
Citation: Ka-man Fong, Shek-yin Au, Ka-lee Lily Chan, Wing-yiu George Ng. Update on management of acute respiratory distress syndrome[J]. AIMS Medical Science, 2018, 5(2): 145-161. doi: 10.3934/medsci.2018.2.145
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During the last decades different dangerous viruses have been recognized which attack the human body and causes many fatal diseases. As an example of these viruses, the human immunodeficiency virus (HIV) which is the causative agent for acquired immunodeficiency syndrome (AIDS). According to global health observatory (GHO, 2018) data of HIV/AIDS published by WHO [1] that says, globally, about 37.9 million HIV-infected people in 2018, 1.7 million newly HIV-infected and 770,000 HIV-related death in the same year. HIV is a retrovirus that infects the susceptible CD$ 4^{+} $T cells which play a central role in immune system defence. During the last decades, mathematical modeling of within-host HIV infection has witnessed a significant development [2]. Nowak and Bangham [3] have introduced the basic HIV infection model which describes the interaction between three compartments, susceptible CD$ 4^{+} $T cells ($ S $), actively HIV-infected cells ($ I $) and free HIV particles ($ V $). Latent viral reservoirs remain one of the major hurdles for eradicating the HIV by current antiviral therapy. Latently HIV-infected cells include HIV virions but do not produce them until they become activated. Mathematical modeling of HIV dynamics with latency can help in predicting the effect of antiviral drug efficacy on HIV progression [4]. Rong and Perelson [5] have incorporated the latently infected cells in the basic HIV model presented in [3] as:
$ {˙S=ρ−αS−η1SV,˙L=(1−β)η1SV−(λ+γ)L,˙I=βη1SV+λL−aI,˙V=bI−εV, $ | (1.1) |
where $ S = S(t), $ $ L = L(t) $, $ I = I(t)\ $and $ V = V(t) $ are the concentrations of susceptible CD$ 4^{+} $T cells, latently HIV-infected cells, actively HIV-infected cells and free HIV particles at time $ t $, respectively. The susceptible CD$ 4^{+} $T cells are produced at specific constant rate $ \rho $. The HIV virions can replicate using virus-to-cell (VTC) transmission. The term $ \eta_{1}SV $ refers to the rate at which new infectious appears by VTC contact between free HIV particles and susceptible CD$ 4^{+} $T cells. Latently HIV-infected cells are transmitted to be active at rate $ \lambda L $. The free HIV particles are generated at rate $ bI $. The natural death rates of the susceptible CD$ 4^{+} $T cells, latently HIV-infected cells, actively HIV-infected cells and free HIV particles are given by $ \alpha S $, $ \gamma L, $ $ aI $ and $ \varepsilon V $, respectively. A fraction $ \beta \in(0, 1) $ of new HIV-infected cells will be active, and the remaining part $ 1-\beta $ will be latent. During the last decades, mathematical modeling and analysis of HIV mono-infection with both latently and actively HIV-infected cells have witnessed a significant development [6,7,8,9,10,11,12].
Model (1.1) assumed that the HIV can only spread by VTC transmission. However, several works have reported that there is another mode of transmission called cell-to-cell (CTC) where the HIV can be transmitted directly from an infected cell to a healthy CD$ 4^{+} $T cell through the formation of virological synapses [13]. Sourisseau et al. [14] have shown that CTC transmission plays an efficient role in the HIV replication. Sigal et al. [15] have demonstrated the importance of CTC transmission in the HIV infection process during the antiviral treatment. Iwami et al. [13] have shown that about 60% of HIV infections are due to CTC transmission. In addition, CTC transmission can increase the HIV fitness by 3.9 times and decrease the production time of HIV particles by 0.9 times [16]. HIV dynamics model with latency and both VTC and CTC transmissions is given by [17,18]:
$ {˙S=ρ−αS−η1SV−η2SI,˙L=(1−β)(η1SV+η2SI)−(λ+γ)L,˙I=β(η1SV+η2SI)+λL−aI,˙V=bI−εV, $ | (1.2) |
where, the term $ \eta_{2}SI $ refers to the rate at which new infectious appears by CTC contact between HIV-infected cells and susceptible CD$ 4^{+} $T cells.
Another example of the dangerous human viruses is called Human T-lymphotropic virus type Ⅰ (HTLV-I) which can lead to two diseases, adult T-cell leukemia (ATL) and HTLV-I-associated myelopathy/tropical spastic paraparesis (HAM/TSP). The discovery of the first human retrovirus HTLV-I is back to 1980, and after 3 years the HIV was determined [19]. HTLV-I is global epidemic that infects about 10-25 million persons [20]. The infection is endemic in the Caribbean, southern Japan, the Middle East, South America, parts of Africa, Melanesia and Papua New Guinea [21]. HTLV-I is a provirus that targets the susceptible CD$ 4^{+} $T cells. HTLV-I can spread to susceptible CD$ 4^{+} $T cells from CTC through the virological synapse. HTLV-infected cells can be divided into two kinds based on the presence of Tax inside the cell or not: (i) Tax$ ^{\bf{-}} $, or latently HTLV-infected cells are resting CD4$ ^{+} $T cells that contain a provirus and do not express Tax, and (ii) Tax$ ^{\bf{+}} $, or actively HTLV-infected cells are activated provirus-carrying CD4$ ^{\bf{+}} $T cells that do express Tax [22]. During the primary infection stage of HTLV-I, the proviral load can reach high level, approximately 30-50% [23]. Unlike in the case of HIV infection, however, only a small percentage of infected individuals develop the disease and 2-5% percent of HTLV-I carriers develop symptoms of ATL and another 0.25-3% develop HAM/TSP [24]. Stilianakis and Seydel [25] have formulated an HTLV-I model to describe the interaction of susceptible CD$ 4^{+} $T cells, latently HTLV-infected cells, Tax-expressing HTLV-infected cells (actively HTLV-infected cells) and leukemia cells (ATL cells) as:
$ {˙S=ρ−αS−η3SY,˙E=η3SY−(ψ+ω)E,˙Y=ψE−(ϑ+δ)Y,˙Z=ϑY+ℓZ(1−ZZmax)−θZ, $ | (1.3) |
where $ S = S(t), $ $ E = E(t), $ $ Y = Y(t) $ and $ Z = Z(t) $ are the concentrations of susceptible CD$ 4^{+} $T cells, latently HTLV-infected cells, Tax-expressing HTLV-infected cells and ATL cells, at time $ t $, respectively. In contrast of HIV, the transmission of HTLV-I can be only from CTC that is the HTLV virions can only survive inside the host CD$ 4^{+} $T cells and cannot be detectable in the plasma. The rate at which new infectious appears by CTC contact between Tax-expressing HTLV-infected cells and susceptible CD$ 4^{+} $T cells is assumed to be $ \eta_{3}SY $. The natural death rate of the latently HTLV-infected cells, Tax-expressing HTLV-infected cells and ATL cells are represented by $ \omega E $, $ \delta Y $ and $ \theta Z $, respectively. The term $ \psi E $ accounts for the rate of latently HTLV-infected cells that become Tax-expressing HTLV-infected cells. $ \vartheta Y $ is the transmission rate at which Tax-expressing HTLV-infected cells convert to ATL cells. The logistic term $ \ell Z\left(1-\frac{Z}{Z_{\max}}\right) $ denotes the proliferation rate of the ATL cells, where $ Z_{\max} $ is the maximal concentration that ATL cells can grow. The parameter $ \ell $ is the maximum proliferation rate constant of ATL cells. Many researchers have been concerned to study mathematical modeling and analysis of HTLV-I mono-infection in several works [26,27,28].
Cytotoxic T lymphocytes (CTLs) are recognized as the significant component of the human immune response against viral infections. CTLs inhibit viral replication and kill the cells which are infected by viruses. In fact, CTLs are necessary and universal to control HIV infection [29]. During the recent years, great efforts have been made to formulate and analyze the within-host HIV mono-infection models under the influence of CTL immune response (see e.g. [2,3]). In [30], latently HIV-infected cells have been included in the HIV dynamics models with CTL immune response. In case of HTLV-I infection, it has been reported in [31] that the CTLs play an effective role in controlling such infection. CTLs can recognize and kill the Tax-expressing HTLV-infected cells, moreover, they can reduce the proviral load. In the literature, several mathematical models have been proposed to describe the dynamics of HTLV-I under the effect of CTL immune response (see e.g. [21,32,33,34,35,36]). In [20,37,38], HTLV-I dynamics models have been presented by incorporating latently HTLV-infected cells and CTL immune response.
Simultaneous infection by HIV and HTLV-I and the etiology of their pathogenic and disease outcomes have become a global health matter over the past 10 years [39]. It is commonly that HIV/HTLV-I co-infection can be endemic in areas where individuals experience high risk attitudes; such as unprotected sexual contact and unsafe injection practices; that cause transmission of contaminated body fluids between individuals. This shed a light on the importance of studying HIV/HTLV-I co-infection [40]. Although CD$ 4^{+} $T cells are the major targets of both HIV and HTLV-I, however, these viruses present a different biological behavior that causes diverse impacts on host immunity and ultimately lead to numerous clinical diseases [41]. It has been reported that the HTLV-I co-infection rate among HIV infected patients as increase as 100 to 500 times in comparison with the general population [42]. In seroepidemiologic studies, it has been recorded that HIV-infected patients are more exposure to be co-infected with HTLV-I, and vice versa compared to the general population [43]. HIV/HTLV-I co-infection is usually found in individuals of specific ethnic or who belonged to geographic origins where these viruses are simultaneously endemic [44]. As an example, the co-infection rates in individuals living in Bahia have reached 16% of HIV-infected patients [45]. The prevalence of dual infection with HIV and HTLV-I has become more widely in several geographical regions throughout the world such as South America, Europe, the Caribbean, Bahia (Brazil), Mozambique (Africa), and Japan [39,43,45,46,47]. HIV and HTLV-I dual infection appears to have an overlap on the course of associated clinical outcomes with both viruses [43]. Several reports have concluded that HIV/HTLV-I co-infected patients were found to have an increase of CD$ 4^{+} $T cells count in comparison with HIV mono-infected patients, although there is no evident to result in a better immune response [41,48]. Indeed, simultaneous infected patients by both viruses with CD$ 4^{+} $T counts greater than 200 cells/mm$ ^{3} $ are more exposure to have other opportunistic infections as compared with HIV mono-infected patients who have similar CD$ 4^{+} $T counts [48]. Studies have reported that higher mortality and shortened survival rates were accompany with co-infected individuals more than mono-infected individuals [46]. Considering the natural history of HIV, many researchers have noted that co-infection with HIV and HTLV-I can accelerate the clinical progression to AIDS. On the other hand, HIV can adjust HTLV-I expression in co-infected individuals which leads them to a higher risk of developing HTLV-I related diseases such as ATL and TSP/HAM [42,43,46].
Great efforts have been made to develop and analyze mathematical models of HIV and HTLV-I mono-infections, however, modeling of HIV/HTLV-I co-infection has not been studied. In fact, such co-infection modeling and its analysis will be needed to help clinicians on estimating the appropriate time to initiate treatment in co-infected patients. Therefore, the aim of the present paper is to formulate a new HIV/HTLV-I co-infection model. We show that the model is well-posed by establishing that the solutions of the model are nonnegative and bounded. We derive a set of threshold parameters which govern the existence and stability of the equilibria of the model. Global stability of all equilibria is proven by constructing suitable Lyapunov functions and utilizing Lyapunov-LaSalle asymptotic stability theorem. We conduct some numerical simulations to illustrate the theoretical results.
The results of this work, such as co-infection model and its analysis will help clinicians estimate the appropriate time for patients with co-infection to begin treatment. On the other hand, this study, from a certain point of view, illustrate the complexity of this co-infection model and the model is helpful to clinic treatment. It is worth mentioning, if we look at research perspectives, that appropriate developments of the model presented in this paper can be focused on the within host modeling of the competition between COVID19 virus and the immune system by a complex dynamics described in [49]. This dynamics which occurs, in human lungs, once the virus, after contagion, has gone over the biological barriers which protect each individuals, see [47].
We set up an ordinary differential equation model that describes the change of concentrations of eight compartments with respect to time $ t $; susceptible CD$ 4^{+} $T cells $ S(t) $, latently HIV-infected cells $ L(t) $, actively HIV-infected cells $ I(t) $, latently HTLV-infected cells $ E(t) $, Tax-expressing HTLV-infected cells $ Y(t) $, free HIV particles $ V(t) $, HIV-specific CTLs $ C^{I}(t) $ and HTLV-specific CTLs $ C^{Y}(t) $. The dynamics of HIV/HTLV-I co-infection is schematically shown in the transfer diagram given in Figure 1. Our proposed model is given by the following form:
$ {˙S=ρ−αS−η1SV−η2SI−η3SY,˙L=(1−β)(η1SV+η2SI)−(λ+γ)L,˙I=β(η1SV+η2SI)+λL−aI−μ1CII,˙E=φη3SY−(ψ+ω)E,˙Y=ψE−δY−μ2CYY,˙V=bI−εV,˙CI=σ1CII−π1CI,˙CY=σ2CYY−π2CY, $ | (2.1) |
where $ (S, L, I, E, Y, V, C^{I}, C^{Y}) = (S(t), L(t), I(t), E(t), Y(t), V(t), C^{I} (t), C^{Y}(t)) $. The term $ \mu_{1}C^{I}I $ is the killing rate of actively HIV-infected cells due to their specific immunity. The term $ \mu_{2}C^{Y}Y $ is the killing rate of Tax-expressing HTLV-infected cells due to their specific immunity. The proliferation and death rates for both effective HIV-specific CTLs and HTLV-specific CTLs are given by $ \sigma_{1}C^{I}I $, $ \sigma_{2} C^{Y}Y $, $ \pi_{1}C^{I} $ and $ \pi_{2}C^{Y} $, respectively. All remaining parameters have the same biological meaning as explained in the previous section. All parameters and their definitions are summarized in Table 1.
Parameter | Description |
$ \rho $ | $ Recruitment rate for the susceptible CD4+T cells $ |
$ \alpha $ | $ Natural mortality rate constant for the susceptible CD4+T cells $ |
$ \eta_{1} $ | $ Virus-cell incidence rate constant between free HIV particles and susceptible CD4+T cells $ |
$ \eta_{2} $ | $ Cell-cell incidence rate constant between HIV-infected cells and susceptible CD4+T cells $ |
$ \eta_{3} $ | $ Cell-cell incidence rate constant between Tax-expressing HTLV-infected cells andsusceptible CD4+T cells $ |
$ \beta \in \left(0, 1\right) $ | $ Fraction coefficient accounts for the probability of new HIV-infected cells could be active, and the remaining part 1−β will be latent $ |
$ \gamma $ | $ Death rate constant of latently HIV-infected cells $ |
$ a $ | $ Death rate constant of actively HIV-infected cells $ |
$ \mu_{1} $ | $ Killing rate constant of actively HIV-infected cells due to HIV-specific CTLs $ |
$ \mu_{2} $ | $ Killing rate constant of Tax-expressing HTLV-infected cells due to HTLV-specific CTLs $ |
$ \varphi \in \left(0, 1\right) $ | $ Probability of new HTLV infections could be enter a latent period $ |
$ \lambda $ | $ Transmission rate constant of latently HIV-infected cells that become actively HIV-infected cells $ |
$ \psi $ | $ Transmission rate constant of latently HTLV-infected cells that become Tax-expressing HTLV-infected cells $ |
$ \omega $ | $ Death rate constant of latently HTLV-infected cells $ |
$ \delta $ | $ Death rate constant of Tax-expressing HTLV-infected cells $ |
$ b $ | $ Generation rate constant of new HIV particles $ |
$ \varepsilon $ | $ Death rate constant of free HIV particles $ |
$ \sigma_{1} $ | $ Proliferation rate constant of HIV-specific CTLs $ |
$ \sigma_{2} $ | $ Proliferation rate constant of HTLV-specific CTLs $ |
$ \pi_{1} $ | $ Decay rate constant of HIV-specific CTLs $ |
$ \pi_{2} $ | $ Decay rate constant of HTLV-specific CTLs $ |
Let $ \Omega_{j} > 0 $, $ j = 1, ..., 5 $ and define
$ Θ={(S,L,I,E,Y,V,CI,CY)∈R8≥0:0≤S(t),L(t),I(t)≤Ω1, 0≤E(t),Y(t)≤Ω2, 0≤V(t)≤Ω3, 0≤CI(t)≤Ω4, 0≤CY(t)≤Ω5}. $ |
Proposition 1. The compact set $ \Theta $ is positively invariant for system (2.1).
Proof. We have
$ ˙S∣S=0=ρ>0, ˙L∣L=0=(1−β)(η1SV+η2SI)≥0 for all S,V,I≥0,˙I∣I=0=βη1SV+λL≥0 for all S,V,L≥0,˙E∣E=0=φη3SY for all S,Y≥0, ˙Y∣Y=0=ψE≥0 for all E≥0,˙V∣V=0=bI≥0 for all I≥0, ˙CI∣CI=0=0, ˙CY∣CY=0=0. $ |
This ensures that, $ (S(t), L(t), I(t), E(t), Y(t), V(t), C^{I}(t), C^{Y} (t))\in \mathbb{R}_{\geq0}^{8} $ for all $ t\geq0 $ when $ (S(0), L(0), I(0), E(0), Y(0), V(0), C^{I}(0), C^{Y}(0))\in \mathbb{R}_{\geq0}^{8} $. To show the boundedness of all state variables, we let
$ \Psi = S+L+I+\frac{1}{\varphi}\left( E+Y\right) +\frac{a}{2b}V+\frac{\mu_{1} }{\sigma_{1}}C^{I}+\frac{\mu_{2}}{\varphi \sigma_{2}}C^{Y}. $ |
Then
$ ˙Ψ=ρ−αS−γL−a2I−ωφE−δφY−aε2bV−μ1π1σ1CI−μ2π2φσ2CY≤ρ−ϕ[S+L)+I+1φ(E+Y)+a2bV+μ1σ1CI+μ2φσ2CY]=ρ−ϕΨ, $ |
where $ \phi = \min \{ \alpha, \gamma, \frac{a}{2}, \omega, \delta, \varepsilon, \pi _{1}, \pi_{2}\} $. Hence, $ 0\leq \Psi(t)\leq \Omega_{1} $ if $ \Psi(0)\leq \Omega _{1} $ for $ t\geq0, $ where $ \Omega_{1} = \frac{\rho}{\phi}. $ Since $ S $, $ L, $ $ I, $ $ E, $ $ Y, $ $ V, $ $ C^{I} $, and $ C^{Y} $ are all nonnegative then $ 0\leq S(t), L(t), I(t)\leq \Omega_{1} $, $ 0\leq E(t), Y(t)\leq \Omega_{2} $, $ 0\leq V(t)\leq \Omega_{3} $, $ 0\leq C^{I}(t)\leq \Omega_{4} $, $ 0\leq C^{Y}(t)\leq \Omega_{5} $ if $ S(0)+L(0)+I(0)+\frac{1}{\varphi}\left(E(0)+Y(0)\right) +\frac{a}{2b}V(0)+\frac{\mu_{1}}{\sigma_{1}}C^{I}(0)+\frac{\mu_{2}} {\varphi \sigma_{2}}C^{Y}(0)\leq \Omega_{1} $, where $ \Omega_{2} = \varphi \Omega_{1} $, $ \Omega_{3} = \dfrac{2b\Omega_{1}}{a} $, $ \Omega_{4} = \dfrac {\sigma_{1}\Omega_{1}}{\mu_{1}} $ and $ \Omega_{5} = \dfrac{\varphi \sigma _{2}\Omega_{1}}{\mu_{2}} $.
In this section, we derive eight threshold parameters which guarantee the existence of the equilibria of the model. Let $ (S, L, I, E, Y, V, C^{I}, C^{Y}) $ be any equilibrium of system (2.1) satisfying the following equations:
$ 0=ρ−αS−η1SV−η2SI−η3SY, $ | (4.1) |
$ 0=(1−β)(η1SV+η2SI)−(λ+γ)L, $ | (4.2) |
$ 0=β(η1SV+η2SI)+λL−aI−μ1CII, $ | (4.3) |
$ 0=φη3SY−(ψ+ω)E, $ | (4.4) |
$ 0=ψE−δY−μ2CYY, $ | (4.5) |
$ 0=bI−εV, $ | (4.6) |
$ 0=(σ1I−π1)CI, $ | (4.7) |
$ 0=(σ2Y−π2)CY. $ | (4.8) |
The straightforward calculation finds that system (2.1) admits eight equilibria.
(ⅰ) Infection-free equilibrium, $Đ _{0} = (S_{0}, 0, 0, 0, 0, 0, 0, 0) $, where $ S_{0} = \rho/\alpha $. This case describes the situation of healthy state where both HIV and HTLV are absent.
(ⅱ) Chronic HIV mono-infection equilibrium with inactive immune response, $Đ _{1} = (S_{1}, L_{1}, I_{1}, 0, 0, V_{1}, 0, 0), $ where
$ S1=aε(γ+λ)(βγ+λ)(η1b+η2ε), L1=aεα(1−β)(βγ+λ)(η1b+η2ε)[S0(βγ+λ)(η1b+η2ε)aε(γ+λ)−1],I1=εαη1b+η2ε[S0(βγ+λ)(η1b+η2ε)aε(γ+λ)−1], V1=αbη1b+η2ε[S0(βγ+λ)(η1b+η2ε)aε(γ+λ)−1]. $ |
Therefore, $Đ _{1} $ exists when
$ \frac{S_{0}\left( \beta \gamma+\lambda \right) \left( \eta_{1}b+\eta _{2}\varepsilon \right) }{a\varepsilon \left( \gamma+\lambda \right) } \gt 1. $ |
At the equilibrium $Đ _{1} $ the chronic HIV mono-infection persists while the immune response is unstimulated. The basic HIV mono-infection reproductive ratio for system (2.1) is defined as:
$ \Re_{1} = \frac{S_{0}\left( \beta \gamma+\lambda \right) \left( \eta_{1} b+\eta_{2}\varepsilon \right) }{a\varepsilon \left( \gamma+\lambda \right) } = \Re_{11}+\Re_{12}, $ |
where
$ \Re_{11} = \frac{S_{0}\eta_{1}b\left( \beta \gamma+\lambda \right) }{a\varepsilon \left( \gamma+\lambda \right) }, \ \ \ \ \ \ \Re _{12} = \frac{S_{0}\eta_{2}\left( \beta \gamma+\lambda \right) }{a\left( \gamma+\lambda \right) }. $ |
The parameter $ \Re_{1} $ determines whether or not a chronic HIV infection can be established. In fact, $ \Re_{11} $ measures the average number of secondary HIV infected generation caused by an existing free HIV particles, while $ \Re_{12} $ measures the average number of secondary HIV infected generation caused by an HIV-infected cell. Therefore, $ \Re_{11} $ and $ \Re_{12} $ are the basic HIV mono-infection reproductive ratio corresponding to VTC and CTC infections, respectively. In terms of $ \Re_{1} $, we can write
$ S_{1} = \dfrac{S_{0}}{\Re_{1}},\text{ }L_{1} = \frac{a\varepsilon \alpha \left( 1-\beta \right) }{\left( \beta \gamma+\lambda \right) \left( \eta_{1} b+\eta_{2}\varepsilon \right) }\left( \Re_{1}-1\right) ,\text{ }I_{1} = \frac{\varepsilon \alpha}{\eta_{1}b+\eta_{2}\varepsilon}\left( \Re _{1}-1\right) ,\text{ }V_{1} = \frac{\alpha b}{\eta_{1}b+\eta_{2}\varepsilon }\left( \Re_{1}-1\right) . $ |
(ⅲ) Chronic HTLV mono-infection equilibrium with inactive immune response, $Đ _{2} = (S_{2}, 0, 0, E_{2}, Y_{2}, 0, 0, 0) $, where
$ S_{2} = \dfrac{\delta \left( \psi+\omega \right) }{\varphi \eta_{3}\psi },\ \ \ E_{2} = \frac{\alpha \delta}{\eta_{3}\psi}\left[ \frac {\varphi \eta_{3}\psi S_{0}}{\delta \left( \psi+\omega \right) }-1\right] ,\text{}Y_{2} = \frac{\alpha}{\eta_{3}}\left[ \frac{\varphi \eta_{3}\psi S_{0}}{\delta \left( \psi+\omega \right) }-1\right] . $ |
Therefore, $Đ _{2} $ exists when
$ \frac{\varphi \eta_{3}\psi S_{0}}{\delta \left( \psi+\omega \right) } \gt 1. $ |
At the equilibrium $Đ _{2} $ the chronic HTLV mono-infection persists while the immune response is unstimulated. The basic HTLV mono-infection reproductive ratio for system (2.1) is defined as:
$ \Re_{2} = \frac{\varphi \eta_{3}\psi S_{0}}{\delta \left( \psi+\omega \right) }. $ |
The parameter $ \Re_{2} $ decides whether or not a chronic HTLV infection can be established. In terms of $ \Re_{2} $, we can write
$ S_{2} = \dfrac{S_{0}}{\Re_{2}}, \ \ \ E_{2} = \frac{\alpha \delta}{\eta _{3}\psi}\left( \Re_{2}-1\right) , \ \ Y_{2} = \frac{\alpha}{\eta_{3} }\left( \Re_{2}-1\right) . $ |
Remark 1. We note that both $ \Re_{1} $ and $ \Re_{2} $ does not depend of parameters $ \sigma_{i} $, $ \pi_{i} $ and $ \mu_{i} $, $ i = 1, 2 $. Therefore, without treatment CTLs will not able to clear HIV or HTLV-I from the body.
(ⅳ) Chronic HIV mono-infection equilibrium with only active HIV-specific CTL, $Đ _{3} = (S_{3}, L_{3}, I_{3}, 0, 0, V_{3}, C_{3}^{I}, 0) $, where
$ S3=εσ1ρπ1(η1b+η2ε)+αεσ1, L3=ρπ1(1−β)(η1b+η2ε)(γ+λ)[π1(η1b+η2ε)+αεσ1], I3=π1σ1,V3=bεI3=bπ1εσ1, CI3=aμ1[σ1ρ(βγ+λ)(η1b+η2ε)a(γ+λ){π1(η1b+η2ε)+αεσ1}−1]. $ |
We note that $Đ _{3} $ exists when $ \dfrac{\sigma_{1}\rho \left(\beta \gamma+\lambda \right) \left(\eta_{1}b+\eta_{2}\varepsilon \right) }{a\left(\gamma+\lambda \right) \left[ \pi_{1}\left(\eta_{1}b+\eta _{2}\varepsilon \right) +\alpha \varepsilon \sigma_{1}\right] } > 1 $. The HIV-specific CTL-mediated immunity reproductive ratio in case of HIV mono-infection is stated as:
$ \Re_{3} = \dfrac{\sigma_{1}\rho \left( \beta \gamma+\lambda \right) \left( \eta_{1}b+\eta_{2}\varepsilon \right) }{a\left( \gamma+\lambda \right) \left[ \pi_{1}\left( \eta_{1}b+\eta_{2}\varepsilon \right) +\alpha \varepsilon \sigma_{1}\right] }. $ |
Thus, $ C_{3}^{I} = \dfrac{a}{\mu_{1}}(\Re_{3}-1) $. The parameter $ \Re_{3} $ determines whether or not the HIV-specific CTL-mediated immune response is stimulated in the absent of HTLV infection.
(ⅴ) Chronic HTLV mono-infection equilibrium with only active HTLV-specific CTL, $Đ _{4} = (S_{4}, 0, 0, E_{4}, Y_{4}, 0, 0, C_{4}^{Y}) $, where
$ S4=σ2ρπ2η3+ασ2, Y4=π2σ2, E4=π2η3ρφ(ψ+ω)(π2η3+ασ2),CY4=δμ2[σ2ρφη3ψδ(ψ+ω)(π2η3+ασ2)−1]. $ |
We note that $Đ _{4} $ exists when $ \dfrac{\sigma_{2}\rho \varphi \eta_{3}\psi }{\delta \left(\psi+\omega \right) \left(\pi_{2}\eta_{3}+\alpha \sigma _{2}\right) } > 1 $. The HTLV-specific CTL-mediated immunity reproductive ratio in case of HTLV mono-infection is stated as:
$ \Re_{4} = \dfrac{\sigma_{2}\rho \varphi \eta_{3}\psi}{\delta \left( \psi +\omega \right) \left( \pi_{2}\eta_{3}+\alpha \sigma_{2}\right) }. $ |
Thus, $ C_{4}^{Y} = \dfrac{\delta}{\mu_{2}}(\Re_{4}-1) $. The parameter $ \Re_{4} $ determines whether or not the HTLV-specific CTL-mediated immune response is stimulated in the absent of HIV infection.
(ⅵ) Chronic HIV/HTLV co-infection equilibrium with only active HIV-specific CTL, $Đ _{5} = (S_{5}, L_{5}, I_{5}, E_{5}, Y_{5}, V_{5}, C_{5}^{I}, 0) $, where
$ S5=δ(ψ+ω)φη3ψ=S2, I5=π1σ1=I3,V5=bπ1εσ1=V3, L5=δπ1(1−β)(ψ+ω)(η1b+η2ε)εη3σ1φψ(γ+λ),E5=δ[π1(η1b+η2ε)+αεσ1]εη3σ1ψ[ρφεη3σ1ψδ(ψ+ω){π1(η1b+η2ε)+αεσ1}−1],Y5=π1(η1b+η2ε)+αεσ1εη3σ1[ρφεη3σ1ψδ(ψ+ω){π1(η1b+η2ε)+αεσ1}−1],CI5=aμ1[δ(η1b+η2ε)(βγ+λ)(ψ+ω)aεφη3ψ(γ+λ)−1]=aμ1(ℜ1/ℜ2−1). $ |
We note that $Đ _{5} $ exists when $ \Re_{1}/\Re_{2} > 1 $ and $ \frac{\rho \varphi \varepsilon \eta_{3}\sigma_{1}\psi}{\delta \left(\psi+\omega \right) \left[ \pi_{1}\left(\eta_{1}b+\eta_{2}\varepsilon \right) +\alpha \varepsilon \sigma_{1}\right] } > 1 $. The HTLV infection reproductive ratio in the presence of HIV infection is stated as:
$ \Re_{5} = \frac{\rho \varphi \varepsilon \eta_{3}\sigma_{1}\psi}{\delta \left( \psi+\omega \right) \left[ \pi_{1}\left( \eta_{1}b+\eta_{2}\varepsilon \right) +\alpha \varepsilon \sigma_{1}\right] }. $ |
Thus, $ E_{5} = \dfrac{\delta \left[ \pi_{1}\left(\eta_{1}b+\eta_{2} \varepsilon \right) +\alpha \varepsilon \sigma_{1}\right] }{\varepsilon \eta _{3}\sigma_{1}\psi}\left(\Re_{5}-1\right), $ $ Y_{5} = \dfrac{\pi_{1}\left(\eta_{1}b+\eta_{2}\varepsilon \right) +\alpha \varepsilon \sigma_{1} }{\varepsilon \eta_{3}\sigma_{1}}\left(\Re_{5}-1\right) $. The parameter $ \Re_{5} $ determines whether or not HIV-infected patients could be co-infected with HTLV.
(ⅶ) Chronic HIV/HTLV co-infection equilibrium with only active HTLV-specific CTL, $Đ _{6} = (S_{6}, L_{6}, I_{6}, E_{6}, Y_{6}, V_{6}, 0, C_{6}^{Y}) $, where
$ S6=aε(γ+λ)(βγ+λ)(η1b+η2ε)=S1, L6=aε(1−β)(π2η3+ασ2)σ2(βγ+λ)(η1b+η2ε)[ρσ2(βγ+λ)(η1b+η2ε)aε(γ+λ)(π2η3+ασ2)−1],I6=ε(π2η3+ασ2)σ2(η1b+η2ε)[ρσ2(βγ+λ)(η1b+η2ε)aε(γ+λ)(π2η3+ασ2)−1], E6=aεφπ2η3(γ+λ)σ2(βγ+λ)(ψ+ω)(η1b+η2ε),Y6=π2σ2=Y4, V6=b(π2η3+ασ2)σ2(η1b+η2ε)[ρσ2(βγ+λ)(η1b+η2ε)aε(γ+λ)(π2η3+ασ2)−1],CY6=δμ2[aεφη3ψ(γ+λ)δ(βγ+λ)(ψ+ω)(η1b+η2ε)−1]=δμ2(ℜ2/ℜ1−1). $ |
We note that $Đ _{6} $ exists when $ \Re_{2}/\Re_{1} > 1 $ and $ \frac{\rho \sigma_{2}\left(\beta \gamma+\lambda \right) \left(\eta_{1}b+\eta _{2}\varepsilon \right) }{a\varepsilon \left(\gamma+\lambda \right) \left(\pi_{2}\eta_{3}+\alpha \sigma_{2}\right) } > 1 $. The HIV infection reproductive ratio in the presence of HTLV infection is stated as:
$ \Re_{6} = \frac{\rho \sigma_{2}\left( \beta \gamma+\lambda \right) \left( \eta_{1}b+\eta_{2}\varepsilon \right) }{a\varepsilon \left( \gamma +\lambda \right) \left( \pi_{2}\eta_{3}+\alpha \sigma_{2}\right) }. $ |
Thus, $ L_{6} = \dfrac{a\varepsilon \left(1-\beta \right) \left(\pi_{2} \eta_{3}+\alpha \sigma_{2}\right) }{\sigma_{2}\left(\beta \gamma +\lambda \right) \left(\eta_{1}b+\eta_{2}\varepsilon \right) }\left(\Re_{6}-1\right), $ $ I_{6} = \dfrac{\varepsilon \left(\pi_{2}\eta_{3} +\alpha \sigma_{2}\right) }{\sigma_{2}\left(\eta_{1}b+\eta_{2} \varepsilon \right) }\left(\Re_{6}-1\right) $, $ V_{6} = \dfrac{b\left(\pi_{2}\eta_{3}+\alpha \sigma_{2}\right) }{\sigma_{2}\left(\eta_{1} b+\eta_{2}\varepsilon \right) }\left(\Re_{6}-1\right) $. The parameter $ \Re_{6} $ determines whether or not HTLV-infected patients could be co-infected with HIV.
(ⅷ) Chronic HIV/HTLV co-infection equilibrium with active HIV-specific CTL and HTLV-specific CTL, $Đ _{7} = (S_{7}, L_{7}, I_{7}, E_{7}, Y_{7}, V_{7}, C_{7}^{I}, C_{7}^{Y}) $, where
$ S7=εσ1σ2ρπ1σ2(η1b+η2ε)+π2η3εσ1+αεσ1σ2, L7=π1σ2ρ(1−β)(η1b+η2ε)(γ+λ)[π1σ2(η1b+η2ε)+π2η3εσ1+αεσ1σ2],E7=π2η3εσ1ρφ(ψ+ω)[π1σ2(η1b+η2ε)+π2η3εσ1+αεσ1σ2], I7=π1σ1=I3=I5, Y7=π2σ2=Y4=Y6,V7=bπ1εσ1=V3=V5, CI7=aμ1[σ1σ2ρ(βγ+λ)(η1b+η2ε)a(γ+λ){π1σ2(η1b+η2ε)+π2η3εσ1+αεσ1σ2}−1],CY7=δμ2[φη3εσ1σ2ρψδ(ψ+ω){π1σ2(η1b+η2ε)+π2η3εσ1+αεσ1σ2}−1]. $ |
It is obvious that $Đ _{7} $ exists when $ \dfrac{\sigma_{1}\sigma_{2} \rho \left(\beta \gamma+\lambda \right) \left(\eta_{1}b+\eta_{2} \varepsilon \right) }{a\left(\gamma+\lambda \right) \left[ \pi_{1} \sigma_{2}\left(\eta_{1}b+\eta_{2}\varepsilon \right) +\pi_{2}\eta _{3}\varepsilon \sigma_{1}+\alpha \varepsilon \sigma_{1}\sigma_{2}\right] } > 1 $ and $ \dfrac{\varphi \eta_{3}\varepsilon \sigma_{1}\sigma_{2}\rho \psi} {\delta \left(\psi+\omega \right) \left[ \pi_{1}\sigma_{2}\left(\eta _{1}b+\eta_{2}\varepsilon \right) +\pi_{2}\eta_{3}\varepsilon \sigma_{1} +\alpha \varepsilon \sigma_{1}\sigma_{2}\right] } > 1 $. Now we define
$ ℜ7=σ1σ2ρ(βγ+λ)(η1b+η2ε)a(γ+λ)[π1σ2(η1b+η2ε)+π2η3εσ1+αεσ1σ2],ℜ8=φη3εσ1σ2ρψδ(ψ+ω)[π1σ2(η1b+η2ε)+π2η3εσ1+αεσ1σ2]. $ |
Clearly, $Đ _{7} $ exists when $ \Re_{7} > 1 $ and $ \Re_{8} > 1 $ and we can write $ C_{7}^{I} = \dfrac{a}{\mu_{1}}\left(\Re_{7}-1\right) $ and $ C_{7}^{Y} = \dfrac{\delta}{\mu_{2}}\left(\Re_{8}-1\right). $ The parameter $ \Re_{7} $ refers to the competed HIV-specific CTL-mediated immunity reproductive ratio in case of HIV/HTLV co-infection. On the other hand, the parameter $ \Re_{8} $ refers to the competed HTLV-specific CTL-mediated immunity reproductive ratio case of HIV/HTLV co-infection.
The eight threshold parameters are given as follows:
$ ℜ1=S0(βγ+λ)(η1b+η2ε)aε(γ+λ), ℜ2=φη3ψS0δ(ψ+ω),ℜ3=σ1ρ(βγ+λ)(η1b+η2ε)a(γ+λ)[π1(η1b+η2ε)+αεσ1], ℜ4=σ2ρφη3ψδ(ψ+ω)(π2η3+ασ2),ℜ5=ρφεη3σ1ψδ(ψ+ω)[π1(η1b+η2ε)+αεσ1], ℜ6=ρσ2(βγ+λ)(η1b+η2ε)aε(γ+λ)(π2η3+ασ2),ℜ7=σ1σ2ρ(βγ+λ)(η1b+η2ε)a(γ+λ)[π1σ2(η1b+η2ε)+π2η3εσ1+αεσ1σ2],ℜ8=φη3εσ1σ2ρψδ(ψ+ω)[π1σ2(η1b+η2ε)+π2η3εσ1+αεσ1σ2]. $ |
According to the above discussion, we sum up the existence conditions for all equilibria in Table 2.
Equilibrium point | $ Definition $ | $ \text{Existence conditions} $ |
$Đ _{0}=(S_{0}, 0, 0, 0, 0, 0, 0, 0) $ | $ Infection-free equilibrium $ | None |
$Đ _{1}=(S_{1}, L_{1}, I_{1}, 0, 0, V_{1}, 0, 0) $ | $ Chronic HIV mono-infection equilibriumwith inactive immune response $ | $ \Re_{1} > 1 $ |
$Đ _{2}=(S_{2}, 0, 0, E_{2}, Y_{2}, 0, 0, 0) $ | $ Chronic HTLV mono-infection equilibriumwith inactive immune response $ | $ \Re_{2} > 1 $ |
$Đ _{3}=(S_{3}, L_{3}, I_{3}, 0, 0, V_{3}, C_{3}^{I}, 0) $ | $ Chronic HIV mono-infection equilibriumwith only active HIV-specific CTL $ | $ \Re_{3} > 1 $ |
$Đ _{4}=(S_{4}, 0, 0, E_{4}, Y_{4}, 0, 0, C_{4}^{Y}) $ | $ Chronic HTLV mono-infection equilibriumwith only active HTLV-specific CTL $ | $ \Re_{4} > 1 $ |
$Đ _{5}=(S_{5}, L_{5}, I_{5}, E_{5}, Y_{5}, V_{5}, C_{5}^{I}, 0) $ | $ Chronic HIV/HTLV co-infection equilibriumwith only active HIV-specific CTL $ | $ \Re_{5} > 1 $ and $ \Re_{1}/\Re_{2} > 1 $ |
$Đ _{6}=(S_{6}, L_{6}, I_{6}, E_{6}, Y_{6}, V_{6}, 0, C_{6}^{Y}) $ | $ Chronic HIV/HTLV co-infection equilibriumwith only active HTLV-specific CTL $ | $ \Re_{6} > 1 $ and $ \Re_{2}/\Re_{1} > 1 $ |
$Đ _{7}=(S_{7}, L_{7}, I_{7}, E_{7}, Y_{7}, V_{7}, C_{7}^{I}, C_{7}^{Y}) $ | $ Chronic HIV/HTLV co-infectionequilibrium with active HIV-specificCTL and HTLV-specific CTL $ | $ \Re_{7} > 1 $ and $ \Re_{8} > 1 $ |
In this section we prove the global asymptotic stability of all equilibria by constructing Lyapunov functional following the method presented in [50]. We will use the arithmetic-geometric mean inequality
$ \frac{1}{n}\sum \limits_{i = 1}^{n}\chi_{i}\geq \sqrt[n]{\prod \limits_{i = 1} ^{n}\chi_{i}}, \ \ \ \chi_{i}\geq0, \ i = 1,2,... $ |
which yields
$ SjS+SILjSjIjL+LIjLjI≥3, j=1,3,5,6,7, $ | (5.1) |
$ SjS+SVIjSjVjI+IVjIjV≥3, j=1,3,5,6,7, $ | (5.2) |
$ SjS+SVLjSjVjL+LIjLjI+IVjIjV≥4, j=1,3,5,6,7, $ | (5.3) |
$ SjS+SYEjSjYjE+EYjEjY≥3, j=2,4,5,6,7. $ | (5.4) |
Theorem 1. If $ \Re_{1}\leq1 $ and $ \Re_{2}\leq1 $, then $Đ _{0} $ is globally asymptotically stable (G.A.S).
Proof. We construct a Lyapunov function $ \Phi_{0}(S, L, I, E, Y, V, C^{I}, C^{Y}) $ as:
$ Φ0=S0ϝ(SS0)+λβγ+λL+γ+λβγ+λI+1φE+ψ+ωφψY+η1S0εV+μ1(γ+λ)σ1(βγ+λ)CI+μ2(ψ+ω)φψσ2CY, $ |
where,
$ \digamma(\upsilon) = \upsilon-1-\ln \upsilon. $ |
Clearly, $ \Phi_{0}(S, L, I, E, Y, V, C^{I}, C^{Y}) > 0 $ for all $ S, L, I, E, Y, V, C^{I}, C^{Y} > 0 $, and $ \Phi_{0}(S_{0}, 0, 0, 0, 0, 0, 0, 0) = 0 $. Calculating $ \frac{d\Phi _{0}}{dt} $ along the solutions of system (2.1) as:
$ dΦ0dt=(1−S0S)[ρ−αS−η1SV−η2SI−η3SY]+λβγ+λ[(1−β)(η1SV+η2SI)−(λ+γ)L]+γ+λβγ+λ[β(η1SV+η2SI)+λL−aI−μ1CII]+1φ[φη3SY−(ψ+ω)E]+ψ+ωφψ[ψE−δY−μ2CYY]+η1S0ε(bI−εV)+μ1(γ+λ)σ1(βγ+λ)[σ1CII−π1CI]+μ2(ψ+ω)φψσ2[σ2CYY−π2CY]=(1−S0S)(ρ−αS)+η2S0I+η3S0Y−a(γ+λ)βγ+λI−δ(ψ+ω)φψY+η1bS0εI−μ1π1(γ+λ)σ1(βγ+λ)CI−μ2π2(ψ+ω)φψσ2CY. $ |
Using $ S_{0} = \rho/\alpha $, we obtain
$ dΦ0dt=−α(S−S0)2S+a(γ+λ)βγ+λ(ℜ1−1)I+δ(ψ+ω)φψ(ℜ2−1)Y−μ1π1(γ+λ)σ1(βγ+λ)CI−μ2π2(ψ+ω)φψσ2CY. $ |
Therefore, $ \frac{d\Phi_{0}}{dt}\leq0 $ for all $ S, I, Y, C^{I}, C^{Y} > 0 $ and $ \frac{d\Phi_{0}}{dt} = 0 $ when $ S = S_{0} $ and $ I = Y = C^{I} = C^{Y} = 0. $ Define $ \Upsilon_{0} = \left \{ (S, L, I, E, Y, V, C^{I}, C^{Y}):\frac {d\Phi_{0}}{dt} = 0\right \} $ and let $ \Upsilon_{0}^{\prime} $ be the largest invariant subset of $ \Upsilon_{0} $. The solutions of system (2.1) converge to $ \Upsilon_{0}^{^{\prime}} $. The set $ \Upsilon _{0}^{^{\prime}} $ includes elements with $ S = S_{0} $ and $ I = Y = C^{I} = C^{Y} = 0 $, and hence $ \dot{S} = \dot{Y} = 0 $. The first and fifth equations of system (2.1) yield
$ 0=˙S=ρ−αS0−η1S0V,0=˙Y=ψE. $ |
Thus, $ V(t) = E(t) = 0 $ for all $ t $. In addition, we have $ \dot{I} = 0 $ and from the third equation of system (2.1) we obtain
$ 0 = \dot{I} = \lambda L, $ |
which yields $ L(t) = 0 $ for all $ t $. Therefore, $ \Upsilon_{0}^{\prime} = \left \{ Đ _{0}\right \} $ and by applying Lyapunov-LaSalle asymptotic stability theorem [51,52,53] we get that $Đ _{0} $ is G.A.S.
Theorem 2. Let $ \Re_{1} > 1 $, $ \Re_{2}/\Re_{1}\leq1 $ and $ \Re_{3}\leq 1 $, then $Đ _{1} $ is G.A.S.
Proof. Define a function $ \Phi_{1}(S, L, I, E, Y, V, C^{I}, C^{Y}) $ as:
$ Φ1=S1ϝ(SS1)+λβγ+λL1ϝ(LL1)+γ+λβγ+λI1ϝ(II1)+1φE+ψ+ωφψY+η1S1εV1ϝ(VV1)+μ1(γ+λ)σ1(βγ+λ)CI+μ2(ψ+ω)φψσ2CY. $ |
Calculating $ \frac{d\Phi_{1}}{dt} $ as:
$ dΦ1dt=(1−S1S)[ρ−αS−η1SV−η2SI−η3SY]+λβγ+λ(1−L1L)[(1−β)(η1SV+η2SI)−(λ+γ)L]+γ+λβγ+λ(1−I1I)[β(η1SV+η2SI)+λL−aI−μ1CII]+1φ[φη3SY−(ψ+ω)E]+ψ+ωφψ[ψE−δY−μ2CYY]+η1S1ε(1−V1V)[bI−εV]+μ1(γ+λ)σ1(βγ+λ)[σ1CII−π1CI]+μ2(ψ+ω)φψσ2[σ2CYY−π2CY]=(1−S1S)(ρ−αS)+η2S1I+η3S1Y−λ(1−β)βγ+λ(η1SV+η2SI)L1L+λ(γ+λ)βγ+λL1−a(γ+λ)βγ+λI−β(γ+λ)βγ+λ(η1SV+η2SI)I1I−λ(γ+λ)βγ+λLI1I+a(γ+λ)βγ+λI1+μ1(γ+λ)βγ+λCII1−δ(ψ+ω)φψY+η1S1bIε−η1S1bIεV1V+η1S1V1−μ1π1(γ+λ)σ1(βγ+λ)CI−μ2π2(ψ+ω)φψσ2CY. $ |
Using the equilibrium conditions for $Đ _{1} $, we get
$ ρ=αS1+η1S1V1+η2S1I1, λ(1−β)βγ+λ(η1S1V1+η2S1I1)=λ(γ+λ)βγ+λL1,η1S1V1+η2S1I1=a(γ+λ)βγ+λI1, V1=bI1ε. $ |
Then, we obtain
$ dΦ1dt=(1−S1S)(αS1−αS)+(η1S1V1+η2S1I1)(1−S1S)+η3S1Y−λ(1−β)βγ+λη1S1V1SVL1S1V1L−λ(1−β)βγ+λη2S1I1SIL1S1I1L+λ(1−β)βγ+λ(η1S1V1+η2S1I1)−β(γ+λ)βγ+λη1S1V1SVI1S1V1I−β(γ+λ)βγ+λη2S1I1SS1−λ(1−β)βγ+λ(η1S1V1+η2S1I1)LI1L1I+η1S1V1+η2S1I1+μ1(γ+λ)βγ+λCII1−δ(ψ+ω)φψY−η1S1V1IV1I1V+η1S1V1−μ1π1(γ+λ)σ1(βγ+λ)CI−μ2π2(ψ+ω)φψσ2CY=−α(S−S1)2S+λ(1−β)βγ+λη1S1V1(4−S1S−SVL1S1V1L−LI1L1I−IV1I1V)+λ(1−β)βγ+λη2S1I1(3−S1S−SIL1S1I1L−LI1L1I)+β(γ+λ)βγ+λη1S1V1(3−S1S−SVI1S1V1I−IV1I1V)+β(γ+λ)βγ+λη2S1I1(2−S1S−SS1)+δ(ψ+ω)φψ(φη3ψS1δ(ψ+ω)−1)Y+μ1(γ+λ)βγ+λ(I1−π1σ1)CI−μ2π2(ψ+ω)φψσ2CY. $ | (5.5) |
Therefore, Eq (5.5) becomes
$ dΦ1dt=−(α+βη2I1(γ+λ)βγ+λ)(S−S1)2S+λ(1−β)βγ+λη1S1V1(4−S1S−SVL1S1V1L−LI1L1I−IV1I1V)+λ(1−β)βγ+λη2S1I1(3−S1S−SIL1S1I1L−LI1L1I)+β(γ+λ)βγ+λη1S1V1(3−S1S−SVI1S1V1I−IV1I1V)+δ(ψ+ω)φψ(ℜ2/ℜ1−1)Y+μ1(γ+λ)[π1(η1b+η2ε)+αεσ1]σ1(βγ+λ)(η1b+η2ε)(ℜ3−1)CI−μ2π2(ψ+ω)φψσ2CY. $ | (5.6) |
Since $ \Re_{2}/\Re_{1}\leq1 $ and $ \Re_{3}\leq1 $, then using inequalities (5.1)–(5.3) we get $ \frac{d\Phi_{1}}{dt}\leq0 $ for all $ S, L, I, Y, V, C^{I}, C^{Y} > 0 $. Moreover, $ \frac{d\Phi_{1}}{dt} = 0 $ when $ S = S_{1} $, $ L = L_{1} $, $ I = I_{1} $, $ V = V_{1} $ and $ Y = C^{I} = C^{Y} = 0. $ The solutions of system (2.1) converge to $ \Upsilon_{1}^{\prime} $ the largest invariant subset of $ \Upsilon_{1} = \left \{ (S, L, I, E, Y, V, C^{I}, C^{Y}):\frac{d\Phi_{1}}{dt} = 0\right \} $. The set $ \Upsilon_{1}^{\prime} $ includes $ Y = 0 $, and then $ \dot{Y} = 0 $. The fifth equation of system (2.1) implies
$ 0 = \dot{Y} = \psi E, $ |
which yields $ E(t) = 0 $ for all $ t. $ Hence, $ \Upsilon_{1}^{\prime} = \{Đ_1\} $ and $Đ _1 $ is G.A.S using Lyapunov-LaSalle asymptotic stability theorem.
Theorem 3. If $ \Re_{2} > 1 $, $ \Re_{1}/\Re_{2}\leq1 $ and $ \Re_{4}\leq1 $, then $Đ _{2} $ is G.A.S.
Proof. We define $ \Phi_{2}(S, L, I, E, Y, V, C^{I}, C^{Y}) $ as:
$ Φ2=S2ϝ(SS2)+λβγ+λL+γ+λβγ+λI+1φE2ϝ(EE2)+ψ+ωφψY2ϝ(YY2)+η1S2εV+μ1(γ+λ)σ1(βγ+λ)CI+μ2(ψ+ω)φψσ2CY. $ |
We calculate $ \frac{d\Phi_{2}}{dt} $ as:
$ dΦ2dt=(1−S2S)[ρ−αS−η1SV−η2SI−η3SY]+λβγ+λ[(1−β)(η1SV+η2SI)−(λ+γ)L]+γ+λβγ+λ[β(η1SV+η2SI)+λL−aI−μ1CII]+1φ(1−E2E)[φη3SY−(ψ+ω)E]+ψ+ωφψ(1−Y2Y)[ψE−δY−μ2CYY]+η1S2ε[bI−εV]+μ1(γ+λ)σ1(βγ+λ)[σ1CII−π1CI]+μ2(ψ+ω)φψσ2[σ2CYY−π2CY]=(1−S2S)(ρ−αS)+η2S2I+η3S2Y−a(γ+λ)βγ+λI−η3SYE2E+ψ+ωφE2−δ(ψ+ω)φψY−ψ+ωφEY2Y+δ(ψ+ω)φψY2+μ2(ψ+ω)φψCYY2+η1S2bIε−μ1π1(γ+λ)σ1(βγ+λ)CI−μ2π2(ψ+ω)φψσ2CY. $ |
Using the equilibrium conditions for $Đ _{2} $:
$ ρ=αS2+η3S2Y2, η3S2Y2=ψ+ωφE2=δ(ψ+ω)φψY2, $ | (5.7) |
we obtain
$ dΦ2dt=(1−S2S)(αS2−αS)+η3S2Y2(1−S2S)+η2S2I−a(γ+λ)βγ+λI−η3S2Y2SYE2S2Y2E+η3S2Y2−η3S2Y2EY2E2Y+η3S2Y2+μ2(ψ+ω)φψCYY2+η1S2bIε−μ1π1(γ+λ)σ1(βγ+λ)CI−μ2π2(ψ+ω)φψσ2CY=−α(S−S2)2S+η3S2Y2(3−S2S−SYE2S2Y2E−EY2E2Y)+a(γ+λ)βγ+λ((η1b+η2ε)(βγ+λ)S2aε(γ+λ)−1)I−μ1π1(γ+λ)σ1(βγ+λ)CI+μ2(ψ+ω)φψ(Y2−π2σ2)CY=−α(S−S2)2S+η3S2Y2(3−S2S−SYE2S2Y2E−EY2E2Y)+a(γ+λ)βγ+λ(ℜ1/ℜ2−1)I−μ1π1(γ+λ)σ1(βγ+λ)CI+μ2(ψ+ω)(ασ2+η3π2)φψη3σ2(ℜ4−1)CY. $ |
Since $ \Re_{1}/\Re_{2}\leq1 $ and $ \Re_{4}\leq1 $, then using inequality (5.4) we get $ \frac{d\Phi_{2}}{dt}\leq0 $ for all $ S, L, I, E, Y, V, C^{I}, C^{Y} > 0 $. In addition, $ \frac{d\Phi_{2}}{dt} = 0 $ when $ S = S_{2}, $ $ E = E_{2}, $ $ Y = Y_{2} $ and $ I = C^{I} = C^{Y} = 0. $ Define $ \Upsilon_{2} = \left \{ (S, L, I, E, Y, V, C^{I}, C^{Y}):\frac{d\Phi_{2}}{dt} = 0\right \} $ and let $ \Upsilon_{2}^{\prime} $ be the largest invariant subset of $ \Upsilon_{2} $. The solutions of system (2.1) converge to $ \Upsilon_{2}^{\prime} $ which includes elements with $ S = S_{2}, $ $ Y = Y_{2}, $ $ I = 0 $, then $ \dot{S} = 0 $. The first equation of system (2.1) gives
$ 0 = \dot{S} = \rho-\alpha S_{2}-\eta_{1}S_{2}V-\eta_{3}S_{2}Y_{2}. $ |
From conditions (5.7) we get $ V(t) = 0 $ for all $ t $. Moreover, we have $ \dot{I} = 0 $ and from the third equation of system (2.1) we obtain
$ 0 = \dot{I} = \lambda L, $ |
which yields $ L(t) = 0 $ for all $ t $. Therefore, $ \Upsilon_{2}^{\prime} = \{ Đ _2 \} $. By applying Lyapunov-LaSalle asymptotic stability theorem we get that $Đ _{2} $ is G.A.S.
Theorem 4. Let $ \Re_{3} > 1 $ and $ \Re_{5}\leq1 $, then $Đ _{3} $ is G.A.S.
Proof. Define a function $ \Phi_{3}(S, L, I, E, Y, V, C^{I}, C^{Y}) $ as:
$ Φ3=S3ϝ(SS3)+λβγ+λL3ϝ(LL3)+γ+λβγ+λI3ϝ(II3)+1φE+ψ+ωφψY+η1S3εV3ϝ(VV3)+μ1(γ+λ)σ1(βγ+λ)CI3ϝ(CICI3)+μ2(ψ+ω)φψσ2CY. $ |
We calculate $ \frac{d\Phi_{3}}{dt} $ as:
$ dΦ3dt=(1−S3S)[ρ−αS−η1SV−η2SI−η3SY]+λβγ+λ(1−L3L)[(1−β)(η1SV+η2SI)−(λ+γ)L]+γ+λβγ+λ(1−I3I)[β(η1SV+η2SI)+λL−aI−μ1CII]+1φ[φη3SY−(ψ+ω)E]+ψ+ωφψ[ψE−δY−μ2CYY]+η1S3ε(1−V3V)[bI−εV]+μ1(γ+λ)σ1(βγ+λ)(1−CI3CI)[σ1CII−π1CI]+μ2(ψ+ω)φψσ2[σ2CYY−π2CY]=(1−S3S)(ρ−αS)+η2S3I+η3S3Y−λ(1−β)βγ+λ(η1SV+η2SI)L3L+λ(γ+λ)βγ+λL3−a(γ+λ)βγ+λI−β(γ+λ)βγ+λ(η1SV+η2SI)I3I−λ(γ+λ)βγ+λLI3I+a(γ+λ)βγ+λI3+μ1(γ+λ)βγ+λCII3−δ(ψ+ω)φψY+η1S3εbI−η1S3εbIV3V+η1S3V3−μ1π1(γ+λ)σ1(βγ+λ)CI−μ1(γ+λ)βγ+λCI3I+μ1π1(γ+λ)σ1(βγ+λ)CI3−μ2π2(ψ+ω)φψσ2CY. $ |
Using the equilibrium conditions for $Đ _{3} $:
$ ρ=αS3+η1S3V3+η2S3I3, λ(1−β)βγ+λ(η1S3V3++η2S3I3)=λ(γ+λ)βγ+λL3,η1S3V3+η2S3I3=a(γ+λ)βγ+λI3+μ1(γ+λ)βγ+λCI3I3, I3=π1σ1, V3=bεI3, $ |
we obtain
$ dΦ3dt=(1−S3S)(αS3−αS)+(η1S3V3+η2S3I3)(1−S3S)+(η3S3−δ(ψ+ω)φψ)Y−λ(1−β)βγ+λη1S3V3SVL3S3V3L−λ(1−β)βγ+λη2S3I3SIL3S3I3L+λ(1−β)βγ+λ(η1S3V3+η2S3I3)−β(γ+λ)βγ+λη1S3V3SVI3S3V3I−β(γ+λ)βγ+λη2S3I3SS3−λ(1−β)βγ+λ(η1S3V3+η2S3I3)LI3L3I+η1S3V3+η2S3I3−η1S3V3IV3I3V+η1S3V3−μ2π2(ψ+ω)φψσ2CY=−α(S−S3)2S+λ(1−β)βγ+λη1S3V3(4−S3S−SVL3S3V3L−LI3L3I−IV3I3V)+λ(1−β)βγ+λη2S3I3(3−S3S−SIL3S3I3L−LI3L3I)+β(γ+λ)βγ+λη1S3V3(3−S3S−SVI3S3V3I−IV3I3V)+β(γ+λ)βγ+λη2S3I3(2−S3S−SS3)+δ(ψ+ω)φψ(φψη3S3δ(ψ+ω)−1)Y−μ2π2(ψ+ω)φψσ2CY=−(α+βη2I3(γ+λ)βγ+λ)(S−S3)2S+λ(1−β)βγ+λη1S3V3(4−S3S−SVL3S3V3L−LI3L3I−IV3I3V)+λ(1−β)βγ+λη2S3I3(3−S3S−SIL3S3I3L−LI3L3I)+β(γ+λ)βγ+λη1S3V3(3−S3S−SVI3S3V3I−IV3I3V)+δ(ψ+ω)φψ(ℜ5−1)Y−μ2π2(ψ+ω)φψσ2CY. $ |
Hence, if $ \Re_{5}\leq1 $, then using inequalities (5.1)–(5.3) we get $ \frac{d\Phi_{3}}{dt}\leq0 $ for all $ S, L, I, E, Y, V, C^{I}, C^{Y} > 0 $. Moreover, $ \frac{d\Phi_{3}}{dt} = 0 $ at $ S = S_{3} $, $ L = L_{3}, $ $ I = I_{3}, $ $ V = V_{3} $ and $ Y = C^{Y} = 0. $ The solutions of system (2.1) converge to $ \Upsilon_{3}^{\prime} $ the largest invariant subset of $ \Upsilon_{3} = \left \{ (S, L, I, E, Y, V, C^{I}, C^{Y}):\frac{d\Phi_{3}} {dt} = 0\right \} $. The set $ \Upsilon_{3}^{\prime} $ contains elements with $ S = S_{3} $, $ L = L_{3}, $ $ I = I_{3}, $ $ V = V_{3}, $ $ Y = 0 $, and then $ \dot{I} = \dot{Y} = 0 $. The third and fifth equations of system (2.1) give
$ 0=˙I=β(η1S3V3+η2S3I3)+λL3−aI3−μ1CII3,0=˙Y=ψE, $ |
which yield $ C^{I}(t) = C_{3}^{I} $ and $ E(t) = 0 $ for all $ t. $ Therefore, $ \Upsilon_{3}^{\prime} = \{ Đ _{3} \} $. By applying Lyapunov-LaSalle asymptotic stability theorem we get that $Đ _{3} $ is G.A.S.
Theorem 5. If $ \Re_{4} > 1 $ and $ \Re_{6}\leq1 $, then $Đ _{4} $ is G.A.S.
Proof. Define $ \Phi_{4}(S, L, I, E, Y, V, C^{I}, C^{Y}) $ as:
$ Φ4=S4ϝ(SS4)+λβγ+λL+γ+λβγ+λI+1φE4ϝ(EE4)+ψ+ωφψY4ϝ(YY4)+η1S4εV+μ1(γ+λ)σ1(βγ+λ)CI+μ2(ψ+ω)φψσ2CY4ϝ(CYCY4). $ |
Calculating $ \frac{d\Phi_{4}}{dt} $ as:
$ dΦ4dt=(1−S4S)[ρ−αS−η1SV−η2SI−η3SY]+λβγ+λ[(1−β)(η1SV+η2SI)−(λ+γ)L]+γ+λβγ+λ[β(η1SV+η2SI)+λL−aI−μ1CII]+1φ(1−E4E)[φη3SY−(ψ+ω)E]+ψ+ωφψ(1−Y4Y)[ψE−δY−μ2CYY]+η1S4ε[bI−εV]+μ1(γ+λ)σ1(βγ+λ)[σ1CII−π1CI]+μ2(ψ+ω)φψσ2(1−CY4CY)[σ2CYY−π2CY]=(1−S4S)(ρ−αS)+η2S4I+η3S4Y−a(γ+λ)βγ+λI−η3SYE4E+ψ+ωφE4−δ(ψ+ω)φψY−ψ+ωφEY4Y+δ(ψ+ω)φψY4+μ2(ψ+ω)φψCYY4+η1S4bIε−μ1π1(γ+λ)σ1(βγ+λ)CI−μ2π2(ψ+ω)φψσ2CY−μ2(ψ+ω)φψCY4Y+μ2π2(ψ+ω)φψσ2CY4. $ |
Using the equilibrium conditions for $Đ _{4} $:
$ ρ=αS4+η3S4Y4, Y4=π2σ2,η3S4Y4=ψ+ωφE4=δ(ψ+ω)φψY4+μ2(ψ+ω)φψCY4Y4. $ |
We obtain
$ dΦ4dt=(1−S4S)(αS4−αS)+η3S4Y4(1−S4S)+η2S4I−a(γ+λ)βγ+λI−η3S4Y4SYE4S4Y4E+η3S4Y4−η3S4Y4EY4E4Y+η3S4Y4+η1S4bIε−μ1π1(γ+λ)σ1(βγ+λ)CI=−α(S−S4)2S+η3S4Y4(3−S4S−SYE4S4Y4E−EY4E4Y)+a(γ+λ)βγ+λ((η1b+η2ε)(βγ+λ)S4aε(γ+λ)−1)I−μ1π1(γ+λ)σ1(βγ+λ)CI=−α(S−S4)2S+η3S4Y4(3−S4S−SYE4S4Y4E−EY4E4Y)+a(γ+λ)βγ+λ(ℜ6−1)I−μ1π1(γ+λ)σ1(βγ+λ)CI. $ |
If $ \Re_{6}\leq1 $, then using inequality (5.4) we get $ \frac{d\Phi _{4}}{dt}\leq0 $ for all $ S, L, I, E, Y, V, C^{I}, C^{Y} > 0 $, where $ \frac{d\Phi_{4} }{dt} = 0 $ at $ S = S_{4} $, $ E = E_{4}, $ $ Y = Y_{4} $ and $ I = C^{I} = 0. $ The solutions of system (2.1) converge to $ \Upsilon_{4}^{\prime} $ the largest invariant subset of $ \Upsilon_{4} = \left \{ (S, L, I, E, Y, V, C^{I}, C^{Y}):\frac{d\Phi_{4}}{dt} = 0\right \} $. The set $ \Upsilon _{4}^{\prime} $ contains elements with $ S = S_{4} $, $ E = E_{4}, $ $ Y = Y_{4} $, $ I = 0 $, and then $ \dot{S} = \dot{Y} = 0 $. The first and fifth equations of system (2.1) imply
$ 0=˙S=ρ−αS4−η1S4V−η3S4Y4,0=˙Y=ψE4−δY4−μ2CYY4, $ |
which yield $ V(t) = 0 $ and $ C^{Y}(t) = C_{4}^{Y} $ for all $ t $. Since $ \dot{I} = 0 $, then from the third equation of system (2.1) we obtain
$ 0 = \dot{I} = \lambda L, $ |
which yields $ L(t) = 0 $ for all $ t. $ Therefore, $ \Upsilon_{4}^{\prime} = \left \{ Đ _{4}\right \} $. By applying Lyapunov-LaSalle asymptotic stability theorem we obtain that $Đ _{4} $ is G.A.S.
Theorem 6. If $ \Re_{5} > 1 $, $ \Re_{8}\leq1 $ and $ \Re_{1}/\Re_{2} > 1, $ then $Đ _{5} $ is G.A.S.
Proof. Define $ \Phi_{5}(S, L, I, E, Y, V, C^{I}, C^{Y}) $ as:
$ Φ5=S5ϝ(SS5)+λβγ+λL5ϝ(LL5)+γ+λβγ+λI5ϝ(II5)+1φE5ϝ(EE5)+ψ+ωφψY5ϝ(YY5)+η1S5εV5ϝ(VV5)+μ1(γ+λ)σ1(βγ+λ)CI5ϝ(CICI5)+μ2(ψ+ω)φψσ2CY. $ |
Calculating $ \frac{d\Phi_{5}}{dt} $ as:
$ dΦ5dt=(1−S5S)[ρ−αS−η1SV−η2SI−η3SY]+λβγ+λ(1−L5L)[(1−β)(η1SV+η2SI)−(λ+γ)L]+γ+λβγ+λ(1−I5I)[β(η1SV+η2SI)+λL−aI−μ1CII]+1φ(1−E5E)[φη3SY−(ψ+ω)E]+ψ+ωφψ(1−Y5Y)[ψE−δY−μ2CYY]+η1S5ε(1−V5V)[bI−εV]+μ1(γ+λ)σ1(βγ+λ)(1−CI5CI)[σ1CII−π1CI]+μ2(ψ+ω)φψσ2[σ2CYY−π2CY]=(1−S5S)(ρ−αS)+η2S5I+η3S5Y−λ(1−β)βγ+λ(η1SV+η2SI)L5L+λ(γ+λ)βγ+λL5−a(γ+λ)βγ+λI−β(γ+λ)βγ+λ(η1SV+η2SI)I5I−λ(γ+λ)βγ+λLI5I+a(γ+λ)βγ+λI5+μ1(γ+λ)βγ+λCII5−η3SYE5E+ψ+ωφE5−δ(ψ+ω)φψY−ψ+ωφEY5Y+δ(ψ+ω)φψY5+μ2(ψ+ω)φψCYY5+η1S5bIε−η1S5V5bIεV+η1S5V5−μ1π1(γ+λ)σ1(βγ+λ)CI−μ1(γ+λ)βγ+λCI5I+μ1π1(γ+λ)σ1(βγ+λ)CI5−μ2π2(ψ+ω)φψσ2CY. $ |
Using the equilibrium conditions for $Đ _{5} $:
$ ρ=αS5+η1S5V5+η2S5I5+η3S5Y5,λ(1−β)βγ+λ(η1S5V5+η2S5I5)=λ(γ+λ)βγ+λL5,η1S5V5+η2S5I5=a(γ+λ)βγ+λI5+μ1(γ+λ)βγ+λCI5I5,η3S5Y5=ψ+ωφE5=δ(ψ+ω)φψY5, I5=π1σ1, V5=bI5ε. $ |
We obtain
$ dΦ5dt=(1−S5S)(αS5−αS)+(η1S5V5+η2S5I5+η3S5Y5)(1−S5S)−λ(1−β)βγ+λη1S5V5SVL5S5V5L−λ(1−β)βγ+λη2S5I5SIL5S5I5L+λ(1−β)βγ+λ(η1S5V5+η2S5I5)−β(γ+λ)βγ+λη1S5V5SVI5S5V5I−β(γ+λ)βγ+λη2S5I5SS5−λ(1−β)βγ+λ(η1S5V5+η2S5I5)LI5L5I+η1S5V5+η2S5I5−η3S5Y5SYE5S5Y5E+η3S5Y5−η3S5Y5EY5E5Y+η3S5Y5−η1S5V5IV5I5V+η1S5V5+μ2(ψ+ω)φψ(Y5−π2σ2)CY=−α(S−S5)2S+λ(1−β)βγ+λη1S5V5(4−S5S−SVL5S5V5L−LI5L5I−IV5I5V)+λ(1−β)βγ+λη2S5I5(3−S5S−SIL5S5I5L−LI5L5I)+β(γ+λ)βγ+λη1S5V5(3−S5S−SVI5S5V5I−IV5I5V)+β(γ+λ)βγ+λη2S5I5(2−S5S−SS5)+η3S5Y5(3−S5S−SYE5S5Y5E−EY5E5Y)+μ2(ψ+ω)φψ(Y5−π2σ2)CY. $ | (5.8) |
Then, Eq (5.8) will be reduced to the form
$ dΦ5dt=−(α+βη2I5(γ+λ)βγ+λ)(S−S5)2S+λ(1−β)βγ+λη1S5V5(4−S5S−SVL5S5V5L−LI5L5I−IV5I5V)+λ(1−β)βγ+λη2S5I5(3−S5S−SIL5S5I5L−LI5L5I)+β(γ+λ)βγ+λη1S5V5(3−S5S−SVI5S5V5I−IV5I5V)+η3S5Y5(3−S5S−SYE5S5Y5E−EY5E5Y)+μ2(ψ+ω)[π1σ2(η1b+η2ε)+π2η3εσ1+αεσ1σ2]φψη3εσ1σ2(ℜ8−1)CY. $ |
If $ \Re_{8}\leq1 $, then using inequalities (5.1)–(5.4) we get $ \frac{d\Phi_{5}}{dt}\leq0 $ for all $ S, L, I, E, Y, V, C^{I}, C^{Y} > 0 $, where $ \frac{d\Phi_{5}}{dt} = 0 $ at $ S = S_{5} $, $ L = L_{5}, $ $ I = I_{5}, $ $ E = E_{5}, $ $ Y = Y_{5}, $ $ V = V_{5} $ and $ C^{Y} = 0. $ The solutions of system (2.1) converge to $ \Upsilon_{5}^{\prime} $ the largest invariant subset of $ \Upsilon_{5} = \left \{ (S, L, I, E, Y, V, C^{I}, C^{Y}):\frac{d\Phi_{5}} {dt} = 0\right \} $. The set $ \Upsilon_{5}^{\prime} $ contains elements with $ S = S_{5} $, $ L = L_{5} $, $ I = I_{5}, $ $ V = V_{5} $, and then $ \dot {I} = 0 $. Third equation of system (2.1) implies
$ 0 = \dot{I} = \beta \left( \eta_{1}S_{5}V_{5}+\eta_{2}S_{5}I_{5}\right) +\lambda L_{5}-aI_{5}-\mu_{1}C^{I}I_{5}, $ |
which gives $ C^{I}(t) = C_{5}^{I} $ for all $ t. $ Therefore, $ \Upsilon_{5} ^{\prime} = \{ Đ _{5} \} $. Applying Lyapunov-LaSalle asymptotic stability theorem we get $Đ _{5} $ is G.A.S.
Theorem 7. If $ \Re_{6} > 1 $, $ \Re_{7}\leq1 $ and $ \Re_{2}/\Re_{1} > 1 $, then $Đ _{6} $ is G.A.S.
Proof. Define $ \Phi_{6}(S, L, I, E, Y, V, C^{I}, C^{Y}) $ as:
$ Φ6=S6ϝ(SS6)+λβγ+λL6ϝ(LL6)+γ+λβγ+λI6ϝ(II6)+1φE6ϝ(EE6)+ψ+ωφψY6ϝ(YY6)+η1S6εV6ϝ(VV6)+μ1(γ+λ)σ1(βγ+λ)CI+μ2(ψ+ω)φψσ2CY6ϝ(CYCY6). $ |
Calculating $ \frac{d\Phi_{6}}{dt} $ as:
$ dΦ6dt=(1−S6S)[ρ−αS−η1SV−η2SI−η3SY]+λβγ+λ(1−L6L)[(1−β)(η1SV+η2SI)−(λ+γ)L]+γ+λβγ+λ(1−I6I)[β(η1SV+η2SI)+λL−aI−μ1CII]+1φ(1−E6E)[φη3SY−(ψ+ω)E]+ψ+ωφψ(1−Y6Y)[ψE−δY−μ2CYY]+η1S6ε(1−V6V)[bI−εV]+μ1(γ+λ)σ1(βγ+λ)[σ1CII−π1CI]+μ2(ψ+ω)φψσ2(1−CY6CY)[σ2CYY−π2CY]=(1−S6S)(ρ−αS)+η2S6I+η3S6Y−λ(1−β)βγ+λ(η1SV+η2SI)L6L+λ(γ+λ)βγ+λL6−a(γ+λ)βγ+λI−β(γ+λ)βγ+λ(η1SV+η2SI)I6I−λ(γ+λ)βγ+λLI6I+a(γ+λ)βγ+λI6+μ1(γ+λ)βγ+λCII6−η3SYE6E+ψ+ωφE6−δ(ψ+ω)φψY−ψ+ωφEY6Y+δ(ψ+ω)φψY6+μ2(ψ+ω)φψCYY6+η1S6bIε−η1S6V6bIεV+η1S6V6−μ1π1(γ+λ)σ1(βγ+λ)CI−μ2π2(ψ+ω)φψσ2CY−μ2(ψ+ω)φψCY6Y+μ2π2(ψ+ω)φψσ2CY6. $ |
Using the equilibrium conditions for $Đ _{6} $:
$ ρ=αS6+η1S6V6+η2S6I6+η3S6Y6,λ(1−β)βγ+λ(η1S6V6+η2S6I6)=λ(γ+λ)βγ+λL6, Y6=π2σ2, V6=bI6ε,η1S6V6+η2S6I6=a(γ+λ)βγ+λI6,η3S6Y6=ψ+ωφE6=δ(ψ+ω)φψY6+μ2(ψ+ω)φψCY6Y6. $ |
We obtain
$ dΦ6dt=(1−S6S)(αS6−αS)+(η1S6V6+η2S6I6+η3S6Y6)(1−S6S)−λ(1−β)βγ+λη1S6V6SVL6S6V6L−λ(1−β)βγ+λη2S6I6SIL6S6I6L+λ(1−β)βγ+λ(η1S6V6+η2S6I6)−β(γ+λ)βγ+λη1S6V6SVI6S6V6I−β(γ+λ)βγ+λη2S6I6SS6−λ(1−β)βγ+λ(η1S6V6+η2S6I6)LI6L6I+η1S6V6+η2S6I6−η3S6Y6SYE6S6Y6E+η3S6Y6−η3S6Y6EY6E6Y+η3S6Y6−η1S6V6IV6I6V+η1S6V6+μ1(γ+λ)βγ+λ(I6−π1σ1)CI=−α(S−S6)2S+λ(1−β)βγ+λη1S6V6(4−S6S−SVL6S6V6L−LI6L6I−IV6I6V)+λ(1−β)βγ+λη2S6I6(3−S6S−SIL6S6I6L−LI6L6I)+β(γ+λ)βγ+λη1S6V6(3−S6S−SVI6S6V6I−IV6I6V)+β(γ+λ)βγ+λη2S6I6(2−S6S−SS6)+η3S6Y6(3−S6S−SYE6S6Y6E−EY6E6Y)+μ1(γ+λ)βγ+λ(I6−π1σ1)CI. $ | (5.9) |
Then, Eq (5.9) will be reduced to the form
$ dΦ6dt=−(α+βη2I6(γ+λ)βγ+λ)(S−S6)2S+λ(1−β)βγ+λη1S6V6(4−S6S−SVL6S6V6L−LI6L6I−IV6I6V)+λ(1−β)βγ+λη2S6I6(3−S6S−SIL6S6I6L−LI6L6I)+β(γ+λ)βγ+λη1S6V6(3−S6S−SVI6S6V6I−IV6I6V)+η3S6Y6(3−S6S−SYE6S6Y6E−EY6E6Y)+μ1(γ+λ)[π1σ2(η1b+η2ε)+π2η3εσ1+αεσ1σ2]σ1σ2(βγ+λ)(η1b+η2ε)(ℜ7−1)CI. $ |
Therefore, if $ \Re_{7}\leq1 $, then using inequalities (5.1)–(5.4) we get $ \frac{d\Phi_{6}}{dt}\leq0 $ for all $ S, L, I, E, Y, V, C^{I}, C^{Y} > 0, $ where $ \frac{d\Phi_{6}}{dt} = 0 $ at $ S = S_{6} $, $ L = L_{6}, $ $ I = I_{6}, $ $ E = E_{6}, $ $ Y = Y_{6}, $ $ V = V_{6} $ and $ C^{I} = 0. $ Define $ \Upsilon _{6} = \left \{ (S, L, I, E, Y, V, C^{I}, C^{Y}):\frac{d\Phi_{6}}{dt} = 0\right \} $ and let $ \Upsilon_{6}^{\prime} $ be the largest invariant subset of $ \Upsilon_{6} $. The solutions of system (2.1) tend to $ \Upsilon_{6}^{\prime} $ which includes elements with $ E = E_{6}, $ $ Y = Y_{6} $, and then $ \dot{Y} = 0 $. The fifth equation of system (2.1) implies
$ 0 = \dot{Y} = \psi E_{6}-\delta Y_{6}-\mu_{2}C^{Y}Y_{6}, $ |
which ensures that $ C^{Y}(t) = C_{6}^{Y} $ for all $ t. $ Therefore, $ \Upsilon _{6}^{\prime} = \left \{ Đ _{6}\right \} $. Applying Lyapunov-LaSalle asymptotic stability theorem we get $Đ _{6} $ is G.A.S.
Theorem 8. If $ \Re_{7} > 1 $ and $ \Re_{8} > 1 $, then $Đ _{7} $ is G.A.S.
Proof. Define $ \Phi_{7}(S, L, I, E, Y, V, C^{I}, C^{Y}) $ as:
$ Φ7=S7ϝ(SS7)+λβγ+λL7ϝ(LL7)+γ+λβγ+λI7ϝ(II7)+1φE7ϝ(EE7)+ψ+ωφψY7ϝ(YY7)+η1S7εV7ϝ(VV7)+μ1(γ+λ)σ1(βγ+λ)CI7ϝ(CICI7)+μ2(ψ+ω)φψσ2CY7ϝ(CYCY7). $ |
Calculating $ \frac{d\Phi_{7}}{dt} $ and after collecting terms we get
$ dΦ7dt=(1−S7S)(ρ−αS)+η2S7I+η3S7Y−λ(1−β)βγ+λ(η1SV+η2SI)L7L+λ(γ+λ)βγ+λL7−a(γ+λ)βγ+λI−β(γ+λ)βγ+λ(η1SV+η2SI)I7I−λ(γ+λ)βγ+λLI7I+a(γ+λ)βγ+λI7+μ1(γ+λ)βγ+λCII7−η3SYE7E+ψ+ωφE7−δ(ψ+ω)φψY−ψ+ωφEY7Y+δ(ψ+ω)φψY7+μ2(ψ+ω)φψCYY7+η1S7bIε−η1S7V7bIεV+η1S7V7−μ1π1(γ+λ)σ1(βγ+λ)CI−μ1(γ+λ)βγ+λCI7I+μ1π1(γ+λ)σ1(βγ+λ)CI7−μ2π2(ψ+ω)φψσ2CY−μ2(ψ+ω)φψCY7Y+μ2π2(ψ+ω)φψσ2CY7. $ |
Using the equilibrium conditions for $Đ _{7} $:
$ ρ=αS7+η1S7V7+η2S7I7+η3S7Y7, λ(1−β)βγ+λ(η1S7V7+η2S7I7)=λ(γ+λ)βγ+λL7,η1S7V7+η2S7I7=a(γ+λ)βγ+λI7+μ1(γ+λ)βγ+λCI7I7, I7=π1σ1, Y7=π2σ2, V7=bI7εη3S7Y7=ψ+ωφE7=δ(ψ+ω)φψY7+μ2(ψ+ω)φψCY7Y7. $ |
We obtain
$ dΦ7dt=(1−S7S)(αS7−αS)+(η1S7V7+η2S7I7+η3S7Y7)(1−S7S)−λ(1−β)βγ+λη1S7V7SVL7S7V7L−λ(1−β)βγ+λη2S7I7SIL7S7I7L+λ(1−β)βγ+λ(η1S7V7+η2S7I7)−β(γ+λ)βγ+λη1S7V7SVI7S7V7I−β(γ+λ)βγ+λη2S7I7SS7−λ(1−β)βγ+λ(η1S7V7+η2S7I7)LI7L7I+η1S7V7+η2S7I7−η3S7Y7SYE7S7Y7E+η3S7Y7−η3S7Y7EY7E7Y+η3S7Y7−η1S7V7IV7I7V+η1S7V7=−(α+βη2I7(γ+λ)βγ+λ)(S−S7)2S+λ(1−β)βγ+λη1S7V7(4−S7S−SVL7S7V7L−LI7L7I−IV7I7V)+λ(1−β)βγ+λη2S7I7(3−S7S−SIL7S7I7L−LI7L7I)+β(γ+λ)βγ+λη1S7V7(3−S7S−SVI7S7V7I−IV7I7V)+η3S7Y7(3−S7S−SYE7S7Y7E−EY7E7Y). $ |
Therefore, using inequalities (5.1)–(5.4) we get $ \frac {d\Phi_{7}}{dt}\leq0 $ for all $ S, L, I, E, Y, V, C^{I}, C^{Y} > 0 $. In addition we have $ \frac{d\Phi_{7}}{dt} = 0 $ at $ S = S_{7} $, $ L = L_{7}, $ $ I = I_{7}, $ $ E = E_{7}, $ $ Y = Y_{7} $ and $ V = V_{7}. $ The solutions of system (2.1) converge to $ \Upsilon_{7}^{\prime} $ the largest invariant subset of $ \Upsilon_{7} = \left \{ (S, L, I, E, Y, V, C^{I}, C^{Y}):\frac{d\Phi_{7}} {dt} = 0\right \} $. The set $ \Upsilon_{7}^{\prime} $ contains elements with $ S = S_{7} $, $ L = L_{7}, $ $ I = I_{7}, $ $ E = E_{7}, $ $ Y = Y_{7} $ and $ V = V_{7} $. Then $ \dot{I} = \dot{Y} = 0 $ and from the third and fifth equations of system (2.1) we get
$ 0=˙I=β(η1S7V7+η2S7I7)+λL7−aI7−μ1CII7,0=˙Y=ψE7−δY7−μ2CYY7, $ |
which ensure that $ C^{I}(t) = C_{7}^{I} $ and $ C^{Y}(t) = C_{7}^{Y} $ for all $ t. $ Therefore, $ \Upsilon_{7}^{\prime} = \left \{ Đ _{7}\right \} $. Applying Lyapunov-LaSalle asymptotic stability theorem we get $Đ _{7} $ is G.A.S.
In Table 3, we summarize the global stability results given in Theorems 1–8.
Equilibrium point | Global stability$ \text{ conditions} $ |
$Đ _{0}=(S_{0}, 0, 0, 0, 0, 0, 0, 0) $ | $ \Re_{1}\leq1 $ and $ \Re_{2}\leq1 $ |
$Đ _{1}=(S_{1}, L_{1}, I_{1}, 0, 0, V_{1}, 0, 0) $ | $ \Re_{1} > 1 $, $ \Re_{2}/\Re _{1}\leq1 $ and $ \Re_{3}\leq1 $ |
$Đ _{2}=(S_{2}, 0, 0, E_{2}, Y_{2}, 0, 0, 0) $ | $ \Re_{2} > 1 $, $ \Re_{1}/\Re_{2}\leq1 $ and $ \Re_{4}\leq1 $ |
$Đ _{3}=(S_{3}, L_{3}, I_{3}, 0, 0, V_{3}, C_{3}^{I}, 0) $ | $ \Re_{3} > 1 $ and $ \Re_{5}\leq1 $ |
$Đ _{4}=(S_{4}, 0, 0, E_{4}, Y_{4}, 0, 0, C_{4}^{Y}) $ | $ \Re_{4} > 1 $ and $ \Re _{6}\leq1 $ |
$Đ _{5}=(S_{5}, L_{5}, I_{5}, E_{5}, Y_{5}, V_{5}, C_{5}^{I}, 0) $ | $ \Re_{5} > 1 $, $ \Re_{8}\leq1 $ and $ \Re_{1}/\Re_{2} > 1 $ |
$Đ _{6}=(S_{6}, L_{6}, I_{6}, E_{6}, Y_{6}, V_{6}, 0, C_{6}^{Y}) $ | $ \Re_{6} > 1 $, $ \Re_{7}\leq1 $ and $ \Re_{2}/\Re_{1} > 1 $ |
$Đ _{7}=(S_{7}, L_{7}, I_{7}, E_{7}, Y_{7}, V_{7}, C_{7}^{I}, C_{7}^{Y}) $ | $ \Re_{7} > 1 $ and $ \Re_{8} > 1 $ |
In this section, we illustrate the results of Theorems 1–8 by performing numerical simulations. Moreover, we study the effect of HTLV-I infection on the HIV mono-infected individuals by making a comparison between the dynamics of HIV mono-infection and HIV/HTLV-I co-infection. Otherwise, we investigate the influence of HIV infection on the HTLV-I mono-infected individuals by conducting a comparison between the dynamics of HTLV-I mono-infection and HIV/HTLV-I co-infection.
To solve system (2.1) numerically we fix the values of some parameters (see Table 4) and the others will be varied.
Parameter | Value | Parameter | Value | Parameter | Value |
$ \rho $ | $ 10 $ | $ \delta $ | $ 0.2 $ | $ \beta $ | $ 0.7 $ |
$ \alpha $ | $ 0.01 $ | $ b $ | $ 5 $ | $ \gamma $ | $ 0.02 $ |
$ \eta_{1} $ | Varied | $ \pi_{1} $ | $ 0.1 $ | $ \sigma_{1} $ | Varied |
$ \eta_{2} $ | Varied | $ \pi_{2} $ | $ 0.1 $ | $ \sigma_{2} $ | Varied |
$ \eta_{3} $ | Varied | $ \mu_{1} $ | $ 0.2 $ | $ \lambda $ | $ 0.2 $ |
$ a $ | $ 0.5 $ | $ \mu_{2} $ | $ 0.2 $ | $ \omega $ | $ 0.01 $ |
$ \varphi $ | $ 0.2 $ | $ \varepsilon $ | $ 2 $ | $ \psi $ | $ 0.003 $ |
In this subsection, we choose the following three different initial conditions for system (2.1):
Initial-1 :$ (S(0), L(0), I(0), E(0), Y(0), V(0), C^{I}(0), C^{Y} (0)) = (600, 1.5, 1.5, 30, 0.3, 5, 1, 3) $,
Initial-2:$ (S(0), L(0), I(0), E(0), Y(0), V(0), C^{I}(0), C^{Y} (0)) = (500, 1, 1, 20, 0.2, 2, 2, 2), $
Initial-3:$ (S(0), L(0), I(0), E(0), Y(0), V(0), C^{I}(0), C^{Y} (0)) = (300, 0.5, 0.5, 10, 0.1, 1.5, 3, 1) $.
Choosing selected values of $ \eta_{1} $, $ \eta_{2} $, $ \eta_{3} $, $ \sigma_{1} $ and $ \sigma_{2} $ under the above initial conditions leads to the following scenarios:
Scenario 1 (Stability of ${\mathit{Đ}} _{0} $): $ \eta _{1} = \eta_{2} = 0.0001, $ $ \eta_{3} = 0.001 $ and $ \sigma_{1} = \sigma _{2} = 0.2 $. For this set of parameters, we have $ \Re_{1} = 0.68 < 1 $ and $ \Re _{2} = 0.23 < 1 $. Figure 2 displays that the trajectories initiating with Initial-1, Initial-2 and Initial-3 reach the equilibrium $Đ _{0} = (1000, 0, 0, 0, 0, 0, 0, 0) $. This shows that $Đ _{0} $ is G.A.S according to Theorem 1. In this situation both HIV and HTLV will be died out.
Scenario 2 (Stability of ${\mathit{Đ}} _{1} $): $ \eta _{1} = 0.0005, $ $ \eta_{2} = 0.0003, $ $ \eta_{3} = 0.0005, $ $ \sigma _{1} = 0.003 $ and $ \sigma_{2} = 0.2 $. With such choice we get $ \Re_{2} = 0.12 < 1 < 3.02 = \Re_{1} $, $ \Re_{3} = 0.49 < 1 $ and hence $ \Re_{2}/\Re_{1} = 0.04 < 1 $. Therefore, the conditions in Table 2 is verified. In fact, the equilibrium point $Đ _{1} $ exists with $Đ _{1} = \left(331.63, 9.11, 13, 0, 0, 32.51, 0, 0\right) $. Figure 3 displays that the trajectories initiating with Initial-1, Initial-2 and Initial-3 tend to $Đ _{1} $. Therefore, the numerical results support Theorem 2. This case corresponds to a chronic HIV mono-infection but with unstimulated CTL-mediated immune response.
Scenario 3 (Stability of ${\mathit{Đ}} _{2} $): $ \eta _{1} = 0.0001, $ $ \eta_{2} = 0.0002, $ $ \eta_{3} = 0.01 $, $ \sigma_{1} = 0.001 $ and $ \sigma_{2} = 0.05 $. Then, we calculate $ \Re_{1} = 0.88 < 1 < 2.31 = \Re_{2} $, $ \Re_{4} = 0.77 < 1 $ and then $ \Re_{1}/\Re_{2} = 0.38 < 1 $. Hence, the conditions in Table 2 is satisfied. The numerical results show that $Đ _{2} = \left(433.33, 0, 0, 87.18, 1.31, 0, 0, 0\right) $ exists. Figure 4 illustrates that the trajectories initiating with Initial-1, Initial-2 and Initial-3 tend to $Đ _{2} $. Thus, the numerical results consistent with Theorem 3. This situation leads to a persistent HTLV mono-infection with unstimulated CTL-mediated immune response.
Scenario 4 (Stability of ${\mathit{Đ}} _{3} $): $ \eta _{1} = 0.001, $ $ \eta_{2} = 0.0001, $ $ \eta_{3} = 0.005 $ and $ \sigma _{1} = \sigma_{2} = 0.01 $. Then, we calculate $ \Re_{3} = 1.41 > 1 $ and $ \Re _{5} = 0.32 < 1 $. Table 2 and Figure 5 show that the trajectories initiating with Initial-1, Initial-2 and Initial-3 tend to $Đ _{3} = \left(277.78, 9.85, 10, 0, 0, 25, 1.01, 0\right) $. Therefore, $Đ _{3} $ is G.A.S and this agrees with Theorem 4. Hence, a chronic HIV mono-infection with HIV-specific CTL-mediated immune response is attained.
Scenario 5 (Stability of ${\mathit{Đ}} _{4} $): $ \eta _{1} = 0.0007, $ $ \eta_{2} = 0.0001, $ $ \eta_{3} = 0.1 $, $ \sigma_{1} = 0.05 $ and $ \sigma_{2} = 0.3 $. Then, we calculate $ \Re_{4} = 5.33 > 1 $ and $ \Re_{6} = 0.83 < 1 $. According to Table 2, $Đ _{4} $ exists with $Đ _{4} = \left(230.77, 0, 0,118.34, 0.33, 0, 0, 4.33\right) $. In Figure 6, we show that the trajectories initiating with Initial-1, Initial-2 and Initial-3 tend to $Đ _{4} $ and then it is G.A.S which agrees with Theorem 5. Hence, a chronic HTLV mono-infection with HTLV-specific CTL-mediated immune response is attained.
Scenario 6 (Stability of ${\mathit{Đ}} _{5} $): $ \eta _{1} = 0.001, $ $ \eta_{2} = 0.0001, $ $ \eta_{3} = 0.01 $, $ \sigma_{1} = 0.05 $ and $ \sigma_{2} = 0.08 $. Then, we calculate $ \Re_{5} = 1.52 > 1 $, $ \Re_{8} = 0.83 < 1 $ and $ \Re_{1}/\Re_{2} = 2.19 > 1 $. Table 2 and the numerical results demonstrated in Figure 7 show that $Đ _{5} = \left(433.33, 3.07, 2, 52.51, 0.79, 5, 2.98, 0\right) $ exists and it is G.A.S and this agrees with Theorem 6. As a result, a chronic co-infection with HIV and HTLV is attained where the HIV-specific CTL-mediated immune response is active and the HTLV-specific CTL-mediated immune response is unstimulated.
Scenario 7 (Stability of ${\mathit{Đ}} _{6} $): $ \eta _{1} = 0.0006, $ $ \eta_{2} = 0.0001, $ $ \eta_{3} = 0.04 $, $ \sigma_{1} = 0.01 $ and $ \sigma_{2} = 0.5 $. We compute $ \Re_{6} = 1.73 > 1 $, $ \Re_{7} = 0.92 < 1 $ and $ \Re_{2}/\Re_{1} = 2.97 > 1 $. Based on the conditions in Table 2, the equilibrium $Đ _{6} = \left(321.26, 5.75, 8.2, 39.54, 0.2, 20.51, 0, 1.97\right) $ exists. Moreover, the numerical results plotted in Figure 8 show that $Đ _{6} $ is G.A.S and this illustrates Theorem 7. As a result, a chronic co-infection with HIV and HTLV is attained where the HTLV-specific CTL-mediated immune response is active and the HIV-specific CTL-mediated immune response is unstimulated.
Scenario 8 (Stability of ${\mathit{Đ}} _{7} $): $ \eta _{1} = 0.0006, $ $ \eta_{2} = 0.0002, $ $ \eta_{3} = 0.04 $, $ \sigma_{1} = 0.05 $ and $ \sigma_{2} = 0.5 $. These data give $ \Re_{7} = 1.55 > 1 $ and $ \Re_{8} = 4.31 > 1 $. According to Table 2, the equilibrium $Đ _{7} $ exists. Figure 9 illustrates that the trajectories initiating with Initial-1, Initial-2 and Initial-3 tend to $Đ _{7} = \left(467.29, 2.17, 2, 57.51, 0.2, 5, 1.36, 3.31\right) $. The numerical results displayed in Figure 9 show that $Đ _{7} $ is G.A.S based on Theorem 8. In this case, a chronic co-infection with HIV and HTLV is attained where both HIV-specific CTL-mediated and HTLV-specific CTL-mediated immune responses are working.
To further confirmation, we calculate the Jacobian matrix $ J = J(S, L, I, E, Y, V, C^{I}, C^{Y}) $ of system (2.1) as in the following form:
$ J = \left( −(α+η1V+η2I+η3Y)0−η2S0−η3S−η1S00(1−β)(η1V+η2I)−(γ+λ)(1−β)η2S00(1−β)η1S00β(η1V+η2I)λβη2S−(a+μ1CI)00βη1S−μ1I0φη3Y00−(ψ+ω)φη3S000000ψ−(δ+μ2CY)00−μ2Y00b00−ε0000σ1CI000σ1I−π100000σ2CY00σ2Y−π2 \right) . $ |
Then, we calculate the eigenvalues $ \lambda_{i}, $ $ i = 1, 2, ..., 8 $ of the matrix $ J $ at each equilibrium. The examined steady will be locally stable if all its eigenvalues satisfy the following condition:
$ \operatorname{Re}(\lambda_{i}) \lt 0,\text{ }i = 1,2,...,8. $ |
We use the parameters $ \eta_{1} $, $ \eta_{2} $, $ \eta_{3} $, $ \sigma_{1} $ and $ \sigma_{2} $ the same as given above to compute all positive equilibria and the corresponding eigenvalues. From the scenarios 1–8, we present in Table 5 the positive equilibria, the real parts of the eigenvalues and whether the equilibrium is locally stable or unstable.
Scenario | The equilibria | $ (\operatorname{Re}(\lambda_{i}), $ $ i=1, 2, ..., 6) $ | Stability |
1 | $ Đ0=(1000,0,0,0,0,0,0,0) $ | $ (−2.19,−0.37,−0.2,−0.1,−0.1,−0.09,−0.01,−0.01) $ | $ stable $ |
2 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(331.63,9.11,13,0,0,32.51,0,0) $ | $ (−2.7,0.51,−0.32,−0.2,−0.1,−0.1,−0.01,−0.01)(−2.3,−0.35,−0.2,−0.1,−0.02,−0.02,−0.06,−0.01) $ | $ unstablestable $ |
3 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ2=(433.33,0,0,87.18,1.31,0,0,0) $ | $ (−2.18,−0.36,−0.23,−0.1,−0.1,−0.03,0.01,−0.01)(−2.09,−0.41,−0.21,−0.16,−0.1,−0.03,−0.01,−0.01) $ | $ unstablestable $ |
4 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(197.7,10.94,15.61,0,0,39.02,0,0)Đ2=(866.67,0,0,20.51,0.31,0,0,0)Đ3=(277.78,9.85,10,0,0,25,1.01,0) $ | $ (−3.22,0.88,−0.31,−0.21,−0.1,−0.1,−0.01,0.002)(−2.36,−0.35,−0.2,−0.03,−0.03,−0.1,0.06,−0.01)(−3.1,0.76,−0.32,−0.21,−0.1,−0.1,−0.01,−0.002)(−2.51,−0.37,−0.2,−0.02,−0.02,−0.1,−0.02,−0.01) $ | $ unstableunstableunstablestable $ |
5 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(277.85,9.85,14.05,0,0,35.12,0,0)Đ2=(43.33,0,0,147.18,2.21,0,0,0)Đ3=(729.93,3.68,2,0,0,5,4.07,0)Đ4=(230.77,0,0,118.34,0.33,0,0,4.33) $ | $ (−2.94,0.61,−0.37,−0.32,0.16,−0.1,−0.1,−0.01)(−2.35,0.6,−0.35,−0.27,−0.1,−0.02,−0.02,0.05)(−2.06,0.56,−0.45,−0.27,−0.2,−0.16,−0.1,−0.01)(−2.99,−0.43,−0.34,−0.03,−0.03,0.12,−0.1,−0.01)(−2.3,−0.99,−0.36,−0.1,−0.06,−0.06,−0.05,−0.01) $ | $ unstableunstableunstableunstablestable $ |
6 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(197.7,10.94,15.61,0,0,39.02,0,0)Đ2=(433.33,0,0,87.18,1.31,0,0,0)Đ3=(657.89,4.67,2,0,0,5,5.82,0)Đ4=(444.44,0,0,85.47,1.25,0,0,0.03)Đ5=(433.33,3.07,2,52.51,0.79,5,2.98,0) $ | $ (−3.22,0.88,−0.31,−0.23,−0.1,−0.1,0.01,−0.01)(−2.36,0.68,−0.35,−0.21,−0.03,−0.03,−0.1,−0.01)(−2.67,−0.33,0.3,−0.21,−0.1,−0.01,−0.01,0.005)(−3.3,−0.45,−0.22,−0.05,−0.05,−0.1,−0.01,0.01)(−2.68,−0.32,0.32,−0.22,−0.1,−0.01,−0.01,−0.005)(−2.83,−0.41,−0.21,−0.03,−0.03,−0.04,−0.01,−0.01) $ | $ unstableunstableunstableunstableunstablestable $ |
7 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(321.26,9.26,13.2,0,0,33.01,0,0)Đ2=(108.33,0,0,137.18,2.06,0,0,0)Đ3=(384.62,8.39,10,0,0,25,0.49,0)Đ4=(555.56,0,0,68.38,0.2,0,0,4.13)Đ6=(321.26,5.75,8.2,39.54,0.2,20.51,0,1.97) $ | $ (−2.84,0.51,−0.32,−0.29,−0.1,−0.1,0.07,−0.01)(−2.35,−0.35,−0.23,−0.1,−0.02,−0.02,0.03,0.02)(−2.13,0.93,−0.42,−0.22,−0.16,−0.1,−0.07,−0.01)(−2.42,−0.36,−0.24,−0.1,−0.01,−0.01,0.03,−0.01)(−2.54,−0.95,−0.33,0.19,−0.1,−0.07,−0.03,−0.01)(−2.35,−0.53,−0.35,−0.02,−0.02,−0.05,−0.02,−0.02) $ | $ unstableunstableunstableunstableunstablestable $ |
8 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(302.36,9.51,13.57,0,0,33.93,0,0)Đ2=(108.33,0,0,137.18,2.06,0,0,0)Đ3=(746.27,3.46,2,0,0,5,3.67,0)Đ4=(555.56,0,0,68.38,0.2,0,0,4.13)Đ6=(302.36,6.21,8.87,37.21,0.2,22.17,0,1.79)Đ7=(467.29,2.17,2,57.51,0.2,5,1.36,3.31) $ | $ (−2.83,0.56,−0.32,−0.29,−0.1,−0.1,0.07,−0.01)(−2.33,0.58,−0.35,−0.23,−0.1,−0.02,−0.02,0.02)(−2.13,0.93,−0.41,−0.22,−0.16,−0.1,−0.07,−0.01)(−2.86,−0.42,−0.27,−0.03,−0.03,−0.1,0.06,−0.01)(−2.53,−0.95,−0.33,0.22,−0.1,−0.07,−0.03,−0.01)(−2.33,−0.5,−0.35,0.34,−0.02,−0.02,−0.05,−0.02)(−2.52,−0.79,−0.38,−0.01,−0.01,−0.06,−0.03,−0.01) $ | $ unstableunstableunstableunstableunstableunstablestable $ |
In this subsection, we study the influence of HTLV-I infection on HIV mono-infection dynamics, and how affect the HIV infection on the dynamics of HTLV-I mono-infection as well.
To investigate the effect of HTLV-I infection on HIV mono-infection dynamics, we make a comparison between model (2.1) and the following HIV mono-infection model:
$ {˙S=ρ−αS−η1SV−η2SI,˙L=(1−β)(η1SV+η2SI)−(λ+γ)L,˙I=βη1(η1SV+η2SI)+λL−aI−μ1CII,˙V=bI−εV,˙CI=σ1CII−π1CI. $ | (6.1) |
We fix parameters $ \eta_{1} = 0.0006 $, $ \eta_{2} = 0.0001 $, $ \sigma_{1} = 0.05 $, and $ \sigma_{2} = 0.5 $ and consider the following initial condition:
Initial-4: $ (S(0), L(0), I(0), E(0), Y(0), V(0), C^{I}(0), C^{Y}(0)) = \left(600, 2.4, 1.8, 60, 0.2, 4.5, 1.8, 3.5\right) $.
We choose two values of the parameter $ \eta_{3} $ as $ \eta_{3} = 0.04 $ (HIV/HTLV-I co-infection), and $ \eta_{3} = 0.0 $ (HIV mono-infection). It can be seen from Figure 10 that when the HIV mono-infected individual is co-infected with HTLV-I then the concentrations of susceptible CD$ 4^{+} $T cells, latently HIV-infected cells and HIV-specific CTLs are decreased. Although, the concentration of free HIV particles tend to the same value in both HIV mono-infection and HIV/HTLV-I co-infection. Indeed, such observation are compatible with the study that has been performed by Vandormael et al. in 2017 [54]. The researchers have not found any worthy differences in the concentration of HIV virus particles in comparison between HIV mono-infected and HIV/HTLV-I co-infected patients.
To investigate the effect of HIV infection on HTLV-I mono-infection dynamics, we make a comparison between model (2.1) and the following HTLV-I mono-infection model:
$ {˙S=ρ−αS−η2SY,˙E=φη2SY−(ψ+ω)E,˙Y=ψE−δY−μ2CYY,˙CY=σ2CYY−π2CY. $ | (6.2) |
We fix parameters $ \eta_{3} = 0.01 $; $ \sigma_{1} = 0.05 $, and $ \sigma_{2} = 0.5 $ and consider the following initial condition:
Initial-5: $ (S(0), L(0), I(0), E(0), Y(0), V(0), C^{I}(0), C^{Y} (0)) = (700, 4, 2, 21, 0.198, 5, 4.5, 0.6) $.
We choose two values of the parameters $ \eta_{1}, $ $ \eta_{2} $ as $ \eta_{1} = $ $ 0.001, $ $ \eta_{2} = 0.0002 $ (HIV/HTLV-I co-infection), and $ \eta_{1} = \eta _{2} = 0.0 $ (HTLV-I mono-infection). It can be seen from Figure 11 that when the HTLV-I mono-infected individual is co-infected with HIV then the concentrations of susceptible CD$ 4^{+} $T cells, latently HTLV-infected cells and HTLV-specific CTLs are decreased. Although, the concentration of Tax-expressing HTLV-infected cells tend to the same value in both HTLV-I mono-infection and HIV/HTLV-I co-infection.
As we discussed in Section 1 that CTLs have significant important in controlling HIV and HTLV-I mono-infections by killing infected cells. Model (2.1) in the absence of CTL immune response leads to a model with competition between HIV and HTLV-I on CD$ 4^{+} $T cells:
$ {˙S=ρ−αS−η1SV−η2SI−η3SY,˙L=(1−β)(η1SV+η2SI)−(λ+γ)L,˙I=β(η1SV+η2SI)+λL−aI,˙E=φη3SY−(ψ+ω)E,˙Y=ψE−δY,˙V=bI−εV. $ | (6.3) |
The system has only three equilibria, infection-free equilibrium, $ \overline{Đ }_{0} = (S_{0}, 0, 0, 0, 0, 0) $, chronic HIV mono-infection equilibrium, $ \overline{Đ}_{1} = (S_{1}, L_{1}, I_{1}, 0, 0, V_{1}) $ and chronic HTLV mono-infection equilibrium, $ \overline{Đ}_{2} = (S_{2}, 0, 0, E_{2}, Y_{2}, 0) $, where $ S_{0} $, $ S_{1} $, $ L_{1} $, $ I_{1} $, $ V_{1} $, $ S_{2} $, $ E_{2} $ and $ Y_{2} $ are given in Section 4. The existence of the these three equilibria is determined by two threshold parameters $ \Re_{1} $ and $ \Re_{2} $ which are defined in Section 4.
Corollary 1. For system (6.3), the following statements hold true.
(ⅰ) If $ \Re_{1}\leq1 $ and $ \Re_{2}\leq1 $, then $ \overline{Đ}_{0} $ is G.A.S.
(ⅱ) If $ \Re_{1} > 1 $ and $ \Re_{2}/\Re_{1}\leq1 $, then $ \overline{Đ}_{1} $ is G.A.S.
(ⅲ) If $ \Re_{2} > 1 $ and $ \Re_{1}/\Re_{2}\leq1 $, then $ \overline{Đ}_{2} $ is G.A.S.
Therefore, the system will tend to one of the three equilibria $ \overline{Đ}_{0} $, $ \overline{Đ}_{1} $ and $ \overline{Đ}_{2} $. The above result says that in the absence of immune response, the competition between HIV and HTLV-I consuming common resources, only one type of viruses with maximum basic reproductive ratio can survive. However, in our proposed model (2.1) involving HIV- and HTLV-specific CTLs, HIV and HTLV-I coexist at equilibrium. We can consider this situation as follows. Since CTL immune responses suppress viral progression, the competition between HIV and HTLV-I is also suppressed and the coexistence of HIV and HTLV-I is occurred [55].
This research work formulates a mathematical model which describes the within host dynamics of HIV/HTLV-I co-infection. The model incorporated the effect of HIV-specific CTLs and HTLV-specific CTLs. HIV has two predominant infection modes: the classical VTC infection and CTC spread. The HTLV-I has two ways of transmission, (ⅰ) horizontal transmission via direct CTC contact, and (ⅱ) vertical transmission through mitotic division of Tax-expressing HTLV-infected cells. We first proved that the model is well-posed by showing that the solutions are nonnegative and bounded. We derived eight threshold parameters that governed the existence and stability of the eight equilibria of the model. We constructed appropriate Lyapunov functions and applied Lyapunov-LaSalle asymptotic stability theorem to prove the global asymptotic stability of all equilibria. We conducted numerical simulations to support and clarify our theoretical results. We studied the effect of HIV infection on HTLV-I mono-infection dynamics and vice versa. The model analysis suggested that co-infected individuals with both viruses will have smaller number of healthy CD$ 4^{+} $T cells in comparison with HIV or HTLV-I mono-infected individuals. We discussed the influence of CTL immune response on the co-infection dynamics.
Our model can be extended in many directions:
● In our model (2.1), we assumed that susceptible CD$ 4^{+} $T cells are produced at a constant rate $ \rho $ and have a linear death rate $ \alpha S $. It would be more reasonable to consider the density dependent production rate. One possibility is to assume a logistic growth for the susceptible CD$ 4^{+} $T cells in the absence of infection. Moreover, the model assumed bilinear incidence rate of infections, $ \eta_{1}SV $, $ \eta_{2}SI $ and $ \eta_{3}SY $. However, such bilinear form may not describe the virus dynamics during the full course of infection. Therefore, it is reasonable to consider other forms of the incidence rate such as: saturated incidence, Beddington-DeAngelis incidence and general incidence [56,57,58].
● Model (2.1) assumed that once susceptible CD$ 4^{+} $T cell is contacted by an HIV or HIV-infected or HTLV-infected cell it becomes latently or actively infected instantaneously. However, such process needs time. Intracellular time delay has a significant effect on the virus dynamics. Delayed viral infection models have been constructed and analyzed in several works (see, e.g., [59,60,61,62,63,64]).
● Model (2.1) assumes that cells and viruses are equally distributed in the domain with no spatial variations. Taking into account spatial variations in the case of HIV/HTLV-I co-infection will be significant [65,66].
We leave these extensions as a future project.
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. (KEP-PhD-20-130-41). The authors, therefore, acknowledge with thanks DSR technical and financial support.
On behalf of all authors, the corresponding author states that there is no conflict of interest.
[1] |
Wheeler AP, Bernard GR (2007) Acute lung injury and the acute respiratory distress syndrome: A clinical review. Lancet 369: 1553–1564. doi: 10.1016/S0140-6736(07)60604-7
![]() |
[2] |
Bellani G, Laffey JG, Pham T, et al. (2016) Epidemiology, Patterns of Care, and Mortality for Patients With Acute Respiratory Distress Syndrome in Intensive Care Units in 50 Countries. JAMA 315: 788–800. doi: 10.1001/jama.2016.0291
![]() |
[3] | Ranieri VM, Rubenfeld GD, Thompson BT, et al. (2012) Acute respiratory distress syndrome: The Berlin Definition. JAMA 307: 2526–2533. |
[4] |
Bernard GR, Artigas A, Brigham KL, et al. (1994) The American-European consensus conference on ARDS: Definitions, mechanisms, relevant outcomes, and clinical trial coordination. Am J Respir Crit Care Med 149: 818–824. doi: 10.1164/ajrccm.149.3.7509706
![]() |
[5] |
Meade MO, Cook RJ, Guyatt GH, et al. (2000) Interobserver variation in interpreting chest radiographs for the diagnosis of acute respiratory distress syndrome. Am J Respir Crit Care Med 161: 85–90. doi: 10.1164/ajrccm.161.1.9809003
![]() |
[6] | Peng JM, Qian CY, Yu XY, et al. (2017) Does training improve diagnostic accuracy and inter-rater agreement in applying the Berlin radiographic definition of acute respiratory distress syndrome? A multicenter prospective study. Crit Care 21: 12. |
[7] |
Gajic O, Dara SI, Mendez JL, et al. (2004) Ventilator-associated lung injury in patients without acute lung injury at the onset of mechanical ventilation. Crit Care Med 32: 1817–1824. doi: 10.1097/01.CCM.0000133019.52531.30
![]() |
[8] |
Anonymous (1999) International consensus conferences in intensive care medicine: Ventilator-associated Lung Injury in ARDS. Am J Respir Crit Care Med 160: 2118–2124. doi: 10.1164/ajrccm.160.6.ats16060
![]() |
[9] |
Pesenti AM (2005) The concept of "baby lung". Intensive Care Med 31: 776–784. doi: 10.1007/s00134-005-2627-z
![]() |
[10] |
Curley GF, Laffey JG, Zhang H, et al. (2016) Biotrauma and Ventilator-Induced Lung Injury: Clinical Implications. Chest 150: 1109–1117. doi: 10.1016/j.chest.2016.07.019
![]() |
[11] |
Thompson BT, Chambers RC, Liu KD (2017) Acute Respiratory Distress Syndrome. N Engl J Med 377: 562–572. doi: 10.1056/NEJMra1608077
![]() |
[12] |
Amato MB, Barbas CS, Medeiros DM, et al. (1998) Effect of a protective-ventilation strategy on mortality in the acute respiratory distress syndrome. N Engl J Med 338: 347–354. doi: 10.1056/NEJM199802053380602
![]() |
[13] |
Ards N (2000) Ventilation with lower tidal volumes as compared with traditional tidal volumes for acute lung injury and the acute respiratory distress syndrome. N Engl J Med 342: 1301–1308. doi: 10.1056/NEJM200005043421801
![]() |
[14] |
Walkey AJ, Goligher E, Del SL, et al. (2017) Low Tidal Volume versus Non-Volume-Limited Strategies for Patients with Acute Respiratory Distress Syndrome: A Systematic Review and Meta-Analysis. Ann Am Thorac Soc 14: S271–S279. doi: 10.1513/AnnalsATS.201704-337OT
![]() |
[15] |
Fan E, Del SL, Goligher EC, et al. (2017) An Official American Thoracic Society/European Society of Intensive Care Medicine/Society of Critical Care Medicine Clinical Practice Guideline: Mechanical Ventilation in Adult Patients with Acute Respiratory Distress Syndrome. Am J Respir Crit Care Med 195: 1253–1263. doi: 10.1164/rccm.201703-0548ST
![]() |
[16] |
Lang JD, Chumley P, Eiserich JP, et al. (2000) Hypercapnia induces injury to alveolar epithelial cells via a nitric oxide-dependent pathway. Am J Physiol Lung Cell Mol Physiol 279: 994–1002. doi: 10.1152/ajplung.2000.279.5.L994
![]() |
[17] |
Morimont P, Batchinsky A, Lambermont B (2015) Update on the role of extracorporeal CO 2 removal as an adjunct to mechanical ventilation in ARDS. Crit Care 19: 117. doi: 10.1186/s13054-015-0799-7
![]() |
[18] | Repessé X, Vieillardbaron A (2017) Hypercapnia during acute respiratory distress syndrome: The tree that hides the forest! J Thorac Dis 9: 1420–1425. |
[19] |
Nin N, Muriel A, Penuelas O, et al. (2017) Severe hypercapnia and outcome of mechanically ventilated patients with moderate or severe acute respiratory distress syndrome. Intensive Care Med 43: 200–208. doi: 10.1007/s00134-016-4611-1
![]() |
[20] | Barnes T, Zochios V, Parhar K (2017) Re-examining permissive hypercapnia in ARDS: A narrative review. Chest. |
[21] | Sahetya SK, Goligher EC, Brower RG (2017) Fifty Years of Research in ARDS. Setting Positive End-Expiratory Pressure in Acute Respiratory Distress Syndrome. Am J Respir Crit Care Med 195: 1429–1438. |
[22] |
Brower RG, Lanken PN, MacIntyre N, et al. (2004) Higher versus lower positive end-expiratory pressures in patients with the acute respiratory distress syndrome. N Engl J Med 351: 327–336. doi: 10.1056/NEJMoa032193
![]() |
[23] |
Meade MO, Cook DJ, Guyatt GH, et al. (2008) Ventilation strategy using low tidal volumes, recruitment maneuvers, and high positive end-expiratory pressure for acute lung injury and acute respiratory distress syndrome: A randomized controlled trial. JAMA 299: 637–645. doi: 10.1001/jama.299.6.637
![]() |
[24] |
Mercat A, Richard JC, Vielle B, et al. (2008) Positive end-expiratory pressure setting in adults with acute lung injury and acute respiratory distress syndrome: A randomized controlled trial. JAMA 299: 646–655. doi: 10.1001/jama.299.6.646
![]() |
[25] | Santa CR, Rojas JI, Nervi R, et al. (2013) High versus low positive end-expiratory pressure (PEEP) levels for mechanically ventilated adult patients with acute lung injury and acute respiratory distress syndrome. Cochrane Database Syst Rev 6: CD009098. |
[26] |
Talmor D, Sarge T, Malhotra A, et al. (2008) Mechanical ventilation guided by esophageal pressure in acute lung injury. N Engl J Med 359: 2095–2104. doi: 10.1056/NEJMoa0708638
![]() |
[27] |
Fish E, Novack V, Bannergoodspeed VM, et al. (2014) The Esophageal Pressure-Guided Ventilation 2 (EPVent2) trial protocol: A multicentre, randomised clinical trial of mechanical ventilation guided by transpulmonary pressure. BMJ Open 4: e006356. doi: 10.1136/bmjopen-2014-006356
![]() |
[28] |
Kacmarek RM, Villar J, Sulemanji D, et al. (2016) Open Lung Approach for the Acute Respiratory Distress Syndrome: A Pilot, Randomized Controlled Trial. Crit Care Med 44: 32–42. doi: 10.1097/CCM.0000000000001383
![]() |
[29] | Writing Group for the Alveolar Recruitment for Acute Respiratory Distress Syndrome Trial I, Cavalcanti AB, Suzumura EA, et al. (2017) Effect of Lung Recruitment and Titrated Positive End-Expiratory Pressure (PEEP) vs. Low PEEP on Mortality in Patients With Acute Respiratory Distress Syndrome: A Randomized Clinical Trial. JAMA 318: 1335–1345. |
[30] |
Luecke T, Corradi F, Pelosi P (2012) Lung imaging for titration of mechanical ventilation. Curr Opin Anaesthesiol 25: 131–140. doi: 10.1097/ACO.0b013e32835003fb
![]() |
[31] |
Jabaudon M, Godet T, Futier E, et al. (2017) Rationale, study design and analysis plan of the lung imaging morphology for ventilator settings in acute respiratory distress syndrome study (LIVE study): Study protocol for a randomised controlled trial. Anaesthesia Crit Care Pain Med 36: 301–306. doi: 10.1016/j.accpm.2017.02.006
![]() |
[32] |
Bugedo G, Retamal J, Bruhn A (2017) Driving pressure: A marker of severity, a safety limit, or a goal for mechanical ventilation? Crit Care 21: 199. doi: 10.1186/s13054-017-1779-x
![]() |
[33] |
Amato MB, Meade MO, Slutsky AS, et al. (2015) Driving pressure and survival in the acute respiratory distress syndrome. N Engl J Med 372: 747–755. doi: 10.1056/NEJMsa1410639
![]() |
[34] |
Estenssoro E, Dubin A, Laffaire E, et al. (2002) Incidence, clinical course, and outcome in 217 patients with acute respiratory distress syndrome. Crit Care Med 30: 2450–2456. doi: 10.1097/00003246-200211000-00008
![]() |
[35] |
Papazian L, Forel JM, Gacouin A, et al. (2010) Neuromuscular blockers in early acute respiratory distress syndrome. N Engl J Med 363: 1107–1116. doi: 10.1056/NEJMoa1005372
![]() |
[36] |
Guerin C, Reignier J, Richard JC, et al. (2013) Prone positioning in severe acute respiratory distress syndrome. N Engl J Med 368: 2159–2168. doi: 10.1056/NEJMoa1214103
![]() |
[37] |
Kassis EB, Loring SH, Talmor D (2016) Mortality and pulmonary mechanics in relation to respiratory system and transpulmonary driving pressures in ARDS. Intensive Care Med 42: 1206–1213. doi: 10.1007/s00134-016-4403-7
![]() |
[38] |
Ferguson ND, Cook DJ, Guyatt GH, et al. (2013) High-frequency oscillation in early acute respiratory distress syndrome. N Engl J Med 368: 795–805. doi: 10.1056/NEJMoa1215554
![]() |
[39] |
Young D, Lamb SE, Shah S, et al. (2013) High-frequency oscillation for acute respiratory distress syndrome. N Engl J Med 368: 806–813. doi: 10.1056/NEJMoa1215716
![]() |
[40] |
Davies SW, Leonard KL, Falls RK, et al. (2015) Lung protective ventilation (ARDSNet) versus airway pressure release ventilation: Ventilatory management in a combined model of acute lung and brain injury. Trauma Acute Care Surg 78: 240–249. doi: 10.1097/TA.0000000000000518
![]() |
[41] |
Mireles-Cabodevila E, Kacmarek RM (2016) Should Airway Pressure Release Ventilation Be the Primary Mode in ARDS? Respir Care 61: 761–773. doi: 10.4187/respcare.04653
![]() |
[42] |
Zhou Y, Jin X, Lv Y, et al. (2017) Early application of airway pressure release ventilation may reduce the duration of mechanical ventilation in acute respiratory distress syndrome. Intensive Care Med 43: 1648–1659. doi: 10.1007/s00134-017-4912-z
![]() |
[43] |
Bellani G, Laffey JG, Pham T, et al. (2017) Noninvasive Ventilation of Patients with Acute Respiratory Distress Syndrome. Insights from the LUNG SAFE Study. Am J Respir Crit Care Med 195: 67–77. doi: 10.1164/rccm.201606-1306OC
![]() |
[44] | Patel BK, Wolfe KS, Pohlman AS, et al. (2016) Effect of Noninvasive Ventilation Delivered by Helmet vs. Face Mask on the Rate of Endotracheal Intubation in Patients With Acute Respiratory Distress Syndrome: A Randomized Clinical Trial. JAMA 315: 2435–2441. |
[45] |
Hernandez G, Roca O, Colinas L (2017) High-flow nasal cannula support therapy: New insights and improving performance. Crit Care 21: 62. doi: 10.1186/s13054-017-1640-2
![]() |
[46] |
Parke RL, Mcguinness SP (2013) Pressures delivered by nasal high flow oxygen during all phases of the respiratory cycle. Respir Care 58: 1621–1624. doi: 10.4187/respcare.02358
![]() |
[47] | Drake MG (2017) High Flow Nasal Cannula Oxygen in Adults: An Evidence-Based Assessment. Ann Am Thorac Soc 15: 145–155. |
[48] |
Frat JP, Thille AW, Mercat A, et al. (2015) High-flow oxygen through nasal cannula in acute hypoxemic respiratory failure. N Engl J Med 372: 2185–2196. doi: 10.1056/NEJMoa1503326
![]() |
[49] |
Kang BJ, Koh Y, Lim CM, et al. (2015) Failure of high-flow nasal cannula therapy may delay intubation and increase mortality. Intensive Care Med 41: 623–632. doi: 10.1007/s00134-015-3693-5
![]() |
[50] | Ni YN, Luo J, Yu H, et al. (2017) Can High-flow Nasal Cannula Reduce the Rate of Endotracheal Intubation in Adult Patients With Acute Respiratory Failure Compared With Conventional Oxygen Therapy and Noninvasive Positive Pressure Ventilation?: A Systematic Review and Meta-analysis. Chest 151: 764–775. |
[51] |
Marik PE, Kaufman D (1996) The effects of neuromuscular paralysis on systemic and splanchnic oxygen utilization in mechanically ventilated patients. Chest 109: 1038–1042. doi: 10.1378/chest.109.4.1038
![]() |
[52] |
Kaisers U, Busch T, Deja M, et al. (2003) Selective pulmonary vasodilation in acute respiratory distress syndrome. Crit Care Med 31: S337–S342. doi: 10.1097/01.CCM.0000057913.45273.1A
![]() |
[53] |
Griffiths MJ, Evans TW (2005) Inhaled nitric oxide therapy in adults. N Engl J Med 353: 2683–2695. doi: 10.1056/NEJMra051884
![]() |
[54] |
Markewitz BA, Michael JR (2000) Inhaled nitric oxide in adults with the acute respiratory distress syndrome. Respir Med 94: 1023–1028. doi: 10.1053/rmed.2000.0928
![]() |
[55] | Gebistorf F, Karam O, Wetterslev J, et al. (2016) Inhaled nitric oxide for acute respiratory distress syndrome (ARDS) in children and adults. Cochrane Database Syst Rev 6: CD002787. |
[56] |
Fuller BM, Mohr NM, Skrupky L, et al. (2015) The use of inhaled prostaglandins in patients with ARDS: A systematic review and meta-analysis. Chest 147: 1510–1522. doi: 10.1378/chest.14-3161
![]() |
[57] |
Bassford CR, Thickett DR, Perkins GD (2012) The rise and fall of beta-agonists in the treatment of ARDS. Crit Care 16: 208. doi: 10.1186/cc11221
![]() |
[58] |
National Heart, Lung, and Blood Institute Acute Respiratory Distress Syndrome (ARDS) Clinical Trials Network, Matthay MA, et al. (2011) Randomized, placebo-controlled clinical trial of an aerosolized beta(2)-agonist for treatment of acute lung injury. Am J Respir Crit Care Med 184: 561–568. doi: 10.1164/rccm.201012-2090OC
![]() |
[59] |
Gao SF, Perkins GD, Gates S, et al. (2012) Effect of intravenous beta-2 agonist treatment on clinical outcomes in acute respiratory distress syndrome (BALTI-2): A multicentre, randomised controlled trial. Lancet 379: 229–235. doi: 10.1016/S0140-6736(11)61623-1
![]() |
[60] |
Lai-Fook SJ, Rodarte JR (1991) Pleural pressure distribution and its relationship to lung volume and interstitial pressure. J Appl Physiol 70: 967–978. doi: 10.1152/jappl.1991.70.3.967
![]() |
[61] |
Pelosi P, D'Andrea L, Vitale G, et al. (1994) Vertical gradient of regional lung inflation in adult respiratory distress syndrome. Am J Respir Crit Care Med 149: 8–13. doi: 10.1164/ajrccm.149.1.8111603
![]() |
[62] |
Tawhai MH, Nash MP, Lin CL, et al. (2009) Supine and prone differences in regional lung density and pleural pressure gradients in the human lung with constant shape. J Appl Physiol 107: 912–920. doi: 10.1152/japplphysiol.00324.2009
![]() |
[63] |
Malbouisson LM, Busch CJ, Puybasset L, et al. (2000) Role of the heart in the loss of aeration characterizing lower lobes in acute respiratory distress syndrome. CT Scan ARDS Study Group. Am J Respir Crit Care Med 161: 2005–2012. doi: 10.1164/ajrccm.161.6.9907067
![]() |
[64] |
Scholten EL, Beitler JR, Prisk GK, et al. (2017) Treatment of ARDS With Prone Positioning. Chest 151: 215–224. doi: 10.1016/j.chest.2016.06.032
![]() |
[65] |
Mutoh T, Guest RJ, Lamm WJ, et al. (1992) Prone position alters the effect of volume overload on regional pleural pressures and improves hypoxemia in pigs in vivo. Am Rev Respir Dis 146: 300–306. doi: 10.1164/ajrccm/146.2.300
![]() |
[66] |
Lamm WJ, Graham MM, Albert RK (1994) Mechanism by which the prone position improves oxygenation in acute lung injury. Am J Respir Crit Care Med 150: 184–193. doi: 10.1164/ajrccm.150.1.8025748
![]() |
[67] | Munshi L, Del LS, Adhikari N, et al. (2017) Prone Position for Acute Respiratory Distress Syndrome. A Systematic Review and Meta-Analysis. Ann Am Thorac Soc 14: S280–S288. |
[68] |
Peek GJ, Mugford M, Tiruvoipati R, et al. (2009) Efficacy and economic assessment of conventional ventilatory support versus extracorporeal membrane oxygenation for severe adult respiratory failure (CESAR): A multicentre randomised controlled trial. Lancet 374: 1351–1363. doi: 10.1016/S0140-6736(09)61069-2
![]() |
[69] |
Dembinski R, Hochhausen N, Terbeck S, et al. (2007) Pumpless extracorporeal lung assist for protective mechanical ventilation in experimental lung injury. Crit Care Med 35: 2359–2366. doi: 10.1097/01.CCM.0000281857.87354.A5
![]() |
[70] |
Schmidt M, Stewart C, Bailey M, et al. (2015) Mechanical ventilation management during extracorporeal membrane oxygenation for acute respiratory distress syndrome: A retrospective international multicenter study. Crit Care Med 43: 654–664. doi: 10.1097/CCM.0000000000000753
![]() |
[71] |
Neto AS, Schmidt M, Azevedo LCP, et al. (2016) Associations between ventilator settings during extracorporeal membrane oxygenation for refractory hypoxemia and outcome in patients with acute respiratory distress syndrome: A pooled individual patient data analysis: Mechanical ventilation during ECMO. Intensive Care Med 42: 1672–1684. doi: 10.1007/s00134-016-4507-0
![]() |
[72] |
Tillmann BW, Klingel ML, Iansavichene AE, et al. (2017) Extracorporeal membrane oxygenation (ECMO) as a treatment strategy for severe acute respiratory distress syndrome (ARDS) in the low tidal volume era: A systematic review. J Crit Care 41: 64–71. doi: 10.1016/j.jcrc.2017.04.041
![]() |
[73] |
Bizzarro MJ, Conrad SA, Kaufman DA, et al. (2011) Infections acquired during extracorporeal membrane oxygenation in neonates, children, and adults. Pediatr Crit Care Med 12: 277–281. doi: 10.1097/PCC.0b013e3181e28894
![]() |
[74] |
Paden ML, Conrad SA, Rycus PT, et al. (2013) Extracorporeal Life Support Organization Registry Report 2012. ASAIO J 59: 202–210. doi: 10.1097/MAT.0b013e3182904a52
![]() |
[75] | Mishra V, Svennevig JL, Bugge JF, et al. (2010) Cost of extracorporeal membrane oxygenation: Evidence from the Rikshospitalet University Hospital, Oslo, Norway. Eur J Cardiothorac Surg 37: 339–342. |
[76] |
Muller T, Lubnow M, Philipp A, et al. (2009) Extracorporeal pumpless interventional lung assist in clinical practice: Determinants of efficacy. Eur Respir J 33: 551–558. doi: 10.1183/09031936.00123608
![]() |
[77] |
Bein T, Aubron C, Papazian L (2017) Focus on ECMO and ECCO2R in ARDS patients. Intensive Care Med 43: 1424–1426. doi: 10.1007/s00134-017-4882-1
![]() |
[78] |
Bein T, Weber-Carstens S, Goldmann A, et al. (2013) Lower tidal volume strategy (approximately 3 ml/kg) combined with extracorporeal CO2 removal versus "conventional" protective ventilation (6 ml/kg) in severe ARDS: The prospective randomized Xtravent-study. Intensive Care Med 39: 847–856. doi: 10.1007/s00134-012-2787-6
![]() |
[79] |
Fitzgerald M, Millar J, Blackwood B, et al. (2014) Extracorporeal carbon dioxide removal for patients with acute respiratory failure secondary to the acute respiratory distress syndrome: A systematic review. Crit Care 18: 222. doi: 10.1186/cc13875
![]() |
[80] |
Calfee CS, Delucchi K, Parsons PE, et al. (2014) Subphenotypes in acute respiratory distress syndrome: Latent class analysis of data from two randomised controlled trials. Lancet Respir Med 2: 611–620. doi: 10.1016/S2213-2600(14)70097-9
![]() |
[81] | Famous KR, Delucchi K, Ware LB, et al. (2017) Acute Respiratory Distress Syndrome Subphenotypes Respond Differently to Randomized Fluid Management Strategy. Am J Respir Crit Care Med 195: 331–338. |
[82] |
Hough CL (2014) Steroids for acute respiratory distress syndrome? Clin Chest Med 35: 781–795. doi: 10.1016/j.ccm.2014.08.014
![]() |
[83] |
Meduri GU, Bridges L, Shih MC, et al. (2016) Prolonged glucocorticoid treatment is associated with improved ARDS outcomes: Analysis of individual patients' data from four randomized trials and trial-level meta-analysis of the updated literature. Intensive Care Med 42: 829–840. doi: 10.1007/s00134-015-4095-4
![]() |
[84] | Network TA (2000) Ketoconazole for early treatment of acute lung injury and acute respiratory distress syndrome: A randomized controlled trial. The ARDS Network. JAMA 283: 1995–2002. |
[85] |
Xiong B, Wang C, Tan J, et al. (2016) Statins for the prevention and treatment of acute lung injury and acute respiratory distress syndrome: A systematic review and meta-analysis. Respirology 21: 1026–1033. doi: 10.1111/resp.12820
![]() |
[86] |
Sabater J, Masclans JR, Sacanell J, et al. (2008) Effects on hemodynamics and gas exchange of omega-3 fatty acid-enriched lipid emulsion in acute respiratory distress syndrome (ARDS): A prospective, randomized, double-blind, parallel group study. Lipids Health Dis 7: 39. doi: 10.1186/1476-511X-7-39
![]() |
[87] |
Raghavendran K, Willson D, Notter RH (2011) Surfactant therapy for acute lung injury and acute respiratory distress syndrome. Crit Care Clin 27: 525–559. doi: 10.1016/j.ccc.2011.04.005
![]() |
[88] |
Shah FA, Girard TD, Yende S (2017) Limiting sedation for patients with acute respiratory distress syndrome-time to wake up. Curr Opin Crit Care 23: 45–51. doi: 10.1097/MCC.0000000000000382
![]() |
[89] |
National Heart L, Wheeler AP, Wiedemann HP, et al. (2006) Comparison of two fluid-management strategies in acute lung injury. N Engl J Med 354: 2564–2575. doi: 10.1056/NEJMoa062200
![]() |
[90] |
Jozwiak M, Silva S, Persichini R, et al. (2013) Extravascular lung water is an independent prognostic factor in patients with acute respiratory distress syndrome. Crit Care Med 41: 472–480. doi: 10.1097/CCM.0b013e31826ab377
![]() |
[91] |
Krzak A, Pleva M, Napolitano LM (2011) Nutrition therapy for ALI and ARDS. Crit Care Clin 27: 647–659. doi: 10.1016/j.ccc.2011.05.004
![]() |
[92] |
Investigators NS, Finfer S, Chittock DR, et al. (2009) Intensive versus conventional glucose control in critically ill patients. N Engl J Med 360: 1283–1297. doi: 10.1056/NEJMoa0810625
![]() |
[93] |
Kortebein P (2009) Rehabilitation for hospital-associated deconditioning. Am J Phys Med Rehabil 88: 66–77. doi: 10.1097/PHM.0b013e3181838f70
![]() |
[94] |
Bailey P, Thomsen GE, Spuhler VJ, et al. (2007) Early activity is feasible and safe in respiratory failure patients. Crit Care Med 35: 139–145. doi: 10.1097/01.CCM.0000251130.69568.87
![]() |
[95] |
Morris PE, Berry MJ, Files DC, et al. (2016) Standardized Rehabilitation and Hospital Length of Stay Among Patients With Acute Respiratory Failure: A Randomized Clinical Trial. JAMA 315: 2694–2702. doi: 10.1001/jama.2016.7201
![]() |
[96] | Jabaudon M, Blondonnet R, Audard J, et al. (2017) Recent directions in personalised acute respiratory distress syndrome medicine. Anaesth Crit Care Pain Med. |
[97] | Shankar-Hari M, Mcauley DF (2017) Acute Respiratory Distress Syndrome Phenotypes and Identifying Treatable Traits. The Dawn of Personalized Medicine for ARDS. Am J Respir Crit Care Med 195: 280–281. |
[98] |
Wilson JG, Liu KD, Zhuo H, et al. (2015) Mesenchymal stem (stromal) cells for treatment of ARDS: A phase 1 clinical trial. Lancet Respir Med 3: 24–32. doi: 10.1016/S2213-2600(14)70291-7
![]() |
[99] | Laffey JG, Matthay MA (2017) Fifty Years of Research in ARDS. Cell-based Therapy for Acute Respiratory Distress Syndrome. Biology and Potential Therapeutic Value. Am J Respir Crit Care Med 196: 266–273. |
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Parameter | Description |
$ \rho $ | $ Recruitment rate for the susceptible CD4+T cells $ |
$ \alpha $ | $ Natural mortality rate constant for the susceptible CD4+T cells $ |
$ \eta_{1} $ | $ Virus-cell incidence rate constant between free HIV particles and susceptible CD4+T cells $ |
$ \eta_{2} $ | $ Cell-cell incidence rate constant between HIV-infected cells and susceptible CD4+T cells $ |
$ \eta_{3} $ | $ Cell-cell incidence rate constant between Tax-expressing HTLV-infected cells andsusceptible CD4+T cells $ |
$ \beta \in \left(0, 1\right) $ | $ Fraction coefficient accounts for the probability of new HIV-infected cells could be active, and the remaining part 1−β will be latent $ |
$ \gamma $ | $ Death rate constant of latently HIV-infected cells $ |
$ a $ | $ Death rate constant of actively HIV-infected cells $ |
$ \mu_{1} $ | $ Killing rate constant of actively HIV-infected cells due to HIV-specific CTLs $ |
$ \mu_{2} $ | $ Killing rate constant of Tax-expressing HTLV-infected cells due to HTLV-specific CTLs $ |
$ \varphi \in \left(0, 1\right) $ | $ Probability of new HTLV infections could be enter a latent period $ |
$ \lambda $ | $ Transmission rate constant of latently HIV-infected cells that become actively HIV-infected cells $ |
$ \psi $ | $ Transmission rate constant of latently HTLV-infected cells that become Tax-expressing HTLV-infected cells $ |
$ \omega $ | $ Death rate constant of latently HTLV-infected cells $ |
$ \delta $ | $ Death rate constant of Tax-expressing HTLV-infected cells $ |
$ b $ | $ Generation rate constant of new HIV particles $ |
$ \varepsilon $ | $ Death rate constant of free HIV particles $ |
$ \sigma_{1} $ | $ Proliferation rate constant of HIV-specific CTLs $ |
$ \sigma_{2} $ | $ Proliferation rate constant of HTLV-specific CTLs $ |
$ \pi_{1} $ | $ Decay rate constant of HIV-specific CTLs $ |
$ \pi_{2} $ | $ Decay rate constant of HTLV-specific CTLs $ |
Equilibrium point | $ Definition $ | $ \text{Existence conditions} $ |
$Đ _{0}=(S_{0}, 0, 0, 0, 0, 0, 0, 0) $ | $ Infection-free equilibrium $ | None |
$Đ _{1}=(S_{1}, L_{1}, I_{1}, 0, 0, V_{1}, 0, 0) $ | $ Chronic HIV mono-infection equilibriumwith inactive immune response $ | $ \Re_{1} > 1 $ |
$Đ _{2}=(S_{2}, 0, 0, E_{2}, Y_{2}, 0, 0, 0) $ | $ Chronic HTLV mono-infection equilibriumwith inactive immune response $ | $ \Re_{2} > 1 $ |
$Đ _{3}=(S_{3}, L_{3}, I_{3}, 0, 0, V_{3}, C_{3}^{I}, 0) $ | $ Chronic HIV mono-infection equilibriumwith only active HIV-specific CTL $ | $ \Re_{3} > 1 $ |
$Đ _{4}=(S_{4}, 0, 0, E_{4}, Y_{4}, 0, 0, C_{4}^{Y}) $ | $ Chronic HTLV mono-infection equilibriumwith only active HTLV-specific CTL $ | $ \Re_{4} > 1 $ |
$Đ _{5}=(S_{5}, L_{5}, I_{5}, E_{5}, Y_{5}, V_{5}, C_{5}^{I}, 0) $ | $ Chronic HIV/HTLV co-infection equilibriumwith only active HIV-specific CTL $ | $ \Re_{5} > 1 $ and $ \Re_{1}/\Re_{2} > 1 $ |
$Đ _{6}=(S_{6}, L_{6}, I_{6}, E_{6}, Y_{6}, V_{6}, 0, C_{6}^{Y}) $ | $ Chronic HIV/HTLV co-infection equilibriumwith only active HTLV-specific CTL $ | $ \Re_{6} > 1 $ and $ \Re_{2}/\Re_{1} > 1 $ |
$Đ _{7}=(S_{7}, L_{7}, I_{7}, E_{7}, Y_{7}, V_{7}, C_{7}^{I}, C_{7}^{Y}) $ | $ Chronic HIV/HTLV co-infectionequilibrium with active HIV-specificCTL and HTLV-specific CTL $ | $ \Re_{7} > 1 $ and $ \Re_{8} > 1 $ |
Equilibrium point | Global stability$ \text{ conditions} $ |
$Đ _{0}=(S_{0}, 0, 0, 0, 0, 0, 0, 0) $ | $ \Re_{1}\leq1 $ and $ \Re_{2}\leq1 $ |
$Đ _{1}=(S_{1}, L_{1}, I_{1}, 0, 0, V_{1}, 0, 0) $ | $ \Re_{1} > 1 $, $ \Re_{2}/\Re _{1}\leq1 $ and $ \Re_{3}\leq1 $ |
$Đ _{2}=(S_{2}, 0, 0, E_{2}, Y_{2}, 0, 0, 0) $ | $ \Re_{2} > 1 $, $ \Re_{1}/\Re_{2}\leq1 $ and $ \Re_{4}\leq1 $ |
$Đ _{3}=(S_{3}, L_{3}, I_{3}, 0, 0, V_{3}, C_{3}^{I}, 0) $ | $ \Re_{3} > 1 $ and $ \Re_{5}\leq1 $ |
$Đ _{4}=(S_{4}, 0, 0, E_{4}, Y_{4}, 0, 0, C_{4}^{Y}) $ | $ \Re_{4} > 1 $ and $ \Re _{6}\leq1 $ |
$Đ _{5}=(S_{5}, L_{5}, I_{5}, E_{5}, Y_{5}, V_{5}, C_{5}^{I}, 0) $ | $ \Re_{5} > 1 $, $ \Re_{8}\leq1 $ and $ \Re_{1}/\Re_{2} > 1 $ |
$Đ _{6}=(S_{6}, L_{6}, I_{6}, E_{6}, Y_{6}, V_{6}, 0, C_{6}^{Y}) $ | $ \Re_{6} > 1 $, $ \Re_{7}\leq1 $ and $ \Re_{2}/\Re_{1} > 1 $ |
$Đ _{7}=(S_{7}, L_{7}, I_{7}, E_{7}, Y_{7}, V_{7}, C_{7}^{I}, C_{7}^{Y}) $ | $ \Re_{7} > 1 $ and $ \Re_{8} > 1 $ |
Parameter | Value | Parameter | Value | Parameter | Value |
$ \rho $ | $ 10 $ | $ \delta $ | $ 0.2 $ | $ \beta $ | $ 0.7 $ |
$ \alpha $ | $ 0.01 $ | $ b $ | $ 5 $ | $ \gamma $ | $ 0.02 $ |
$ \eta_{1} $ | Varied | $ \pi_{1} $ | $ 0.1 $ | $ \sigma_{1} $ | Varied |
$ \eta_{2} $ | Varied | $ \pi_{2} $ | $ 0.1 $ | $ \sigma_{2} $ | Varied |
$ \eta_{3} $ | Varied | $ \mu_{1} $ | $ 0.2 $ | $ \lambda $ | $ 0.2 $ |
$ a $ | $ 0.5 $ | $ \mu_{2} $ | $ 0.2 $ | $ \omega $ | $ 0.01 $ |
$ \varphi $ | $ 0.2 $ | $ \varepsilon $ | $ 2 $ | $ \psi $ | $ 0.003 $ |
Scenario | The equilibria | $ (\operatorname{Re}(\lambda_{i}), $ $ i=1, 2, ..., 6) $ | Stability |
1 | $ Đ0=(1000,0,0,0,0,0,0,0) $ | $ (−2.19,−0.37,−0.2,−0.1,−0.1,−0.09,−0.01,−0.01) $ | $ stable $ |
2 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(331.63,9.11,13,0,0,32.51,0,0) $ | $ (−2.7,0.51,−0.32,−0.2,−0.1,−0.1,−0.01,−0.01)(−2.3,−0.35,−0.2,−0.1,−0.02,−0.02,−0.06,−0.01) $ | $ unstablestable $ |
3 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ2=(433.33,0,0,87.18,1.31,0,0,0) $ | $ (−2.18,−0.36,−0.23,−0.1,−0.1,−0.03,0.01,−0.01)(−2.09,−0.41,−0.21,−0.16,−0.1,−0.03,−0.01,−0.01) $ | $ unstablestable $ |
4 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(197.7,10.94,15.61,0,0,39.02,0,0)Đ2=(866.67,0,0,20.51,0.31,0,0,0)Đ3=(277.78,9.85,10,0,0,25,1.01,0) $ | $ (−3.22,0.88,−0.31,−0.21,−0.1,−0.1,−0.01,0.002)(−2.36,−0.35,−0.2,−0.03,−0.03,−0.1,0.06,−0.01)(−3.1,0.76,−0.32,−0.21,−0.1,−0.1,−0.01,−0.002)(−2.51,−0.37,−0.2,−0.02,−0.02,−0.1,−0.02,−0.01) $ | $ unstableunstableunstablestable $ |
5 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(277.85,9.85,14.05,0,0,35.12,0,0)Đ2=(43.33,0,0,147.18,2.21,0,0,0)Đ3=(729.93,3.68,2,0,0,5,4.07,0)Đ4=(230.77,0,0,118.34,0.33,0,0,4.33) $ | $ (−2.94,0.61,−0.37,−0.32,0.16,−0.1,−0.1,−0.01)(−2.35,0.6,−0.35,−0.27,−0.1,−0.02,−0.02,0.05)(−2.06,0.56,−0.45,−0.27,−0.2,−0.16,−0.1,−0.01)(−2.99,−0.43,−0.34,−0.03,−0.03,0.12,−0.1,−0.01)(−2.3,−0.99,−0.36,−0.1,−0.06,−0.06,−0.05,−0.01) $ | $ unstableunstableunstableunstablestable $ |
6 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(197.7,10.94,15.61,0,0,39.02,0,0)Đ2=(433.33,0,0,87.18,1.31,0,0,0)Đ3=(657.89,4.67,2,0,0,5,5.82,0)Đ4=(444.44,0,0,85.47,1.25,0,0,0.03)Đ5=(433.33,3.07,2,52.51,0.79,5,2.98,0) $ | $ (−3.22,0.88,−0.31,−0.23,−0.1,−0.1,0.01,−0.01)(−2.36,0.68,−0.35,−0.21,−0.03,−0.03,−0.1,−0.01)(−2.67,−0.33,0.3,−0.21,−0.1,−0.01,−0.01,0.005)(−3.3,−0.45,−0.22,−0.05,−0.05,−0.1,−0.01,0.01)(−2.68,−0.32,0.32,−0.22,−0.1,−0.01,−0.01,−0.005)(−2.83,−0.41,−0.21,−0.03,−0.03,−0.04,−0.01,−0.01) $ | $ unstableunstableunstableunstableunstablestable $ |
7 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(321.26,9.26,13.2,0,0,33.01,0,0)Đ2=(108.33,0,0,137.18,2.06,0,0,0)Đ3=(384.62,8.39,10,0,0,25,0.49,0)Đ4=(555.56,0,0,68.38,0.2,0,0,4.13)Đ6=(321.26,5.75,8.2,39.54,0.2,20.51,0,1.97) $ | $ (−2.84,0.51,−0.32,−0.29,−0.1,−0.1,0.07,−0.01)(−2.35,−0.35,−0.23,−0.1,−0.02,−0.02,0.03,0.02)(−2.13,0.93,−0.42,−0.22,−0.16,−0.1,−0.07,−0.01)(−2.42,−0.36,−0.24,−0.1,−0.01,−0.01,0.03,−0.01)(−2.54,−0.95,−0.33,0.19,−0.1,−0.07,−0.03,−0.01)(−2.35,−0.53,−0.35,−0.02,−0.02,−0.05,−0.02,−0.02) $ | $ unstableunstableunstableunstableunstablestable $ |
8 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(302.36,9.51,13.57,0,0,33.93,0,0)Đ2=(108.33,0,0,137.18,2.06,0,0,0)Đ3=(746.27,3.46,2,0,0,5,3.67,0)Đ4=(555.56,0,0,68.38,0.2,0,0,4.13)Đ6=(302.36,6.21,8.87,37.21,0.2,22.17,0,1.79)Đ7=(467.29,2.17,2,57.51,0.2,5,1.36,3.31) $ | $ (−2.83,0.56,−0.32,−0.29,−0.1,−0.1,0.07,−0.01)(−2.33,0.58,−0.35,−0.23,−0.1,−0.02,−0.02,0.02)(−2.13,0.93,−0.41,−0.22,−0.16,−0.1,−0.07,−0.01)(−2.86,−0.42,−0.27,−0.03,−0.03,−0.1,0.06,−0.01)(−2.53,−0.95,−0.33,0.22,−0.1,−0.07,−0.03,−0.01)(−2.33,−0.5,−0.35,0.34,−0.02,−0.02,−0.05,−0.02)(−2.52,−0.79,−0.38,−0.01,−0.01,−0.06,−0.03,−0.01) $ | $ unstableunstableunstableunstableunstableunstablestable $ |
Parameter | Description |
$ \rho $ | $ Recruitment rate for the susceptible CD4+T cells $ |
$ \alpha $ | $ Natural mortality rate constant for the susceptible CD4+T cells $ |
$ \eta_{1} $ | $ Virus-cell incidence rate constant between free HIV particles and susceptible CD4+T cells $ |
$ \eta_{2} $ | $ Cell-cell incidence rate constant between HIV-infected cells and susceptible CD4+T cells $ |
$ \eta_{3} $ | $ Cell-cell incidence rate constant between Tax-expressing HTLV-infected cells andsusceptible CD4+T cells $ |
$ \beta \in \left(0, 1\right) $ | $ Fraction coefficient accounts for the probability of new HIV-infected cells could be active, and the remaining part 1−β will be latent $ |
$ \gamma $ | $ Death rate constant of latently HIV-infected cells $ |
$ a $ | $ Death rate constant of actively HIV-infected cells $ |
$ \mu_{1} $ | $ Killing rate constant of actively HIV-infected cells due to HIV-specific CTLs $ |
$ \mu_{2} $ | $ Killing rate constant of Tax-expressing HTLV-infected cells due to HTLV-specific CTLs $ |
$ \varphi \in \left(0, 1\right) $ | $ Probability of new HTLV infections could be enter a latent period $ |
$ \lambda $ | $ Transmission rate constant of latently HIV-infected cells that become actively HIV-infected cells $ |
$ \psi $ | $ Transmission rate constant of latently HTLV-infected cells that become Tax-expressing HTLV-infected cells $ |
$ \omega $ | $ Death rate constant of latently HTLV-infected cells $ |
$ \delta $ | $ Death rate constant of Tax-expressing HTLV-infected cells $ |
$ b $ | $ Generation rate constant of new HIV particles $ |
$ \varepsilon $ | $ Death rate constant of free HIV particles $ |
$ \sigma_{1} $ | $ Proliferation rate constant of HIV-specific CTLs $ |
$ \sigma_{2} $ | $ Proliferation rate constant of HTLV-specific CTLs $ |
$ \pi_{1} $ | $ Decay rate constant of HIV-specific CTLs $ |
$ \pi_{2} $ | $ Decay rate constant of HTLV-specific CTLs $ |
Equilibrium point | $ Definition $ | $ \text{Existence conditions} $ |
$Đ _{0}=(S_{0}, 0, 0, 0, 0, 0, 0, 0) $ | $ Infection-free equilibrium $ | None |
$Đ _{1}=(S_{1}, L_{1}, I_{1}, 0, 0, V_{1}, 0, 0) $ | $ Chronic HIV mono-infection equilibriumwith inactive immune response $ | $ \Re_{1} > 1 $ |
$Đ _{2}=(S_{2}, 0, 0, E_{2}, Y_{2}, 0, 0, 0) $ | $ Chronic HTLV mono-infection equilibriumwith inactive immune response $ | $ \Re_{2} > 1 $ |
$Đ _{3}=(S_{3}, L_{3}, I_{3}, 0, 0, V_{3}, C_{3}^{I}, 0) $ | $ Chronic HIV mono-infection equilibriumwith only active HIV-specific CTL $ | $ \Re_{3} > 1 $ |
$Đ _{4}=(S_{4}, 0, 0, E_{4}, Y_{4}, 0, 0, C_{4}^{Y}) $ | $ Chronic HTLV mono-infection equilibriumwith only active HTLV-specific CTL $ | $ \Re_{4} > 1 $ |
$Đ _{5}=(S_{5}, L_{5}, I_{5}, E_{5}, Y_{5}, V_{5}, C_{5}^{I}, 0) $ | $ Chronic HIV/HTLV co-infection equilibriumwith only active HIV-specific CTL $ | $ \Re_{5} > 1 $ and $ \Re_{1}/\Re_{2} > 1 $ |
$Đ _{6}=(S_{6}, L_{6}, I_{6}, E_{6}, Y_{6}, V_{6}, 0, C_{6}^{Y}) $ | $ Chronic HIV/HTLV co-infection equilibriumwith only active HTLV-specific CTL $ | $ \Re_{6} > 1 $ and $ \Re_{2}/\Re_{1} > 1 $ |
$Đ _{7}=(S_{7}, L_{7}, I_{7}, E_{7}, Y_{7}, V_{7}, C_{7}^{I}, C_{7}^{Y}) $ | $ Chronic HIV/HTLV co-infectionequilibrium with active HIV-specificCTL and HTLV-specific CTL $ | $ \Re_{7} > 1 $ and $ \Re_{8} > 1 $ |
Equilibrium point | Global stability$ \text{ conditions} $ |
$Đ _{0}=(S_{0}, 0, 0, 0, 0, 0, 0, 0) $ | $ \Re_{1}\leq1 $ and $ \Re_{2}\leq1 $ |
$Đ _{1}=(S_{1}, L_{1}, I_{1}, 0, 0, V_{1}, 0, 0) $ | $ \Re_{1} > 1 $, $ \Re_{2}/\Re _{1}\leq1 $ and $ \Re_{3}\leq1 $ |
$Đ _{2}=(S_{2}, 0, 0, E_{2}, Y_{2}, 0, 0, 0) $ | $ \Re_{2} > 1 $, $ \Re_{1}/\Re_{2}\leq1 $ and $ \Re_{4}\leq1 $ |
$Đ _{3}=(S_{3}, L_{3}, I_{3}, 0, 0, V_{3}, C_{3}^{I}, 0) $ | $ \Re_{3} > 1 $ and $ \Re_{5}\leq1 $ |
$Đ _{4}=(S_{4}, 0, 0, E_{4}, Y_{4}, 0, 0, C_{4}^{Y}) $ | $ \Re_{4} > 1 $ and $ \Re _{6}\leq1 $ |
$Đ _{5}=(S_{5}, L_{5}, I_{5}, E_{5}, Y_{5}, V_{5}, C_{5}^{I}, 0) $ | $ \Re_{5} > 1 $, $ \Re_{8}\leq1 $ and $ \Re_{1}/\Re_{2} > 1 $ |
$Đ _{6}=(S_{6}, L_{6}, I_{6}, E_{6}, Y_{6}, V_{6}, 0, C_{6}^{Y}) $ | $ \Re_{6} > 1 $, $ \Re_{7}\leq1 $ and $ \Re_{2}/\Re_{1} > 1 $ |
$Đ _{7}=(S_{7}, L_{7}, I_{7}, E_{7}, Y_{7}, V_{7}, C_{7}^{I}, C_{7}^{Y}) $ | $ \Re_{7} > 1 $ and $ \Re_{8} > 1 $ |
Parameter | Value | Parameter | Value | Parameter | Value |
$ \rho $ | $ 10 $ | $ \delta $ | $ 0.2 $ | $ \beta $ | $ 0.7 $ |
$ \alpha $ | $ 0.01 $ | $ b $ | $ 5 $ | $ \gamma $ | $ 0.02 $ |
$ \eta_{1} $ | Varied | $ \pi_{1} $ | $ 0.1 $ | $ \sigma_{1} $ | Varied |
$ \eta_{2} $ | Varied | $ \pi_{2} $ | $ 0.1 $ | $ \sigma_{2} $ | Varied |
$ \eta_{3} $ | Varied | $ \mu_{1} $ | $ 0.2 $ | $ \lambda $ | $ 0.2 $ |
$ a $ | $ 0.5 $ | $ \mu_{2} $ | $ 0.2 $ | $ \omega $ | $ 0.01 $ |
$ \varphi $ | $ 0.2 $ | $ \varepsilon $ | $ 2 $ | $ \psi $ | $ 0.003 $ |
Scenario | The equilibria | $ (\operatorname{Re}(\lambda_{i}), $ $ i=1, 2, ..., 6) $ | Stability |
1 | $ Đ0=(1000,0,0,0,0,0,0,0) $ | $ (−2.19,−0.37,−0.2,−0.1,−0.1,−0.09,−0.01,−0.01) $ | $ stable $ |
2 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(331.63,9.11,13,0,0,32.51,0,0) $ | $ (−2.7,0.51,−0.32,−0.2,−0.1,−0.1,−0.01,−0.01)(−2.3,−0.35,−0.2,−0.1,−0.02,−0.02,−0.06,−0.01) $ | $ unstablestable $ |
3 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ2=(433.33,0,0,87.18,1.31,0,0,0) $ | $ (−2.18,−0.36,−0.23,−0.1,−0.1,−0.03,0.01,−0.01)(−2.09,−0.41,−0.21,−0.16,−0.1,−0.03,−0.01,−0.01) $ | $ unstablestable $ |
4 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(197.7,10.94,15.61,0,0,39.02,0,0)Đ2=(866.67,0,0,20.51,0.31,0,0,0)Đ3=(277.78,9.85,10,0,0,25,1.01,0) $ | $ (−3.22,0.88,−0.31,−0.21,−0.1,−0.1,−0.01,0.002)(−2.36,−0.35,−0.2,−0.03,−0.03,−0.1,0.06,−0.01)(−3.1,0.76,−0.32,−0.21,−0.1,−0.1,−0.01,−0.002)(−2.51,−0.37,−0.2,−0.02,−0.02,−0.1,−0.02,−0.01) $ | $ unstableunstableunstablestable $ |
5 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(277.85,9.85,14.05,0,0,35.12,0,0)Đ2=(43.33,0,0,147.18,2.21,0,0,0)Đ3=(729.93,3.68,2,0,0,5,4.07,0)Đ4=(230.77,0,0,118.34,0.33,0,0,4.33) $ | $ (−2.94,0.61,−0.37,−0.32,0.16,−0.1,−0.1,−0.01)(−2.35,0.6,−0.35,−0.27,−0.1,−0.02,−0.02,0.05)(−2.06,0.56,−0.45,−0.27,−0.2,−0.16,−0.1,−0.01)(−2.99,−0.43,−0.34,−0.03,−0.03,0.12,−0.1,−0.01)(−2.3,−0.99,−0.36,−0.1,−0.06,−0.06,−0.05,−0.01) $ | $ unstableunstableunstableunstablestable $ |
6 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(197.7,10.94,15.61,0,0,39.02,0,0)Đ2=(433.33,0,0,87.18,1.31,0,0,0)Đ3=(657.89,4.67,2,0,0,5,5.82,0)Đ4=(444.44,0,0,85.47,1.25,0,0,0.03)Đ5=(433.33,3.07,2,52.51,0.79,5,2.98,0) $ | $ (−3.22,0.88,−0.31,−0.23,−0.1,−0.1,0.01,−0.01)(−2.36,0.68,−0.35,−0.21,−0.03,−0.03,−0.1,−0.01)(−2.67,−0.33,0.3,−0.21,−0.1,−0.01,−0.01,0.005)(−3.3,−0.45,−0.22,−0.05,−0.05,−0.1,−0.01,0.01)(−2.68,−0.32,0.32,−0.22,−0.1,−0.01,−0.01,−0.005)(−2.83,−0.41,−0.21,−0.03,−0.03,−0.04,−0.01,−0.01) $ | $ unstableunstableunstableunstableunstablestable $ |
7 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(321.26,9.26,13.2,0,0,33.01,0,0)Đ2=(108.33,0,0,137.18,2.06,0,0,0)Đ3=(384.62,8.39,10,0,0,25,0.49,0)Đ4=(555.56,0,0,68.38,0.2,0,0,4.13)Đ6=(321.26,5.75,8.2,39.54,0.2,20.51,0,1.97) $ | $ (−2.84,0.51,−0.32,−0.29,−0.1,−0.1,0.07,−0.01)(−2.35,−0.35,−0.23,−0.1,−0.02,−0.02,0.03,0.02)(−2.13,0.93,−0.42,−0.22,−0.16,−0.1,−0.07,−0.01)(−2.42,−0.36,−0.24,−0.1,−0.01,−0.01,0.03,−0.01)(−2.54,−0.95,−0.33,0.19,−0.1,−0.07,−0.03,−0.01)(−2.35,−0.53,−0.35,−0.02,−0.02,−0.05,−0.02,−0.02) $ | $ unstableunstableunstableunstableunstablestable $ |
8 | $ Đ0=(1000,0,0,0,0,0,0,0)Đ1=(302.36,9.51,13.57,0,0,33.93,0,0)Đ2=(108.33,0,0,137.18,2.06,0,0,0)Đ3=(746.27,3.46,2,0,0,5,3.67,0)Đ4=(555.56,0,0,68.38,0.2,0,0,4.13)Đ6=(302.36,6.21,8.87,37.21,0.2,22.17,0,1.79)Đ7=(467.29,2.17,2,57.51,0.2,5,1.36,3.31) $ | $ (−2.83,0.56,−0.32,−0.29,−0.1,−0.1,0.07,−0.01)(−2.33,0.58,−0.35,−0.23,−0.1,−0.02,−0.02,0.02)(−2.13,0.93,−0.41,−0.22,−0.16,−0.1,−0.07,−0.01)(−2.86,−0.42,−0.27,−0.03,−0.03,−0.1,0.06,−0.01)(−2.53,−0.95,−0.33,0.22,−0.1,−0.07,−0.03,−0.01)(−2.33,−0.5,−0.35,0.34,−0.02,−0.02,−0.05,−0.02)(−2.52,−0.79,−0.38,−0.01,−0.01,−0.06,−0.03,−0.01) $ | $ unstableunstableunstableunstableunstableunstablestable $ |