi(s,0)=0\) for \(s, p\in(0,\infty)\), and
$$ \frac{\partial\chi(s,p)}{\partial s}=\frac{p}{1+\eta p} ,\quad\quad\frac{\partial\chi(s,p)}{\partial p}= \frac{s}{ ( 1+\eta p ) ^{2}},\quad\quad\frac{\partial\chi(s,0)}{ \partial p}=s. $$
Then \(\frac{\partial\chi(s,p)}{\partial s}>0\), \(\frac{\partial \chi(s,p)}{\partial p}>0\), and \(\frac{\partial\chi(s,0)}{\partial p}>0\) for \(s,p\in(0,\infty)\). Therefore, hypothesis (H1) is satisfied. In addition,
$$\begin{aligned}& \chi(s,p)=\frac{sp}{1+\eta p}\leq sp=p\frac{\partial\chi(s,0)}{ \partial p}, \\& \biggl( \frac{\partial\chi(s,0)}{\partial p} \biggr) ^{\prime}=1>0\quad\text{for all }s>0. \end{aligned}$$
It follows that (H2) is satisfied. Clearly, hypothesis (H3) holds true. Moreover,
$$ \frac{\partial}{\partial p} \biggl( \frac{\chi(s,p)}{g_{4}(p)} \biggr) =\frac{-\eta s}{(1+\eta p)^{2}}< 0. $$
Therefore, hypothesis (H4) holds true and Theorems 13 are applicable. The parameters \(R_{0}\) and \(R_{1}\) for this application are given by
$$\begin{aligned}& R_{0} =\frac{(1-\varepsilon_{1})(1-\varepsilon_{2}) \{ \overline{N}(a_{1}\overline{\lambda}_{1}e^{-\mu_{1}\tau_{1}}+(a _{1}+\beta_{2})\overline{\lambda}_{2}e^{-\mu_{2}\tau_{2}})+ \overline{M}\overline{\lambda}_{3}e^{-\mu_{3}\tau_{3}}(a _{1}+\beta_{2}) \} }{\beta_{5}(a_{1}+\beta_{2})}s_{0}, \\& R_{1} =\frac{(1-\varepsilon_{1})(1-\varepsilon_{2}) \{ \overline{N}(a_{1}\overline{\lambda}_{1}e^{-\mu_{1}\tau_{1}}+(a _{1}+\beta_{2})\overline{\lambda}_{2}e^{-\mu_{2}\tau_{2}})+ \overline{M}\overline{\lambda}_{3}e^{-\mu_{3}\tau_{3}}(a _{1}+\beta_{2}) \} }{\beta_{5}(a_{1}+\beta_{2})}\frac{s_{2}}{1+ \eta p_{2}}. \end{aligned}$$

Remark

There are several forms of the general function \(\chi(s,p)\) where (H1)–(H4) can be satisfied such as:
  1. (i)

    Holling-type incidence \(\chi(s,p)=\frac{sp}{1+\eta_{1}s}\),

     
  2. (ii)

    Beddington–DeAngelis incidence \(\chi(s,p)=\frac{sp}{1+\eta_{1}s+ \eta_{2}p}\),

     
  3. (iii)

    Crowley–Martin incidence \(\chi(s,p)=\frac{sp}{(1+\eta_{1}s)(1+ \eta_{2}p)}\),

     
  4. (iv)

    Hill-type incidence \(\chi(s,p)=\frac{s^{m}p}{\eta^{m}+s^{m}}\).

     
Now we are ready to perform some numerical simulations for system (66)–(71). The data of system (66)–(71) are provided in Table 1. We let \(\tau_{1}=\tau_{2}=\tau_{3}=\tau\).
Table 1

The data of example (66)–(71)

Parameter

Value

ρ

10

\(\beta_{1}\)

0.01

ω

0.0001

\(s_{\max}\)

1200

q

0.5

\(\beta_{2}\)

0.02

\(\beta_{3}\)

0.36

\(\beta_{4}\)

0.031

\(\beta_{5}\)

3.0

\(\beta_{6}\)

0.1

\(\mu_{1}\)

1

\(\mu_{2}\)

1

\(\mu_{3}\)

1

η

0.01

\(a_{1}\)

0.2

6

3

\(\varepsilon _{1}\), \(\varepsilon_{2}\), τ

Varied

\(\lambda _{i}\), i = 1,2,3

Varied

r

Varied

Effect of the parameters \(\lambda_{i}\) and r on the stability of the steady states

To discuss our global results, we choose three different initial conditions:
IC1:: 

\((s(0),w(0),y(0),u(0),p(0),x(0))=(900,5,5,15,3,3)\).

IC2:: 

\((s(0),w(0),y(0),u(0),p(0),x(0))=(700,7,8,30,5,5)\).

IC3:: 

\((s(0),w(0),y(0),u(0),p(0),x(0))=(500,15,18,60,12,7)\).

Let us address three cases for the parameters \(\lambda_{i}\), \(i=1,2,3\), and r. We assume that \(\varepsilon_{1}=\varepsilon_{2}=0\) (there is no treatment) and \(\tau_{i}=0\) (there is no time delay).

Case (I): Choose \(\lambda_{i}=0.0000625\) and \(r=0.005\), which gives \(R_{0}=0.3016<1\) and \(R_{1}=0.1917<1\). Therefore, based on Lemma 2 and Theorem 1, the system has unique steady state, that is, \(\Pi_{0}\), and it is GAS. As we can see from Figs. 16, the concentration of the uninfected CD4+ T cells is increased and approaches its normal value before infection, that is, \(s_{0}=1001.98\); while concentrations of the other compartments converge to zero for all the three initial conditions. As a result, the HIV-1 is removed from the plasma.
Figure 1
Figure 1

The concentration of uninfected CD4+ T cells for system (66)–(71)

Figure 2
Figure 2

The concentration of latently infected cells for system (66)–(71)

Figure 3
Figure 3

The concentration of short-lived productively infected cells for system (66)–(71)

Figure 4
Figure 4

The concentration of long-lived productively infected cells for system (66)–(71)

Figure 5
Figure 5

The concentration of free virus particles for system (66)–(71)

Figure 6
Figure 6

The concentration of B cells for system (66)–(71)

Case (II): By taking \(\lambda_{i}=0.000625\) and \(r=0.005\). For these values, \(R_{1}=0.6095<1<R_{0}=3.0163\). Consequently, based on Lemma 2 and Theorem 2, the humoral-inactivated infection steady state \(\Pi_{1}\) is positive and is GAS. Figures 16 confirm that the numerical results support the theoretical results presented in Theorem 2. It can be observed that the variables of the model eventually converge to \(\Pi_{1}=(366.317,9.72758,11.3488,69.0344, 10.3112,0.0)\) for all the three initial conditions. This case corresponds to a chronic HIV-1 infection in the absence of immune response.

Case (III): \(\lambda_{i}=0.000625\) and \(r=0.05\). Then we calculate \(R_{0}=3.0163>1\) and \(R_{1}=2.1648>1\). According to Lemma 2 and Theorem 3, the humoral-activated infection steady state \(\Pi_{2}\) is positive and is GAS. We can see from Figs. 16 that there is a consistency between the numerical results and theoretical results of Theorem 3. The states of the system converge to \(\Pi_{2}=(734.586,4.09194,4.77393,29.0396,2.0,7.01238)\) for all the three initial conditions. In this case the humoral immune response is activated and can control the disease.

Effect of the HAART on the HIV dynamics

We take \(\tau_{i}=0\), \(\lambda_{i}=0.000625\), and \(r=0.05\). We choose the initial conditions \((s(0),w(0), y(0),u(0),p(0),x(0))=(850,5,6,17,1.5,5)\). In Figs. 712 we show the effect of the drug efficacy parameters \(\varepsilon_{1}\) and \(\varepsilon_{2}\) on the HIV dynamics. Also, we can observe that, as the drug efficacy parameters \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are increased, the concentration of uninfected cells is increased, while the concentrations of free virus particles and the three types of infected cells are decreased. Table 2 shows that the values of \(R_{0}\) and \(R_{1}\) are decreased as \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are increased.
Figure 7
Figure 7

The concentration of uninfected CD4+ T cells for system (66)–(71)

Figure 8
Figure 8

The concentration of latently infected cells for system (66)–(71)

Figure 9
Figure 9

The concentration of short-lived productively infected cells for system (66)–(71)

Figure 10
Figure 10

The concentration of long-lived productively infected cells for system (66)–(71)

Figure 11
Figure 11

The concentration of free virus particles for system (66)–(71)

Figure 12
Figure 12

The concentration of B cells for system (66)–(71)

Table 2

The values of steady states \(R_{0}\) and \(R_{1}\) for model (66)–(71) with different values of \(\varepsilon_{1}\) and \(\varepsilon_{2}\)

Drug

Steady states

\(R_{0}\)

\(R_{1}\)

\(\varepsilon_{1}=0\), \(\varepsilon_{2}=0\)

\(\Pi _{2}=(734.586,4.09194,4.77393,29.0396,2.0,7.01238)\)

3.01635

2.16484

\(\varepsilon_{1}=0.1\), \(\varepsilon_{2}=0.2\)

\(\Pi _{2}=(734.586,3.68275,4.29654,26.1356,2.0,3.36892)\)

2.17177

1.55868

\(\varepsilon_{1}=0.3\), \(\varepsilon_{2}=0.4\)

\(\Pi _{1}=(801.457,2.14802,2.50603,15.244,1.36614,0)\)

1.26687

0.909233

\(\varepsilon_{1}=0.5\), \(\varepsilon_{2}=0.6\)

\(\Pi _{0}=(1001.98,0,0,0,0,0)\)

0.60327

0.432968

Let us define the overall HAART effect as \(\varepsilon_{e}=\varepsilon_{1}+ \varepsilon_{2}-\varepsilon_{1}\varepsilon_{2}\) [13]. If \(\varepsilon_{e}=0\), then the HAART has no effect, if \(\varepsilon_{e}=1\), the HIV growth is completely halted. Consequently, the parameter \(R_{0}\) is given by
$$ R_{0}(\varepsilon_{e})=\frac{(1-\varepsilon_{e}) \{ \overline{N}(a _{1}\overline{\lambda}_{1}e^{-\mu_{1}\tau_{1}}+(a_{1}+\beta _{2})\overline{\lambda}_{2}e^{-\mu_{2}\tau_{2}})+\overline{M}\overline {\lambda}_{3}e^{-\mu_{3}\tau_{3}}(a_{1}+\beta_{2}) \} }{\beta _{5}(a_{1}+\beta_{2})}s_{0}. $$
Since the goal is to clear the viruses from the body, we have to determine the drug efficacy that makes \(R_{0}(\varepsilon_{e})\leq1\) for system (66)–(71). In this case, we get the critical drug efficacy (i.e., the efficacy needed in order to stabilize the system around the uninfected steady state). For the model (66)–(71), \(\Pi_{0}\) is GAS when \(R_{0}(\varepsilon _{e})\leq1\), i.e.,
$$ \varepsilon_{e}^{\mathrm {crit}}\leq\varepsilon_{e}< 1, \quad \quad\varepsilon_{e} ^{\mathrm {crit}}=\max \biggl\{ 0, \frac{R_{0}(0)-1}{R_{0}(0)} \biggr\} . $$
Using the data in Table 1, we have \(\varepsilon_{e}^{\mathrm {crit}}=0.668473\).

Effect of the time delay on the stability of the system

Choosing \(\varepsilon_{1}=\varepsilon_{2}=0\), \(\lambda_{i}=0.000625\), and \(r=0.05\). The initial conditions are considered to be \((s(0),w(0),y(0),u(0),p(0),x(0))=(850,2,3,17,1,5)\). Figures 1318 and Table 3 show the effect of the time delay parameter τ on the stability of \(\Pi_{0}\), \(\Pi_{1}\), and \(\Pi_{2}\). Clearly, the parameter τ has similar effect as the drug efficacy parameters \(\varepsilon_{1}\) and \(\varepsilon_{2}\).
Figure 13
Figure 13

The concentration of uninfected CD4+ T cells for system (66)–(71)

Figure 14
Figure 14

The concentration of latently infected cells for system (66)–(71)

Figure 15
Figure 15

The concentration of short-lived productively infected cells for system (66)–(71)

Figure 16
Figure 16

The concentration of long-lived productively infected cells for system (66)–(71)

Figure 17
Figure 17

The concentration of free virus particles for system (66)–(71)

Figure 18
Figure 18

The concentration of B cells for system (66)–(71)

Table 3

The values of steady states \(R_{0}\) and \(R_{1}\) for model (66)–(71) with different values of τ

Drug

Steady states

\(R_{0}\)

\(R_{1}\)

τ = 0

\(\Pi_{2}=(734.586,4.09194,4.77393,29.0396,2.0,7.01238)\)

3.01635

2.16484

τ = 0.5

\(\Pi_{2}=(734.586,2.48189,2.89554,17.6134,2.0,1.89241)\)

1.82951

1.31304

τ = 0.9

\(\Pi_{1}=(826.24,1.09341,1.27564,7.75968,1.15901,0)\)

1.22636

0.880158

τ = 1.5

\(\Pi_{0}=(1001.98,0,0,0,0,0)\)

0.673038

0.483041

τ = 2.5

\(\Pi_{0}=(1001.98,0,0,0,0,0)\)

0.247597

0.177701

5 Conclusion

In this paper, we have proposed and analyzed two general nonlinear HIV infection models with humoral immune response. We have considered three types of infected cells: latently infected cells, long-lived productively infected cells, and short-lived productively infected cells. We have incorporated three discrete or distributed time delays into the models. We have considered more general nonlinear functions for the HIV-target incidence rate, production/proliferation, and removal rates of the cells and HIV. We have derived a set of conditions on these general functions and have determined two threshold parameters: the basic reproduction number \(R_{0}\) and the humoral immune response activation number \(R_{1}\). We have proved the nonnegativity and ultimate boundedness of the model’s solutions and the existence and stability of the model’s steady states. Using Lyapunov functionals, we have established the global stability of the three steady states of the models. We have presented an example and performed some numerical simulations to support our theoretical results.

Declarations

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for technical and financial support.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
(2)
Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt
(3)
Department of Mathematics, Faculty of Science, King Khalid University, Abha, Saudi Arabia

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