Global Dynamics of Viral Infection with Two Distinct Populations of Antibodies

Abstract

This paper presents two viral infection models that describe dynamics of the virus under the effect of two distinct types of antibodies. The first model considers the population of five compartments, target cells, infected cells, free virus particles, antibodies type-1 and antibodies type-2. The presence of two types of antibodies can be a result of secondary viral infection. In the second model, we incorporate the latently infected cells. We assume that the antibody responsiveness is given by a combination of the self-regulating antibody response and the predator–prey-like antibody response. For both models, we verify the nonnegativity and boundedness of their solutions, then we outline all possible equilibria and prove the global stability by constructing proper Lyapunov functions. The stability of the uninfected equilibrium ${\mathcal{EQ}}^0$ and infected equilibrium ${\mathcal{EQ}}^{*}$ is determined by the basic reproduction number $R_0$. The theoretical findings are verified through numerical simulations. According to the outcomes, the trajectories of the solutions approach ${\mathcal{EQ}}^0$ and ${\mathcal{EQ}}^{*}$ when $R_0\le 1$ and $R_0>1$, respectively. We study the sensitivity analysis to show how the values of all the parameters of the suggested model affect $R_0$ under the given data. The impact of including the self-regulating antibody response and latently infected cells in the viral infection model is discussed. We showed that the presence of the self-regulating antibody response reduces $R_0$ and makes the system more stabilizable around ${\mathcal{EQ}}^0$. Moreover, we established that neglecting the latently infected cells in the viral infection modeling leads to the design of an overflow of antiviral drug therapy.

1. Introduction

Recently, the world witnessed the spread of many infectious viruses, which caused many deaths, and which negatively affected public health and the global economy. Scientists and researchers in various fields have made great efforts to study how to confront these viruses and eliminate the diseases resulting from them. Mathematical modeling is considered an effective and powerful method for understanding the within-host (or between-host) dynamics of several human viruses. Mathematical models have been used to provide useful insights for the development of new antiviral drug treatments. Examples of these viruses are as follows:

• Chronic viruses including: hepatitis C virus (HCV) [1,2], hepatitis B virus (HBV) [3,4], human T-lymphotropic virus type I (HTLV-I) [5,6,7] and human immunodeficiency virus type 1 (HIV-1) [8,9,10].
• Respiratory viruses including: coronaviruses [11,12,13,14,15,16,17,18] and the influenza virus [19,20].
• Viruses that cause vector-borne diseases including: dengue virus (DENV) [21,22], chikungunya virus [23] and Zika virus [24].

The basic model of viral infection was introduced by Nowak and Bangham [8] by considering the interaction of three compartments, target cells (x), infected cells (y) and free viruses (v). When foreign bodies enter the human body, the immune system becomes active against these bodies. Antibodies and cytotoxic T lymphocytes (CTLs) play major roles in fighting viruses. Antibodies are produced from the B cells and neutralize viruses. CTLs are responsible for killing the infected cells. Wodarz et al. [25] incorporated the influence of antibodies into the viral infection model as in the following:

$$\begin{array}{cc}\hfill \mathrm{Target}\mathrm{cells}:& \frac{dx}{dt}=\underset{\mathrm{Production}\mathrm{of}\mathrm{target}\mathrm{cells}}{\underbrace{\varphi }}-\underset{\mathrm{Natural}\mathrm{death}}{\underbrace{\varrho x}}-\underset{\mathrm{Infectious}\mathrm{transmission}}{\underbrace{\kappa xv}},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \mathrm{Infected}\mathrm{cells}:& \frac{dy}{dt}=\underset{\mathrm{Infectious}\mathrm{transmission}}{\underbrace{\kappa xv}}-\underset{\mathrm{Natural}\mathrm{death}}{\underbrace{\delta y}},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \mathrm{Viruses}:& \frac{dv}{dt}=\underset{\mathrm{Burst}\mathrm{size}}{\underbrace{\eta y}}-\underset{\mathrm{Natural}\mathrm{death}}{\underbrace{\rho v}}-\underset{\mathrm{Neutralization}\mathrm{of}\mathrm{viruses}}{\underbrace{\beta vz}},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \mathrm{Antibodies}:& \frac{dz}{dt}=\underset{\mathrm{Antibody}\mathrm{responsiveness}}{\underbrace{\lambda vz}}-\underset{\mathrm{Natural}\mathrm{death}}{\underbrace{\psi z}},\hfill \end{array}$$

(4)

where $z=z\left(t\right)$ is the concentration of antibodies. Parameters $\varphi$, $\varrho$, $\kappa$, $\delta$, $\eta$, $\rho$, $\beta$, $\lambda$ and $\psi$ are positive. The model was extended by including, (i) time delay [26,27,28,29], (ii) cell-to-cell transmission [30], (iii) latently infected cells [31] and (iv) age structure [32,33,34].

Equation (4) can be generalized as:

$$\frac{dz}{dt}=\Theta (v,z)-\psi z,$$

where $\Theta (v,z)$ is a general function that represents the antibody responsiveness. Several forms of $\Theta (v,z)$ have been presented in the literature such as:

F1.

Self-regulating antibody response, $\Theta (v,z)=\gamma$, where $\gamma >0$ [10,15];

F2.

Linear immune response, $\Theta (v,z)=\vartheta v$, where $\vartheta >0$ [35];

F3.

Predator–prey-like antibody response, $\Theta (v,z)=\lambda vz$, where $\lambda >0$ [10,26,27,28,29,30,31,35,36,37,38];

F4.

Combination of forms F1–F3, $\Theta (v,z)=\gamma +\vartheta v+\lambda vz$ [10];

F5.

Saturated antibody expansion, $\Theta (v,z)=\frac{\lambda vz}{\tau +z}$, where $\tau >0$ [32,34].

All the mathematical models given above are formulated for primary viral infection. In this case, it is assumed that the virus is attacked by one type of antibody. It may happen that there are two distinct populations of antibodies against a viral infection. For example, in the case of DENV infection, there are four DENV serotypes that can infect the human [39]. When a person is infected for the second time with another type of DENV serotype, there are two types of antibodies, a heterologous (nonspecific) antibody previously formed on the primary infection and a homologous (strain-specific) antibody against the new DENV serotype of the secondary infection. Gujarati and Ambika [40] formulated a secondary DENV infection model with two types of antibodies. Elaiw and Alofi [37] and Raezah [38] extended the model presented [40] by considering the spatial dependence. In [37,38], the antibody responsiveness is given by $\Theta (v,z)=\lambda vz$, where the antibodies are available just when the infection occurs. On the other hand, it was assumed in [10,15,41] that the antibodies are available even if there is no infection.

The aim of this paper is to develop two virus dynamics models with two types of antibodies, $z_1$ and $z_2$. In the case of secondary viral infection, $z_1$ and $z_2$ can be the populations of nonspecific and strain-specific antibodies, respectively. We assume that both types of antibodies, $z_1$ and $z_2$, are at levels ${\gamma }_1/{\psi }_1$ and ${\gamma }_2/{\psi }_2$, respectively, in the absence of infection, with ${\gamma }_1$ and ${\gamma }_2$ being the sources [10,15,42]. We model this by assuming the expansion of the populations $z_1$ and $z_2$ at rates ${\lambda }_{1}v{z}_1$ and ${\lambda }_{2}v{z}_2$, respectively. In this case, the antibody responsiveness function is given by the self-regulating antibody response and the predator–prey-like antibody response i.e., $\Theta (v,z_i)={\gamma }_i+{\lambda }_{i}v{z}_i$, $i=1,2$. The second model is an extension of the first one by considering two classes of viral-infected cells, latently infected cells (which contain the viruses but are not producing them) and actively infected cells (which produce the viruses). For the proposed models, basic properties are established such as the solutions’ nonnegativity and boundedness. We derive the basic reproduction number $R_0$ (or $R_0^L)$ and the prospective equilibrium points with the conditions of their existence. We examine the global stability of the equilibria by using appropriate Lyapunov functions and applying the Lyapunov–LaSalle asymptotic stability (L-LAS) theorem. Finally, some numerical simulations are carried out to test the key parameters’ impact on the models’ dynamics and to ensure the theoretical results. We also study the sensitivity analysis to show how the values of all the parameters of the suggested model affect $R_0$ (or $R_0^L)$ under the given data.

2. Model with Two Antibodies

We formulate a virus dynamics model with two types of antibodies $z_1$ and $z_2$:

$$\begin{array}{cc}\hfill \frac{dx}{dt}& =\varphi -\varrho x-\kappa xv,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \frac{dy}{dt}& =\kappa xv-\delta y,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \frac{dv}{dt}& =\eta y-\rho v-{\beta }_{1}v{z}_1-{\beta }_{2}v{z}_2,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill \frac{d{z}_1}{dt}& ={\gamma }_1+{\lambda }_{1}v{z}_1-{\psi }_1{z}_1,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \frac{d{z}_2}{dt}& ={\gamma }_2+{\lambda }_{2}v{z}_2-{\psi }_2{z}_2,\hfill \end{array}$$

(9)

where $z_i=z_i\left(t\right)$ is the concentration of antibodies type-i, $i=1,2$. In the following, we study the basic and global properties of the model.

2.1. Preliminary Results

First, we determine a bounded region for the model’s solutions to illustrate the biological acceptability of the proposed model. In particular, the concentrations of cells and viruses should not become negative or unbounded. Let ${\ell }_i>0$, $i=1,2,3,4$ be defined as:

$${\ell }_1=\frac{\varphi }{\sigma }+\frac{\delta {\beta }_1{\gamma }_1}{2\eta {\lambda }_1\sigma }+\frac{\delta {\beta }_2{\gamma }_2}{2\eta {\lambda }_2\sigma },{\ell }_2=\frac{2\eta }{\delta }{\ell }_1,{\ell }_3=\frac{2\eta {\lambda }_1}{\delta {\beta }_1}{\ell }_1\mathrm{and}{\ell }_4=\frac{2\eta {\lambda }_2}{\delta {\beta }_2}{\ell }_1,$$

where $\sigma =min\{\varrho ,\frac{1}{2}\delta ,\rho ,{\psi }_1,{\psi }_2\}$. We define a region $\Xi$ as:

$$\Xi =\left\{(x,y,v,z_1,z_2)\in {\mathbb{R}}_{\ge 0}^5:0\le x,y\le {\ell }_1,0\le v\le {\ell }_2,0\le z_1\le {\ell }_3,0\le z_2\le {\ell }_4\right\},$$

and $\mathring{\Xi }$ be the interior of $\Xi$.

Lemma 1.

Solutions of system (5)–(9) are positively invariant and bounded in Ξ.

Proof. 

We have

$$\begin{array}{cc}\hfill \frac{dx}{dt}& {\mid }_{x=0}=\varphi >0,\hfill \\ \hfill \frac{dy}{dt}& {\mid }_{y=0}=\kappa xv\ge 0\mathrm{for}x,v\ge 0,\hfill \\ \hfill \frac{dv}{dt}& {\mid }_{v=0}=\eta y\ge 0\mathrm{for}y\ge 0,\hfill \\ \hfill \frac{d{z}_1}{dt}& {\mid }_{z_1=0}={\gamma }_1>0,\hfill \\ \hfill \frac{d{z}_2}{dt}& {\mid }_{z_2=0}={\gamma }_2>0.\hfill \end{array}$$

Thus, all solutions of system (5)–(9) with initial $(x\left(0\right),y\left(0\right),v\left(0\right),z_1\left(0\right),z_2\left(0\right))\in {\mathbb{R}}_{\ge 0}^5$ satisfy $(x\left(t\right),y\left(t\right),v\left(t\right),z_1\left(t\right),z_2\left(t\right))\in {\mathbb{R}}_{\ge 0}^5$ (see Proposition B.7 of [43]). Define a function $T\left(t\right)$ as:

$$T=x+y+\frac{\delta }{2\eta }v+\frac{\delta {\beta }_1}{2\eta {\lambda }_1}{z}_1+\frac{\delta {\beta }_2}{2\eta {\lambda }_2}{z}_2,$$

then

$$\begin{array}{cc}\hfill \frac{dT}{dt}& =\varphi -\varrho x-\kappa xv+\kappa xv-\delta y+\frac{\delta }{2\eta }\left(\eta y-\rho v-{\beta }_{1}v{z}_1-{\beta }_{2}v{z}_2\right)\hfill \\ \hfill & +\frac{\delta {\beta }_1}{2\eta {\lambda }_1}\left({\gamma }_1+{\lambda }_{1}v{z}_1-{\psi }_1{z}_1\right)+\frac{\delta {\beta }_2}{2\eta {\lambda }_2}\left({\gamma }_2+{\lambda }_{2}v{z}_2-{\psi }_2{z}_2\right)\hfill \\ \hfill & =\varphi +\frac{\delta {\beta }_1{\gamma }_1}{2\eta {\lambda }_1}+\frac{\delta {\beta }_2{\gamma }_2}{2\eta {\lambda }_2}-\varrho x-\frac{\delta }{2}y-\frac{\delta \rho }{2\eta }v-\frac{\delta {\beta }_1{\psi }_1}{2\eta {\lambda }_1}{z}_1-\frac{\delta {\beta }_2{\psi }_2}{2\eta {\lambda }_2}{z}_2\hfill \\ \hfill & \le \varphi +\frac{\delta {\beta }_1{\gamma }_1}{2\eta {\lambda }_1}+\frac{\delta {\beta }_2{\gamma }_2}{2\eta {\lambda }_2}-\sigma \left(x+y+\frac{\delta }{2\eta }v+\frac{\delta {\beta }_1}{2\eta {\lambda }_1}{z}_1+\frac{\delta {\beta }_2}{2\eta {\lambda }_2}{z}_2\right)\hfill \\ \hfill & =\varphi +\frac{\delta {\beta }_1{\gamma }_1}{2\eta {\lambda }_1}+\frac{\delta {\beta }_2{\gamma }_2}{2\eta {\lambda }_2}-\sigma T.\hfill \end{array}$$

Hence, $T\left(t\right)\le {\ell }_1$, if $T\left(0\right)\le {\ell }_1$. This implies that $0\le x\left(t\right),y\left(t\right)\le {\ell }_1$, $0\le v\left(t\right)\le {\ell }_2$, $0\le z_1\left(t\right)\le {\ell }_3$ and $0\le z_2\left(t\right)\le {\ell }_4$ if $0\le x\left(0\right)+y\left(0\right)+\frac{\delta }{2\eta }v\left(0\right)+\frac{\delta {\beta }_1}{2\eta {\lambda }_1}{z}_1\left(0\right)+\frac{\delta {\beta }_2}{2\eta {\lambda }_2}{z}_2\left(0\right)\le {\ell }_1$. □

Lemma 2.

There exists a threshold parameter $R_0>0$ such that (i) if $R_0\le 1,$ then there exists a unique uninfected equilibrium ${\mathcal{EQ}}^0$, (ii) if $R_0>1,$ then there exists an infected equilibrium ${\mathcal{EQ}}^{*}$ besides ${\mathcal{EQ}}^0$.

Proof. 

Let the R.H.S. of Equations (5)–(9) be zero

$$\begin{array}{cc}\hfill 0& =\varphi -\varrho x-\kappa xv,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill 0& =\kappa xv-\delta y,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill 0& =\eta y-\rho v-{\beta }_{1}v{z}_1-{\beta }_{2}v{z}_2,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill 0& ={\gamma }_1+{\lambda }_{1}v{z}_1-{\psi }_1{z}_1,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill 0& ={\gamma }_2+{\lambda }_{2}v{z}_2-{\psi }_2{z}_2.\hfill \end{array}$$

(14)

From Equations (10), (11), (13) and (14), we have

$$\begin{array}{ll}x=\frac{\varphi }{\varrho +\kappa v},& y=\frac{\kappa \varphi v}{\delta (\varrho +\kappa v)},\\ z_1=\frac{{\gamma }_1}{{\psi }_1-{\lambda }_{1}v},& z_2=\frac{{\gamma }_2}{{\psi }_2-{\lambda }_{2}v}.\end{array}$$

(15)

Substituting in Equation (12), we obtain

$$\left(-\rho +\frac{\eta \kappa \varphi }{\delta \varrho +\delta v\kappa }+\frac{{\beta }_1{\gamma }_1}{{\lambda }_{1}v-{\psi }_1}+\frac{{\beta }_2{\gamma }_2}{{\lambda }_{2}v-{\psi }_2}\right)v=0.$$

(16)

Equation (16) has two possibilities, the first is $v=0$ which gives the infection-free equilibrium ${\mathcal{EQ}}^0(x^0,0,0,z_1^0,z_2^0)$, where $x^0=\frac{\varphi }{\varrho }$, $z_1^0=\frac{{\gamma }_1}{{\psi }_1}$ and $z_2^0=\frac{{\gamma }_2}{{\psi }_2}$. The other possibility of Equation (16) is $v\ne 0$ and

$$-\rho +\frac{\eta \kappa \varphi }{\delta \varrho +\delta v\kappa }+\frac{{\beta }_1{\gamma }_1}{{\lambda }_{1}v-{\psi }_1}+\frac{{\beta }_2{\gamma }_2}{{\lambda }_{2}v-{\psi }_2}=0,$$

which gives

$$\frac{A{v}^3+B{v}^2+Cv+D}{(\delta \varrho +\delta v\kappa )({\lambda }_{1}v-{\psi }_1)({\lambda }_{2}v-{\psi }_2)}=0,$$

(17)

where

$$\begin{array}{cc}\hfill A& =\delta \rho {\lambda }_1{\lambda }_2\kappa ,\hfill \\ \hfill B& =\delta \rho \varrho {\lambda }_1{\lambda }_2-\eta {\lambda }_1{\lambda }_2\kappa \varphi -\delta {\lambda }_2{\beta }_1\kappa {\gamma }_1-\delta {\lambda }_1{\beta }_2\kappa {\gamma }_2-\delta \rho {\lambda }_2\kappa {\psi }_1-\delta \rho {\lambda }_1\kappa {\psi }_2,\hfill \\ \hfill C& =-\delta \varrho {\lambda }_2{\beta }_1{\gamma }_1-\delta \varrho {\lambda }_1{\beta }_2{\gamma }_2-\delta \rho \varrho {\lambda }_2{\psi }_1+\eta {\lambda }_2\kappa \varphi {\psi }_1+\delta {\beta }_2\kappa {\gamma }_2{\psi }_1\hfill \\ \hfill & -\delta \rho \varrho {\lambda }_1{\psi }_2+\eta {\lambda }_1\kappa \varphi {\psi }_2+\delta {\beta }_1\kappa {\gamma }_1{\psi }_2+\delta \rho \kappa {\psi }_1{\psi }_2,\hfill \\ \hfill D& =\delta \varrho {\beta }_2{\gamma }_2{\psi }_1+\delta \varrho {\beta }_1{\gamma }_1{\psi }_2+\delta \rho \varrho {\psi }_1{\psi }_2-\eta \kappa \varphi {\psi }_1{\psi }_2.\hfill \end{array}$$

We define a function $H\left(v\right)=A{v}^3+B{v}^2+Cv+D$, then we obtain

$$\begin{array}{cc}\hfill H\left(0\right)& =-\delta \varrho ({\beta }_2{\gamma }_2{\psi }_1+{\beta }_1{\gamma }_1{\psi }_2+\rho {\psi }_1{\psi }_2)\left(\frac{\eta \kappa \varphi }{\delta \varrho \rho \left(\frac{{\beta }_1{\gamma }_1}{{\psi }_1\rho }+\frac{{\beta }_2{\gamma }_2}{{\psi }_2\rho }+1\right)}-1\right),\hfill \\ \hfill H\left(\frac{{\psi }_1}{{\lambda }_1}\right)& =\frac{\delta {\beta }_1{\gamma }_1{\lambda }_2(\varrho {\lambda }_1+\kappa {\psi }_1)}{{\lambda }_1}\left(\frac{{\psi }_2}{{\lambda }_2}-\frac{{\psi }_1}{{\lambda }_1}\right),\hfill \\ \hfill H\left(\frac{{\psi }_2}{{\lambda }_2}\right)& =\frac{\delta {\beta }_2{\gamma }_2{\lambda }_1(\varrho {\lambda }_2+\kappa {\psi }_2)}{{\lambda }_2}\left(\frac{{\psi }_1}{{\lambda }_1}-\frac{{\psi }_2}{{\lambda }_2}\right),\hfill \\ \hfill \underset{v\to \infty }{lim}H\left(v\right)& =\infty .\hfill \end{array}$$

We have $H\left(0\right)<0$ if the following condition is satisfied

$$\frac{\eta \kappa \varphi }{\delta \varrho \rho \left(\frac{{\beta }_1{\gamma }_1}{{\psi }_1\rho }+\frac{{\beta }_2{\gamma }_2}{{\psi }_2\rho }+1\right)}>1.$$

(18)

Observe that

$$\begin{array}{cc}\hfill \frac{{\psi }_2}{{\lambda }_2}& >\frac{{\psi }_1}{{\lambda }_1}⟹H\left(\frac{{\psi }_1}{{\lambda }_1}\right)>0\mathrm{and}H\left(\frac{{\psi }_2}{{\lambda }_2}\right)<0,\hfill \\ \hfill \frac{{\psi }_2}{{\lambda }_2}& <\frac{{\psi }_1}{{\lambda }_1}⟹H\left(\frac{{\psi }_1}{{\lambda }_1}\right)<0\mathrm{and}H\left(\frac{{\psi }_2}{{\lambda }_2}\right)>0.\hfill \end{array}$$

Hence,

$$H\left(min\left\{\frac{{\psi }_1}{{\lambda }_1},\frac{{\psi }_2}{{\lambda }_2}\right\}\right)>0\mathrm{and}H\left(max\left\{\frac{{\psi }_1}{{\lambda }_1},\frac{{\psi }_2}{{\lambda }_2}\right\}\right)<0.$$

If condition (18) holds, then Equation (17) has three positive roots

$$\begin{array}{cc}\hfill v^{*}& \in \left(0,min\left\{\frac{{\psi }_1}{{\lambda }_1},\frac{{\psi }_2}{{\lambda }_2}\right\}\right),\hfill \\ \hfill \overline{v}& \in \left(min\left\{\frac{{\psi }_1}{{\lambda }_1},\frac{{\psi }_2}{{\lambda }_2}\right\},max\left\{\frac{{\psi }_1}{{\lambda }_1},\frac{{\psi }_2}{{\lambda }_2}\right\}\right),\hfill \\ \hfill \tilde{v}& \in \left(max\left\{\frac{{\psi }_1}{{\lambda }_1},\frac{{\psi }_2}{{\lambda }_2}\right\},\infty \right).\hfill \end{array}$$

We note from Equation (15) that the solution $\overline{v}$ gives $z_1<0$ or $z_2<0$; moreover, $\tilde{v}$ yields $z_1<0$ and $z_2<0$. Therefore, the only acceptable solution is $v^{*}$, which gives

$$\begin{array}{cc}\hfill x^{*}=\frac{\varphi }{\varrho +\kappa v^{*}}>0,& y^{*}=\frac{\kappa \varphi v^{*}}{\delta (\varrho +\kappa v^{*})}>0,\hfill \\ \hfill z_1^{*}=\frac{{\gamma }_1}{{\psi }_1-{\lambda }_1{v}^{*}}>0,& z_2^{*}=\frac{{\gamma }_2}{{\psi }_2-{\lambda }_2{v}^{*}}>0.\hfill \end{array}$$

We define the basic reproduction number $R_0$ as:

$$R_0=\frac{\eta \kappa \varphi }{\delta \varrho \rho \left(\frac{{\beta }_1{\gamma }_1}{{\psi }_1\rho }+\frac{{\beta }_2{\gamma }_2}{{\psi }_2\rho }+1\right)}.$$

Then, the infected equilibrium ${\mathcal{EQ}}^{*}(x^{*},y^{*},v^{*},z_1^{*},z_2^{*})$ exists when $R_0>1$. Here, $R_0$ represents the total number of those newly infected that arise from any one infected cell at the beginning of infection [10]. □

2.2. Global Stability

In this section, we prove the global stability of the two equilibria of system (5)–(9). The global stability of these models is established using a Lyapunov approach, which is closely related to the one given by Korobeinikov [44], Korobeinikov and Wake [45] and Elaiw [46] by constructing explicit Lyapunov functions, which are extensions and modified forms of the Lyapunov functions given in [44,45,46].

Define a function $W_j(x,y,v,z_1,z_2)$ and let ${\tilde{\Gamma }}_j$ be the largest invariant subset of ${\Gamma }_j=\left\{(x,y,v,z_1,z_2):\frac{d{W}_j}{dt}=0\right\}$, $j=0,1$. We use the arithmetic mean–geometric mean inequality [47]:

$$\frac{1}{n}\sum _{i=1}^n{k}_i\ge \sqrt[n]{{\displaystyle \prod _{i=1}^n}{k}_i},k_i\ge 0,i=1,2,\dots ,n.$$

(19)

Theorem 1.

(i) 

The uninfected equilibrium ${\mathcal{EQ}}^0(x^0,0,0,z_1^0,z_2^0)$ is globally asymptotically stable (G.A.S) in Ξ if $R_0\le 1$,

(ii) 

${\mathcal{EQ}}^0$ is unstable if $R_0>1$.

Proof. 

(i)

Define

$$\begin{array}{cc}\hfill W_0& =x^0\left(\frac{x}{x^0}-1-ln\left(\frac{x}{x^0}\right)\right)+y+\frac{\delta }{\eta }v+\frac{\delta {\beta }_1}{\eta {\lambda }_1}{z}_1^0\left(\frac{z_1}{z_1^0}-1-ln\left(\frac{z_1}{z_1^0}\right)\right)\hfill \\ \hfill & +\frac{\delta {\beta }_2}{\eta {\lambda }_2}{z}_2^0\left(\frac{z_2}{z_2^0}-1-ln\left(\frac{z_2}{z_2^0}\right)\right).\hfill \end{array}$$

We observe that $W_0(x,y,v,z_1,z_2)>0$ for all $(x,y,v,z_1,z_2)>0$ and $W_0(x^0,0,0,z_1^0,z_2^0)=0$. Calculating $\frac{d{W}_0}{dt}$ along the solutions of (5)–(9) as:

$$\begin{array}{cc}\hfill \frac{d{W}_0}{dt}& =\left(1-\frac{x^0}{x}\right)\left(\varphi -\varrho x-\kappa xv\right)+\kappa xv-\delta y\hfill \\ \hfill & +\frac{\delta }{\eta }\left(\eta y-\rho v-{\beta }_{1}v{z}_1-{\beta }_{2}v{z}_2\right)+\frac{\delta {\beta }_1}{\eta {\lambda }_1}\left(1-\frac{z_1^0}{z_1}\right)({\gamma }_1+{\lambda }_{1}v{z}_1-{\psi }_1{z}_1)\hfill \\ \hfill & +\frac{\delta {\beta }_2}{\eta {\lambda }_2}\left(1-\frac{z_2^0}{z_2}\right)({\gamma }_2+{\lambda }_{2}v{z}_2-{\psi }_2{z}_2)\hfill \\ \hfill & =\left(1-\frac{x^0}{x}\right)\left(\varphi -\varrho x\right)+\kappa x^{0}v-\frac{\delta \rho }{\eta }v\hfill \\ \hfill & +\frac{\delta {\beta }_1}{\eta {\lambda }_1}\left(1-\frac{z_1^0}{z_1}\right)({\gamma }_1-{\psi }_1{z}_1)-\frac{\delta {\beta }_1}{\eta }{z}_1^{0}v\hfill \\ \hfill & +\frac{\delta {\beta }_2}{\eta {\lambda }_2}\left(1-\frac{z_2^0}{z_2}\right)({\gamma }_2-{\psi }_2{z}_2)-\frac{\delta {\beta }_2}{\eta }{z}_2^{0}v.\hfill \end{array}$$

Using $\varphi =\varrho x^0$, ${\gamma }_1={\psi }_1{z}_1^0$ and ${\gamma }_2={\psi }_2{z}_2^0$, we obtain

$$\begin{array}{cc}\hfill \frac{d{W}_0}{dt}& =-{\frac{\varrho \left(x-x^0\right)}{x}}^2-\frac{\delta {\beta }_1{\psi }_1}{\eta {\lambda }_1}\frac{{\left(z_1-z_1^0\right)}^2}{z_1}-\frac{\delta {\beta }_2{\psi }_2}{\eta {\lambda }_2}\frac{{\left(z_2-z_2^0\right)}^2}{z_2}\hfill \\ \hfill & +\left(\kappa x^0-\frac{\delta \rho }{\eta }-\frac{\delta {\beta }_1{z}_1^0}{\eta }-\frac{\delta {\beta }_2{z}_2^0}{\eta }\right)v\hfill \\ \hfill & =-{\frac{\varrho \left(x-x^0\right)}{x}}^2-\frac{\delta {\beta }_1{\psi }_1}{\eta {\lambda }_1}\frac{{\left(z_1-z_1^0\right)}^2}{z_1}-\frac{\delta {\beta }_2{\psi }_2}{\eta {\lambda }_2}\frac{{\left(z_2-z_2^0\right)}^2}{z_2}\hfill \\ \hfill & +\frac{\kappa \varphi }{\varrho R_0}\left(R_0-1\right)v.\hfill \end{array}$$

If $R_0\le 1$, then $\frac{d{W}_0}{dt}\le 0$, for all $x,y,v,z_1,z_2\in (0,\infty )$. Moreover, $\frac{d{W}_0}{dt}=0$ when $x\left(t\right)=x^0$, $z_1\left(t\right)=z_1^0$, $z_2\left(t\right)=z_2^0$ and $v\left(t\right)=0$ for all t. The solutions of system (5)–(9) converge to ${\tilde{\Gamma }}_0$ [48]. The set ${\tilde{\Gamma }}_0$ has elements satisfying $x\left(t\right)=x^0$, $z_1\left(t\right)=z_1^0$, $z_2\left(t\right)=z_2^0$ and $v\left(t\right)=0$. We find from Equation (7) that

$$0=\frac{dv\left(t\right)}{dt}=\eta y\left(t\right)⟹y\left(t\right)=0.$$

Then, ${\tilde{\Gamma }}_0$ is the singleton $\left\{{\mathcal{EQ}}^0\right\}$. Applying L-LAS theorem [49,50,51], we obtain that ${\mathcal{EQ}}^0$ is G.A.S in $\Xi$.

(ii)

The Jacobian matrix $J_1=J_1(x,y,v,z_1,z_2)$ of system (5)–(9) is calculated as:

$$J_1=\left(\begin{array}{ccccc}-\varrho -v\kappa & 0& -\kappa x& 0& 0\\ \kappa v& -\delta & \kappa x& 0& 0\\ 0& \eta & -\rho -{\beta }_1{z}_1-{\beta }_2{z}_2& -{\beta }_{1}v& -{\beta }_{2}v\\ 0& 0& {\lambda }_1{z}_1& {\lambda }_{1}v-{\psi }_1& 0\\ 0& 0& {\lambda }_2{z}_2& 0& {\lambda }_{2}v-{\psi }_2\end{array}\right).$$

Then, the characteristic equation at the equilibrium ${\mathcal{EQ}}^0$ is given by

$$det(J_1-\xi I)=(\xi +\varrho )(\xi +{\psi }_1)(\xi +{\psi }_2)\left(P_2{\xi }^2+P_1\xi +P_0\right)=0,$$

(20)

where $\xi$ is the eigenvalue and

$$\begin{array}{cc}\hfill P_2& =\varrho {\psi }_1{\psi }_2,\hfill \\ \hfill P_1& =\varrho {\beta }_2{\gamma }_2{\psi }_1+\varrho {\beta }_1{\gamma }_1{\psi }_2+\delta \varrho {\psi }_1{\psi }_2+\rho \varrho {\psi }_1{\psi }_2,\hfill \\ \hfill P_0& =\delta \varrho {\beta }_2{\gamma }_2{\psi }_1+\delta \varrho {\beta }_1{\gamma }_1{\psi }_2+\delta \rho \varrho {\psi }_1{\psi }_2-\eta \kappa \varphi {\psi }_1{\psi }_2\hfill \\ \hfill & =\delta \varrho ({\beta }_2{\gamma }_2{\psi }_1+{\beta }_1{\gamma }_1{\psi }_2+\rho {\psi }_1{\psi }_2)(1-R_0).\hfill \end{array}$$

Clearly if $R_0>1$, then $P_0<0$ and Equation (20) has a positive root, and hence, ${\mathcal{EQ}}^0$ is unstable.

□

The result of Theorem 1 established that, if there are some control strategies (such as using some types of antiviral drug therapies) which make $R_1\le 1$, then the viruses will be cleared from the body regardless of the initial states.

Theorem 2.

The infected equilibrium ${\mathcal{EQ}}^{*}(x^{*},y^{*},v^{*},z_1^{*},z_2^{*})$ is G.A.S in $\mathring{\Xi }$ if $R_0>1$.

Proof. 

Define

$$\begin{array}{cc}\hfill W_1& =x^{*}\left(\frac{x}{x^{*}}-1-ln\left(\frac{x}{x^{*}}\right)\right)+y^{*}\left(\frac{y}{y^{*}}-1-ln\left(\frac{y}{y^{*}}\right)\right)\hfill \\ \hfill & +\frac{\delta }{\eta }{v}^{*}\left(\frac{v}{v^{*}}-1-ln\left(\frac{v}{v^{*}}\right)\right)+\frac{\delta {\beta }_1}{\eta {\lambda }_1}{z}_1^{*}\left(\frac{z_1}{z_1^{*}}-1-ln\left(\frac{z_1}{z_1^{*}}\right)\right)\hfill \\ \hfill & +\frac{\delta {\beta }_2}{\eta {\lambda }_2}{z}_2^{*}\left(\frac{z_2}{z_2^{*}}-1-ln\left(\frac{z_2}{z_2^{*}}\right)\right).\hfill \end{array}$$

Calculating $\frac{d{W}_1}{dt}$ along the trajectories of (5)–(9):

$$\begin{array}{cc}\hfill \frac{d{W}_1}{dt}& =\left(1-\frac{x^{*}}{x}\right)\left(\varphi -\varrho x-\kappa xv\right)+\left(1-\frac{y^{*}}{y}\right)\left(\kappa xv-\delta y\right)\hfill \\ \hfill & +\frac{\delta }{\eta }\left(1-\frac{v^{*}}{v}\right)\left(\eta y-\rho v-{\beta }_{1}v{z}_1-{\beta }_{2}v{z}_2\right)\hfill \\ \hfill & +\frac{\delta {\beta }_1}{\eta {\lambda }_1}\left(1-\frac{z_1^{*}}{z_1}\right)({\gamma }_1+{\lambda }_{1}v{z}_1-{\psi }_1{z}_1)+\frac{\delta {\beta }_2}{\eta {\lambda }_2}\left(1-\frac{z_2^{*}}{z_2}\right)({\gamma }_2+{\lambda }_{2}v{z}_2-{\psi }_2{z}_2)\hfill \\ \hfill & =\left(1-\frac{x^{*}}{x}\right)(\varphi -\varrho x)+\kappa x^{*}v-\frac{\kappa xv{y}^{*}}{y}+\delta y^{*}\hfill \\ \hfill & -\frac{\delta \rho }{\eta }v-\delta \frac{v^{*}y}{v}+\frac{\delta \rho }{\eta }{v}^{*}+\frac{\delta }{\eta }{\beta }_1{v}^{*}{z}_1+\frac{\delta }{\eta }{\beta }_2{v}^{*}{z}_2\hfill \\ \hfill & +\frac{\delta {\beta }_1}{\eta {\lambda }_1}\left(1-\frac{z_1^{*}}{z_1}\right)({\gamma }_1-{\psi }_1{z}_1)-\frac{\delta {\beta }_1}{\eta }{z}_1^{*}v\hfill \\ \hfill & +\frac{\delta {\beta }_2}{\eta {\lambda }_2}\left(1-\frac{z_2^{*}}{z_2}\right)({\gamma }_2-{\psi }_2{z}_2)-\frac{\delta {\beta }_2}{\eta }{z}_2^{*}v.\hfill \end{array}$$

Applying the equilibrium conditions

$$\begin{array}{cc}\hfill \varphi & =\varrho x^{*}+\kappa x^{*}{v}^{*},\hfill \\ \hfill \delta y^{*}& =\kappa x^{*}{v}^{*},\hfill \\ \hfill \eta y^{*}& =\rho v^{*}+{\beta }_1{v}^{*}{z}_1^{*}+{\beta }_2{v}^{*}{z}_2^{*},\hfill \\ \hfill {\gamma }_1& ={\psi }_1{z}_1^{*}-{\lambda }_1{v}^{*}{z}_1^{*},\hfill \\ \hfill {\gamma }_2& ={\psi }_2{z}_2^{*}-{\lambda }_2{v}^{*}{z}_2^{*},\hfill \end{array}$$

we obtain

$$\kappa x^{*}v-\frac{\delta \rho }{\eta }v-\frac{\delta {\beta }_1}{\eta }{z}_1^{*}v-\frac{\delta {\beta }_2}{\eta }{z}_2^{*}v=\left(\kappa x^{*}{v}^{*}-\frac{\delta \rho }{\eta }{v}^{*}-\frac{\delta {\beta }_1}{\eta }{z}_1^{*}{v}^{*}-\frac{\delta {\beta }_2}{\eta }{z}_2^{*}{v}^{*}\right)\frac{v}{v^{*}}=0,$$

and

$$\begin{array}{cc}\hfill \frac{d{W}_1}{dt}& =\left(1-\frac{x^{*}}{x}\right)(\varrho x^{*}+\delta y^{*}-\varrho x)-\frac{\kappa xv{y}^{*}}{y}+\delta y^{*}\hfill \\ \hfill & -\delta \frac{v^{*}y}{v}+\frac{\delta \rho }{\eta }{v}^{*}+\frac{\delta }{\eta }{\beta }_1{v}^{*}{z}_1+\frac{\delta }{\eta }{\beta }_2{v}^{*}{z}_2\hfill \\ \hfill & +\frac{\delta {\beta }_1}{\eta {\lambda }_1}\left(1-\frac{z_1^{*}}{z_1}\right)({\psi }_1{z}_1^{*}-{\lambda }_1{v}^{*}{z}_1^{*}-{\psi }_1{z}_1)\hfill \\ \hfill & +\frac{\delta {\beta }_2}{\eta {\lambda }_2}\left(1-\frac{z_2^{*}}{z_2}\right)({\psi }_2{z}_2^{*}-{\lambda }_2{v}^{*}{z}_2^{*}-{\psi }_2{z}_2)\hfill \\ \hfill & =-\frac{\varrho {(x-x^{*})}^2}{x}+\delta y^{*}\left(1-\frac{x^{*}}{x}\right)-\delta y^{*}\frac{xv{y}^{*}}{x^{*}{v}^{*}y}+2\delta y^{*}\hfill \\ \hfill & -\delta y^{*}\frac{v^{*}y}{v{y}^{*}}-\frac{\delta {\beta }_1{\psi }_1}{\eta {\lambda }_1}\frac{{\left(z_1-z_1^{*}\right)}^2}{z_1}-\frac{\delta {\beta }_1}{\eta }{v}^{*}{z}_1^{*}\left(2-\frac{z_1^{*}}{z_1}-\frac{z_1}{z_1^{*}}\right)\hfill \\ \hfill & -\frac{\delta {\beta }_2{\psi }_2}{\eta {\lambda }_2}\frac{{\left(z_2-z_2^{*}\right)}^2}{z_2}-\frac{\delta {\beta }_2}{\eta }{v}^{*}{z}_2^{*}\left(2-\frac{z_2^{*}}{z_2}-\frac{z_2}{z_2^{*}}\right)\hfill \\ \hfill & =-\frac{\varrho {(x-x^{*})}^2}{x}+\delta y^{*}\left[3-\frac{x^{*}}{x}-\frac{xv{y}^{*}}{x^{*}{v}^{*}y}-\frac{v^{*}y}{v{y}^{*}}\right]\hfill \\ \hfill & -\frac{\delta {\beta }_1{\psi }_1}{\eta {\lambda }_1}\frac{{\left(z_1-z_1^{*}\right)}^2}{z_1}+\frac{\delta {\beta }_1}{\eta }{v}^{*}\frac{{\left(z_1-z_1^{*}\right)}^2}{z_1}\hfill \\ \hfill & -\frac{\delta {\beta }_2{\psi }_2}{\eta {\lambda }_2}\frac{{\left(z_2-z_2^{*}\right)}^2}{z_2}+\frac{\delta {\beta }_2}{\eta }{v}^{*}\frac{{\left(z_2-z_2^{*}\right)}^2}{z_2}\hfill \\ \hfill & =-\frac{\varrho {(x-x^{*})}^2}{x}+\delta y^{*}\left[3-\frac{x^{*}}{x}-\frac{xv{y}^{*}}{x^{*}{v}^{*}y}-\frac{v^{*}y}{v{y}^{*}}\right]\hfill \\ \hfill & +\frac{\delta {\beta }_1}{\eta }\frac{{\left(z_1-z_1^{*}\right)}^2}{z_1}\left(v^{*}-\frac{{\psi }_1}{{\lambda }_1}\right)+\frac{\delta {\beta }_2}{\eta }\frac{{\left(z_2-z_2^{*}\right)}^2}{z_2}\left(v^{*}-\frac{{\psi }_2}{{\lambda }_2}\right).\hfill \end{array}$$

From the equilibrium conditions, we have $v^{*}-\frac{{\psi }_1}{{\lambda }_1}=-\frac{{\gamma }_1}{{\lambda }_1{z}_1^{*}}$ and $v^{*}-\frac{{\psi }_2}{{\lambda }_2}=-\frac{{\gamma }_2}{{\lambda }_2{z}_2^{*}}$. It follows that

$$\begin{array}{cc}\hfill \frac{d{W}_1}{dt}& =-\frac{\varrho {(x-x^{*})}^2}{x}-\frac{\delta {\beta }_1{\gamma }_1}{\eta {\lambda }_1}\frac{{\left(z_1-z_1^{*}\right)}^2}{z_1^{*}{z}_1}-\frac{\delta {\beta }_2{\gamma }_2}{\eta {\lambda }_2}\frac{{\left(z_2-z_2^{*}\right)}^2}{z_2^{*}{z}_2}\hfill \\ \hfill & +\delta y^{*}\left[3-\frac{x^{*}}{x}-\frac{xv{y}^{*}}{x^{*}{v}^{*}y}-\frac{v^{*}y}{v{y}^{*}}\right].\hfill \end{array}$$

Applying inequality (19) we obtain

$$\frac{1}{3}\left(\frac{x^{*}}{x}+\frac{xv{y}^{*}}{x^{*}{v}^{*}y}+\frac{v^{*}y}{v{y}^{*}}\right)\ge 1.$$

Consequently, $\frac{d{W}_1}{dt}\le 0$ for all $x,y,v,z_1,z_2>0$. Moreover, $\frac{d{W}_1}{dt}=0$ when $(x,y,v,z_1,z_2)=(x^{*},y^{*},v^{*},z_1^{*},z_2^{*})$ and thus ${\tilde{\Gamma }}_1=\left\{{\mathcal{EQ}}^{*}\right\}$. L-LAS theorem implies that ${\mathcal{EQ}}^{*}$ is G.A.S in $\mathring{\Xi }$. □

Theorem 2 suggests that, if $R_1>1$, then a chronic viral infection will be established regardless of the initial states.

Remark 1.

In some cases, one can consider a more complex within-host virus dynamics model with n-compartments (Target cells, infected cells, free viruses, immune cells, etc.), $x_1,x_2,\dots ,x_n$. However, it seems that the following Lyapunov function can be successfully applied to establish the global stability of an equilibrium $\mathcal{EQ}({\widehat{x}}_1,{\widehat{x}}_2,\dots ,{\widehat{x}}_n)$:

$$W(x_1,x_2,\dots ,x_n)=\sum _{i=1}^n{\alpha }_i{M}_i\left(x_i\right),$$

where ${\alpha }_i$ are positive constants and

$$M_i\left(x_i\right)=\left\{\begin{array}{cc}{\widehat{x}}_i\left(\frac{x_i}{{\widehat{x}}_i}-1-ln\left(\frac{x_i}{{\widehat{x}}_i}\right)\right)\hfill & \mathit{for}{\widehat{x}}_i>0,\\ x_i\hfill & \mathit{for}{\widehat{x}}_i=0.\end{array}\right.$$

3. Model with Latency

In this section, we consider two populations of viral-infected cells, latently infected cells and actively infected cells. We assume that a fraction $\theta \in (0,1)$ of infected cells become active and the remaining ($1-\theta$) become latent. Let $u=u\left(t\right)$ be the concentration of latently virus-infected cells at time t. The model with latency can be formulated as:

$$\begin{array}{cc}\hfill \frac{dx}{dt}& =\varphi -\varrho x-\kappa xv,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \frac{du}{dt}& =(1-\theta )\kappa xv-(\mu +\phi )u,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \frac{dy}{dt}& =\theta \kappa xv+\mu u-\delta y,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \frac{dv}{dt}& =\eta y-\rho v-{\beta }_{1}v{z}_1-{\beta }_{2}v{z}_2,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \frac{d{z}_1}{dt}& ={\gamma }_1+{\lambda }_{1}v{z}_1-{\psi }_1{z}_1,\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill \frac{d{z}_2}{dt}& ={\gamma }_2+{\lambda }_{2}v{z}_2-{\psi }_2{z}_2.\hfill \end{array}$$

(26)

The latently infected cells are activated at rate $\mu u$ and die at rate $\phi u$.

3.1. Preliminary Results

Let ${\ell }_i^L>0$, $i=1,2,3,4$ be defined as:

$${\ell }_1^L=\frac{\varphi }{{\sigma }^L}+\frac{\delta {\beta }_1{\gamma }_1}{2\eta {\lambda }_1{\sigma }^L}+\frac{\delta {\beta }_2{\gamma }_2}{2\eta {\lambda }_2{\sigma }^L},{\ell }_2^L=\frac{2\eta }{\delta }{\ell }_1^L,{\ell }_3^L=\frac{2\eta {\lambda }_1}{\delta {\beta }_1}{\ell }_1^L\mathrm{and}{\ell }_4^L=\frac{2\eta {\lambda }_2}{\delta {\beta }_2}{\ell }_1^L,$$

where ${\sigma }^L=min\{\varrho ,\phi ,\frac{1}{2}\delta ,\rho ,{\psi }_1,{\psi }_2\}$. Define a domain ${\Xi }^L$ as:

$${\Xi }^L=\left\{(x,u,y,v,z_1,z_2)\in {\mathbb{R}}_{\ge 0}^6:0\le x,u,y\le {\ell }_1^L,0\le v\le {\ell }_2^L,0\le z_1\le {\ell }_3^L,0\le z_2\le {\ell }_4^L\right\}.$$

Lemma 3.

Solutions of system (21)–(26) are positively invariant and bounded in ${\Xi }^L$.

Proof. 

We have

$$\begin{array}{cc}\hfill \frac{dx}{dt}& {\mid }_{x=0}=\varphi >0,\hfill \\ \hfill \frac{du}{dt}& {\mid }_{u=0}=(1-\theta )\kappa xv\ge 0,\mathrm{for}x,v\ge 0,\hfill \\ \hfill \frac{dy}{dt}& {\mid }_{y=0}=\theta \kappa xv+\mu u\ge 0,\mathrm{for}x,v,u\ge 0,\hfill \\ \hfill \frac{dv}{dt}& {\mid }_{v=0}=\eta y\ge 0,\mathrm{for}y\ge 0,\hfill \\ \hfill \frac{d{z}_1}{dt}& {\mid }_{z_1=0}={\gamma }_1>0,\hfill \\ \hfill \frac{d{z}_2}{dt}& {\mid }_{z_2=0}={\gamma }_2>0.\hfill \end{array}$$

Hence, all solutions of system (21)–(26) with initial $(x\left(0\right),u\left(0\right),y\left(0\right),v\left(0\right),z_1\left(0\right),z_2\left(0\right))\in {\mathbb{R}}_{\ge 0}^6$ satisfy $(x\left(t\right),u\left(t\right),y\left(t\right),v\left(t\right),z_1\left(t\right),z_2\left(t\right))\in {\mathbb{R}}_{\ge 0}^6$. Let

$$T^L=x+u+y+\frac{\delta }{2\eta }v+\frac{\delta {\beta }_1}{2\eta {\lambda }_1}{z}_1+\frac{\delta {\beta }_2}{2\eta {\lambda }_2}{z}_2,$$

then

$$\begin{array}{cc}\hfill \frac{d{T}^L}{dt}& =\varphi -\varrho x-\kappa xv+(1-\theta )\kappa xv-(\mu +\phi )u+\theta \kappa xv+\mu u-\delta y\hfill \\ \hfill & +\frac{\delta }{2\eta }\left(\eta y-\rho v-{\beta }_{1}v{z}_1-{\beta }_{2}v{z}_2\right)\hfill \\ \hfill & +\frac{\delta {\beta }_1}{2\eta {\lambda }_1}\left({\gamma }_1+{\lambda }_{1}v{z}_1-{\psi }_1{z}_1\right)+\frac{\delta {\beta }_2}{2\eta {\lambda }_2}\left({\gamma }_2+{\lambda }_{2}v{z}_2-{\psi }_2{z}_2\right)\hfill \\ \hfill & =\varphi +\frac{\delta {\beta }_1{\gamma }_1}{2\eta {\lambda }_1}+\frac{\delta {\beta }_2{\gamma }_2}{2\eta {\lambda }_2}-\varrho x-\phi u-\frac{\delta }{2}y-\frac{\delta \rho }{2\eta }v-\frac{\delta {\beta }_1{\psi }_1}{2\eta {\lambda }_1}{z}_1-\frac{\delta {\beta }_2{\psi }_2}{2\eta {\lambda }_2}{z}_2\hfill \\ \hfill & \le \varphi +\frac{\delta {\beta }_1{\gamma }_1}{2\eta {\lambda }_1}+\frac{\delta {\beta }_2{\gamma }_2}{2\eta {\lambda }_2}-{\sigma }^L\left(x+u+y+\frac{\delta }{2\eta }v+\frac{\delta {\beta }_1}{2\eta {\lambda }_1}{z}_1+\frac{\delta {\beta }_2}{2\eta {\lambda }_2}{z}_2\right)\hfill \\ \hfill & =\varphi +\frac{\delta {\beta }_1{\gamma }_1}{2\eta {\lambda }_1}+\frac{\delta {\beta }_2{\gamma }_2}{2\eta {\lambda }_2}-{\sigma }^L{T}^L,\hfill \end{array}$$

Hence, $T^L\left(t\right)\le {\ell }_1^L$ if $T^L\left(0\right)\le {\ell }_1^L$. This implies that $0\le x\left(t\right),u\left(t\right),y\left(t\right)\le {\ell }_1^L$, $0\le v\left(t\right)\le {\ell }_2^L$, $0\le z_1\left(t\right)\le {\ell }_3^L$ and $0\le z_2\left(t\right)\le {\ell }_4^L$ if $0\le x\left(0\right)+u\left(0\right)+y\left(0\right)+\frac{\delta }{2\eta }v\left(0\right)+\frac{\delta {\beta }_1}{2\eta {\lambda }_1}{z}_1\left(0\right)+\frac{\delta {\beta }_2}{2\eta {\lambda }_2}{z}_2\left(0\right)\le {\ell }_1^L$. □

Lemma 4.

There exists a threshold parameter $R_0^L>0$ such that (i) if $R_0^L\le 1,$ then there exists a unique uninfected equilibrium ${\mathcal{EQ}}^0$, (ii) if $R_0^L>1,$ then there exists an infected equilibrium ${\mathcal{EQ}}^{*}$ as well as ${\mathcal{EQ}}^0$.

Proof. 

The equilibrium points of system (21)–(26) are calculated by solving the following equations:

$$\begin{array}{cc}\hfill 0& =\varphi -\varrho x-\kappa xv,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill 0& =(1-\theta )\kappa xv-(\mu +\phi )u,\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill 0& =\theta \kappa xv+\mu u-\delta y,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill 0& =\eta y-\rho v-{\beta }_{1}v{z}_1-{\beta }_{2}v{z}_2,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill 0& ={\gamma }_1+{\lambda }_{1}v{z}_1-{\psi }_1{z}_1,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill 0& ={\gamma }_2+{\lambda }_{2}v{z}_2-{\psi }_2{z}_2.\hfill \end{array}$$

(32)

From Equations (27)–(29), (31) and (32), we have

$$\begin{array}{cc}\hfill x& =\frac{\varphi }{\varrho +\kappa v},u=\frac{(1-\theta )\kappa \varphi v}{(\mu +\phi )(\varrho +\kappa v)},y=\frac{\kappa \varphi (\mu +\theta \phi )v}{\delta (\mu +\phi )(\varrho +\kappa v)},\hfill \\ \hfill z_1& =\frac{{\gamma }_1}{{\psi }_1-{\lambda }_{1}v},z_2=\frac{{\gamma }_2}{{\psi }_2-{\lambda }_{2}v}.\hfill \end{array}$$

(33)

Substituting in Equation (30) we obtain

$$\left(-\rho +\frac{{\beta }_1{\gamma }_1}{{\lambda }_{1}v-{\psi }_1}+\frac{{\beta }_2{\gamma }_2}{{\lambda }_{2}v-{\psi }_2}+\frac{\eta \kappa \varphi (\mu +\theta \phi )}{\delta (\varrho +\kappa v)(\mu +\phi )}\right)v=0.$$

(34)

Equation (34) has two possibilities, the first is $v=0$, which gives the infection-free equilibrium ${\mathcal{EQ}}^0(x^0,0,0,0,z_1^0,z_2^0)$, where $x^0=\frac{\varphi }{\varrho }$, $z_1^0=\frac{{\gamma }_1}{{\psi }_1}$ and $z_2^0=\frac{{\gamma }_2}{{\psi }_2}$. The other possibility of Equation (34) is $v\ne 0$ and

$$-\rho +\frac{{\beta }_1{\gamma }_1}{{\lambda }_{1}v-{\psi }_1}+\frac{{\beta }_2{\gamma }_2}{{\lambda }_{2}v-{\psi }_2}+\frac{\eta \kappa \varphi (\mu +\theta \phi )}{\delta (\varrho +\kappa v)(\mu +\phi )}=0,$$

$$⟹\frac{A^L{v}^3+B^L{v}^2+C^{L}v+D^L}{\delta (\mu +\phi )(\delta \varrho +\delta \kappa v)({\lambda }_{1}v-{\psi }_1)({\lambda }_{2}v-{\psi }_2)}=0,$$

(35)

where

$$\begin{array}{cc}\hfill A^L& =\delta \rho {\lambda }_1{\lambda }_2\kappa (\mu +\phi ),\hfill \\ \hfill B^L& =\delta (\mu +\phi )(\rho \varrho {\lambda }_1{\lambda }_2-{\lambda }_2{\beta }_1\kappa {\gamma }_1-{\lambda }_1{\beta }_2\kappa {\gamma }_2-\rho {\lambda }_2\kappa {\psi }_1-\rho {\lambda }_1\kappa {\psi }_2)-\eta {\lambda }_1{\lambda }_2\kappa \varphi (\mu +\theta \phi ),\hfill \\ \hfill C^L& =-\delta (\mu +\phi )(\varrho ({\lambda }_2{\beta }_1{\gamma }_1+{\lambda }_1{\beta }_2{\gamma }_2+\rho {\lambda }_2{\psi }_1+\rho {\lambda }_1{\psi }_2)-\kappa ({\beta }_2{\gamma }_2{\psi }_1+{\beta }_1{\gamma }_1{\psi }_2+\rho {\psi }_1{\psi }_2))\hfill \\ \hfill & +\eta \kappa \varphi ({\lambda }_2{\psi }_1+{\lambda }_1{\psi }_2)(\mu +\theta \phi ),\hfill \\ \hfill D^L& =\delta \varrho (\mu +\phi )({\beta }_2{\gamma }_2{\psi }_1+{\beta }_1{\gamma }_1{\psi }_2+\rho {\psi }_1{\psi }_2)-\eta \kappa \varphi {\psi }_1{\psi }_2(\mu +\theta \phi ).\hfill \end{array}$$

Let $H^L\left(v\right)=A^L{v}^3+B^L{v}^2+C^{L}v+D^L$, then we obtain

$$\begin{array}{cc}\hfill H^L\left(0\right)& =-\delta \varrho (\mu +\phi )({\beta }_2{\gamma }_2{\psi }_1+{\beta }_1{\gamma }_1{\psi }_2+\rho {\psi }_1{\psi }_2)\left(\frac{\eta \kappa \varphi (\mu +\theta \phi )}{\delta \varrho \rho (\mu +\phi )\left(\frac{{\beta }_1{\gamma }_1}{{\psi }_1\rho }+\frac{{\beta }_2{\gamma }_2}{{\psi }_2\rho }+1\right)}-1\right),\hfill \\ \hfill H^L\left(\frac{{\psi }_1}{{\lambda }_1}\right)& =\frac{\delta {\beta }_1{\gamma }_1{\lambda }_2(\mu +\phi )(\varrho {\lambda }_1+\kappa {\psi }_1)}{{\lambda }_1}\left(\frac{{\psi }_2}{{\lambda }_2}-\frac{{\psi }_1}{{\lambda }_1}\right),\hfill \\ \hfill H^L\left(\frac{{\psi }_2}{{\lambda }_2}\right)& =\frac{\delta {\beta }_2{\gamma }_2{\lambda }_1(\mu +\phi )(\varrho {\lambda }_2+\kappa {\psi }_2)}{{\lambda }_2}\left(\frac{{\psi }_1}{{\lambda }_1}-\frac{{\psi }_2}{{\lambda }_2}\right),\hfill \\ \hfill \underset{v\to \infty }{lim}{H}^L\left(v\right)& =\infty .\hfill \end{array}$$

We have $H^L\left(0\right)<0$ when

$$\frac{\eta \kappa \varphi (\mu +\theta \phi )}{\delta \varrho \rho (\mu +\phi )\left(\frac{{\beta }_1{\gamma }_1}{{\psi }_1\rho }+\frac{{\beta }_2{\gamma }_2}{{\psi }_2\rho }+1\right)}>1.$$

(36)

We note that

$$\begin{array}{cc}\hfill \frac{{\psi }_2}{{\lambda }_2}& >\frac{{\psi }_1}{{\lambda }_1}⟹H^L\left(\frac{{\psi }_1}{{\lambda }_1}\right)>0\mathrm{and}{H}^L\left(\frac{{\psi }_2}{{\lambda }_2}\right)<0,\hfill \\ \hfill \frac{{\psi }_2}{{\lambda }_2}& <\frac{{\psi }_1}{{\lambda }_1}⟹H^L\left(\frac{{\psi }_1}{{\lambda }_1}\right)<0\mathrm{and}{H}^L\left(\frac{{\psi }_2}{{\lambda }_2}\right)>0.\hfill \end{array}$$

$$H^L\left(min\left\{\frac{{\psi }_1}{{\lambda }_1},\frac{{\psi }_2}{{\lambda }_2}\right\}\right)>0\mathrm{and}{H}^L\left(max\left\{\frac{{\psi }_1}{{\lambda }_1},\frac{{\psi }_2}{{\lambda }_2}\right\}\right)<0.$$

If condition (36) is satisfied, then Equation (35) has three positive roots

$$\begin{array}{cc}\hfill v^{*}& \in (0,min\left\{\frac{{\psi }_1}{{\lambda }_1},\frac{{\psi }_2}{{\lambda }_2}\right\}),\hfill \\ \hfill \overline{v}& \in (min\left\{\frac{{\psi }_1}{{\lambda }_1},\frac{{\psi }_2}{{\lambda }_2}\right\},max\left\{\frac{{\psi }_1}{{\lambda }_1},\frac{{\psi }_2}{{\lambda }_2}\right\}),\hfill \\ \hfill \tilde{v}& \in (max\left\{\frac{{\psi }_1}{{\lambda }_1},\frac{{\psi }_2}{{\lambda }_2}\right\},\infty ).\hfill \end{array}$$

From Equation (33), the solution $\overline{v}$ yields $z_1<0$ or $z_2<0$; moreover, $\tilde{v}$ gives $z_1<0$ and $z_2<0$. Therefore, the only acceptable solution is $v^{*}$, which provides

$$\begin{array}{cc}\hfill x^{*}& =\frac{\varphi }{\varrho +\kappa v^{*}}>0,u^{*}=\frac{(1-\theta )\kappa \varphi v^{*}}{(\mu +\phi )(\varrho +\kappa v^{*})}>0,y^{*}=\frac{\kappa \varphi (\mu +\theta \phi )v^{*}}{\delta (\mu +\phi )(\varrho +\kappa v^{*})}>0,\hfill \\ \hfill z_1^{*}& =\frac{{\gamma }_1}{{\psi }_1-{\lambda }_1{v}^{*}}>0,z_2^{*}=\frac{{\gamma }_2}{{\psi }_2-{\lambda }_2{v}^{*}}>0.\hfill \end{array}$$

The basic reproduction number of model (21)–(26) can be defined as:

$$R_0^L=\frac{\eta \kappa \varphi (\mu +\theta \phi )}{\delta \varrho \rho (\mu +\phi )\left(\frac{{\beta }_1{\gamma }_1}{{\psi }_1\rho }+\frac{{\beta }_2{\gamma }_2}{{\psi }_2\rho }+1\right)}.$$

Then the infected equilibrium with immunity ${\mathcal{EQ}}^{*}(x^{*},u^{*},y^{*},v^{*},z_1^{*},z_2^{*})$ exists when $R_0^L>1.$ □

3.2. Global Stability

Define a function $W_j^L(x,u,y,v,z_1,z_2)$ and let ${\tilde{\Gamma }}_j^L$ be the largest invariant subset of ${\Gamma }_j^L=\left\{(x,u,y,v,z_1,z_2):\frac{d{W}_j^L}{dt}=0\right\}$, $j=0,1$.

Theorem 3.

(i) 

If $R_0^L\le 1,$ then the uninfected equilibrium ${\mathcal{EQ}}^0(x^0,0,0,0,z_1^0,z_2^0)$ of system (21)–(26) is G.A.S in ${\Xi }^L$,

(ii) 

if $R_0^L>1$, then ${\mathcal{EQ}}^0$ is unstable.

Proof. 

(i)

Define

$$\begin{array}{cc}\hfill W_0^L& =x^0\left(\frac{x}{x^0}-1-ln\left(\frac{x}{x^0}\right)\right)+\frac{\mu }{\mu +\theta \phi }u+\frac{\mu +\phi }{\mu +\theta \phi }y+\frac{\delta (\mu +\phi )}{\eta (\mu +\theta \phi )}v\hfill \\ \hfill & +\frac{\delta {\beta }_1(\mu +\phi )}{\eta {\lambda }_1(\mu +\theta \phi )}{z}_1^0\left(\frac{z_1}{z_1^0}-1-ln\left(\frac{z_1}{z_1^0}\right)\right)+\frac{\delta {\beta }_2(\mu +\phi )}{\eta {\lambda }_2(\mu +\theta \phi )}{z}_2^0\left(\frac{z_2}{z_2^0}-1-ln\left(\frac{z_2}{z_2^0}\right)\right).\hfill \end{array}$$

Observe that $W_0^L(x,u,y,v,z_1,z_2)>0$ for all $(x,u,y,v,z_1,z_2)>0$ and $W_0^L(x^0,0,0,0,$$z_1^0,z_2^0)=0.$ Calculating $\frac{d{W}_0^L}{dt}$ along the solutions of (21)–(26) as:

$$\begin{array}{cc}\hfill \frac{d{W}_0^L}{dt}& =\left(1-\frac{x^0}{x}\right)\left(\varphi -\varrho x-\kappa xv\right)+\frac{\mu }{\mu +\theta \phi }\left((1-\theta )\kappa xv-(\mu +\phi )u\right)\hfill \\ \hfill & +\frac{\mu +\phi }{\mu +\theta \phi }\left(\theta \kappa xv+\mu u-\delta y\right)+\frac{\delta (\mu +\phi )}{\eta (\mu +\theta \phi )}\left(\eta y-\rho v-{\beta }_{1}v{z}_1-{\beta }_{2}v{z}_2\right)\hfill \\ \hfill & +\frac{\delta {\beta }_1(\mu +\phi )}{\eta {\lambda }_1(\mu +\theta \phi )}\left(1-\frac{z_1^0}{z_1}\right)({\gamma }_1+{\lambda }_{1}v{z}_1-{\psi }_1{z}_1)\hfill \\ \hfill & +\frac{\delta {\beta }_2(\mu +\phi )}{\eta {\lambda }_2(\mu +\theta \phi )}\left(1-\frac{z_2^0}{z_2}\right)({\gamma }_2+{\lambda }_{2}v{z}_2-{\psi }_2{z}_2)\hfill \\ \hfill & =\left(1-\frac{x^0}{x}\right)\left(\varphi -\varrho x\right)+\kappa x^{0}v-\frac{\delta \rho (\mu +\phi )}{\eta (\mu +\theta \phi )}v\hfill \\ \hfill & +\frac{\delta {\beta }_1(\mu +\phi )}{\eta {\lambda }_1(\mu +\theta \phi )}\left(1-\frac{z_1^0}{z_1}\right)({\gamma }_1-{\psi }_1{z}_1)-\frac{\delta {\beta }_1(\mu +\phi )}{\eta (\mu +\theta \phi )}{z}_1^{0}v\hfill \\ \hfill & +\frac{\delta {\beta }_2(\mu +\phi )}{\eta {\lambda }_2(\mu +\theta \phi )}\left(1-\frac{z_2^0}{z_2}\right)({\gamma }_2-{\psi }_2{z}_2)-\frac{\delta {\beta }_2(\mu +\phi )}{\eta (\mu +\theta \phi )}{z}_2^{0}v.\hfill \end{array}$$

Using $\varphi =\varrho x^0$, ${\gamma }_1={\psi }_1{z}_1^0$ and ${\gamma }_2={\psi }_2{z}_2^0$ we obtain

$$\begin{array}{cc}\hfill \frac{d{W}_0^L}{dt}& =-{\frac{\varrho \left(x-x^0\right)}{x}}^2-\frac{\delta {\beta }_1{\psi }_1(\mu +\phi )}{\eta {\lambda }_1(\mu +\theta \phi )}\frac{{\left(z_1-z_1^0\right)}^2}{z_1}-\frac{\delta {\beta }_2{\psi }_2(\mu +\phi )}{\eta {\lambda }_2(\mu +\theta \phi )}\frac{{\left(z_2-z_2^0\right)}^2}{z_2}\hfill \\ \hfill & +\left(\kappa x^0-\frac{\delta \rho (\mu +\phi )}{\eta (\mu +\theta \phi )}-\frac{\delta {\beta }_1(\mu +\phi )z_1^0}{\eta (\mu +\theta \phi )}-\frac{\delta {\beta }_2(\mu +\phi )z_2^0}{\eta (\mu +\theta \phi )}\right)v\hfill \\ \hfill & =-{\frac{\varrho \left(x-x^0\right)}{x}}^2-\frac{\delta {\beta }_1{\psi }_1(\mu +\phi )}{\eta {\lambda }_1(\mu +\theta \phi )}\frac{{\left(z_1-z_1^0\right)}^2}{z_1}-\frac{\delta {\beta }_2{\psi }_2(\mu +\phi )}{\eta {\lambda }_2(\mu +\theta \phi )}\frac{{\left(z_2-z_2^0\right)}^2}{z_2}\hfill \\ \hfill & +\frac{\kappa \varphi }{\varrho R_0^L}\left(R_0^L-1\right)v.\hfill \end{array}$$

Therefore, if $R_0^L\le 1$, then $\frac{d{W}_0^L}{dt}\le 0$ for all $(x,u,y,v,z_1,z_2)\in (0,\infty )$. Moreover, $\frac{d{W}_0^L}{dt}=0$ when $x\left(t\right)=x^0$, $z_1\left(t\right)=z_1^0$, $z_2\left(t\right)=z_2^0$ and $v\left(t\right)=0$ for all t. The solutions of system (21)–(26) converge to ${\tilde{\Gamma }}_0^L$, which contains elements that satisfy $x\left(t\right)=x^0$, $z_1\left(t\right)=z_1^0$, $z_2\left(t\right)=z_2^0$ and $v\left(t\right)=0$. It follows from Equation (24) that

$$0=\frac{dv\left(t\right)}{dt}=\eta y\left(t\right)⟹y\left(t\right)=0.$$

Furthermore, from Equation (23) we have

$$0=\frac{dy\left(t\right)}{dt}=\mu u\left(t\right)⟹u\left(t\right)=0.$$

Hence, ${\tilde{\Gamma }}_0^L=\left\{{\mathcal{EQ}}^0\right\}$ and L-LAS theorem provides that ${\mathcal{EQ}}^0$ is G.A.S in ${\Xi }^L$.

(ii)

The Jacobian matrix $J_2=J_2(x,u,y,v,z_1,z_2)$ of system (21)–(26) is calculated as:

$$J_2=\left(\begin{array}{cccccc}-\varrho -\kappa v& 0& 0& -\kappa x& 0& 0\\ (1-\theta )\kappa v& -\mu -\phi & 0& (1-\theta )\kappa x& 0& 0\\ \theta \kappa v& \mu & -\delta & \theta \kappa x& 0& 0\\ 0& 0& \eta & -\rho -{\beta }_1{z}_1-{\beta }_2{z}_2& -{\beta }_{1}v& -{\beta }_{2}v\\ 0& 0& 0& {\lambda }_{1}z1& {\lambda }_{1}v-{\psi }_1& 0\\ 0& 0& 0& {\lambda }_2{z}_2& 0& {\lambda }_{2}v-{\psi }_2\end{array}\right).$$

Then, the characteristic equation at the equilibrium ${\mathcal{EQ}}^0$ is given by

$$det(J_2-\xi I)=(\xi +\varrho )(\xi +{\psi }_1)(\xi +{\psi }_2)\left({\xi }^3+P_2^L{\xi }^2+P_1^L\xi +P_0^L\right)=0,$$

(37)

where $\xi$ is the eigenvalue and

$$\begin{array}{cc}\hfill P_2^L& =\delta +\mu +\rho +\phi +\frac{{\beta }_1{\gamma }_1}{{\psi }_1}+\frac{{\beta }_2{\gamma }_2}{{\psi }_2},\hfill \\ \hfill P_1^L& =\delta (\mu +\phi )-\frac{\eta \theta \kappa \varphi }{\varrho }+(\delta +\mu +\phi )\left(\rho +\frac{{\beta }_1{\gamma }_1}{{\psi }_1}+\frac{{\beta }_2{\gamma }_2}{{\psi }_2}\right),\hfill \\ \hfill P_0^L& =-\frac{\mu \eta \kappa \varphi }{\varrho }-\frac{\eta \theta \kappa \varphi \phi }{\varrho }+\delta (\mu +\phi )\left(\rho +\frac{{\beta }_1{\gamma }_1}{{\psi }_1}+\frac{{\beta }_2{\gamma }_2}{{\psi }_2}\right)\hfill \\ \hfill & =\frac{\delta (\mu +\phi )({\beta }_2{\gamma }_2{\psi }_1+{\beta }_1{\gamma }_1{\psi }_2+\rho {\psi }_1{\psi }_2)}{{\psi }_1{\psi }_2}(1-R_0^L).\hfill \end{array}$$

Clearly, if $R_0^L>1$, then $P_0^L<0$ and Equation (37) has a positive root, and hence, ${\mathcal{EQ}}^0$ is unstable.

□

Theorem 4.

The infected equilibrium ${\mathcal{EQ}}^{*}(x^{*},u^{*},y^{*},v^{*},z_1^{*},z_2^{*})$ of system (21)–(26) is G.A.S in ${\mathring{\Xi }}^L$ if $R_0>1$.

Proof. 

Define

$$\begin{array}{cc}\hfill W_1^L& =x^{*}\left(\frac{x}{x^{*}}-1-ln\left(\frac{x}{x^{*}}\right)\right)+\frac{\mu }{\mu +\theta \phi }{u}^{*}\left(\frac{u}{u^{*}}-1-ln\left(\frac{u}{u^{*}}\right)\right)\hfill \\ \hfill & +\frac{\mu +\phi }{\mu +\theta \phi }{y}^{*}\left(\frac{y}{y^{*}}-1-ln\left(\frac{y}{y^{*}}\right)\right)+\frac{\delta (\mu +\phi )}{\eta (\mu +\theta \phi )}{v}^{*}\left(\frac{v}{v^{*}}-1-ln\left(\frac{v}{v^{*}}\right)\right)\hfill \\ \hfill & +\frac{\delta {\beta }_1(\mu +\phi )}{\eta {\lambda }_1(\mu +\theta \phi )}{z}_1^{*}\left(\frac{z_1}{z_1^{*}}-1-ln\left(\frac{z_1}{z_1^{*}}\right)\right)+\frac{\delta {\beta }_2(\mu +\phi )}{\eta {\lambda }_2(\mu +\theta \phi )}{z}_2^{*}\left(\frac{z_2}{z_2^{*}}-1-ln\left(\frac{z_2}{z_2^{*}}\right)\right).\hfill \end{array}$$

Calculating $\frac{d{W}_1^L}{dt}$ along the trajectories of (21)–(26):

$$\begin{array}{cc}\hfill \frac{d{W}_1^L}{dt}& =\left(1-\frac{x^{*}}{x}\right)\left(\varphi -\varrho x-\kappa xv\right)+\frac{\mu }{\mu +\theta \phi }\left(1-\frac{u^{*}}{u}\right)\left((1-\theta )\kappa xv-(\mu +\phi )u\right)\hfill \\ \hfill & +\frac{\mu +\phi }{\mu +\theta \phi }\left(1-\frac{y^{*}}{y}\right)\left(\theta \kappa xv+\mu u-\delta y\right)\hfill \\ \hfill & +\frac{\delta (\mu +\phi )}{\eta (\mu +\theta \phi )}\left(1-\frac{v^{*}}{v}\right)\left(\eta y-\rho v-{\beta }_{1}v{z}_1-{\beta }_{2}v{z}_2\right)\hfill \\ \hfill & +\frac{\delta {\beta }_1(\mu +\phi )}{\eta {\lambda }_1(\mu +\theta \phi )}\left(1-\frac{z_1^{*}}{z_1}\right)({\gamma }_1+{\lambda }_{1}v{z}_1-{\psi }_1{z}_1)\hfill \\ \hfill & +\frac{\delta {\beta }_2(\mu +\phi )}{\eta {\lambda }_2(\mu +\theta \phi )}\left(1-\frac{z_2^{*}}{z_2}\right)({\gamma }_2+{\lambda }_{2}v{z}_2-{\psi }_2{z}_2)\hfill \\ \hfill & =\left(1-\frac{x^{*}}{x}\right)(\varphi -\varrho x)+\kappa x^{*}v-\frac{\mu }{\mu +\theta \phi }\frac{(1-\theta )\kappa xv{u}^{*}}{u}+\frac{\mu (\mu +\phi )}{\mu +\theta \phi }{u}^{*}\hfill \\ \hfill & -\frac{\mu +\phi }{\mu +\theta \phi }\frac{\theta \kappa xv{y}^{*}}{y}-\frac{\mu (\mu +\phi )}{\mu +\theta \phi }\frac{y^{*}u}{y}+\frac{\mu +\phi }{\mu +\theta \phi }\delta y^{*}\hfill \\ \hfill & -\frac{\delta \rho (\mu +\phi )}{\eta (\mu +\theta \phi )}v-\frac{\delta (\mu +\phi )}{\mu +\theta \phi }\frac{y{v}^{*}}{v}+\frac{\delta \rho (\mu +\phi )}{\eta (\mu +\theta \phi )}{v}^{*}+\frac{\delta {\beta }_1(\mu +\phi )}{\eta (\mu +\theta \phi )}{v}^{*}{z}_1+\frac{\delta {\beta }_2(\mu +\phi )}{\eta (\mu +\theta \phi )}{v}^{*}{z}_2\hfill \\ \hfill & +\frac{\delta {\beta }_1(\mu +\phi )}{\eta {\lambda }_1(\mu +\theta \phi )}\left(1-\frac{z_1^{*}}{z_1}\right)({\gamma }_1-{\psi }_1{z}_1)-\frac{\delta {\beta }_1(\mu +\phi )}{\eta (\mu +\theta \phi )}{z}_1^{*}v\hfill \\ \hfill & +\frac{\delta {\beta }_2(\mu +\phi )}{\eta {\lambda }_2(\mu +\theta \phi )}\left(1-\frac{z_2^{*}}{z_2}\right)({\gamma }_2-{\psi }_2{z}_2)-\frac{\delta {\beta }_2(\mu +\phi )}{\eta (\mu +\theta \phi )}{z}_2^{*}v.\hfill \end{array}$$

The equilibrium conditions of ${\mathcal{EQ}}^{*}$ imply

$$\begin{array}{cc}\hfill \varphi & =\varrho x^{*}+\kappa x^{*}{v}^{*},\hfill \\ \hfill \frac{\mu (\mu +\phi )}{\mu +\theta \phi }{u}^{*}& =\frac{(1-\theta )\mu }{\mu +\theta \phi }\kappa x^{*}{v}^{*}\hfill \\ \hfill \frac{\mu +\phi }{\mu +\theta \phi }\delta y^{*}& =\kappa x^{*}{v}^{*},\hfill \\ \hfill \eta y^{*}& =\rho v^{*}+{\beta }_1{v}^{*}{z}_1^{*}+{\beta }_2{v}^{*}{z}_2^{*},\hfill \\ \hfill {\gamma }_1& ={\psi }_1{z}_1^{*}-{\lambda }_1{v}^{*}{z}_1^{*},\hfill \\ \hfill {\gamma }_2& ={\psi }_2{z}_2^{*}-{\lambda }_2{v}^{*}{z}_2^{*}.\hfill \end{array}$$

Then, we obtain

$$\left(\kappa x^{*}{v}^{*}-\frac{\delta \rho (\mu +\phi )}{\eta (\mu +\theta \phi )}{v}^{*}-\frac{\delta {\beta }_1(\mu +\phi )}{\eta (\mu +\theta \phi )}{z}_1^{*}{v}^{*}-\frac{\delta {\beta }_2(\mu +\phi )}{\eta (\mu +\theta \phi )}{z}_2^{*}{v}^{*}\right)\frac{v}{v^{*}}=0.$$

and

$$\begin{array}{cc}\hfill \frac{d{W}_1^L}{dt}& =\left(1-\frac{x^{*}}{x}\right)(\varrho x^{*}+\kappa x^{*}{v}^{*}-\varrho x)-\frac{\mu (1-\theta )}{\mu +\theta \phi }\kappa x^{*}{v}^{*}\frac{xv{u}^{*}}{x^{*}{v}^{*}u}+\frac{(1-\theta )\mu }{\mu +\theta \phi }\kappa x^{*}{v}^{*}\hfill \\ \hfill & -\frac{\theta (\mu +\phi )}{\mu +\theta \phi }\kappa x^{*}{v}^{*}\frac{xv{y}^{*}}{x^{*}{v}^{*}y}-\frac{(1-\theta )\mu }{\mu +\theta \phi }\kappa x^{*}{v}^{*}\frac{y^{*}u}{y{u}^{*}}+\kappa x^{*}{v}^{*}\hfill \\ \hfill & +\frac{\delta \rho (\mu +\phi )}{\eta (\mu +\theta \phi )}{v}^{*}-\kappa x^{*}{v}^{*}\frac{y{v}^{*}}{y^{*}v}+\frac{\delta {\beta }_1(\mu +\phi )}{\eta (\mu +\theta \phi )}{v}^{*}{z}_1+\frac{\delta {\beta }_2(\mu +\phi )}{\eta (\mu +\theta \phi )}{v}^{*}{z}_2\hfill \\ \hfill & +\frac{\delta {\beta }_1(\mu +\phi )}{\eta {\lambda }_1(\mu +\theta \phi )}\left(1-\frac{z_1^{*}}{z_1}\right)({\psi }_1{z}_1^{*}-{\lambda }_1{v}^{*}{z}_1^{*}-{\psi }_1{z}_1)\hfill \\ \hfill & +\frac{\delta {\beta }_2(\mu +\phi )}{\eta {\lambda }_2(\mu +\theta \phi )}\left(1-\frac{z_2^{*}}{z_2}\right)({\psi }_2{z}_2^{*}-{\lambda }_2{v}^{*}{z}_2^{*}-{\psi }_2{z}_2).\hfill \end{array}$$

With more simplifications we obtain

$$\begin{array}{cc}\hfill \frac{d{W}_1^L}{dt}& =\left(1-\frac{x^{*}}{x}\right)(\varrho x^{*}-\varrho x)+\kappa x^{*}{v}^{*}\left(1-\frac{x^{*}}{x}\right)-\frac{\mu (1-\theta )}{\mu +\theta \phi }\kappa x^{*}{v}^{*}\frac{xv{u}^{*}}{x^{*}{v}^{*}u}\hfill \\ \hfill & +\frac{(1-\theta )\mu }{\mu +\theta \phi }\kappa x^{*}{v}^{*}-\frac{\theta (\mu +\phi )}{\mu +\theta \phi }\kappa x^{*}{v}^{*}\frac{xv{y}^{*}}{x^{*}{v}^{*}y}-\frac{(1-\theta )\mu }{\mu +\theta \phi }\kappa x^{*}{v}^{*}\frac{y^{*}u}{y{u}^{*}}+\kappa x^{*}{v}^{*}\hfill \\ \hfill & +\kappa x^{*}{v}^{*}-\frac{\delta {\beta }_1(\mu +\phi )}{\eta (\mu +\theta \phi )}{z}_1^{*}{v}^{*}-\frac{\delta {\beta }_2(\mu +\phi )}{\eta (\mu +\theta \phi )}{z}_2^{*}{v}^{*}-\kappa x^{*}{v}^{*}\frac{y{v}^{*}}{y^{*}v}+\frac{\delta {\beta }_1(\mu +\phi )}{\eta (\mu +\theta \phi )}{v}^{*}{z}_1\hfill \\ \hfill & +\frac{\delta {\beta }_2(\mu +\phi )}{\eta (\mu +\theta \phi )}{v}^{*}{z}_2+\frac{\delta {\beta }_1(\mu +\phi )}{\eta {\lambda }_1(\mu +\theta \phi )}\left(1-\frac{z_1^{*}}{z_1}\right)({\psi }_1{z}_1^{*}-{\psi }_1{z}_1)-\frac{\delta {\beta }_1(\mu +\phi )}{\eta (\mu +\theta \phi )}{v}^{*}{z}_1^{*}\left(1-\frac{z_1^{*}}{z_1}\right)\hfill \\ \hfill & +\frac{\delta {\beta }_2(\mu +\phi )}{\eta {\lambda }_2(\mu +\theta \phi )}\left(1-\frac{z_2^{*}}{z_2}\right)({\psi }_2{z}_2^{*}-{\psi }_2{z}_2)-\frac{\delta {\beta }_2(\mu +\phi )}{\eta (\mu +\theta \phi )}{v}^{*}{z}_2^{*}\left(1-\frac{z_2^{*}}{z_2}\right)\hfill \\ \hfill & =-\frac{\varrho {(x-x^{*})}^2}{x}+\frac{(1-\theta )\mu }{\mu +\theta \phi }\kappa x^{*}{v}^{*}\left[4-\frac{x^{*}}{x}-\frac{xv{u}^{*}}{x^{*}{v}^{*}u}-\frac{y^{*}u}{y{u}^{*}}-\frac{y{v}^{*}}{y^{*}v}\right]\hfill \\ \hfill & +\frac{\theta (\mu +\phi )}{\mu +\theta \phi }\kappa x^{*}{v}^{*}\left[3-\frac{x^{*}}{x}-\frac{xv{y}^{*}}{x^{*}{v}^{*}y}-\frac{y{v}^{*}}{y^{*}v}\right]\hfill \\ \hfill & -\frac{\delta {\beta }_1{\psi }_1(\mu +\phi )}{\eta {\lambda }_1(\mu +\theta \phi )}\frac{{(z_1-z_1^{*})}^2}{z_1}+\frac{\delta {\beta }_1(\mu +\phi )}{\eta (\mu +\theta \phi )}{v}^{*}\frac{{(z_1-z_1^{*})}^2}{z_1}\hfill \\ \hfill & -\frac{\delta {\beta }_2{\psi }_2(\mu +\phi )}{\eta {\lambda }_2(\mu +\theta \phi )}\frac{{(z_2-z_2^{*})}^2}{z_2}+\frac{\delta {\beta }_2(\mu +\phi )}{\eta (\mu +\theta \phi )}{v}^{*}\frac{{(z_2-z_2^{*})}^2}{z_2}\hfill \end{array}$$

The equilibrium conditions give $v^{*}-\frac{{\psi }_1}{{\lambda }_1}=-\frac{{\gamma }_1}{{\lambda }_1{z}_1^{*}}$ and $v^{*}-\frac{{\psi }_2}{{\lambda }_2}=-\frac{{\gamma }_2}{{\lambda }_2{z}_2^{*}}$. Hence

$$\begin{array}{cc}\hfill \frac{d{W}_1^L}{dt}& =-\frac{\varrho {(x-x^{*})}^2}{x}-\frac{\delta {\beta }_1(\mu +\phi ){\gamma }_1}{\eta {\lambda }_1(\mu +\theta \phi )}\frac{{(z_1-z_1^{*})}^2}{z_1{z}_1^{*}}-\frac{\delta {\beta }_2(\mu +\phi ){\gamma }_2}{\eta {\lambda }_2(\mu +\theta \phi )}\frac{{(z_2-z_2^{*})}^2}{z_2{z}_2^{*}}\hfill \\ \hfill & +\frac{(1-\theta )\mu }{\mu +\theta \phi }\kappa x^{*}{v}^{*}\left[4-\frac{x^{*}}{x}-\frac{xv{u}^{*}}{x^{*}{v}^{*}u}-\frac{y^{*}u}{y{u}^{*}}-\frac{y{v}^{*}}{y^{*}v}\right]\hfill \\ \hfill & +\frac{\theta (\mu +\phi )}{\mu +\theta \phi }\kappa x^{*}{v}^{*}\left[3-\frac{x^{*}}{x}-\frac{xv{y}^{*}}{x^{*}{v}^{*}y}-\frac{y{v}^{*}}{y^{*}v}\right].\hfill \end{array}$$

(38)

From inequality (19) we obtain $\frac{d{W}_1^L}{dt}\le 0$ for all $x,u,y,v,z_1,z_2>0$. Furthermore, $\frac{d{W}_1^L}{dt}=0$ when $(x,u,y,v,z_1,z_2)=(x^{*},u^{*},y^{*},v^{*},z_1^{*},z_2^{*})$. This gives ${\tilde{\Gamma }}_1^L=\left\{{\mathcal{EQ}}^{*}\right\}$, and according to L-LAS theorem, ${\mathcal{EQ}}^{*}$ is G.A.S in ${\mathring{\Xi }}^L$. □

4. Numerical Simulations

4.1. Numerical Simulations for Model (5)–(9)

In this subsection, we present numerical simulation to confirm the global stability of equilibria for system (5)–(9) using the values given in

Table 1. The values of the parameters of model (5)–(9).

Parameter	Value	Parameter	Value	Parameter	Value
$\varphi$	10	$\rho$	3	${\lambda }_1$	0.01
$\varrho$	$0.01$	${\beta }_1$	0.3	${\lambda }_2$	0.002
$\kappa$	Varied	${\beta }_2$	0.1	${\psi }_1$	0.1
$\delta$	0.5	${\gamma }_1$	5	${\psi }_2$	0.1
$\eta$	10	${\gamma }_2$	3

.

Stability of Equilibria

We simulate system (5)–(9) using the following initials:

IS-1:$(x,y,v,z_1,z_2)\left(0\right)=(500,5,2,60,20)$,

IS-2:$(x,y,v,z_1,z_2)\left(0\right)=(300,10,4,80,30),$

IS-3:$(x,y,v,z_1,z_2)\left(0\right)=(100,20,8,100,50)$.

We note that, due to the difficulty of obtaining real data from viral-infected patients with two types of active antibodies, we therefore take arbitrary initial conditions in solving the system numerically. The global stability results given in Theorems 1 and 2 ensure that the solutions of system (5)–(9) converge to either uninfected equilibrium ${\mathcal{EQ}}^0$ (when $R_0\le 1$) or infected equilibrium ${\mathcal{EQ}}^{*}$ (when $R_0>1$) regardless of the chosen initial values.

Selecting different values of $\kappa$ under the above initial conditions leads to the following cases:

Case 1 (Stability of ${\mathcal{EQ}}^0$): We take $\kappa =0.0005$. For this case, we have $R_0=0.48<1.$ Figure 1 shows that the solutions with initials IS-1, IS-2 and IS-3 reach the uninfected equilibrium ${\mathcal{EQ}}^0=\left(1000,0,0,50,30\right)$. According to the Theorem 1, ${\mathcal{EQ}}^0$ is G.A.S. In this case, the virus will die out and the number of target cells will return to its normal level.

Case 2 (Stability of ${\mathcal{EQ}}^{*}$): By choosing $\kappa =0.005$, we obtain $R_0=4.8>1$. Figure 2 displays that the solutions starting with initials IS-1, IS-2 and IS-3 tend to the infected equilibrium ${\mathcal{EQ}}^{*}=\left(321.94,13.56,4.21,86.39,32.76\right)$. Lemma 2 and Theorem 2 state that ${\mathcal{EQ}}^{*}$ exists and it is G.A.S. This strategy will lead to chronic viral infection.

4.2. Numerical Simulations for Model (21)–(26)

We present a numerical simulation for system (21)–(26). We consider the values of the parameters given in Table 1 and the following parameters: $\theta =0.3,$ $\mu =0.4$ and $\phi =0.1$.

4.2.1. Sensitivity Analysis

When modeling complicated interactions, sensitivity analysis is particularly important in pathology and epidemiology [52]. Our ability to control the spread of disease or crime can be determined by analyzing sensitivities. It is possible to calculate sensitivity indexes in three ways: directly through direct differentiation, using a Latin hypercube sampling method or by linearizing the model and solving the resulting equations [52,53].

The indices can be expressed analytically in this study using direct differentiation. By using partial derivatives, you can calculate the sensitivity index when variables vary based on parameters. In terms of the parameter, the normalized forward sensitivity index of $R_0^L$ is expressed as follows:

$${\Omega }_{\Lambda }=\frac{\Lambda }{R_0^L}\frac{\partial R_0^L}{\partial \Lambda },$$

(39)

where $\Lambda$ is a given parameter. The sensitivity indices for each parameter included in $R_0^L$ are calculated using Equation (39). For instance, the sensitivity index of the parameter value with respect to $\mu$ is computed as:

$${\Omega }_{\mu }=\frac{\mu }{R_0^L}\frac{\partial R_0^L}{\partial \mu }=-\frac{\mu \phi (\theta -1)}{(\mu +\phi )(\theta \phi +\mu )}.$$

Table 2. Sensitivity index of $R_0^L$.

Parameter $\Lambda$	Value of ${\Omega }_{\Lambda }$	Parameter $\Lambda$	Value of ${\Omega }_{\Lambda }$	Parameter $\Lambda$	Value of ${\Omega }_{\Lambda }$
$\varphi$	1	$\delta$	$-1$	${\gamma }_2$	$-0.143$
$\varrho$	$-1$	$\eta$	1	${\lambda }_1$	0
$\kappa$	1	$\rho$	$-0.143$	${\lambda }_2$	0
$\theta$	$0.07$	${\beta }_1$	$-0.7143$	${\psi }_1$	$0.714$
$\mu$	$0.13$	${\beta }_2$	$-0.143$	${\psi }_2$	$0.143$
$\phi$	$-0.13$	${\gamma }_1$	$-0.714$

and Figure 3 display the value of the sensitivity index of $R_0^L$ based on the parameter values in Table 1. Clearly, $\varphi$, $\kappa$, $\theta$, $\mu$, $\eta$, ${\psi }_1$ and ${\psi }_2$ have positive indices. In terms of sensitivity, $\varphi$, $\kappa$ and $\eta$ are the most important parameters and $\theta$ is the least important. In this case, there is a positive relationship between the endemicity of the disease and the increase in the values of these parameters $\varphi$, $\kappa$, $\theta$, $\mu$, $\eta$, ${\psi }_1$ and ${\psi }_2$, while keeping other parameters constant. The remaining indices are negative, i.e., the value of $R_0^L$ decreases as the values of $\varrho$, $\phi$, $\delta$, $\rho$, ${\beta }_1$, ${\beta }_2$, ${\gamma }_1$ and ${\gamma }_2$ increase. The parameters of antibody responsiveness, ${\lambda }_1$ and ${\lambda }_2$, do not affect $R_0^L$.

4.2.2. Stability of Equilibria

We choose three different initial states as follows:

IS-1:$(x,u,y,v,z_1,z_2)\left(0\right)=(500,5,5,2,60,20)$,

IS-2:$(x,u,y,v,z_1,z_2)\left(0\right)=(300,9,10,4,80,30),$

IS-3:$(x,u,y,v,z_1,z_2)\left(0\right)=(100,12,20,8,100,50)$.

Selecting different values of $\kappa$ under the above initial states leads to the following cases:

Case 1 (Stability of ${\mathcal{EQ}}^0$): We consider $\kappa =0.0005$. For this case, we obtain $R_0^L=0.41<1.$ Figure 4 illustrates that the solutions with initials IS-1, IS-2 and IS-3 reach the equilibrium ${\mathcal{EQ}}^0=\left(1000,0,0,0,50,30\right).$ This shows that ${\mathcal{EQ}}^0$ is G.A.S according to Theorem 7. In this situation, the infected cells and virus particles will die out.

Case 2 (Stability of ${\mathcal{EQ}}^{*}$): We select $\kappa =0.005.$ This gives $R_0^L=4.1>1$. In Figure 5, we show that the solutions starting with initials IS-1, IS-2 and IS-3 tend to ${\mathcal{EQ}}^{*}=\left(350,9.1,11.17,3.71,79.54,32.41\right)$ and then it is G.A.S which agrees with Theorem 8.

5. Discussions

In this section, we discuss the effect of including the self-regulating antibody response and latently infected cells on the virus dynamics. To address the influence of including the self-regulating antibody response in the viral infection model. We compare the basic reproduction number of system (1)–(4) where the self-regulating antibody response is not included and that of system (5)–(9). The basic reproduction number of model (1)–(4) can be calculated as:

$${\widehat{R}}_0=\frac{\eta \kappa \varphi }{\delta \varrho \rho }.$$

Clearly,

$$R_0=\frac{\eta \kappa \varphi }{\delta \varrho \rho \left(\frac{{\beta }_1{\gamma }_1}{{\psi }_1\rho }+\frac{{\beta }_2{\gamma }_2}{{\psi }_2\rho }+1\right)}=\frac{{\widehat{R}}_0}{\frac{{\beta }_1{\gamma }_1}{{\psi }_1\rho }+\frac{{\beta }_2{\gamma }_2}{{\psi }_2\rho }+1}<{\widehat{R}}_0.$$

Therefore, the presence of the self-regulating antibody response reduces the basic reproduction number $R_0$ and then makes the system more stabilizable around the uninfected equilibrium ${\mathcal{EQ}}^0$.

To discuss the impact of incorporating the latently infected cells in the model, we include the effect of an antiviral drug therapy of type reverse transcriptase inhibitor (RTI) with drug efficacy $\omega \in \left[0,1\right]$. Systems (5)–(9) and (21)–(26) under the effect of RTI are obtained by replacing the parameter $\kappa$ by $(1-\omega )\kappa$. Consequently, the basic reproduction numbers $R_0$ and $R_0^L$ become

$$\begin{array}{cc}\hfill R_0\left(\omega \right)& =\frac{(1-\omega )\eta \kappa \varphi }{\delta \varrho \rho \left(\frac{{\beta }_1{\gamma }_1}{{\psi }_1\rho }+\frac{{\beta }_2{\gamma }_2}{{\psi }_2\rho }+1\right)}=(1-\omega )R_0,\hfill \\ \hfill R_0^L\left(\omega \right)& =\frac{(1-\omega )\eta \kappa \varphi (\mu +\theta \phi )}{\delta \varrho \rho (\mu +\phi )\left(\frac{{\beta }_1{\gamma }_1}{{\psi }_1\rho }+\frac{{\beta }_2{\gamma }_2}{{\psi }_2\rho }+1\right)}=(1-\omega )R_0^L.\hfill \end{array}$$

Let us calculate the minimum drug efficacies ${\omega }_{min}$ and ${\omega }_{min}^L$ that make

$$\begin{array}{cc}\hfill R_0\left(\omega \right)& \le 1,\mathrm{for}\mathrm{all}{\omega }_{min}\le \omega \le 1,\hfill \\ \hfill R_0^L\left(\omega \right)& \le 1,\mathrm{for}\mathrm{all}{\omega }_{min}^L\le \omega \le 1,\hfill \end{array}$$

where

$${\omega }_{min}=max\left\{1-\frac{1}{R_0},0\right\},{\omega }_{min}^L=max\left\{1-\frac{1}{R_0^L},0\right\}.$$

We have

$$R_0^L=\frac{\eta \kappa \varphi (\mu +\theta \phi )}{\delta \varrho \rho (\mu +\phi )\left(\frac{{\beta }_1{\gamma }_1}{{\psi }_1\rho }+\frac{{\beta }_2{\gamma }_2}{{\psi }_2\rho }+1\right)}=\frac{\mu +\theta \phi }{\mu +\phi }{R}_0<R_0.$$

By comparison, we obtain ${\omega }_{min}^L<{\omega }_{min}$. This means that including the latently infected cells will reduce the antiviral drug dose needed to stabilize the system around the uninfected equilibrium ${\mathcal{EQ}}^0$ and clear the virus from the body. As a result, ignoring the latently infected cells in the viral infection model will yield the design of overflow antiviral drug doses.

6. Conclusions

In this paper, two viral infection models with two types of antibodies were considered. The presence of two types of antibodies can be a result of secondary viral infection. We included the latently infected cells in the second model. We assumed that the antibody responsiveness is given by a combination of the self-regulating antibody response and the predator–prey-like antibody response. We proved that the proposed model solutions are nonnegative and bounded. We showed that the model has two possible equilibrium points, uninfected equilibrium ${\mathcal{EQ}}^0$ and infected equilibrium ${\mathcal{EQ}}^{*}$ depending on the basic reproduction number $R_0$. This number governs the dynamic behavior of the model; if $R_0\le 1$, then the point ${\mathcal{EQ}}^0$ is G.A.S, and if $R_0>1$, then the point ${\mathcal{EQ}}^{*}$ is G.A.S. In order to verify the theoretical results, we conducted some numerical computations. We also studied the sensitivity analysis to show how the values of all the parameters of the suggested model affect $R_0$ under given data. We discussed the impact of including the self-regulating antibody response and latently infected cells in the viral infection model. We noted that the presence of the self-regulating antibody response reduces $R_0$ and makes the system more stabilizable around ${\mathcal{EQ}}^0$. Moreover, we established that neglecting the latently infected cells in the viral infection modeling leads to the design of an overflow of antiviral drug therapy.

Our proposed models can be extended in different directions by incorporating (i) time delays [9,18], (ii) stochastic interactions [54], (iii) diffusion effect [55,56,57], (iv) viral mutations [13] and (v) both antibody and CTL immune responses [16]. These extensions will be left for future works.